http://imagej.273.s1.nabble.com/fitting-y-f-x-data-to-arbitrary-functions-in-ImageJ-tp5020273p5020298.html
> 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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> ImageJ mailing list:
http://imagej.nih.gov/ij/list.html