fitting y=f(x) data to arbitrary functions in ImageJ?

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fitting y=f(x) data to arbitrary functions in ImageJ?

Kenneth Sloan-3
I have some simple data: samples of y=f(x) at regularly spaced discrete values for x.  The data
is born as a simple array of y values, but I can turn that into (for example) a polyline, if that
will help.  I'm currently doing that to draw the data as an Overlay.  The size of the y array is between 500 and 1000. (think 1 y value for every integer x-coordinate in an image - some y values may be recorded as "missing").

Is there an ImageJ tool that will fit (more or less) arbitrary functions to this data?  Approximately 8 parameters.

The particular function I have in mind at the moment is a difference of Gaussians.  The Gaussians
most likely have the same location (in x) - but this is not guaranteed, and I'd prefer
to use this as a sanity check rather than impose it as a constraint.

Note that the context is a Java plugin - not a macro.

If not in ImageJ, perhaps someone could point me at a Java package that will do this.  I am far
from an expert in curve fitting, so please be gentle.

If not, I can do it in R - but I prefer to do it "on the fly" while displaying the image, and the Overlay.

--
Kenneth Sloan
[hidden email]
Vision is the art of seeing what is invisible to others.

--
ImageJ mailing list: http://imagej.nih.gov/ij/list.html
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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Fred Damen
Below is a routine to fit MRI Inversion Recover data for T1.

Note:
a) CurveFitter comes with ImageJ.
b) Calls are made to UserFunction once for each x.
c) If your initial guess is not close it does not seem to converge.

Bonus question for Java Gurus:
How to declare the user function as a variable, call it, and pass it to
another function?

Enjoy,

Fred

   private double[] nonlinearFit(double[] x, double[] y) {

      CurveFitter cf = new CurveFitter(x, y);
      double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};

      cf.setMaxIterations(200);
      cf.doCustomFit(new UserFunction() {
         @Override
         public double userFunction(double[] p, double x) {
            return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
            }
         }, params.length, "", params, null, false);
      //IJ.log(cf.getResultString());

      return cf.getParams();
      }


On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:

> I have some simple data: samples of y=f(x) at regularly spaced discrete values
> for x.  The data
> is born as a simple array of y values, but I can turn that into (for example)
> a polyline, if that
> will help.  I'm currently doing that to draw the data as an Overlay.  The size
> of the y array is between 500 and 1000. (think 1 y value for every integer
> x-coordinate in an image - some y values may be recorded as "missing").
>
> Is there an ImageJ tool that will fit (more or less) arbitrary functions to
> this data?  Approximately 8 parameters.
>
> The particular function I have in mind at the moment is a difference of
> Gaussians.  The Gaussians
> most likely have the same location (in x) - but this is not guaranteed, and
> I'd prefer
> to use this as a sanity check rather than impose it as a constraint.
>
> Note that the context is a Java plugin - not a macro.
>
> If not in ImageJ, perhaps someone could point me at a Java package that will
> do this.  I am far
> from an expert in curve fitting, so please be gentle.
>
> If not, I can do it in R - but I prefer to do it "on the fly" while displaying
> the image, and the Overlay.
>
> --
> Kenneth Sloan
> [hidden email]
> Vision is the art of seeing what is invisible to others.
>
> --
> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>

--
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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Michael Schmid-3
Hi Kenneth,

Concerning fitting an 8-parameter function:

If the fit is not linear (as in the case of a difference of Gaussians),
having 8 fit parameters is a rather ambitious task, and there is a high
probability that the fit will run into a local minimum or some point
that looks like a local minimum to the fitting program.

It would be best to reduce the number of parameters, e.g. using a fixed
ratio between the two sigma values in the Difference of Gaussians.

You also need some reasonable guess for the initial values of the
parameters.

For the ImageJ CurveFitter, if there are many parameters it is very
important to specify roughly how much the parameters can vary, these are
the 'initialParamVariations'
If cf is the CurveFitter, you will have
     cf.doCustomFit(UserFunction userFunction, int numParams,
         String formula, double[] initialParams,
         double[] initialParamVariations, boolean showSettings

For the initialParamVariations, use e.g. 1/10th of how much the
respective parameter might deviate from the initial guess (only the
order of magnitude is important).

If you have many parameters and your function can be written as, e.g.
     a + b*function(x; c,d,e...)
or
     a + b*x + function(x; c,d,e)

you should also specify these parameters via
     cf.setOffsetMultiplySlopeParams(int offsetParam, int multiplyParam,
int slopeParam)
where 'offsetParam' is the number of the parameter that is only an
offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1 for
'b' in the first example above, or 'slopeParam' would be 1 for 'b' in
the second type of function above. You cannot have a 'multiplyParam' and
a 'slopeParam' at the same time, set the unused one to -1.

Specifying an offsetParam and multiplyParam (or slopeParam) makes the
CurveFitter calculate these parameters via linear regression, so the
actual minimization does not include these parameters. In other words,
you get fewer parameters, which makes the fitting much more likely to
succeed.

In my experience, if you end up with 3-4 parameters (not counting the
parameters eliminated by setOffsetMultiplySlopeParams), there is a good
chance that the fit will work very well, with 5-6 parameters it gets
difficult, and above 6 parameters the chance to get the correct result
is rather low.

If you need to control the minimization process in detail, before
starting the fit, you can use
     Minimizer minimizer = cf.getMinimizer()
to get access to the Minimizer that will be used and you can use the
Minimizer's methods to control its behavior (e.g. allow it to do more
steps than by default by minimizer.setMaxIterations, setting it to try
more restarts, use different error values for more/less accurate
convergence, etc.

Best see the documentation in the source code, e.g.
   https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
   https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java

-------------

 > Bonus question for Java Gurus:
 > How to declare the user function as a variable, call it,
 > and pass it to another function?

public class MyFunction implements UserFunction {...
   public double userFunction(double[] params, double x) {
     return params[0]+params[1]*x;
   }
}

public class PassingClass { ...
   UserFunction exampleFunction = new MyFunction(...);
   otherClass.doFitting(xData, yData, exampleFunction)
}

Public class OtherClass { ...
   public void doFitting(double[] xData, double[] yData,
         UserFunction userFunction) {
     CurveFitter cf = new CurveFitter(xData, yData);
     cf.doCustomFit(userFunction, /*numParams=*/2, null,
         null, null, false);
   }
}

-------------

Best,

Michael
________________________________________________________________


On 13/03/2018 18:56, Fred Damen wrote:

> Below is a routine to fit MRI Inversion Recover data for T1.
>
> Note:
> a) CurveFitter comes with ImageJ.
> b) Calls are made to UserFunction once for each x.
> c) If your initial guess is not close it does not seem to converge.
>
> Bonus question for Java Gurus:
> How to declare the user function as a variable, call it, and pass it to
> another function?
>
> Enjoy,
>
> Fred
>
>     private double[] nonlinearFit(double[] x, double[] y) {
>
>        CurveFitter cf = new CurveFitter(x, y);
>        double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>
>        cf.setMaxIterations(200);
>        cf.doCustomFit(new UserFunction() {
>           @Override
>           public double userFunction(double[] p, double x) {
>              return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>              }
>           }, params.length, "", params, null, false);
>        //IJ.log(cf.getResultString());
>
>        return cf.getParams();
>        }
>
>
> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>> I have some simple data: samples of y=f(x) at regularly spaced discrete values
>> for x.  The data
>> is born as a simple array of y values, but I can turn that into (for example)
>> a polyline, if that
>> will help.  I'm currently doing that to draw the data as an Overlay.  The size
>> of the y array is between 500 and 1000. (think 1 y value for every integer
>> x-coordinate in an image - some y values may be recorded as "missing").
>>
>> Is there an ImageJ tool that will fit (more or less) arbitrary functions to
>> this data?  Approximately 8 parameters.
>>
>> The particular function I have in mind at the moment is a difference of
>> Gaussians.  The Gaussians
>> most likely have the same location (in x) - but this is not guaranteed, and
>> I'd prefer
>> to use this as a sanity check rather than impose it as a constraint.
>>
>> Note that the context is a Java plugin - not a macro.
>>
>> If not in ImageJ, perhaps someone could point me at a Java package that will
>> do this.  I am far
>> from an expert in curve fitting, so please be gentle.
>>
>> If not, I can do it in R - but I prefer to do it "on the fly" while displaying
>> the image, and the Overlay.
>>
>> --
>> Kenneth Sloan
>> [hidden email]
>> Vision is the art of seeing what is invisible to others.
>>
>> --
>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>
>
> --
> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>

--
ImageJ mailing list: http://imagej.nih.gov/ij/list.html
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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Kenneth Sloan-3
Thanks for the replies.  Looks good.

As for reducing the number of parameters - well, yes - except that all the parameters are important.  
The good news is that many of them can be estimated well enough to produce very good initial values.

For example, the centers of the two Gaussians are almost certainly very close to each other, and their location is already "known" to pretty good precision.  Amplitudes fall in a narrow range.  Even the spreads have fairly standard values based on previous work.  I'm looking for fine-grained optimization of the values, not diving in with zero a priori information.  And, if the fitter produces centers that deviate too much from the "known" values, that will be a strong signal that the model is inadequate.

On the other hand, it's not at all clear that the data are perfectly symmetric about the center of the two Gaussians - so, I may end up doing one fit to the left, and one to the right.

Mostly, I'm doing this to compare with directly measuring 5 distinct features along the curve.  Previous work fit the function to the data and computed the locations of these features using analytic means (finding the 1 min, 2 max and min&max slope - or, said another way, the min, max, and zero-crossings of the first derivative).  So far, I have had no difficulty extracting these features directly from the raw data.  But, I don't want to count on that always being the case, and I do want to compare newly acquired data with the previous methods.

My suspicion is that the direct method will work at least as well as the curve-fitting method - but I have
to make a legitimate effort to do good curve fits in order to demonstrate that.  Science, you know...

Careful reading of the above will reveal why I'm not an expert in curve fitting techniques.  I really
appreciate the help.  Thanks again.


--
Kenneth Sloan
[hidden email]
Vision is the art of seeing what is invisible to others.





> On 13 Mar 2018, at 14:30 , Michael Schmid <[hidden email]> wrote:
>
> Hi Kenneth,
>
> Concerning fitting an 8-parameter function:
>
> If the fit is not linear (as in the case of a difference of Gaussians), having 8 fit parameters is a rather ambitious task, and there is a high probability that the fit will run into a local minimum or some point that looks like a local minimum to the fitting program.
>
> It would be best to reduce the number of parameters, e.g. using a fixed ratio between the two sigma values in the Difference of Gaussians.
>
> You also need some reasonable guess for the initial values of the parameters.
>
> For the ImageJ CurveFitter, if there are many parameters it is very important to specify roughly how much the parameters can vary, these are the 'initialParamVariations'
> If cf is the CurveFitter, you will have
>    cf.doCustomFit(UserFunction userFunction, int numParams,
>        String formula, double[] initialParams,
>        double[] initialParamVariations, boolean showSettings
>
> For the initialParamVariations, use e.g. 1/10th of how much the respective parameter might deviate from the initial guess (only the order of magnitude is important).
>
> If you have many parameters and your function can be written as, e.g.
>    a + b*function(x; c,d,e...)
> or
>    a + b*x + function(x; c,d,e)
>
> you should also specify these parameters via
>    cf.setOffsetMultiplySlopeParams(int offsetParam, int multiplyParam, int slopeParam)
> where 'offsetParam' is the number of the parameter that is only an offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1 for 'b' in the first example above, or 'slopeParam' would be 1 for 'b' in the second type of function above. You cannot have a 'multiplyParam' and a 'slopeParam' at the same time, set the unused one to -1.
>
> Specifying an offsetParam and multiplyParam (or slopeParam) makes the CurveFitter calculate these parameters via linear regression, so the actual minimization does not include these parameters. In other words, you get fewer parameters, which makes the fitting much more likely to succeed.
>
> In my experience, if you end up with 3-4 parameters (not counting the parameters eliminated by setOffsetMultiplySlopeParams), there is a good chance that the fit will work very well, with 5-6 parameters it gets difficult, and above 6 parameters the chance to get the correct result is rather low.
>
> If you need to control the minimization process in detail, before starting the fit, you can use
>    Minimizer minimizer = cf.getMinimizer()
> to get access to the Minimizer that will be used and you can use the Minimizer's methods to control its behavior (e.g. allow it to do more steps than by default by minimizer.setMaxIterations, setting it to try more restarts, use different error values for more/less accurate convergence, etc.
>
> Best see the documentation in the source code, e.g.
>  https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
>  https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java
>
> -------------
>
> > Bonus question for Java Gurus:
> > How to declare the user function as a variable, call it,
> > and pass it to another function?
>
> public class MyFunction implements UserFunction {...
>  public double userFunction(double[] params, double x) {
>    return params[0]+params[1]*x;
>  }
> }
>
> public class PassingClass { ...
>  UserFunction exampleFunction = new MyFunction(...);
>  otherClass.doFitting(xData, yData, exampleFunction)
> }
>
> Public class OtherClass { ...
>  public void doFitting(double[] xData, double[] yData,
>        UserFunction userFunction) {
>    CurveFitter cf = new CurveFitter(xData, yData);
>    cf.doCustomFit(userFunction, /*numParams=*/2, null,
>        null, null, false);
>  }
> }
>
> -------------
>
> Best,
>
> Michael
> ________________________________________________________________
>
>
> On 13/03/2018 18:56, Fred Damen wrote:
>> Below is a routine to fit MRI Inversion Recover data for T1.
>> Note:
>> a) CurveFitter comes with ImageJ.
>> b) Calls are made to UserFunction once for each x.
>> c) If your initial guess is not close it does not seem to converge.
>> Bonus question for Java Gurus:
>> How to declare the user function as a variable, call it, and pass it to
>> another function?
>> Enjoy,
>> Fred
>>    private double[] nonlinearFit(double[] x, double[] y) {
>>       CurveFitter cf = new CurveFitter(x, y);
>>       double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>       cf.setMaxIterations(200);
>>       cf.doCustomFit(new UserFunction() {
>>          @Override
>>          public double userFunction(double[] p, double x) {
>>             return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>             }
>>          }, params.length, "", params, null, false);
>>       //IJ.log(cf.getResultString());
>>       return cf.getParams();
>>       }
>> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>>> I have some simple data: samples of y=f(x) at regularly spaced discrete values
>>> for x.  The data
>>> is born as a simple array of y values, but I can turn that into (for example)
>>> a polyline, if that
>>> will help.  I'm currently doing that to draw the data as an Overlay.  The size
>>> of the y array is between 500 and 1000. (think 1 y value for every integer
>>> x-coordinate in an image - some y values may be recorded as "missing").
>>>
>>> Is there an ImageJ tool that will fit (more or less) arbitrary functions to
>>> this data?  Approximately 8 parameters.
>>>
>>> The particular function I have in mind at the moment is a difference of
>>> Gaussians.  The Gaussians
>>> most likely have the same location (in x) - but this is not guaranteed, and
>>> I'd prefer
>>> to use this as a sanity check rather than impose it as a constraint.
>>>
>>> Note that the context is a Java plugin - not a macro.
>>>
>>> If not in ImageJ, perhaps someone could point me at a Java package that will
>>> do this.  I am far
>>> from an expert in curve fitting, so please be gentle.
>>>
>>> If not, I can do it in R - but I prefer to do it "on the fly" while displaying
>>> the image, and the Overlay.
>>>
>>> --
>>> Kenneth Sloan
>>> [hidden email]
>>> Vision is the art of seeing what is invisible to others.
>>>
>>> --
>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>
>> --
>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>
> --
> ImageJ mailing list: http://imagej.nih.gov/ij/list.html

--
ImageJ mailing list: http://imagej.nih.gov/ij/list.html
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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Kenneth Sloan-3
Here's a progress report, following up on all the good advice I received on CurveFitter.  Again, my thanks to all who replied.  It turned out to be simple as pi.

The attached plot is kind of busy (it's mostly for debugging), but it makes the point.

There is a sampled function t = f(x) drawn in BLACK.  It is mostly obscured by a smoothed version
of the same data, drawn over it in RED.  The DoG curve is drawn in GREEN.

I'm lazy, so I did the minimal amount of work necessary to start off with, planning on adding tweaks as needed The GREEN curve is the result of asking CurveFitter to converge with no hints whatsoever.  The userFunction has 7 parameters: mu1, sigma1, mu2, sigma2, Amplitude1, Amplitude2, offset.  

I'm very pleased with the result, so I think I'll just accept this result and move on!

The BLACK circles, and the RED and GREEN arrows show the locations of 5 key points on the curve.
They are, from left to right:
        A - max(t)        
        B - min(dt/dx)
        C - min(t)
        D - max(dt/dx)
        E - max(t)

Point C is found first, restricting the ranges for B and D.  B and D further restrict the ranges of A and E.

The BLACK results come from a simple search of the original data.  (well, not all THAT simple)

The RED results come from the same search, after smoothing and re-sampling the data

The GREEN results come from the same search, after creating a sampled version of the DoG model

I told you I was lazy - I should use calculus to find the key points on the DoG model, but the explicit search of a sampled signal was already written, so...  Note that a major plus of the DoG model is that these points are all unique.  

I'm a bit disappointed that the smoothing didn't do more to the curve - I used a Gaussian kernel that was only 7 wide (which I lifted from a Google search).  Time to bite the bullet and remember how to generate larger kernels.  I am pleased to note that the smoothing did fix one obviously wrong answer. Now, the question is whether to believe the RED or the GREEN answers.  Moving on!


--
Kenneth Sloan
[hidden email]
Vision is the art of seeing what is invisible to others.





> On 13 Mar 2018, at 18:04 , Kenneth Sloan <[hidden email]> wrote:
>
> Thanks for the replies.  Looks good.
>
> As for reducing the number of parameters - well, yes - except that all the parameters are important.  
> The good news is that many of them can be estimated well enough to produce very good initial values.
>
> For example, the centers of the two Gaussians are almost certainly very close to each other, and their location is already "known" to pretty good precision.  Amplitudes fall in a narrow range.  Even the spreads have fairly standard values based on previous work.  I'm looking for fine-grained optimization of the values, not diving in with zero a priori information.  And, if the fitter produces centers that deviate too much from the "known" values, that will be a strong signal that the model is inadequate.
>
> On the other hand, it's not at all clear that the data are perfectly symmetric about the center of the two Gaussians - so, I may end up doing one fit to the left, and one to the right.
>
> Mostly, I'm doing this to compare with directly measuring 5 distinct features along the curve.  Previous work fit the function to the data and computed the locations of these features using analytic means (finding the 1 min, 2 max and min&max slope - or, said another way, the min, max, and zero-crossings of the first derivative).  So far, I have had no difficulty extracting these features directly from the raw data.  But, I don't want to count on that always being the case, and I do want to compare newly acquired data with the previous methods.
>
> My suspicion is that the direct method will work at least as well as the curve-fitting method - but I have
> to make a legitimate effort to do good curve fits in order to demonstrate that.  Science, you know...
>
> Careful reading of the above will reveal why I'm not an expert in curve fitting techniques.  I really
> appreciate the help.  Thanks again.
>
>
> --
> Kenneth Sloan
> [hidden email]
> Vision is the art of seeing what is invisible to others.
>
>
>
>
>
>> On 13 Mar 2018, at 14:30 , Michael Schmid <[hidden email]> wrote:
>>
>> Hi Kenneth,
>>
>> Concerning fitting an 8-parameter function:
>>
>> If the fit is not linear (as in the case of a difference of Gaussians), having 8 fit parameters is a rather ambitious task, and there is a high probability that the fit will run into a local minimum or some point that looks like a local minimum to the fitting program.
>>
>> It would be best to reduce the number of parameters, e.g. using a fixed ratio between the two sigma values in the Difference of Gaussians.
>>
>> You also need some reasonable guess for the initial values of the parameters.
>>
>> For the ImageJ CurveFitter, if there are many parameters it is very important to specify roughly how much the parameters can vary, these are the 'initialParamVariations'
>> If cf is the CurveFitter, you will have
>>   cf.doCustomFit(UserFunction userFunction, int numParams,
>>       String formula, double[] initialParams,
>>       double[] initialParamVariations, boolean showSettings
>>
>> For the initialParamVariations, use e.g. 1/10th of how much the respective parameter might deviate from the initial guess (only the order of magnitude is important).
>>
>> If you have many parameters and your function can be written as, e.g.
>>   a + b*function(x; c,d,e...)
>> or
>>   a + b*x + function(x; c,d,e)
>>
>> you should also specify these parameters via
>>   cf.setOffsetMultiplySlopeParams(int offsetParam, int multiplyParam, int slopeParam)
>> where 'offsetParam' is the number of the parameter that is only an offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1 for 'b' in the first example above, or 'slopeParam' would be 1 for 'b' in the second type of function above. You cannot have a 'multiplyParam' and a 'slopeParam' at the same time, set the unused one to -1.
>>
>> Specifying an offsetParam and multiplyParam (or slopeParam) makes the CurveFitter calculate these parameters via linear regression, so the actual minimization does not include these parameters. In other words, you get fewer parameters, which makes the fitting much more likely to succeed.
>>
>> In my experience, if you end up with 3-4 parameters (not counting the parameters eliminated by setOffsetMultiplySlopeParams), there is a good chance that the fit will work very well, with 5-6 parameters it gets difficult, and above 6 parameters the chance to get the correct result is rather low.
>>
>> If you need to control the minimization process in detail, before starting the fit, you can use
>>   Minimizer minimizer = cf.getMinimizer()
>> to get access to the Minimizer that will be used and you can use the Minimizer's methods to control its behavior (e.g. allow it to do more steps than by default by minimizer.setMaxIterations, setting it to try more restarts, use different error values for more/less accurate convergence, etc.
>>
>> Best see the documentation in the source code, e.g.
>> https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
>> https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java
>>
>> -------------
>>
>>> Bonus question for Java Gurus:
>>> How to declare the user function as a variable, call it,
>>> and pass it to another function?
>>
>> public class MyFunction implements UserFunction {...
>> public double userFunction(double[] params, double x) {
>>   return params[0]+params[1]*x;
>> }
>> }
>>
>> public class PassingClass { ...
>> UserFunction exampleFunction = new MyFunction(...);
>> otherClass.doFitting(xData, yData, exampleFunction)
>> }
>>
>> Public class OtherClass { ...
>> public void doFitting(double[] xData, double[] yData,
>>       UserFunction userFunction) {
>>   CurveFitter cf = new CurveFitter(xData, yData);
>>   cf.doCustomFit(userFunction, /*numParams=*/2, null,
>>       null, null, false);
>> }
>> }
>>
>> -------------
>>
>> Best,
>>
>> Michael
>> ________________________________________________________________
>>
>>
>> On 13/03/2018 18:56, Fred Damen wrote:
>>> Below is a routine to fit MRI Inversion Recover data for T1.
>>> Note:
>>> a) CurveFitter comes with ImageJ.
>>> b) Calls are made to UserFunction once for each x.
>>> c) If your initial guess is not close it does not seem to converge.
>>> Bonus question for Java Gurus:
>>> How to declare the user function as a variable, call it, and pass it to
>>> another function?
>>> Enjoy,
>>> Fred
>>>   private double[] nonlinearFit(double[] x, double[] y) {
>>>      CurveFitter cf = new CurveFitter(x, y);
>>>      double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>      cf.setMaxIterations(200);
>>>      cf.doCustomFit(new UserFunction() {
>>>         @Override
>>>         public double userFunction(double[] p, double x) {
>>>            return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>            }
>>>         }, params.length, "", params, null, false);
>>>      //IJ.log(cf.getResultString());
>>>      return cf.getParams();
>>>      }
>>> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>>>> I have some simple data: samples of y=f(x) at regularly spaced discrete values
>>>> for x.  The data
>>>> is born as a simple array of y values, but I can turn that into (for example)
>>>> a polyline, if that
>>>> will help.  I'm currently doing that to draw the data as an Overlay.  The size
>>>> of the y array is between 500 and 1000. (think 1 y value for every integer
>>>> x-coordinate in an image - some y values may be recorded as "missing").
>>>>
>>>> Is there an ImageJ tool that will fit (more or less) arbitrary functions to
>>>> this data?  Approximately 8 parameters.
>>>>
>>>> The particular function I have in mind at the moment is a difference of
>>>> Gaussians.  The Gaussians
>>>> most likely have the same location (in x) - but this is not guaranteed, and
>>>> I'd prefer
>>>> to use this as a sanity check rather than impose it as a constraint.
>>>>
>>>> Note that the context is a Java plugin - not a macro.
>>>>
>>>> If not in ImageJ, perhaps someone could point me at a Java package that will
>>>> do this.  I am far
>>>> from an expert in curve fitting, so please be gentle.
>>>>
>>>> If not, I can do it in R - but I prefer to do it "on the fly" while displaying
>>>> the image, and the Overlay.
>>>>
>>>> --
>>>> Kenneth Sloan
>>>> [hidden email]
>>>> Vision is the art of seeing what is invisible to others.
>>>>
>>>> --
>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>
>>> --
>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>
>> --
>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>


--
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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Michael Schmid-3
Hi Kenneth,

looking at your curves, I have the impression the 'negative' peak does
not look like a Gaussian - its peak is sharper and the tails are wider
than for a Gaussian.
If this is typical for your data, you might try other shapes like a
Lorentzian,
   y = a/(1+(x-c)^2/b^2)
If the Lorentzian is too sharp and its tails are too strong, you could
also try other function.
Both the inflection points and the maxima will change depending on the
fit function, so it makes sense to try which function fits best.
(Obviously, this should be done with many curves, not just one).

Michael
________________________________________________________________


On 2018-03-17 19:00, Kenneth Sloan wrote:

> Here's a progress report, following up on all the good advice I
> received on CurveFitter.  Again, my thanks to all who replied.  It
> turned out to be simple as pi.
>
> The attached plot is kind of busy (it's mostly for debugging), but it
> makes the point.
>
> There is a sampled function t = f(x) drawn in BLACK.  It is mostly
> obscured by a smoothed version
> of the same data, drawn over it in RED.  The DoG curve is drawn in
> GREEN.
>
> I'm lazy, so I did the minimal amount of work necessary to start off
> with, planning on adding tweaks as needed The GREEN curve is the
> result of asking CurveFitter to converge with no hints whatsoever.
> The userFunction has 7 parameters: mu1, sigma1, mu2, sigma2,
> Amplitude1, Amplitude2, offset.
>
> I'm very pleased with the result, so I think I'll just accept this
> result and move on!
>
> The BLACK circles, and the RED and GREEN arrows show the locations of
> 5 key points on the curve.
> They are, from left to right:
> A - max(t)
> B - min(dt/dx)
> C - min(t)
> D - max(dt/dx)
> E - max(t)
>
> Point C is found first, restricting the ranges for B and D.  B and D
> further restrict the ranges of A and E.
>
> The BLACK results come from a simple search of the original data.
> (well, not all THAT simple)
>
> The RED results come from the same search, after smoothing and
> re-sampling the data
>
> The GREEN results come from the same search, after creating a sampled
> version of the DoG model
>
> I told you I was lazy - I should use calculus to find the key points
> on the DoG model, but the explicit search of a sampled signal was
> already written, so...  Note that a major plus of the DoG model is
> that these points are all unique.
>
> I'm a bit disappointed that the smoothing didn't do more to the curve
> - I used a Gaussian kernel that was only 7 wide (which I lifted from a
> Google search).  Time to bite the bullet and remember how to generate
> larger kernels.  I am pleased to note that the smoothing did fix one
> obviously wrong answer. Now, the question is whether to believe the
> RED or the GREEN answers.  Moving on!
>
>
> --
> Kenneth Sloan
> [hidden email]
> Vision is the art of seeing what is invisible to others.
>
>
>
>
>
>> On 13 Mar 2018, at 18:04 , Kenneth Sloan <[hidden email]>
>> wrote:
>>
>> Thanks for the replies.  Looks good.
>>
>> As for reducing the number of parameters - well, yes - except that all
>> the parameters are important.
>> The good news is that many of them can be estimated well enough to
>> produce very good initial values.
>>
>> For example, the centers of the two Gaussians are almost certainly
>> very close to each other, and their location is already "known" to
>> pretty good precision.  Amplitudes fall in a narrow range.  Even the
>> spreads have fairly standard values based on previous work.  I'm
>> looking for fine-grained optimization of the values, not diving in
>> with zero a priori information.  And, if the fitter produces centers
>> that deviate too much from the "known" values, that will be a strong
>> signal that the model is inadequate.
>>
>> On the other hand, it's not at all clear that the data are perfectly
>> symmetric about the center of the two Gaussians - so, I may end up
>> doing one fit to the left, and one to the right.
>>
>> Mostly, I'm doing this to compare with directly measuring 5 distinct
>> features along the curve.  Previous work fit the function to the data
>> and computed the locations of these features using analytic means
>> (finding the 1 min, 2 max and min&max slope - or, said another way,
>> the min, max, and zero-crossings of the first derivative).  So far, I
>> have had no difficulty extracting these features directly from the raw
>> data.  But, I don't want to count on that always being the case, and I
>> do want to compare newly acquired data with the previous methods.
>>
>> My suspicion is that the direct method will work at least as well as
>> the curve-fitting method - but I have
>> to make a legitimate effort to do good curve fits in order to
>> demonstrate that.  Science, you know...
>>
>> Careful reading of the above will reveal why I'm not an expert in
>> curve fitting techniques.  I really
>> appreciate the help.  Thanks again.
>>
>>
>> --
>> Kenneth Sloan
>> [hidden email]
>> Vision is the art of seeing what is invisible to others.
>>
>>
>>
>>
>>
>>> On 13 Mar 2018, at 14:30 , Michael Schmid <[hidden email]>
>>> wrote:
>>>
>>> Hi Kenneth,
>>>
>>> Concerning fitting an 8-parameter function:
>>>
>>> If the fit is not linear (as in the case of a difference of
>>> Gaussians), having 8 fit parameters is a rather ambitious task, and
>>> there is a high probability that the fit will run into a local
>>> minimum or some point that looks like a local minimum to the fitting
>>> program.
>>>
>>> It would be best to reduce the number of parameters, e.g. using a
>>> fixed ratio between the two sigma values in the Difference of
>>> Gaussians.
>>>
>>> You also need some reasonable guess for the initial values of the
>>> parameters.
>>>
>>> For the ImageJ CurveFitter, if there are many parameters it is very
>>> important to specify roughly how much the parameters can vary, these
>>> are the 'initialParamVariations'
>>> If cf is the CurveFitter, you will have
>>>   cf.doCustomFit(UserFunction userFunction, int numParams,
>>>       String formula, double[] initialParams,
>>>       double[] initialParamVariations, boolean showSettings
>>>
>>> For the initialParamVariations, use e.g. 1/10th of how much the
>>> respective parameter might deviate from the initial guess (only the
>>> order of magnitude is important).
>>>
>>> If you have many parameters and your function can be written as, e.g.
>>>   a + b*function(x; c,d,e...)
>>> or
>>>   a + b*x + function(x; c,d,e)
>>>
>>> you should also specify these parameters via
>>>   cf.setOffsetMultiplySlopeParams(int offsetParam, int multiplyParam,
>>> int slopeParam)
>>> where 'offsetParam' is the number of the parameter that is only an
>>> offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1
>>> for 'b' in the first example above, or 'slopeParam' would be 1 for
>>> 'b' in the second type of function above. You cannot have a
>>> 'multiplyParam' and a 'slopeParam' at the same time, set the unused
>>> one to -1.
>>>
>>> Specifying an offsetParam and multiplyParam (or slopeParam) makes the
>>> CurveFitter calculate these parameters via linear regression, so the
>>> actual minimization does not include these parameters. In other
>>> words, you get fewer parameters, which makes the fitting much more
>>> likely to succeed.
>>>
>>> In my experience, if you end up with 3-4 parameters (not counting the
>>> parameters eliminated by setOffsetMultiplySlopeParams), there is a
>>> good chance that the fit will work very well, with 5-6 parameters it
>>> gets difficult, and above 6 parameters the chance to get the correct
>>> result is rather low.
>>>
>>> If you need to control the minimization process in detail, before
>>> starting the fit, you can use
>>>   Minimizer minimizer = cf.getMinimizer()
>>> to get access to the Minimizer that will be used and you can use the
>>> Minimizer's methods to control its behavior (e.g. allow it to do more
>>> steps than by default by minimizer.setMaxIterations, setting it to
>>> try more restarts, use different error values for more/less accurate
>>> convergence, etc.
>>>
>>> Best see the documentation in the source code, e.g.
>>> https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
>>> https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java
>>>
>>> -------------
>>>
>>>> Bonus question for Java Gurus:
>>>> How to declare the user function as a variable, call it,
>>>> and pass it to another function?
>>>
>>> public class MyFunction implements UserFunction {...
>>> public double userFunction(double[] params, double x) {
>>>   return params[0]+params[1]*x;
>>> }
>>> }
>>>
>>> public class PassingClass { ...
>>> UserFunction exampleFunction = new MyFunction(...);
>>> otherClass.doFitting(xData, yData, exampleFunction)
>>> }
>>>
>>> Public class OtherClass { ...
>>> public void doFitting(double[] xData, double[] yData,
>>>       UserFunction userFunction) {
>>>   CurveFitter cf = new CurveFitter(xData, yData);
>>>   cf.doCustomFit(userFunction, /*numParams=*/2, null,
>>>       null, null, false);
>>> }
>>> }
>>>
>>> -------------
>>>
>>> Best,
>>>
>>> Michael
>>> ________________________________________________________________
>>>
>>>
>>> On 13/03/2018 18:56, Fred Damen wrote:
>>>> Below is a routine to fit MRI Inversion Recover data for T1.
>>>> Note:
>>>> a) CurveFitter comes with ImageJ.
>>>> b) Calls are made to UserFunction once for each x.
>>>> c) If your initial guess is not close it does not seem to converge.
>>>> Bonus question for Java Gurus:
>>>> How to declare the user function as a variable, call it, and pass it
>>>> to
>>>> another function?
>>>> Enjoy,
>>>> Fred
>>>>   private double[] nonlinearFit(double[] x, double[] y) {
>>>>      CurveFitter cf = new CurveFitter(x, y);
>>>>      double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>>      cf.setMaxIterations(200);
>>>>      cf.doCustomFit(new UserFunction() {
>>>>         @Override
>>>>         public double userFunction(double[] p, double x) {
>>>>            return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>>            }
>>>>         }, params.length, "", params, null, false);
>>>>      //IJ.log(cf.getResultString());
>>>>      return cf.getParams();
>>>>      }
>>>> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>>>>> I have some simple data: samples of y=f(x) at regularly spaced
>>>>> discrete values
>>>>> for x.  The data
>>>>> is born as a simple array of y values, but I can turn that into
>>>>> (for example)
>>>>> a polyline, if that
>>>>> will help.  I'm currently doing that to draw the data as an
>>>>> Overlay.  The size
>>>>> of the y array is between 500 and 1000. (think 1 y value for every
>>>>> integer
>>>>> x-coordinate in an image - some y values may be recorded as
>>>>> "missing").
>>>>>
>>>>> Is there an ImageJ tool that will fit (more or less) arbitrary
>>>>> functions to
>>>>> this data?  Approximately 8 parameters.
>>>>>
>>>>> The particular function I have in mind at the moment is a
>>>>> difference of
>>>>> Gaussians.  The Gaussians
>>>>> most likely have the same location (in x) - but this is not
>>>>> guaranteed, and
>>>>> I'd prefer
>>>>> to use this as a sanity check rather than impose it as a
>>>>> constraint.
>>>>>
>>>>> Note that the context is a Java plugin - not a macro.
>>>>>
>>>>> If not in ImageJ, perhaps someone could point me at a Java package
>>>>> that will
>>>>> do this.  I am far
>>>>> from an expert in curve fitting, so please be gentle.
>>>>>
>>>>> If not, I can do it in R - but I prefer to do it "on the fly" while
>>>>> displaying
>>>>> the image, and the Overlay.
>>>>>
>>>>> --
>>>>> Kenneth Sloan
>>>>> [hidden email]
>>>>> Vision is the art of seeing what is invisible to others.
>>>>>
>>>>> --
>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>>
>>>> --
>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>
>>> --
>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>
>
>
> --
> ImageJ mailing list: http://imagej.nih.gov/ij/list.html

--
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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Kenneth Sloan-3
I agree - there are lots of possible models.  This was an attempt to directly replicate (with slightly different inputs) previous work in the literature, so the first task was to exactly match the model used there.  It looks like I can check off that box and move on to the next step.

DoG is a favorite for most folk who learned neuroscience or human vision in the 1970's.  So, it's a perfectly respectable starting point.  But...it's just a starting point.

One problem with this model is the implicit assumption that there ought to be some symmetry.  In this example, it's quite clear that the two Gaussians have different means (and CurveFitter confirms this).  One thing I've been thinking about is fitting various models to only HALF the curve.  Once this choice is made, notice that the possible models get dramatically simpler.  My first thought is simply a cubic, perhaps with the derivative constrained to be 0.0 at the center point - perhaps clipped at the point where the derivative once again becomes 0.0.  We're mostly interested in the central "pit".
 
But...and this is a big but...people sometimes confuse the measurement model with a model of causality.  The main point of this fitting is to find consistent measurements for the 5 points shown.  They lead to descriptors such as the DEPTH and WIDTH of the central "pit".  Depending on your taste, it might be argued that the plot demonstrates two different methods that produce highly similar results.  The question might become - in less "normal" examples, where the two methods (the GREEN and the RED) differ, which one should you believe?  And also - how significant are the differences in this one example, compared to the distribution of results in a large population.

On the other hand, the negative Gaussian may turn out to be what we really want to measure, perhaps considering the broader positive Gaussian as "noise".  That would make the DoG model very relevant.

Finally, there are biological justifications for using a model like the DoG, based on mechanisms that might have given rise to the original data.  Other models might fit the raw data better, but they might not have as much explanatory power.

As you say - more curves!  I'm about to have a fairly large dataset to play with, so that's coming.
Again, one motivation for today is to be prepared to produce answers for this new dataset that are
compatible with previously reported literature.

Given my druthers, I'm pretty happy with simply smoothing the raw data and directly measuring the 5 key points.  There are minor difficulties having to do with the fact that the answers are not necessarily unique - but these are tractable.  My primary motivation in fitting the DoG was to match previous methods.

My congratulations to the authors of CurveFitter.  It was easy to use (given your guidance), and is clearly capable of handling MUCH more difficult tasks than this one.  It did just fine, with absolutely zero hints.
Thanks again for your help.

--
Kenneth Sloan
[hidden email]
Vision is the art of seeing what is invisible to others.





> On 17 Mar 2018, at 14:05 , Michael Schmid <[hidden email]> wrote:
>
> Hi Kenneth,
>
> looking at your curves, I have the impression the 'negative' peak does not look like a Gaussian - its peak is sharper and the tails are wider than for a Gaussian.
> If this is typical for your data, you might try other shapes like a Lorentzian,
>  y = a/(1+(x-c)^2/b^2)
> If the Lorentzian is too sharp and its tails are too strong, you could also try other function.
> Both the inflection points and the maxima will change depending on the fit function, so it makes sense to try which function fits best.
> (Obviously, this should be done with many curves, not just one).
>
> Michael
> ________________________________________________________________
>
>
> On 2018-03-17 19:00, Kenneth Sloan wrote:
>> Here's a progress report, following up on all the good advice I
>> received on CurveFitter.  Again, my thanks to all who replied.  It
>> turned out to be simple as pi.
>> The attached plot is kind of busy (it's mostly for debugging), but it
>> makes the point.
>> There is a sampled function t = f(x) drawn in BLACK.  It is mostly
>> obscured by a smoothed version
>> of the same data, drawn over it in RED.  The DoG curve is drawn in GREEN.
>> I'm lazy, so I did the minimal amount of work necessary to start off
>> with, planning on adding tweaks as needed The GREEN curve is the
>> result of asking CurveFitter to converge with no hints whatsoever.
>> The userFunction has 7 parameters: mu1, sigma1, mu2, sigma2,
>> Amplitude1, Amplitude2, offset.
>> I'm very pleased with the result, so I think I'll just accept this
>> result and move on!
>> The BLACK circles, and the RED and GREEN arrows show the locations of
>> 5 key points on the curve.
>> They are, from left to right:
>> A - max(t)
>> B - min(dt/dx)
>> C - min(t)
>> D - max(dt/dx)
>> E - max(t)
>> Point C is found first, restricting the ranges for B and D.  B and D
>> further restrict the ranges of A and E.
>> The BLACK results come from a simple search of the original data.
>> (well, not all THAT simple)
>> The RED results come from the same search, after smoothing and
>> re-sampling the data
>> The GREEN results come from the same search, after creating a sampled
>> version of the DoG model
>> I told you I was lazy - I should use calculus to find the key points
>> on the DoG model, but the explicit search of a sampled signal was
>> already written, so...  Note that a major plus of the DoG model is
>> that these points are all unique.
>> I'm a bit disappointed that the smoothing didn't do more to the curve
>> - I used a Gaussian kernel that was only 7 wide (which I lifted from a
>> Google search).  Time to bite the bullet and remember how to generate
>> larger kernels.  I am pleased to note that the smoothing did fix one
>> obviously wrong answer. Now, the question is whether to believe the
>> RED or the GREEN answers.  Moving on!
>> --
>> Kenneth Sloan
>> [hidden email]
>> Vision is the art of seeing what is invisible to others.
>>> On 13 Mar 2018, at 18:04 , Kenneth Sloan <[hidden email]> wrote:
>>> Thanks for the replies.  Looks good.
>>> As for reducing the number of parameters - well, yes - except that all the parameters are important.
>>> The good news is that many of them can be estimated well enough to produce very good initial values.
>>> For example, the centers of the two Gaussians are almost certainly very close to each other, and their location is already "known" to pretty good precision.  Amplitudes fall in a narrow range.  Even the spreads have fairly standard values based on previous work.  I'm looking for fine-grained optimization of the values, not diving in with zero a priori information.  And, if the fitter produces centers that deviate too much from the "known" values, that will be a strong signal that the model is inadequate.
>>> On the other hand, it's not at all clear that the data are perfectly symmetric about the center of the two Gaussians - so, I may end up doing one fit to the left, and one to the right.
>>> Mostly, I'm doing this to compare with directly measuring 5 distinct features along the curve.  Previous work fit the function to the data and computed the locations of these features using analytic means (finding the 1 min, 2 max and min&max slope - or, said another way, the min, max, and zero-crossings of the first derivative).  So far, I have had no difficulty extracting these features directly from the raw data.  But, I don't want to count on that always being the case, and I do want to compare newly acquired data with the previous methods.
>>> My suspicion is that the direct method will work at least as well as the curve-fitting method - but I have
>>> to make a legitimate effort to do good curve fits in order to demonstrate that.  Science, you know...
>>> Careful reading of the above will reveal why I'm not an expert in curve fitting techniques.  I really
>>> appreciate the help.  Thanks again.
>>> --
>>> Kenneth Sloan
>>> [hidden email]
>>> Vision is the art of seeing what is invisible to others.
>>>> On 13 Mar 2018, at 14:30 , Michael Schmid <[hidden email]> wrote:
>>>> Hi Kenneth,
>>>> Concerning fitting an 8-parameter function:
>>>> If the fit is not linear (as in the case of a difference of Gaussians), having 8 fit parameters is a rather ambitious task, and there is a high probability that the fit will run into a local minimum or some point that looks like a local minimum to the fitting program.
>>>> It would be best to reduce the number of parameters, e.g. using a fixed ratio between the two sigma values in the Difference of Gaussians.
>>>> You also need some reasonable guess for the initial values of the parameters.
>>>> For the ImageJ CurveFitter, if there are many parameters it is very important to specify roughly how much the parameters can vary, these are the 'initialParamVariations'
>>>> If cf is the CurveFitter, you will have
>>>>  cf.doCustomFit(UserFunction userFunction, int numParams,
>>>>      String formula, double[] initialParams,
>>>>      double[] initialParamVariations, boolean showSettings
>>>> For the initialParamVariations, use e.g. 1/10th of how much the respective parameter might deviate from the initial guess (only the order of magnitude is important).
>>>> If you have many parameters and your function can be written as, e.g.
>>>>  a + b*function(x; c,d,e...)
>>>> or
>>>>  a + b*x + function(x; c,d,e)
>>>> you should also specify these parameters via
>>>>  cf.setOffsetMultiplySlopeParams(int offsetParam, int multiplyParam, int slopeParam)
>>>> where 'offsetParam' is the number of the parameter that is only an offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1 for 'b' in the first example above, or 'slopeParam' would be 1 for 'b' in the second type of function above. You cannot have a 'multiplyParam' and a 'slopeParam' at the same time, set the unused one to -1.
>>>> Specifying an offsetParam and multiplyParam (or slopeParam) makes the CurveFitter calculate these parameters via linear regression, so the actual minimization does not include these parameters. In other words, you get fewer parameters, which makes the fitting much more likely to succeed.
>>>> In my experience, if you end up with 3-4 parameters (not counting the parameters eliminated by setOffsetMultiplySlopeParams), there is a good chance that the fit will work very well, with 5-6 parameters it gets difficult, and above 6 parameters the chance to get the correct result is rather low.
>>>> If you need to control the minimization process in detail, before starting the fit, you can use
>>>>  Minimizer minimizer = cf.getMinimizer()
>>>> to get access to the Minimizer that will be used and you can use the Minimizer's methods to control its behavior (e.g. allow it to do more steps than by default by minimizer.setMaxIterations, setting it to try more restarts, use different error values for more/less accurate convergence, etc.
>>>> Best see the documentation in the source code, e.g.
>>>> https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
>>>> https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java
>>>> -------------
>>>>> Bonus question for Java Gurus:
>>>>> How to declare the user function as a variable, call it,
>>>>> and pass it to another function?
>>>> public class MyFunction implements UserFunction {...
>>>> public double userFunction(double[] params, double x) {
>>>>  return params[0]+params[1]*x;
>>>> }
>>>> }
>>>> public class PassingClass { ...
>>>> UserFunction exampleFunction = new MyFunction(...);
>>>> otherClass.doFitting(xData, yData, exampleFunction)
>>>> }
>>>> Public class OtherClass { ...
>>>> public void doFitting(double[] xData, double[] yData,
>>>>      UserFunction userFunction) {
>>>>  CurveFitter cf = new CurveFitter(xData, yData);
>>>>  cf.doCustomFit(userFunction, /*numParams=*/2, null,
>>>>      null, null, false);
>>>> }
>>>> }
>>>> -------------
>>>> Best,
>>>> Michael
>>>> ________________________________________________________________
>>>> On 13/03/2018 18:56, Fred Damen wrote:
>>>>> Below is a routine to fit MRI Inversion Recover data for T1.
>>>>> Note:
>>>>> a) CurveFitter comes with ImageJ.
>>>>> b) Calls are made to UserFunction once for each x.
>>>>> c) If your initial guess is not close it does not seem to converge.
>>>>> Bonus question for Java Gurus:
>>>>> How to declare the user function as a variable, call it, and pass it to
>>>>> another function?
>>>>> Enjoy,
>>>>> Fred
>>>>>  private double[] nonlinearFit(double[] x, double[] y) {
>>>>>     CurveFitter cf = new CurveFitter(x, y);
>>>>>     double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>>>     cf.setMaxIterations(200);
>>>>>     cf.doCustomFit(new UserFunction() {
>>>>>        @Override
>>>>>        public double userFunction(double[] p, double x) {
>>>>>           return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>>>           }
>>>>>        }, params.length, "", params, null, false);
>>>>>     //IJ.log(cf.getResultString());
>>>>>     return cf.getParams();
>>>>>     }
>>>>> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>>>>>> I have some simple data: samples of y=f(x) at regularly spaced discrete values
>>>>>> for x.  The data
>>>>>> is born as a simple array of y values, but I can turn that into (for example)
>>>>>> a polyline, if that
>>>>>> will help.  I'm currently doing that to draw the data as an Overlay.  The size
>>>>>> of the y array is between 500 and 1000. (think 1 y value for every integer
>>>>>> x-coordinate in an image - some y values may be recorded as "missing").
>>>>>> Is there an ImageJ tool that will fit (more or less) arbitrary functions to
>>>>>> this data?  Approximately 8 parameters.
>>>>>> The particular function I have in mind at the moment is a difference of
>>>>>> Gaussians.  The Gaussians
>>>>>> most likely have the same location (in x) - but this is not guaranteed, and
>>>>>> I'd prefer
>>>>>> to use this as a sanity check rather than impose it as a constraint.
>>>>>> Note that the context is a Java plugin - not a macro.
>>>>>> If not in ImageJ, perhaps someone could point me at a Java package that will
>>>>>> do this.  I am far
>>>>>> from an expert in curve fitting, so please be gentle.
>>>>>> If not, I can do it in R - but I prefer to do it "on the fly" while displaying
>>>>>> the image, and the Overlay.
>>>>>> --
>>>>>> Kenneth Sloan
>>>>>> [hidden email]
>>>>>> Vision is the art of seeing what is invisible to others.
>>>>>> --
>>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>> --
>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>> --
>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>> --
>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>
> --
> ImageJ mailing list: http://imagej.nih.gov/ij/list.html

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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Fred Damen
In reply to this post by Michael Schmid-3
Greetings,

First, thanks for expounding on CurveFitter.

I have a few questions and a bug.

a) Is there any way to keep the fitting within a given range for each
parameter?  On fits to some data the parameters returned are obnoxiously bad.
The input data resembles data that fits reasonably, i.e., I do not suspect
that the initial parameters are outside the local minimum.  This is not using
setOffsetMultiplySlopeParams.

b) When using the method setOffsetMultiplySlopeParams and passing a value to
doCustionFit's initialParamVariations parameter, set to what is documented as
the default value that would be used if null is passed, the fitting is
effectively a linear fit; see example below.  A reasonable fit is had if the
call to setOffsetMultiplySlopeParams is commented out.  If this worked what
kind of improvement would I expect to see in efficiency or goodnees-of-fit?

b) When using the method setOffsetMultiplySlopeParams and passing a null value
to doCustionFit's initialParamVariations parameter, a null pointer exception
is thrown:
ImageJ 1.50h; Java 1.8.0_91 [64-bit]; Linux 4.5.5-300.fc24.x86_64; 287MB of
1820MB (15%)

java.lang.NullPointerException
        at ij.measure.CurveFitter.modifyInitialParamsAndVariations(CurveFitter.java:835)
        at ij.measure.CurveFitter.doFit(CurveFitter.java:178)
        at ij.measure.CurveFitter.doCustomFit(CurveFitter.java:283)
        at My_Plugin2.nonlinearFit(My_Plugin2.java:38)
        at My_Plugin2.run(My_Plugin2.java:14)
        at ij.plugin.PlugInExecuter.runCompiledPlugin(Compiler.java:318)
        at ij.plugin.PlugInExecuter.run(Compiler.java:307)
        at java.lang.Thread.run(Thread.java:745)

Thanks in advance,

Fred

----------------------------
import ij.*;
import ij.process.*;
import ij.gui.*;
import java.awt.*;
import ij.plugin.*;
import ij.plugin.frame.*;
import ij.measure.*;

public class My_Plugin2 implements PlugIn {

  public void run(String arg) {
    double[] x = {0.10,0.25,0.50,0.75,1.00,1.25,1.50,1.75,2.00,5.00};
    double[] y =
{1123.175,838.206,469.320,453.003,725.135,1094.360,1450.741,1787.361,2128.119,4670.120};
    double[] p = nonlinearFit(x,y);
    Plot plot = new Plot("","x","y");
    plot.setLineWidth(2);
    plot.setColor(Color.black);
    plot.addPoints(x,y,PlotWindow.X);
    double[] y2 = new double[y.length];
    for(int i=0; i<y.length; i++)
       y2[i] = Math.abs( p[1] + p[2]*Math.exp(-x[i]/p[0]) );
    plot.setColor(Color.blue);
    plot.addPoints(x,y2,PlotWindow.LINE);
    plot.show();

    }

   private double[] nonlinearFit(double[] x, double[] y) {

      CurveFitter cf = new CurveFitter(x, y);
      double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
      double[] ipv = new double[params.length];
      for(int i=0; i<ipv.length; i++)
         ipv[i] = Math.abs(params[i]*0.1);

      cf.setMaxIterations(200);
      cf.setOffsetMultiplySlopeParams(1, 2, -1);
      cf.doCustomFit(new UserFunction() {
         @Override
         public double userFunction(double[] p, double x) {
            return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
            }
         }, params.length, "", params, ipv, false);
      //IJ.log(cf.getResultString());

      return cf.getParams();
      }
}



On Tue, March 13, 2018 2:30 pm, Michael Schmid wrote:

> Hi Kenneth,
>
> Concerning fitting an 8-parameter function:
>
> If the fit is not linear (as in the case of a difference of Gaussians),
> having 8 fit parameters is a rather ambitious task, and there is a high
> probability that the fit will run into a local minimum or some point
> that looks like a local minimum to the fitting program.
>
> It would be best to reduce the number of parameters, e.g. using a fixed
> ratio between the two sigma values in the Difference of Gaussians.
>
> You also need some reasonable guess for the initial values of the
> parameters.
>
> For the ImageJ CurveFitter, if there are many parameters it is very
> important to specify roughly how much the parameters can vary, these are
> the 'initialParamVariations'
> If cf is the CurveFitter, you will have
>      cf.doCustomFit(UserFunction userFunction, int numParams,
>          String formula, double[] initialParams,
>          double[] initialParamVariations, boolean showSettings
>
> For the initialParamVariations, use e.g. 1/10th of how much the
> respective parameter might deviate from the initial guess (only the
> order of magnitude is important).
>
> If you have many parameters and your function can be written as, e.g.
>      a + b*function(x; c,d,e...)
> or
>      a + b*x + function(x; c,d,e)
>
> you should also specify these parameters via
>      cf.setOffsetMultiplySlopeParams(int offsetParam, int multiplyParam,
> int slopeParam)
> where 'offsetParam' is the number of the parameter that is only an
> offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1 for
> 'b' in the first example above, or 'slopeParam' would be 1 for 'b' in
> the second type of function above. You cannot have a 'multiplyParam' and
> a 'slopeParam' at the same time, set the unused one to -1.
>
> Specifying an offsetParam and multiplyParam (or slopeParam) makes the
> CurveFitter calculate these parameters via linear regression, so the
> actual minimization does not include these parameters. In other words,
> you get fewer parameters, which makes the fitting much more likely to
> succeed.
>
> In my experience, if you end up with 3-4 parameters (not counting the
> parameters eliminated by setOffsetMultiplySlopeParams), there is a good
> chance that the fit will work very well, with 5-6 parameters it gets
> difficult, and above 6 parameters the chance to get the correct result
> is rather low.
>
> If you need to control the minimization process in detail, before
> starting the fit, you can use
>      Minimizer minimizer = cf.getMinimizer()
> to get access to the Minimizer that will be used and you can use the
> Minimizer's methods to control its behavior (e.g. allow it to do more
> steps than by default by minimizer.setMaxIterations, setting it to try
> more restarts, use different error values for more/less accurate
> convergence, etc.
>
> Best see the documentation in the source code, e.g.
>    https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
>    https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java
>
> -------------
>
>  > Bonus question for Java Gurus:
>  > How to declare the user function as a variable, call it,
>  > and pass it to another function?
>
> public class MyFunction implements UserFunction {...
>    public double userFunction(double[] params, double x) {
>      return params[0]+params[1]*x;
>    }
> }
>
> public class PassingClass { ...
>    UserFunction exampleFunction = new MyFunction(...);
>    otherClass.doFitting(xData, yData, exampleFunction)
> }
>
> Public class OtherClass { ...
>    public void doFitting(double[] xData, double[] yData,
>          UserFunction userFunction) {
>      CurveFitter cf = new CurveFitter(xData, yData);
>      cf.doCustomFit(userFunction, /*numParams=*/2, null,
>          null, null, false);
>    }
> }
>
> -------------
>
> Best,
>
> Michael
> ________________________________________________________________
>
>
> On 13/03/2018 18:56, Fred Damen wrote:
>> Below is a routine to fit MRI Inversion Recover data for T1.
>>
>> Note:
>> a) CurveFitter comes with ImageJ.
>> b) Calls are made to UserFunction once for each x.
>> c) If your initial guess is not close it does not seem to converge.
>>
>> Bonus question for Java Gurus:
>> How to declare the user function as a variable, call it, and pass it to
>> another function?
>>
>> Enjoy,
>>
>> Fred
>>
>>     private double[] nonlinearFit(double[] x, double[] y) {
>>
>>        CurveFitter cf = new CurveFitter(x, y);
>>        double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>
>>        cf.setMaxIterations(200);
>>        cf.doCustomFit(new UserFunction() {
>>           @Override
>>           public double userFunction(double[] p, double x) {
>>              return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>              }
>>           }, params.length, "", params, null, false);
>>        //IJ.log(cf.getResultString());
>>
>>        return cf.getParams();
>>        }
>>
>>
>> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>>> I have some simple data: samples of y=f(x) at regularly spaced discrete
>>> values
>>> for x.  The data
>>> is born as a simple array of y values, but I can turn that into (for
>>> example)
>>> a polyline, if that
>>> will help.  I'm currently doing that to draw the data as an Overlay.  The
>>> size
>>> of the y array is between 500 and 1000. (think 1 y value for every integer
>>> x-coordinate in an image - some y values may be recorded as "missing").
>>>
>>> Is there an ImageJ tool that will fit (more or less) arbitrary functions to
>>> this data?  Approximately 8 parameters.
>>>
>>> The particular function I have in mind at the moment is a difference of
>>> Gaussians.  The Gaussians
>>> most likely have the same location (in x) - but this is not guaranteed, and
>>> I'd prefer
>>> to use this as a sanity check rather than impose it as a constraint.
>>>
>>> Note that the context is a Java plugin - not a macro.
>>>
>>> If not in ImageJ, perhaps someone could point me at a Java package that
>>> will
>>> do this.  I am far
>>> from an expert in curve fitting, so please be gentle.
>>>
>>> If not, I can do it in R - but I prefer to do it "on the fly" while
>>> displaying
>>> the image, and the Overlay.
>>>
>>> --
>>> Kenneth Sloan
>>> [hidden email]
>>> Vision is the art of seeing what is invisible to others.
>>>
>>> --
>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>
>>
>> --
>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>
>
> --
> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>

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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Michael Schmid-3
Hi Fred,

concerning (1), restricting parameter ranges:
You can have a function that returns NaN if the parameter enters an
invalid range. Then the CurveFitter will avoid this range.
Convergence will be rather bad if the best fit lies very close to a 'NaN
boundary', however.

 > On fits to some data the parameters returned are obnoxiously bad.
What is the fit function? maybe I have some idea how to improve the
situation.

Concering (2) setOffsetMultiplySlopeParams:
Parameter numbers that are not set should be -1. Note that 0 refers to
parameter 'a', 1 to 'b', etc. Since you can specify only two out of the
three arguments, at least one of (multiplyParam or slopeParam) them has
to be -1.
If you let me know the fit function, I can tell you the
setOffsetMultiplySlopeParams call should look like.

Concerning (3), NullPointerException:
I'll have a look at it.
Anyhow, if you have problems with the fit not converging properly, I
would strongly suggest setting initial parameter that give at least the
order of magnitude of the initial parameters. Better, specify also the
initialParamVariations (e.g. for each parameter 1/10 of the range that
it may have).


Michael
________________________________________________________________
On 22/03/2018 22:07, Fred Damen wrote:

> Greetings,
>
> First, thanks for expounding on CurveFitter.
>
> I have a few questions and a bug.
>
> a) Is there any way to keep the fitting within a given range for each
> parameter?  On fits to some data the parameters returned are obnoxiously bad.
> The input data resembles data that fits reasonably, i.e., I do not suspect
> that the initial parameters are outside the local minimum.  This is not using
> setOffsetMultiplySlopeParams.
>
> b) When using the method setOffsetMultiplySlopeParams and passing a value to
> doCustionFit's initialParamVariations parameter, set to what is documented as
> the default value that would be used if null is passed, the fitting is
> effectively a linear fit; see example below.  A reasonable fit is had if the
> call to setOffsetMultiplySlopeParams is commented out.  If this worked what
> kind of improvement would I expect to see in efficiency or goodnees-of-fit?
>
> b) When using the method setOffsetMultiplySlopeParams and passing a null value
> to doCustionFit's initialParamVariations parameter, a null pointer exception
> is thrown:
> ImageJ 1.50h; Java 1.8.0_91 [64-bit]; Linux 4.5.5-300.fc24.x86_64; 287MB of
> 1820MB (15%)
>
> java.lang.NullPointerException
> at ij.measure.CurveFitter.modifyInitialParamsAndVariations(CurveFitter.java:835)
> at ij.measure.CurveFitter.doFit(CurveFitter.java:178)
> at ij.measure.CurveFitter.doCustomFit(CurveFitter.java:283)
> at My_Plugin2.nonlinearFit(My_Plugin2.java:38)
> at My_Plugin2.run(My_Plugin2.java:14)
> at ij.plugin.PlugInExecuter.runCompiledPlugin(Compiler.java:318)
> at ij.plugin.PlugInExecuter.run(Compiler.java:307)
> at java.lang.Thread.run(Thread.java:745)
>
> Thanks in advance,
>
> Fred
>
> ----------------------------
> import ij.*;
> import ij.process.*;
> import ij.gui.*;
> import java.awt.*;
> import ij.plugin.*;
> import ij.plugin.frame.*;
> import ij.measure.*;
>
> public class My_Plugin2 implements PlugIn {
>
>    public void run(String arg) {
>      double[] x = {0.10,0.25,0.50,0.75,1.00,1.25,1.50,1.75,2.00,5.00};
>      double[] y =
> {1123.175,838.206,469.320,453.003,725.135,1094.360,1450.741,1787.361,2128.119,4670.120};
>      double[] p = nonlinearFit(x,y);
>      Plot plot = new Plot("","x","y");
>      plot.setLineWidth(2);
>      plot.setColor(Color.black);
>      plot.addPoints(x,y,PlotWindow.X);
>      double[] y2 = new double[y.length];
>      for(int i=0; i<y.length; i++)
>         y2[i] = Math.abs( p[1] + p[2]*Math.exp(-x[i]/p[0]) );
>      plot.setColor(Color.blue);
>      plot.addPoints(x,y2,PlotWindow.LINE);
>      plot.show();
>
>      }
>
>     private double[] nonlinearFit(double[] x, double[] y) {
>
>        CurveFitter cf = new CurveFitter(x, y);
>        double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>        double[] ipv = new double[params.length];
>        for(int i=0; i<ipv.length; i++)
>           ipv[i] = Math.abs(params[i]*0.1);
>
>        cf.setMaxIterations(200);
>        cf.setOffsetMultiplySlopeParams(1, 2, -1);
>        cf.doCustomFit(new UserFunction() {
>           @Override
>           public double userFunction(double[] p, double x) {
>              return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>              }
>           }, params.length, "", params, ipv, false);
>        //IJ.log(cf.getResultString());
>
>        return cf.getParams();
>        }
> }
>
>
>
> On Tue, March 13, 2018 2:30 pm, Michael Schmid wrote:
>> Hi Kenneth,
>>
>> Concerning fitting an 8-parameter function:
>>
>> If the fit is not linear (as in the case of a difference of Gaussians),
>> having 8 fit parameters is a rather ambitious task, and there is a high
>> probability that the fit will run into a local minimum or some point
>> that looks like a local minimum to the fitting program.
>>
>> It would be best to reduce the number of parameters, e.g. using a fixed
>> ratio between the two sigma values in the Difference of Gaussians.
>>
>> You also need some reasonable guess for the initial values of the
>> parameters.
>>
>> For the ImageJ CurveFitter, if there are many parameters it is very
>> important to specify roughly how much the parameters can vary, these are
>> the 'initialParamVariations'
>> If cf is the CurveFitter, you will have
>>       cf.doCustomFit(UserFunction userFunction, int numParams,
>>           String formula, double[] initialParams,
>>           double[] initialParamVariations, boolean showSettings
>>
>> For the initialParamVariations, use e.g. 1/10th of how much the
>> respective parameter might deviate from the initial guess (only the
>> order of magnitude is important).
>>
>> If you have many parameters and your function can be written as, e.g.
>>       a + b*function(x; c,d,e...)
>> or
>>       a + b*x + function(x; c,d,e)
>>
>> you should also specify these parameters via
>>       cf.setOffsetMultiplySlopeParams(int offsetParam, int multiplyParam,
>> int slopeParam)
>> where 'offsetParam' is the number of the parameter that is only an
>> offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1 for
>> 'b' in the first example above, or 'slopeParam' would be 1 for 'b' in
>> the second type of function above. You cannot have a 'multiplyParam' and
>> a 'slopeParam' at the same time, set the unused one to -1.
>>
>> Specifying an offsetParam and multiplyParam (or slopeParam) makes the
>> CurveFitter calculate these parameters via linear regression, so the
>> actual minimization does not include these parameters. In other words,
>> you get fewer parameters, which makes the fitting much more likely to
>> succeed.
>>
>> In my experience, if you end up with 3-4 parameters (not counting the
>> parameters eliminated by setOffsetMultiplySlopeParams), there is a good
>> chance that the fit will work very well, with 5-6 parameters it gets
>> difficult, and above 6 parameters the chance to get the correct result
>> is rather low.
>>
>> If you need to control the minimization process in detail, before
>> starting the fit, you can use
>>       Minimizer minimizer = cf.getMinimizer()
>> to get access to the Minimizer that will be used and you can use the
>> Minimizer's methods to control its behavior (e.g. allow it to do more
>> steps than by default by minimizer.setMaxIterations, setting it to try
>> more restarts, use different error values for more/less accurate
>> convergence, etc.
>>
>> Best see the documentation in the source code, e.g.
>>     https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
>>     https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java
>>
>> -------------
>>
>>   > Bonus question for Java Gurus:
>>   > How to declare the user function as a variable, call it,
>>   > and pass it to another function?
>>
>> public class MyFunction implements UserFunction {...
>>     public double userFunction(double[] params, double x) {
>>       return params[0]+params[1]*x;
>>     }
>> }
>>
>> public class PassingClass { ...
>>     UserFunction exampleFunction = new MyFunction(...);
>>     otherClass.doFitting(xData, yData, exampleFunction)
>> }
>>
>> Public class OtherClass { ...
>>     public void doFitting(double[] xData, double[] yData,
>>           UserFunction userFunction) {
>>       CurveFitter cf = new CurveFitter(xData, yData);
>>       cf.doCustomFit(userFunction, /*numParams=*/2, null,
>>           null, null, false);
>>     }
>> }
>>
>> -------------
>>
>> Best,
>>
>> Michael
>> ________________________________________________________________
>>
>>
>> On 13/03/2018 18:56, Fred Damen wrote:
>>> Below is a routine to fit MRI Inversion Recover data for T1.
>>>
>>> Note:
>>> a) CurveFitter comes with ImageJ.
>>> b) Calls are made to UserFunction once for each x.
>>> c) If your initial guess is not close it does not seem to converge.
>>>
>>> Bonus question for Java Gurus:
>>> How to declare the user function as a variable, call it, and pass it to
>>> another function?
>>>
>>> Enjoy,
>>>
>>> Fred
>>>
>>>      private double[] nonlinearFit(double[] x, double[] y) {
>>>
>>>         CurveFitter cf = new CurveFitter(x, y);
>>>         double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>
>>>         cf.setMaxIterations(200);
>>>         cf.doCustomFit(new UserFunction() {
>>>            @Override
>>>            public double userFunction(double[] p, double x) {
>>>               return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>               }
>>>            }, params.length, "", params, null, false);
>>>         //IJ.log(cf.getResultString());
>>>
>>>         return cf.getParams();
>>>         }
>>>
>>>
>>> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>>>> I have some simple data: samples of y=f(x) at regularly spaced discrete
>>>> values
>>>> for x.  The data
>>>> is born as a simple array of y values, but I can turn that into (for
>>>> example)
>>>> a polyline, if that
>>>> will help.  I'm currently doing that to draw the data as an Overlay.  The
>>>> size
>>>> of the y array is between 500 and 1000. (think 1 y value for every integer
>>>> x-coordinate in an image - some y values may be recorded as "missing").
>>>>
>>>> Is there an ImageJ tool that will fit (more or less) arbitrary functions to
>>>> this data?  Approximately 8 parameters.
>>>>
>>>> The particular function I have in mind at the moment is a difference of
>>>> Gaussians.  The Gaussians
>>>> most likely have the same location (in x) - but this is not guaranteed, and
>>>> I'd prefer
>>>> to use this as a sanity check rather than impose it as a constraint.
>>>>
>>>> Note that the context is a Java plugin - not a macro.
>>>>
>>>> If not in ImageJ, perhaps someone could point me at a Java package that
>>>> will
>>>> do this.  I am far
>>>> from an expert in curve fitting, so please be gentle.
>>>>
>>>> If not, I can do it in R - but I prefer to do it "on the fly" while
>>>> displaying
>>>> the image, and the Overlay.
>>>>
>>>> --
>>>> Kenneth Sloan
>>>> [hidden email]
>>>> Vision is the art of seeing what is invisible to others.
>>>>
>>>> --
>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>
>>>
>>> --
>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>
>>
>> --
>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>
>
> --
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>

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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Fred Damen
Greetings Michael,

Thanks for the reply.

I must have missed the NaN trick in the documentation.

The fit function (UserFunction), is:
           public double userFunction(double[] p, double x) {
              return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
              }

which was in the example plugin below my signature, which includes sample data
and plot for the fit mentioned in 2 and can produce the exception if the
variable 'ipv' is replaced with the null value.

Note that without the call to setOffsetMultiplySlopeParams the fits are
acceptable almost all of the time,  i.e., only about 10-20 apparently valid
datum fail with obnoxious results out of 40%*64x64 fits.

My main interest in using the method setOffsetMultiplySlopeParams is
efficiency.  Right now on a fast computer it takes about 2-5 seconds to
process a slice.

Thanks,

Fred

On Fri, March 23, 2018 9:00 am, Michael Schmid wrote:

> Hi Fred,
>
> concerning (1), restricting parameter ranges:
> You can have a function that returns NaN if the parameter enters an
> invalid range. Then the CurveFitter will avoid this range.
> Convergence will be rather bad if the best fit lies very close to a 'NaN
> boundary', however.
>
>  > On fits to some data the parameters returned are obnoxiously bad.
> What is the fit function? maybe I have some idea how to improve the
> situation.
>
> Concering (2) setOffsetMultiplySlopeParams:
> Parameter numbers that are not set should be -1. Note that 0 refers to
> parameter 'a', 1 to 'b', etc. Since you can specify only two out of the
> three arguments, at least one of (multiplyParam or slopeParam) them has
> to be -1.
> If you let me know the fit function, I can tell you the
> setOffsetMultiplySlopeParams call should look like.
>
> Concerning (3), NullPointerException:
> I'll have a look at it.
> Anyhow, if you have problems with the fit not converging properly, I
> would strongly suggest setting initial parameter that give at least the
> order of magnitude of the initial parameters. Better, specify also the
> initialParamVariations (e.g. for each parameter 1/10 of the range that
> it may have).
>
>
> Michael
> ________________________________________________________________
> On 22/03/2018 22:07, Fred Damen wrote:
>> Greetings,
>>
>> First, thanks for expounding on CurveFitter.
>>
>> I have a few questions and a bug.
>>
>> a) Is there any way to keep the fitting within a given range for each
>> parameter?  On fits to some data the parameters returned are obnoxiously
>> bad.
>> The input data resembles data that fits reasonably, i.e., I do not suspect
>> that the initial parameters are outside the local minimum.  This is not
>> using
>> setOffsetMultiplySlopeParams.
>>
>> b) When using the method setOffsetMultiplySlopeParams and passing a value to
>> doCustionFit's initialParamVariations parameter, set to what is documented
>> as
>> the default value that would be used if null is passed, the fitting is
>> effectively a linear fit; see example below.  A reasonable fit is had if the
>> call to setOffsetMultiplySlopeParams is commented out.  If this worked what
>> kind of improvement would I expect to see in efficiency or goodnees-of-fit?
>>
>> b) When using the method setOffsetMultiplySlopeParams and passing a null
>> value
>> to doCustionFit's initialParamVariations parameter, a null pointer exception
>> is thrown:
>> ImageJ 1.50h; Java 1.8.0_91 [64-bit]; Linux 4.5.5-300.fc24.x86_64; 287MB of
>> 1820MB (15%)
>>
>> java.lang.NullPointerException
>> at
>> ij.measure.CurveFitter.modifyInitialParamsAndVariations(CurveFitter.java:835)
>> at ij.measure.CurveFitter.doFit(CurveFitter.java:178)
>> at ij.measure.CurveFitter.doCustomFit(CurveFitter.java:283)
>> at My_Plugin2.nonlinearFit(My_Plugin2.java:38)
>> at My_Plugin2.run(My_Plugin2.java:14)
>> at ij.plugin.PlugInExecuter.runCompiledPlugin(Compiler.java:318)
>> at ij.plugin.PlugInExecuter.run(Compiler.java:307)
>> at java.lang.Thread.run(Thread.java:745)
>>
>> Thanks in advance,
>>
>> Fred
>>
>> ----------------------------
>> import ij.*;
>> import ij.process.*;
>> import ij.gui.*;
>> import java.awt.*;
>> import ij.plugin.*;
>> import ij.plugin.frame.*;
>> import ij.measure.*;
>>
>> public class My_Plugin2 implements PlugIn {
>>
>>    public void run(String arg) {
>>      double[] x = {0.10,0.25,0.50,0.75,1.00,1.25,1.50,1.75,2.00,5.00};
>>      double[] y =
>> {1123.175,838.206,469.320,453.003,725.135,1094.360,1450.741,1787.361,2128.119,4670.120};
>>      double[] p = nonlinearFit(x,y);
>>      Plot plot = new Plot("","x","y");
>>      plot.setLineWidth(2);
>>      plot.setColor(Color.black);
>>      plot.addPoints(x,y,PlotWindow.X);
>>      double[] y2 = new double[y.length];
>>      for(int i=0; i<y.length; i++)
>>         y2[i] = Math.abs( p[1] + p[2]*Math.exp(-x[i]/p[0]) );
>>      plot.setColor(Color.blue);
>>      plot.addPoints(x,y2,PlotWindow.LINE);
>>      plot.show();
>>
>>      }
>>
>>     private double[] nonlinearFit(double[] x, double[] y) {
>>
>>        CurveFitter cf = new CurveFitter(x, y);
>>        double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>        double[] ipv = new double[params.length];
>>        for(int i=0; i<ipv.length; i++)
>>           ipv[i] = Math.abs(params[i]*0.1);
>>
>>        cf.setMaxIterations(200);
>>        cf.setOffsetMultiplySlopeParams(1, 2, -1);
>>        cf.doCustomFit(new UserFunction() {
>>           @Override
>>           public double userFunction(double[] p, double x) {
>>              return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>              }
>>           }, params.length, "", params, ipv, false);
>>        //IJ.log(cf.getResultString());
>>
>>        return cf.getParams();
>>        }
>> }
>>
>>
>>
>> On Tue, March 13, 2018 2:30 pm, Michael Schmid wrote:
>>> Hi Kenneth,
>>>
>>> Concerning fitting an 8-parameter function:
>>>
>>> If the fit is not linear (as in the case of a difference of Gaussians),
>>> having 8 fit parameters is a rather ambitious task, and there is a high
>>> probability that the fit will run into a local minimum or some point
>>> that looks like a local minimum to the fitting program.
>>>
>>> It would be best to reduce the number of parameters, e.g. using a fixed
>>> ratio between the two sigma values in the Difference of Gaussians.
>>>
>>> You also need some reasonable guess for the initial values of the
>>> parameters.
>>>
>>> For the ImageJ CurveFitter, if there are many parameters it is very
>>> important to specify roughly how much the parameters can vary, these are
>>> the 'initialParamVariations'
>>> If cf is the CurveFitter, you will have
>>>       cf.doCustomFit(UserFunction userFunction, int numParams,
>>>           String formula, double[] initialParams,
>>>           double[] initialParamVariations, boolean showSettings
>>>
>>> For the initialParamVariations, use e.g. 1/10th of how much the
>>> respective parameter might deviate from the initial guess (only the
>>> order of magnitude is important).
>>>
>>> If you have many parameters and your function can be written as, e.g.
>>>       a + b*function(x; c,d,e...)
>>> or
>>>       a + b*x + function(x; c,d,e)
>>>
>>> you should also specify these parameters via
>>>       cf.setOffsetMultiplySlopeParams(int offsetParam, int multiplyParam,
>>> int slopeParam)
>>> where 'offsetParam' is the number of the parameter that is only an
>>> offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1 for
>>> 'b' in the first example above, or 'slopeParam' would be 1 for 'b' in
>>> the second type of function above. You cannot have a 'multiplyParam' and
>>> a 'slopeParam' at the same time, set the unused one to -1.
>>>
>>> Specifying an offsetParam and multiplyParam (or slopeParam) makes the
>>> CurveFitter calculate these parameters via linear regression, so the
>>> actual minimization does not include these parameters. In other words,
>>> you get fewer parameters, which makes the fitting much more likely to
>>> succeed.
>>>
>>> In my experience, if you end up with 3-4 parameters (not counting the
>>> parameters eliminated by setOffsetMultiplySlopeParams), there is a good
>>> chance that the fit will work very well, with 5-6 parameters it gets
>>> difficult, and above 6 parameters the chance to get the correct result
>>> is rather low.
>>>
>>> If you need to control the minimization process in detail, before
>>> starting the fit, you can use
>>>       Minimizer minimizer = cf.getMinimizer()
>>> to get access to the Minimizer that will be used and you can use the
>>> Minimizer's methods to control its behavior (e.g. allow it to do more
>>> steps than by default by minimizer.setMaxIterations, setting it to try
>>> more restarts, use different error values for more/less accurate
>>> convergence, etc.
>>>
>>> Best see the documentation in the source code, e.g.
>>>     https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
>>>     https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java
>>>
>>> -------------
>>>
>>>   > Bonus question for Java Gurus:
>>>   > How to declare the user function as a variable, call it,
>>>   > and pass it to another function?
>>>
>>> public class MyFunction implements UserFunction {...
>>>     public double userFunction(double[] params, double x) {
>>>       return params[0]+params[1]*x;
>>>     }
>>> }
>>>
>>> public class PassingClass { ...
>>>     UserFunction exampleFunction = new MyFunction(...);
>>>     otherClass.doFitting(xData, yData, exampleFunction)
>>> }
>>>
>>> Public class OtherClass { ...
>>>     public void doFitting(double[] xData, double[] yData,
>>>           UserFunction userFunction) {
>>>       CurveFitter cf = new CurveFitter(xData, yData);
>>>       cf.doCustomFit(userFunction, /*numParams=*/2, null,
>>>           null, null, false);
>>>     }
>>> }
>>>
>>> -------------
>>>
>>> Best,
>>>
>>> Michael
>>> ________________________________________________________________
>>>
>>>
>>> On 13/03/2018 18:56, Fred Damen wrote:
>>>> Below is a routine to fit MRI Inversion Recover data for T1.
>>>>
>>>> Note:
>>>> a) CurveFitter comes with ImageJ.
>>>> b) Calls are made to UserFunction once for each x.
>>>> c) If your initial guess is not close it does not seem to converge.
>>>>
>>>> Bonus question for Java Gurus:
>>>> How to declare the user function as a variable, call it, and pass it to
>>>> another function?
>>>>
>>>> Enjoy,
>>>>
>>>> Fred
>>>>
>>>>      private double[] nonlinearFit(double[] x, double[] y) {
>>>>
>>>>         CurveFitter cf = new CurveFitter(x, y);
>>>>         double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>>
>>>>         cf.setMaxIterations(200);
>>>>         cf.doCustomFit(new UserFunction() {
>>>>            @Override
>>>>            public double userFunction(double[] p, double x) {
>>>>               return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>>               }
>>>>            }, params.length, "", params, null, false);
>>>>         //IJ.log(cf.getResultString());
>>>>
>>>>         return cf.getParams();
>>>>         }
>>>>
>>>>
>>>> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>>>>> I have some simple data: samples of y=f(x) at regularly spaced discrete
>>>>> values
>>>>> for x.  The data
>>>>> is born as a simple array of y values, but I can turn that into (for
>>>>> example)
>>>>> a polyline, if that
>>>>> will help.  I'm currently doing that to draw the data as an Overlay.  The
>>>>> size
>>>>> of the y array is between 500 and 1000. (think 1 y value for every
>>>>> integer
>>>>> x-coordinate in an image - some y values may be recorded as "missing").
>>>>>
>>>>> Is there an ImageJ tool that will fit (more or less) arbitrary functions
>>>>> to
>>>>> this data?  Approximately 8 parameters.
>>>>>
>>>>> The particular function I have in mind at the moment is a difference of
>>>>> Gaussians.  The Gaussians
>>>>> most likely have the same location (in x) - but this is not guaranteed,
>>>>> and
>>>>> I'd prefer
>>>>> to use this as a sanity check rather than impose it as a constraint.
>>>>>
>>>>> Note that the context is a Java plugin - not a macro.
>>>>>
>>>>> If not in ImageJ, perhaps someone could point me at a Java package that
>>>>> will
>>>>> do this.  I am far
>>>>> from an expert in curve fitting, so please be gentle.
>>>>>
>>>>> If not, I can do it in R - but I prefer to do it "on the fly" while
>>>>> displaying
>>>>> the image, and the Overlay.
>>>>>
>>>>> --
>>>>> Kenneth Sloan
>>>>> [hidden email]
>>>>> Vision is the art of seeing what is invisible to others.
>>>>>
>>>>> --
>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>>
>>>>
>>>> --
>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>
>>>
>>> --
>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>
>>
>> --
>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>
>
> --
> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>

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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Michael Schmid-3
Hi Fred,

sorry, I did not read what was below!

Your function is already a built-in fitting function, named 'Exponential
with Offset':
   y = a*exp(-bx) + c

Using the built-in function, you need not care about initial values,
setOffsetMultiplySlopeParams, etc.
By the way, the test data in your example do not look like something
where one would typically use that fit function...

Michael
________________________________________________________________

On 23/03/2018 17:21, Fred Damen wrote:

> Greetings Michael,
>
> Thanks for the reply.
>
> I must have missed the NaN trick in the documentation.
>
> The fit function (UserFunction), is:
>             public double userFunction(double[] p, double x) {
>                return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>                }
>
> which was in the example plugin below my signature, which includes sample data
> and plot for the fit mentioned in 2 and can produce the exception if the
> variable 'ipv' is replaced with the null value.
>
> Note that without the call to setOffsetMultiplySlopeParams the fits are
> acceptable almost all of the time,  i.e., only about 10-20 apparently valid
> datum fail with obnoxious results out of 40%*64x64 fits.
>
> My main interest in using the method setOffsetMultiplySlopeParams is
> efficiency.  Right now on a fast computer it takes about 2-5 seconds to
> process a slice.
>
> Thanks,
>
> Fred
>
> On Fri, March 23, 2018 9:00 am, Michael Schmid wrote:
>> Hi Fred,
>>
>> concerning (1), restricting parameter ranges:
>> You can have a function that returns NaN if the parameter enters an
>> invalid range. Then the CurveFitter will avoid this range.
>> Convergence will be rather bad if the best fit lies very close to a 'NaN
>> boundary', however.
>>
>>   > On fits to some data the parameters returned are obnoxiously bad.
>> What is the fit function? maybe I have some idea how to improve the
>> situation.
>>
>> Concering (2) setOffsetMultiplySlopeParams:
>> Parameter numbers that are not set should be -1. Note that 0 refers to
>> parameter 'a', 1 to 'b', etc. Since you can specify only two out of the
>> three arguments, at least one of (multiplyParam or slopeParam) them has
>> to be -1.
>> If you let me know the fit function, I can tell you the
>> setOffsetMultiplySlopeParams call should look like.
>>
>> Concerning (3), NullPointerException:
>> I'll have a look at it.
>> Anyhow, if you have problems with the fit not converging properly, I
>> would strongly suggest setting initial parameter that give at least the
>> order of magnitude of the initial parameters. Better, specify also the
>> initialParamVariations (e.g. for each parameter 1/10 of the range that
>> it may have).
>>
>>
>> Michael
>> ________________________________________________________________
>> On 22/03/2018 22:07, Fred Damen wrote:
>>> Greetings,
>>>
>>> First, thanks for expounding on CurveFitter.
>>>
>>> I have a few questions and a bug.
>>>
>>> a) Is there any way to keep the fitting within a given range for each
>>> parameter?  On fits to some data the parameters returned are obnoxiously
>>> bad.
>>> The input data resembles data that fits reasonably, i.e., I do not suspect
>>> that the initial parameters are outside the local minimum.  This is not
>>> using
>>> setOffsetMultiplySlopeParams.
>>>
>>> b) When using the method setOffsetMultiplySlopeParams and passing a value to
>>> doCustionFit's initialParamVariations parameter, set to what is documented
>>> as
>>> the default value that would be used if null is passed, the fitting is
>>> effectively a linear fit; see example below.  A reasonable fit is had if the
>>> call to setOffsetMultiplySlopeParams is commented out.  If this worked what
>>> kind of improvement would I expect to see in efficiency or goodnees-of-fit?
>>>
>>> b) When using the method setOffsetMultiplySlopeParams and passing a null
>>> value
>>> to doCustionFit's initialParamVariations parameter, a null pointer exception
>>> is thrown:
>>> ImageJ 1.50h; Java 1.8.0_91 [64-bit]; Linux 4.5.5-300.fc24.x86_64; 287MB of
>>> 1820MB (15%)
>>>
>>> java.lang.NullPointerException
>>> at
>>> ij.measure.CurveFitter.modifyInitialParamsAndVariations(CurveFitter.java:835)
>>> at ij.measure.CurveFitter.doFit(CurveFitter.java:178)
>>> at ij.measure.CurveFitter.doCustomFit(CurveFitter.java:283)
>>> at My_Plugin2.nonlinearFit(My_Plugin2.java:38)
>>> at My_Plugin2.run(My_Plugin2.java:14)
>>> at ij.plugin.PlugInExecuter.runCompiledPlugin(Compiler.java:318)
>>> at ij.plugin.PlugInExecuter.run(Compiler.java:307)
>>> at java.lang.Thread.run(Thread.java:745)
>>>
>>> Thanks in advance,
>>>
>>> Fred
>>>
>>> ----------------------------
>>> import ij.*;
>>> import ij.process.*;
>>> import ij.gui.*;
>>> import java.awt.*;
>>> import ij.plugin.*;
>>> import ij.plugin.frame.*;
>>> import ij.measure.*;
>>>
>>> public class My_Plugin2 implements PlugIn {
>>>
>>>     public void run(String arg) {
>>>       double[] x = {0.10,0.25,0.50,0.75,1.00,1.25,1.50,1.75,2.00,5.00};
>>>       double[] y =
>>> {1123.175,838.206,469.320,453.003,725.135,1094.360,1450.741,1787.361,2128.119,4670.120};
>>>       double[] p = nonlinearFit(x,y);
>>>       Plot plot = new Plot("","x","y");
>>>       plot.setLineWidth(2);
>>>       plot.setColor(Color.black);
>>>       plot.addPoints(x,y,PlotWindow.X);
>>>       double[] y2 = new double[y.length];
>>>       for(int i=0; i<y.length; i++)
>>>          y2[i] = Math.abs( p[1] + p[2]*Math.exp(-x[i]/p[0]) );
>>>       plot.setColor(Color.blue);
>>>       plot.addPoints(x,y2,PlotWindow.LINE);
>>>       plot.show();
>>>
>>>       }
>>>
>>>      private double[] nonlinearFit(double[] x, double[] y) {
>>>
>>>         CurveFitter cf = new CurveFitter(x, y);
>>>         double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>         double[] ipv = new double[params.length];
>>>         for(int i=0; i<ipv.length; i++)
>>>            ipv[i] = Math.abs(params[i]*0.1);
>>>
>>>         cf.setMaxIterations(200);
>>>         cf.setOffsetMultiplySlopeParams(1, 2, -1);
>>>         cf.doCustomFit(new UserFunction() {
>>>            @Override
>>>            public double userFunction(double[] p, double x) {
>>>               return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>               }
>>>            }, params.length, "", params, ipv, false);
>>>         //IJ.log(cf.getResultString());
>>>
>>>         return cf.getParams();
>>>         }
>>> }
>>>
>>>
>>>
>>> On Tue, March 13, 2018 2:30 pm, Michael Schmid wrote:
>>>> Hi Kenneth,
>>>>
>>>> Concerning fitting an 8-parameter function:
>>>>
>>>> If the fit is not linear (as in the case of a difference of Gaussians),
>>>> having 8 fit parameters is a rather ambitious task, and there is a high
>>>> probability that the fit will run into a local minimum or some point
>>>> that looks like a local minimum to the fitting program.
>>>>
>>>> It would be best to reduce the number of parameters, e.g. using a fixed
>>>> ratio between the two sigma values in the Difference of Gaussians.
>>>>
>>>> You also need some reasonable guess for the initial values of the
>>>> parameters.
>>>>
>>>> For the ImageJ CurveFitter, if there are many parameters it is very
>>>> important to specify roughly how much the parameters can vary, these are
>>>> the 'initialParamVariations'
>>>> If cf is the CurveFitter, you will have
>>>>        cf.doCustomFit(UserFunction userFunction, int numParams,
>>>>            String formula, double[] initialParams,
>>>>            double[] initialParamVariations, boolean showSettings
>>>>
>>>> For the initialParamVariations, use e.g. 1/10th of how much the
>>>> respective parameter might deviate from the initial guess (only the
>>>> order of magnitude is important).
>>>>
>>>> If you have many parameters and your function can be written as, e.g.
>>>>        a + b*function(x; c,d,e...)
>>>> or
>>>>        a + b*x + function(x; c,d,e)
>>>>
>>>> you should also specify these parameters via
>>>>        cf.setOffsetMultiplySlopeParams(int offsetParam, int multiplyParam,
>>>> int slopeParam)
>>>> where 'offsetParam' is the number of the parameter that is only an
>>>> offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1 for
>>>> 'b' in the first example above, or 'slopeParam' would be 1 for 'b' in
>>>> the second type of function above. You cannot have a 'multiplyParam' and
>>>> a 'slopeParam' at the same time, set the unused one to -1.
>>>>
>>>> Specifying an offsetParam and multiplyParam (or slopeParam) makes the
>>>> CurveFitter calculate these parameters via linear regression, so the
>>>> actual minimization does not include these parameters. In other words,
>>>> you get fewer parameters, which makes the fitting much more likely to
>>>> succeed.
>>>>
>>>> In my experience, if you end up with 3-4 parameters (not counting the
>>>> parameters eliminated by setOffsetMultiplySlopeParams), there is a good
>>>> chance that the fit will work very well, with 5-6 parameters it gets
>>>> difficult, and above 6 parameters the chance to get the correct result
>>>> is rather low.
>>>>
>>>> If you need to control the minimization process in detail, before
>>>> starting the fit, you can use
>>>>        Minimizer minimizer = cf.getMinimizer()
>>>> to get access to the Minimizer that will be used and you can use the
>>>> Minimizer's methods to control its behavior (e.g. allow it to do more
>>>> steps than by default by minimizer.setMaxIterations, setting it to try
>>>> more restarts, use different error values for more/less accurate
>>>> convergence, etc.
>>>>
>>>> Best see the documentation in the source code, e.g.
>>>>      https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
>>>>      https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java
>>>>
>>>> -------------
>>>>
>>>>    > Bonus question for Java Gurus:
>>>>    > How to declare the user function as a variable, call it,
>>>>    > and pass it to another function?
>>>>
>>>> public class MyFunction implements UserFunction {...
>>>>      public double userFunction(double[] params, double x) {
>>>>        return params[0]+params[1]*x;
>>>>      }
>>>> }
>>>>
>>>> public class PassingClass { ...
>>>>      UserFunction exampleFunction = new MyFunction(...);
>>>>      otherClass.doFitting(xData, yData, exampleFunction)
>>>> }
>>>>
>>>> Public class OtherClass { ...
>>>>      public void doFitting(double[] xData, double[] yData,
>>>>            UserFunction userFunction) {
>>>>        CurveFitter cf = new CurveFitter(xData, yData);
>>>>        cf.doCustomFit(userFunction, /*numParams=*/2, null,
>>>>            null, null, false);
>>>>      }
>>>> }
>>>>
>>>> -------------
>>>>
>>>> Best,
>>>>
>>>> Michael
>>>> ________________________________________________________________
>>>>
>>>>
>>>> On 13/03/2018 18:56, Fred Damen wrote:
>>>>> Below is a routine to fit MRI Inversion Recover data for T1.
>>>>>
>>>>> Note:
>>>>> a) CurveFitter comes with ImageJ.
>>>>> b) Calls are made to UserFunction once for each x.
>>>>> c) If your initial guess is not close it does not seem to converge.
>>>>>
>>>>> Bonus question for Java Gurus:
>>>>> How to declare the user function as a variable, call it, and pass it to
>>>>> another function?
>>>>>
>>>>> Enjoy,
>>>>>
>>>>> Fred
>>>>>
>>>>>       private double[] nonlinearFit(double[] x, double[] y) {
>>>>>
>>>>>          CurveFitter cf = new CurveFitter(x, y);
>>>>>          double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>>>
>>>>>          cf.setMaxIterations(200);
>>>>>          cf.doCustomFit(new UserFunction() {
>>>>>             @Override
>>>>>             public double userFunction(double[] p, double x) {
>>>>>                return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>>>                }
>>>>>             }, params.length, "", params, null, false);
>>>>>          //IJ.log(cf.getResultString());
>>>>>
>>>>>          return cf.getParams();
>>>>>          }
>>>>>
>>>>>
>>>>> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>>>>>> I have some simple data: samples of y=f(x) at regularly spaced discrete
>>>>>> values
>>>>>> for x.  The data
>>>>>> is born as a simple array of y values, but I can turn that into (for
>>>>>> example)
>>>>>> a polyline, if that
>>>>>> will help.  I'm currently doing that to draw the data as an Overlay.  The
>>>>>> size
>>>>>> of the y array is between 500 and 1000. (think 1 y value for every
>>>>>> integer
>>>>>> x-coordinate in an image - some y values may be recorded as "missing").
>>>>>>
>>>>>> Is there an ImageJ tool that will fit (more or less) arbitrary functions
>>>>>> to
>>>>>> this data?  Approximately 8 parameters.
>>>>>>
>>>>>> The particular function I have in mind at the moment is a difference of
>>>>>> Gaussians.  The Gaussians
>>>>>> most likely have the same location (in x) - but this is not guaranteed,
>>>>>> and
>>>>>> I'd prefer
>>>>>> to use this as a sanity check rather than impose it as a constraint.
>>>>>>
>>>>>> Note that the context is a Java plugin - not a macro.
>>>>>>
>>>>>> If not in ImageJ, perhaps someone could point me at a Java package that
>>>>>> will
>>>>>> do this.  I am far
>>>>>> from an expert in curve fitting, so please be gentle.
>>>>>>
>>>>>> If not, I can do it in R - but I prefer to do it "on the fly" while
>>>>>> displaying
>>>>>> the image, and the Overlay.
>>>>>>
>>>>>> --
>>>>>> Kenneth Sloan
>>>>>> [hidden email]
>>>>>> Vision is the art of seeing what is invisible to others.
>>>>>>
>>>>>> --
>>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>>>
>>>>>
>>>>> --
>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>>
>>>>
>>>> --
>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>
>>>
>>> --
>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>
>>
>> --
>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>
>
> --
> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>

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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Herbie
In reply to this post by Fred Damen
Sorry to intrude,

but the function in question is

Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) )

where the absolute value may make the situation more complicated.

Best regards

Herbie

::::::::::::::::::::::::::::::::::::::::
Am 23.03.18 um 17:21 schrieb Fred Damen:

> Greetings Michael,
>
> Thanks for the reply.
>
> I must have missed the NaN trick in the documentation.
>
> The fit function (UserFunction), is:
>             public double userFunction(double[] p, double x) {
>                return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>                }
>
> which was in the example plugin below my signature, which includes sample data
> and plot for the fit mentioned in 2 and can produce the exception if the
> variable 'ipv' is replaced with the null value.
>
> Note that without the call to setOffsetMultiplySlopeParams the fits are
> acceptable almost all of the time,  i.e., only about 10-20 apparently valid
> datum fail with obnoxious results out of 40%*64x64 fits.
>
> My main interest in using the method setOffsetMultiplySlopeParams is
> efficiency.  Right now on a fast computer it takes about 2-5 seconds to
> process a slice.
>
> Thanks,
>
> Fred
>
> On Fri, March 23, 2018 9:00 am, Michael Schmid wrote:
>> Hi Fred,
>>
>> concerning (1), restricting parameter ranges:
>> You can have a function that returns NaN if the parameter enters an
>> invalid range. Then the CurveFitter will avoid this range.
>> Convergence will be rather bad if the best fit lies very close to a 'NaN
>> boundary', however.
>>
>>   > On fits to some data the parameters returned are obnoxiously bad.
>> What is the fit function? maybe I have some idea how to improve the
>> situation.
>>
>> Concering (2) setOffsetMultiplySlopeParams:
>> Parameter numbers that are not set should be -1. Note that 0 refers to
>> parameter 'a', 1 to 'b', etc. Since you can specify only two out of the
>> three arguments, at least one of (multiplyParam or slopeParam) them has
>> to be -1.
>> If you let me know the fit function, I can tell you the
>> setOffsetMultiplySlopeParams call should look like.
>>
>> Concerning (3), NullPointerException:
>> I'll have a look at it.
>> Anyhow, if you have problems with the fit not converging properly, I
>> would strongly suggest setting initial parameter that give at least the
>> order of magnitude of the initial parameters. Better, specify also the
>> initialParamVariations (e.g. for each parameter 1/10 of the range that
>> it may have).
>>
>>
>> Michael
>> ________________________________________________________________
>> On 22/03/2018 22:07, Fred Damen wrote:
>>> Greetings,
>>>
>>> First, thanks for expounding on CurveFitter.
>>>
>>> I have a few questions and a bug.
>>>
>>> a) Is there any way to keep the fitting within a given range for each
>>> parameter?  On fits to some data the parameters returned are obnoxiously
>>> bad.
>>> The input data resembles data that fits reasonably, i.e., I do not suspect
>>> that the initial parameters are outside the local minimum.  This is not
>>> using
>>> setOffsetMultiplySlopeParams.
>>>
>>> b) When using the method setOffsetMultiplySlopeParams and passing a value to
>>> doCustionFit's initialParamVariations parameter, set to what is documented
>>> as
>>> the default value that would be used if null is passed, the fitting is
>>> effectively a linear fit; see example below.  A reasonable fit is had if the
>>> call to setOffsetMultiplySlopeParams is commented out.  If this worked what
>>> kind of improvement would I expect to see in efficiency or goodnees-of-fit?
>>>
>>> b) When using the method setOffsetMultiplySlopeParams and passing a null
>>> value
>>> to doCustionFit's initialParamVariations parameter, a null pointer exception
>>> is thrown:
>>> ImageJ 1.50h; Java 1.8.0_91 [64-bit]; Linux 4.5.5-300.fc24.x86_64; 287MB of
>>> 1820MB (15%)
>>>
>>> java.lang.NullPointerException
>>> at
>>> ij.measure.CurveFitter.modifyInitialParamsAndVariations(CurveFitter.java:835)
>>> at ij.measure.CurveFitter.doFit(CurveFitter.java:178)
>>> at ij.measure.CurveFitter.doCustomFit(CurveFitter.java:283)
>>> at My_Plugin2.nonlinearFit(My_Plugin2.java:38)
>>> at My_Plugin2.run(My_Plugin2.java:14)
>>> at ij.plugin.PlugInExecuter.runCompiledPlugin(Compiler.java:318)
>>> at ij.plugin.PlugInExecuter.run(Compiler.java:307)
>>> at java.lang.Thread.run(Thread.java:745)
>>>
>>> Thanks in advance,
>>>
>>> Fred
>>>
>>> ----------------------------
>>> import ij.*;
>>> import ij.process.*;
>>> import ij.gui.*;
>>> import java.awt.*;
>>> import ij.plugin.*;
>>> import ij.plugin.frame.*;
>>> import ij.measure.*;
>>>
>>> public class My_Plugin2 implements PlugIn {
>>>
>>>     public void run(String arg) {
>>>       double[] x = {0.10,0.25,0.50,0.75,1.00,1.25,1.50,1.75,2.00,5.00};
>>>       double[] y =
>>> {1123.175,838.206,469.320,453.003,725.135,1094.360,1450.741,1787.361,2128.119,4670.120};
>>>       double[] p = nonlinearFit(x,y);
>>>       Plot plot = new Plot("","x","y");
>>>       plot.setLineWidth(2);
>>>       plot.setColor(Color.black);
>>>       plot.addPoints(x,y,PlotWindow.X);
>>>       double[] y2 = new double[y.length];
>>>       for(int i=0; i<y.length; i++)
>>>          y2[i] = Math.abs( p[1] + p[2]*Math.exp(-x[i]/p[0]) );
>>>       plot.setColor(Color.blue);
>>>       plot.addPoints(x,y2,PlotWindow.LINE);
>>>       plot.show();
>>>
>>>       }
>>>
>>>      private double[] nonlinearFit(double[] x, double[] y) {
>>>
>>>         CurveFitter cf = new CurveFitter(x, y);
>>>         double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>         double[] ipv = new double[params.length];
>>>         for(int i=0; i<ipv.length; i++)
>>>            ipv[i] = Math.abs(params[i]*0.1);
>>>
>>>         cf.setMaxIterations(200);
>>>         cf.setOffsetMultiplySlopeParams(1, 2, -1);
>>>         cf.doCustomFit(new UserFunction() {
>>>            @Override
>>>            public double userFunction(double[] p, double x) {
>>>               return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>               }
>>>            }, params.length, "", params, ipv, false);
>>>         //IJ.log(cf.getResultString());
>>>
>>>         return cf.getParams();
>>>         }
>>> }
>>>
>>>
>>>
>>> On Tue, March 13, 2018 2:30 pm, Michael Schmid wrote:
>>>> Hi Kenneth,
>>>>
>>>> Concerning fitting an 8-parameter function:
>>>>
>>>> If the fit is not linear (as in the case of a difference of Gaussians),
>>>> having 8 fit parameters is a rather ambitious task, and there is a high
>>>> probability that the fit will run into a local minimum or some point
>>>> that looks like a local minimum to the fitting program.
>>>>
>>>> It would be best to reduce the number of parameters, e.g. using a fixed
>>>> ratio between the two sigma values in the Difference of Gaussians.
>>>>
>>>> You also need some reasonable guess for the initial values of the
>>>> parameters.
>>>>
>>>> For the ImageJ CurveFitter, if there are many parameters it is very
>>>> important to specify roughly how much the parameters can vary, these are
>>>> the 'initialParamVariations'
>>>> If cf is the CurveFitter, you will have
>>>>        cf.doCustomFit(UserFunction userFunction, int numParams,
>>>>            String formula, double[] initialParams,
>>>>            double[] initialParamVariations, boolean showSettings
>>>>
>>>> For the initialParamVariations, use e.g. 1/10th of how much the
>>>> respective parameter might deviate from the initial guess (only the
>>>> order of magnitude is important).
>>>>
>>>> If you have many parameters and your function can be written as, e.g.
>>>>        a + b*function(x; c,d,e...)
>>>> or
>>>>        a + b*x + function(x; c,d,e)
>>>>
>>>> you should also specify these parameters via
>>>>        cf.setOffsetMultiplySlopeParams(int offsetParam, int multiplyParam,
>>>> int slopeParam)
>>>> where 'offsetParam' is the number of the parameter that is only an
>>>> offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1 for
>>>> 'b' in the first example above, or 'slopeParam' would be 1 for 'b' in
>>>> the second type of function above. You cannot have a 'multiplyParam' and
>>>> a 'slopeParam' at the same time, set the unused one to -1.
>>>>
>>>> Specifying an offsetParam and multiplyParam (or slopeParam) makes the
>>>> CurveFitter calculate these parameters via linear regression, so the
>>>> actual minimization does not include these parameters. In other words,
>>>> you get fewer parameters, which makes the fitting much more likely to
>>>> succeed.
>>>>
>>>> In my experience, if you end up with 3-4 parameters (not counting the
>>>> parameters eliminated by setOffsetMultiplySlopeParams), there is a good
>>>> chance that the fit will work very well, with 5-6 parameters it gets
>>>> difficult, and above 6 parameters the chance to get the correct result
>>>> is rather low.
>>>>
>>>> If you need to control the minimization process in detail, before
>>>> starting the fit, you can use
>>>>        Minimizer minimizer = cf.getMinimizer()
>>>> to get access to the Minimizer that will be used and you can use the
>>>> Minimizer's methods to control its behavior (e.g. allow it to do more
>>>> steps than by default by minimizer.setMaxIterations, setting it to try
>>>> more restarts, use different error values for more/less accurate
>>>> convergence, etc.
>>>>
>>>> Best see the documentation in the source code, e.g.
>>>>      https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
>>>>      https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java
>>>>
>>>> -------------
>>>>
>>>>    > Bonus question for Java Gurus:
>>>>    > How to declare the user function as a variable, call it,
>>>>    > and pass it to another function?
>>>>
>>>> public class MyFunction implements UserFunction {...
>>>>      public double userFunction(double[] params, double x) {
>>>>        return params[0]+params[1]*x;
>>>>      }
>>>> }
>>>>
>>>> public class PassingClass { ...
>>>>      UserFunction exampleFunction = new MyFunction(...);
>>>>      otherClass.doFitting(xData, yData, exampleFunction)
>>>> }
>>>>
>>>> Public class OtherClass { ...
>>>>      public void doFitting(double[] xData, double[] yData,
>>>>            UserFunction userFunction) {
>>>>        CurveFitter cf = new CurveFitter(xData, yData);
>>>>        cf.doCustomFit(userFunction, /*numParams=*/2, null,
>>>>            null, null, false);
>>>>      }
>>>> }
>>>>
>>>> -------------
>>>>
>>>> Best,
>>>>
>>>> Michael
>>>> ________________________________________________________________
>>>>
>>>>
>>>> On 13/03/2018 18:56, Fred Damen wrote:
>>>>> Below is a routine to fit MRI Inversion Recover data for T1.
>>>>>
>>>>> Note:
>>>>> a) CurveFitter comes with ImageJ.
>>>>> b) Calls are made to UserFunction once for each x.
>>>>> c) If your initial guess is not close it does not seem to converge.
>>>>>
>>>>> Bonus question for Java Gurus:
>>>>> How to declare the user function as a variable, call it, and pass it to
>>>>> another function?
>>>>>
>>>>> Enjoy,
>>>>>
>>>>> Fred
>>>>>
>>>>>       private double[] nonlinearFit(double[] x, double[] y) {
>>>>>
>>>>>          CurveFitter cf = new CurveFitter(x, y);
>>>>>          double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>>>
>>>>>          cf.setMaxIterations(200);
>>>>>          cf.doCustomFit(new UserFunction() {
>>>>>             @Override
>>>>>             public double userFunction(double[] p, double x) {
>>>>>                return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>>>                }
>>>>>             }, params.length, "", params, null, false);
>>>>>          //IJ.log(cf.getResultString());
>>>>>
>>>>>          return cf.getParams();
>>>>>          }
>>>>>
>>>>>
>>>>> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>>>>>> I have some simple data: samples of y=f(x) at regularly spaced discrete
>>>>>> values
>>>>>> for x.  The data
>>>>>> is born as a simple array of y values, but I can turn that into (for
>>>>>> example)
>>>>>> a polyline, if that
>>>>>> will help.  I'm currently doing that to draw the data as an Overlay.  The
>>>>>> size
>>>>>> of the y array is between 500 and 1000. (think 1 y value for every
>>>>>> integer
>>>>>> x-coordinate in an image - some y values may be recorded as "missing").
>>>>>>
>>>>>> Is there an ImageJ tool that will fit (more or less) arbitrary functions
>>>>>> to
>>>>>> this data?  Approximately 8 parameters.
>>>>>>
>>>>>> The particular function I have in mind at the moment is a difference of
>>>>>> Gaussians.  The Gaussians
>>>>>> most likely have the same location (in x) - but this is not guaranteed,
>>>>>> and
>>>>>> I'd prefer
>>>>>> to use this as a sanity check rather than impose it as a constraint.
>>>>>>
>>>>>> Note that the context is a Java plugin - not a macro.
>>>>>>
>>>>>> If not in ImageJ, perhaps someone could point me at a Java package that
>>>>>> will
>>>>>> do this.  I am far
>>>>>> from an expert in curve fitting, so please be gentle.
>>>>>>
>>>>>> If not, I can do it in R - but I prefer to do it "on the fly" while
>>>>>> displaying
>>>>>> the image, and the Overlay.
>>>>>>
>>>>>> --
>>>>>> Kenneth Sloan
>>>>>> [hidden email]
>>>>>> Vision is the art of seeing what is invisible to others.
>>>>>>
>>>>>> --
>>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>>>
>>>>>
>>>>> --
>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>>
>>>>
>>>> --
>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>
>>>
>>> --
>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>
>>
>> --
>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>
>
> --
> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>

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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Fred Damen
Thanks Herbie for spotting this.

Yep, I'm an idiot.  This is why the setOffsetMultiplySlopeParams does not
help.  The actual data is complex and I only have the magnitude version.

Some times the hardest things to see are staring you in the face,

Fred

On Fri, March 23, 2018 12:30 pm, Herbie wrote:

> Sorry to intrude,
>
> but the function in question is
>
> Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) )
>
> where the absolute value may make the situation more complicated.
>
> Best regards
>
> Herbie
>
> ::::::::::::::::::::::::::::::::::::::::
> Am 23.03.18 um 17:21 schrieb Fred Damen:
>> Greetings Michael,
>>
>> Thanks for the reply.
>>
>> I must have missed the NaN trick in the documentation.
>>
>> The fit function (UserFunction), is:
>>             public double userFunction(double[] p, double x) {
>>                return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>                }
>>
>> which was in the example plugin below my signature, which includes sample
>> data
>> and plot for the fit mentioned in 2 and can produce the exception if the
>> variable 'ipv' is replaced with the null value.
>>
>> Note that without the call to setOffsetMultiplySlopeParams the fits are
>> acceptable almost all of the time,  i.e., only about 10-20 apparently valid
>> datum fail with obnoxious results out of 40%*64x64 fits.
>>
>> My main interest in using the method setOffsetMultiplySlopeParams is
>> efficiency.  Right now on a fast computer it takes about 2-5 seconds to
>> process a slice.
>>
>> Thanks,
>>
>> Fred
>>
>> On Fri, March 23, 2018 9:00 am, Michael Schmid wrote:
>>> Hi Fred,
>>>
>>> concerning (1), restricting parameter ranges:
>>> You can have a function that returns NaN if the parameter enters an
>>> invalid range. Then the CurveFitter will avoid this range.
>>> Convergence will be rather bad if the best fit lies very close to a 'NaN
>>> boundary', however.
>>>
>>>   > On fits to some data the parameters returned are obnoxiously bad.
>>> What is the fit function? maybe I have some idea how to improve the
>>> situation.
>>>
>>> Concering (2) setOffsetMultiplySlopeParams:
>>> Parameter numbers that are not set should be -1. Note that 0 refers to
>>> parameter 'a', 1 to 'b', etc. Since you can specify only two out of the
>>> three arguments, at least one of (multiplyParam or slopeParam) them has
>>> to be -1.
>>> If you let me know the fit function, I can tell you the
>>> setOffsetMultiplySlopeParams call should look like.
>>>
>>> Concerning (3), NullPointerException:
>>> I'll have a look at it.
>>> Anyhow, if you have problems with the fit not converging properly, I
>>> would strongly suggest setting initial parameter that give at least the
>>> order of magnitude of the initial parameters. Better, specify also the
>>> initialParamVariations (e.g. for each parameter 1/10 of the range that
>>> it may have).
>>>
>>>
>>> Michael
>>> ________________________________________________________________
>>> On 22/03/2018 22:07, Fred Damen wrote:
>>>> Greetings,
>>>>
>>>> First, thanks for expounding on CurveFitter.
>>>>
>>>> I have a few questions and a bug.
>>>>
>>>> a) Is there any way to keep the fitting within a given range for each
>>>> parameter?  On fits to some data the parameters returned are obnoxiously
>>>> bad.
>>>> The input data resembles data that fits reasonably, i.e., I do not suspect
>>>> that the initial parameters are outside the local minimum.  This is not
>>>> using
>>>> setOffsetMultiplySlopeParams.
>>>>
>>>> b) When using the method setOffsetMultiplySlopeParams and passing a value
>>>> to
>>>> doCustionFit's initialParamVariations parameter, set to what is documented
>>>> as
>>>> the default value that would be used if null is passed, the fitting is
>>>> effectively a linear fit; see example below.  A reasonable fit is had if
>>>> the
>>>> call to setOffsetMultiplySlopeParams is commented out.  If this worked
>>>> what
>>>> kind of improvement would I expect to see in efficiency or
>>>> goodnees-of-fit?
>>>>
>>>> b) When using the method setOffsetMultiplySlopeParams and passing a null
>>>> value
>>>> to doCustionFit's initialParamVariations parameter, a null pointer
>>>> exception
>>>> is thrown:
>>>> ImageJ 1.50h; Java 1.8.0_91 [64-bit]; Linux 4.5.5-300.fc24.x86_64; 287MB
>>>> of
>>>> 1820MB (15%)
>>>>
>>>> java.lang.NullPointerException
>>>> at
>>>> ij.measure.CurveFitter.modifyInitialParamsAndVariations(CurveFitter.java:835)
>>>> at ij.measure.CurveFitter.doFit(CurveFitter.java:178)
>>>> at ij.measure.CurveFitter.doCustomFit(CurveFitter.java:283)
>>>> at My_Plugin2.nonlinearFit(My_Plugin2.java:38)
>>>> at My_Plugin2.run(My_Plugin2.java:14)
>>>> at ij.plugin.PlugInExecuter.runCompiledPlugin(Compiler.java:318)
>>>> at ij.plugin.PlugInExecuter.run(Compiler.java:307)
>>>> at java.lang.Thread.run(Thread.java:745)
>>>>
>>>> Thanks in advance,
>>>>
>>>> Fred
>>>>
>>>> ----------------------------
>>>> import ij.*;
>>>> import ij.process.*;
>>>> import ij.gui.*;
>>>> import java.awt.*;
>>>> import ij.plugin.*;
>>>> import ij.plugin.frame.*;
>>>> import ij.measure.*;
>>>>
>>>> public class My_Plugin2 implements PlugIn {
>>>>
>>>>     public void run(String arg) {
>>>>       double[] x = {0.10,0.25,0.50,0.75,1.00,1.25,1.50,1.75,2.00,5.00};
>>>>       double[] y =
>>>> {1123.175,838.206,469.320,453.003,725.135,1094.360,1450.741,1787.361,2128.119,4670.120};
>>>>       double[] p = nonlinearFit(x,y);
>>>>       Plot plot = new Plot("","x","y");
>>>>       plot.setLineWidth(2);
>>>>       plot.setColor(Color.black);
>>>>       plot.addPoints(x,y,PlotWindow.X);
>>>>       double[] y2 = new double[y.length];
>>>>       for(int i=0; i<y.length; i++)
>>>>          y2[i] = Math.abs( p[1] + p[2]*Math.exp(-x[i]/p[0]) );
>>>>       plot.setColor(Color.blue);
>>>>       plot.addPoints(x,y2,PlotWindow.LINE);
>>>>       plot.show();
>>>>
>>>>       }
>>>>
>>>>      private double[] nonlinearFit(double[] x, double[] y) {
>>>>
>>>>         CurveFitter cf = new CurveFitter(x, y);
>>>>         double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>>         double[] ipv = new double[params.length];
>>>>         for(int i=0; i<ipv.length; i++)
>>>>            ipv[i] = Math.abs(params[i]*0.1);
>>>>
>>>>         cf.setMaxIterations(200);
>>>>         cf.setOffsetMultiplySlopeParams(1, 2, -1);
>>>>         cf.doCustomFit(new UserFunction() {
>>>>            @Override
>>>>            public double userFunction(double[] p, double x) {
>>>>               return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>>               }
>>>>            }, params.length, "", params, ipv, false);
>>>>         //IJ.log(cf.getResultString());
>>>>
>>>>         return cf.getParams();
>>>>         }
>>>> }
>>>>
>>>>
>>>>
>>>> On Tue, March 13, 2018 2:30 pm, Michael Schmid wrote:
>>>>> Hi Kenneth,
>>>>>
>>>>> Concerning fitting an 8-parameter function:
>>>>>
>>>>> If the fit is not linear (as in the case of a difference of Gaussians),
>>>>> having 8 fit parameters is a rather ambitious task, and there is a high
>>>>> probability that the fit will run into a local minimum or some point
>>>>> that looks like a local minimum to the fitting program.
>>>>>
>>>>> It would be best to reduce the number of parameters, e.g. using a fixed
>>>>> ratio between the two sigma values in the Difference of Gaussians.
>>>>>
>>>>> You also need some reasonable guess for the initial values of the
>>>>> parameters.
>>>>>
>>>>> For the ImageJ CurveFitter, if there are many parameters it is very
>>>>> important to specify roughly how much the parameters can vary, these are
>>>>> the 'initialParamVariations'
>>>>> If cf is the CurveFitter, you will have
>>>>>        cf.doCustomFit(UserFunction userFunction, int numParams,
>>>>>            String formula, double[] initialParams,
>>>>>            double[] initialParamVariations, boolean showSettings
>>>>>
>>>>> For the initialParamVariations, use e.g. 1/10th of how much the
>>>>> respective parameter might deviate from the initial guess (only the
>>>>> order of magnitude is important).
>>>>>
>>>>> If you have many parameters and your function can be written as, e.g.
>>>>>        a + b*function(x; c,d,e...)
>>>>> or
>>>>>        a + b*x + function(x; c,d,e)
>>>>>
>>>>> you should also specify these parameters via
>>>>>        cf.setOffsetMultiplySlopeParams(int offsetParam, int
>>>>> multiplyParam,
>>>>> int slopeParam)
>>>>> where 'offsetParam' is the number of the parameter that is only an
>>>>> offset (in the examples above, 0 for 'a', 'multiplyParam' would be 1 for
>>>>> 'b' in the first example above, or 'slopeParam' would be 1 for 'b' in
>>>>> the second type of function above. You cannot have a 'multiplyParam' and
>>>>> a 'slopeParam' at the same time, set the unused one to -1.
>>>>>
>>>>> Specifying an offsetParam and multiplyParam (or slopeParam) makes the
>>>>> CurveFitter calculate these parameters via linear regression, so the
>>>>> actual minimization does not include these parameters. In other words,
>>>>> you get fewer parameters, which makes the fitting much more likely to
>>>>> succeed.
>>>>>
>>>>> In my experience, if you end up with 3-4 parameters (not counting the
>>>>> parameters eliminated by setOffsetMultiplySlopeParams), there is a good
>>>>> chance that the fit will work very well, with 5-6 parameters it gets
>>>>> difficult, and above 6 parameters the chance to get the correct result
>>>>> is rather low.
>>>>>
>>>>> If you need to control the minimization process in detail, before
>>>>> starting the fit, you can use
>>>>>        Minimizer minimizer = cf.getMinimizer()
>>>>> to get access to the Minimizer that will be used and you can use the
>>>>> Minimizer's methods to control its behavior (e.g. allow it to do more
>>>>> steps than by default by minimizer.setMaxIterations, setting it to try
>>>>> more restarts, use different error values for more/less accurate
>>>>> convergence, etc.
>>>>>
>>>>> Best see the documentation in the source code, e.g.
>>>>>      https://github.com/imagej/imagej1/blob/master/ij/measure/CurveFitter.java
>>>>>      https://github.com/imagej/imagej1/blob/master/ij/measure/Minimizer.java
>>>>>
>>>>> -------------
>>>>>
>>>>>    > Bonus question for Java Gurus:
>>>>>    > How to declare the user function as a variable, call it,
>>>>>    > and pass it to another function?
>>>>>
>>>>> public class MyFunction implements UserFunction {...
>>>>>      public double userFunction(double[] params, double x) {
>>>>>        return params[0]+params[1]*x;
>>>>>      }
>>>>> }
>>>>>
>>>>> public class PassingClass { ...
>>>>>      UserFunction exampleFunction = new MyFunction(...);
>>>>>      otherClass.doFitting(xData, yData, exampleFunction)
>>>>> }
>>>>>
>>>>> Public class OtherClass { ...
>>>>>      public void doFitting(double[] xData, double[] yData,
>>>>>            UserFunction userFunction) {
>>>>>        CurveFitter cf = new CurveFitter(xData, yData);
>>>>>        cf.doCustomFit(userFunction, /*numParams=*/2, null,
>>>>>            null, null, false);
>>>>>      }
>>>>> }
>>>>>
>>>>> -------------
>>>>>
>>>>> Best,
>>>>>
>>>>> Michael
>>>>> ________________________________________________________________
>>>>>
>>>>>
>>>>> On 13/03/2018 18:56, Fred Damen wrote:
>>>>>> Below is a routine to fit MRI Inversion Recover data for T1.
>>>>>>
>>>>>> Note:
>>>>>> a) CurveFitter comes with ImageJ.
>>>>>> b) Calls are made to UserFunction once for each x.
>>>>>> c) If your initial guess is not close it does not seem to converge.
>>>>>>
>>>>>> Bonus question for Java Gurus:
>>>>>> How to declare the user function as a variable, call it, and pass it to
>>>>>> another function?
>>>>>>
>>>>>> Enjoy,
>>>>>>
>>>>>> Fred
>>>>>>
>>>>>>       private double[] nonlinearFit(double[] x, double[] y) {
>>>>>>
>>>>>>          CurveFitter cf = new CurveFitter(x, y);
>>>>>>          double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
>>>>>>
>>>>>>          cf.setMaxIterations(200);
>>>>>>          cf.doCustomFit(new UserFunction() {
>>>>>>             @Override
>>>>>>             public double userFunction(double[] p, double x) {
>>>>>>                return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
>>>>>>                }
>>>>>>             }, params.length, "", params, null, false);
>>>>>>          //IJ.log(cf.getResultString());
>>>>>>
>>>>>>          return cf.getParams();
>>>>>>          }
>>>>>>
>>>>>>
>>>>>> On Tue, March 13, 2018 11:06 am, Kenneth Sloan wrote:
>>>>>>> I have some simple data: samples of y=f(x) at regularly spaced discrete
>>>>>>> values
>>>>>>> for x.  The data
>>>>>>> is born as a simple array of y values, but I can turn that into (for
>>>>>>> example)
>>>>>>> a polyline, if that
>>>>>>> will help.  I'm currently doing that to draw the data as an Overlay.
>>>>>>> The
>>>>>>> size
>>>>>>> of the y array is between 500 and 1000. (think 1 y value for every
>>>>>>> integer
>>>>>>> x-coordinate in an image - some y values may be recorded as "missing").
>>>>>>>
>>>>>>> Is there an ImageJ tool that will fit (more or less) arbitrary
>>>>>>> functions
>>>>>>> to
>>>>>>> this data?  Approximately 8 parameters.
>>>>>>>
>>>>>>> The particular function I have in mind at the moment is a difference of
>>>>>>> Gaussians.  The Gaussians
>>>>>>> most likely have the same location (in x) - but this is not guaranteed,
>>>>>>> and
>>>>>>> I'd prefer
>>>>>>> to use this as a sanity check rather than impose it as a constraint.
>>>>>>>
>>>>>>> Note that the context is a Java plugin - not a macro.
>>>>>>>
>>>>>>> If not in ImageJ, perhaps someone could point me at a Java package that
>>>>>>> will
>>>>>>> do this.  I am far
>>>>>>> from an expert in curve fitting, so please be gentle.
>>>>>>>
>>>>>>> If not, I can do it in R - but I prefer to do it "on the fly" while
>>>>>>> displaying
>>>>>>> the image, and the Overlay.
>>>>>>>
>>>>>>> --
>>>>>>> Kenneth Sloan
>>>>>>> [hidden email]
>>>>>>> Vision is the art of seeing what is invisible to others.
>>>>>>>
>>>>>>> --
>>>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>>>>
>>>>>>
>>>>>> --
>>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>>>
>>>>>
>>>>> --
>>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>>
>>>>
>>>> --
>>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>>
>>>
>>> --
>>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>>
>>
>> --
>> ImageJ mailing list: http://imagej.nih.gov/ij/list.html
>>
>
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Re: fitting y=f(x) data to arbitrary functions in ImageJ?

Michael Schmid-3
Hi Herbie, Fred,

oops, yes, I missed the 'abs'!

So one can't use the built-in version, and fitting certainly becomes tricky!
One should probably rewrite the problem as
   y = a * (1 + b*exp(-x/c))
or, if the c parameter is known to be always positive, the following may
be better:
   y = a * (1 + b*exp(-x/(c*c)))

Then, the pre-exponential factor has to be calculated as a*b.

Like this, the ImageJ CurveFitter can eliminate the factor 'a' as a
'multiplyParameter' via linear regression, and it becomes a
two-parameter fit.

If the data typically look like the example data, with a minimum where
the argument of the 'abs' passes through zero,
   0.1    1123.175
   0.25    838.206
   0.5    469.32
   0.75    453.003
   1    725.135
   1.25    1094.36
   1.5    1450.741
   1.75    1787.361
   2    2128.119
   5    4670.12
then one should take the minimum and find initial b and c values where
the function becomes zero at the minimum (here named xOfMin):
   1 + b*exp(-xOfMin/c) = 0
or
   b = -exp(xOfMin/c)

If there is no prior knowledge for c, this parameter might be taken
equal to the difference between the highest and lowest x value.
(For the second fit function with c*c, replace 'c' with 'c*c' everywhere.)

Michael
________________________________________________________________
On 23/03/2018 18:30, Herbie wrote:
 > Sorry to intrude,
 >
 > but the function in question is
 >
 > Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) )
 >
 > where the absolute value may make the situation more complicated.
 >
 > Best regards
 >
 > Herbie
 >
 > ::::::::::::::::::::::::::::::::::::::::
 > Am 23.03.18 um 17:21 schrieb Fred Damen:
 >> Greetings Michael,
 >>
 >> Thanks for the reply.
 >>
 >> I must have missed the NaN trick in the documentation.
 >>
 >> The fit function (UserFunction), is:
 >>             public double userFunction(double[] p, double x) {
 >>                return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
 >>                }
 >>
 >> which was in the example plugin below my signature, which includes
sample data
 >> and plot for the fit mentioned in 2 and can produce the exception if the
 >> variable 'ipv' is replaced with the null value.
 >>
 >> Note that without the call to setOffsetMultiplySlopeParams the fits are
 >> acceptable almost all of the time,  i.e., only about 10-20
apparently valid
 >> datum fail with obnoxious results out of 40%*64x64 fits.
 >>
 >> My main interest in using the method setOffsetMultiplySlopeParams is
 >> efficiency.  Right now on a fast computer it takes about 2-5 seconds to
 >> process a slice.
 >>
 >> Thanks,
 >>
 >> Fred
 >>
 >> On Fri, March 23, 2018 9:00 am, Michael Schmid wrote:
 >>> Hi Fred,
 >>>
 >>> concerning (1), restricting parameter ranges:
 >>> You can have a function that returns NaN if the parameter enters an
 >>> invalid range. Then the CurveFitter will avoid this range.
 >>> Convergence will be rather bad if the best fit lies very close to a
'NaN
 >>> boundary', however.
 >>>
 >>>   > On fits to some data the parameters returned are obnoxiously bad.
 >>> What is the fit function? maybe I have some idea how to improve the
 >>> situation.
 >>>
 >>> Concering (2) setOffsetMultiplySlopeParams:
 >>> Parameter numbers that are not set should be -1. Note that 0 refers to
 >>> parameter 'a', 1 to 'b', etc. Since you can specify only two out of the
 >>> three arguments, at least one of (multiplyParam or slopeParam) them has
 >>> to be -1.
 >>> If you let me know the fit function, I can tell you the
 >>> setOffsetMultiplySlopeParams call should look like.
 >>>
 >>> Concerning (3), NullPointerException:
 >>> I'll have a look at it.
 >>> Anyhow, if you have problems with the fit not converging properly, I
 >>> would strongly suggest setting initial parameter that give at least the
 >>> order of magnitude of the initial parameters. Better, specify also the
 >>> initialParamVariations (e.g. for each parameter 1/10 of the range that
 >>> it may have).
 >>>
 >>>
 >>> Michael
 >>> ________________________________________________________________
 >>> On 22/03/2018 22:07, Fred Damen wrote:
 >>>> Greetings,
 >>>>
 >>>> First, thanks for expounding on CurveFitter.
 >>>>
 >>>> I have a few questions and a bug.
 >>>>
 >>>> a) Is there any way to keep the fitting within a given range for each
 >>>> parameter?  On fits to some data the parameters returned are
obnoxiously
 >>>> bad.
 >>>> The input data resembles data that fits reasonably, i.e., I do not
suspect
 >>>> that the initial parameters are outside the local minimum.  This
is not
 >>>> using
 >>>> setOffsetMultiplySlopeParams.
 >>>>
 >>>> b) When using the method setOffsetMultiplySlopeParams and passing
a value to
 >>>> doCustionFit's initialParamVariations parameter, set to what is
documented
 >>>> as
 >>>> the default value that would be used if null is passed, the fitting is
 >>>> effectively a linear fit; see example below.  A reasonable fit is
had if the
 >>>> call to setOffsetMultiplySlopeParams is commented out.  If this
worked what
 >>>> kind of improvement would I expect to see in efficiency or
goodnees-of-fit?
 >>>>
 >>>> b) When using the method setOffsetMultiplySlopeParams and passing
a null
 >>>> value
 >>>> to doCustionFit's initialParamVariations parameter, a null pointer
exception
 >>>> is thrown:
 >>>> ImageJ 1.50h; Java 1.8.0_91 [64-bit]; Linux 4.5.5-300.fc24.x86_64;
287MB of
 >>>> 1820MB (15%)
 >>>>
 >>>> java.lang.NullPointerException
 >>>>     at
 >>>>
ij.measure.CurveFitter.modifyInitialParamsAndVariations(CurveFitter.java:835)
 >>>>     at ij.measure.CurveFitter.doFit(CurveFitter.java:178)
 >>>>     at ij.measure.CurveFitter.doCustomFit(CurveFitter.java:283)
 >>>>     at My_Plugin2.nonlinearFit(My_Plugin2.java:38)
 >>>>     at My_Plugin2.run(My_Plugin2.java:14)
 >>>>     at ij.plugin.PlugInExecuter.runCompiledPlugin(Compiler.java:318)
 >>>>     at ij.plugin.PlugInExecuter.run(Compiler.java:307)
 >>>>     at java.lang.Thread.run(Thread.java:745)
 >>>>
 >>>> Thanks in advance,
 >>>>
 >>>> Fred
 >>>>
 >>>> ----------------------------
 >>>> import ij.*;
 >>>> import ij.process.*;
 >>>> import ij.gui.*;
 >>>> import java.awt.*;
 >>>> import ij.plugin.*;
 >>>> import ij.plugin.frame.*;
 >>>> import ij.measure.*;
 >>>>
 >>>> public class My_Plugin2 implements PlugIn {
 >>>>
 >>>>     public void run(String arg) {
 >>>>       double[] x =
{0.10,0.25,0.50,0.75,1.00,1.25,1.50,1.75,2.00,5.00};
 >>>>       double[] y =
 >>>>
{1123.175,838.206,469.320,453.003,725.135,1094.360,1450.741,1787.361,2128.119,4670.120};
 >>>>       double[] p = nonlinearFit(x,y);
 >>>>       Plot plot = new Plot("","x","y");
 >>>>       plot.setLineWidth(2);
 >>>>       plot.setColor(Color.black);
 >>>>       plot.addPoints(x,y,PlotWindow.X);
 >>>>       double[] y2 = new double[y.length];
 >>>>       for(int i=0; i<y.length; i++)
 >>>>          y2[i] = Math.abs( p[1] + p[2]*Math.exp(-x[i]/p[0]) );
 >>>>       plot.setColor(Color.blue);
 >>>>       plot.addPoints(x,y2,PlotWindow.LINE);
 >>>>       plot.show();
 >>>>
 >>>>       }
 >>>>
 >>>>      private double[] nonlinearFit(double[] x, double[] y) {
 >>>>
 >>>>         CurveFitter cf = new CurveFitter(x, y);
 >>>>         double[] params = {1, 2*y[0], -(y[0]+y[y.length-1])};
 >>>>         double[] ipv = new double[params.length];
 >>>>         for(int i=0; i<ipv.length; i++)
 >>>>            ipv[i] = Math.abs(params[i]*0.1);
 >>>>
 >>>>         cf.setMaxIterations(200);
 >>>>         cf.setOffsetMultiplySlopeParams(1, 2, -1);
 >>>>         cf.doCustomFit(new UserFunction() {
 >>>>            @Override
 >>>>            public double userFunction(double[] p, double x) {
 >>>>               return Math.abs( p[1] + p[2]*Math.exp(-x/p[0]) );
 >>>>               }
 >>>>            }, params.length, "", params, ipv, false);
 >>>>         //IJ.log(cf.getResultString());
 >>>>
 >>>>         return cf.getParams();
 >>>>         }
 >>>> }

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