Gauss2D Fit: 'cor' parameter

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Hi,

I'm using the Gauss2D function to fit data contained in an image.
In this function, we have the possibility to access to a 'cor' parameter, which corresponds to a cross-correlation parameter.
What does it represent exactly. I can't find this formula on the litterature.

The second question is maybe a lack of knowledge but if in my image I've got something like a PSF that looks like an ellipse. If I want to fit this PSF with a function that could reveal the fact that the ellipse could be rotated compared to the axes, how could I do?

Thanks for your help.


Sébastien
mailfert wrote:
I'm using the Gauss2D function to fit data contained in an image.
In this function, we have the possibility to access to a 'cor' parameter, which corresponds to a cross-correlation parameter.
What does it represent exactly. I can't find this formula on the litterature.

http://en.wikipedia.org/wiki/Multivariate_normal_distribution
But note that Igor fits a function that is not normalized as a probability distribution (that is, the total integral is not equal to 1.0).
Quote:
The second question is maybe a lack of knowledge but if in my image I've got something like a PSF that looks like an ellipse. If I want to fit this PSF with a function that could reveal the fact that the ellipse could be rotated compared to the axes, how could I do?

I assume that you mean that the projection of the PSF onto the XY plane has an elliptical outline, and the distribution of Z values is Gaussian. The ellipse at an angle is what the cor fit coefficient is for. If you are looking for a fitting function that uses the angle with the X axis as a fit parameter, you will need to find the formula and create a user-defined fitting function.

John Weeks
WaveMetrics, Inc.
support@wavemetrics.com

July 22, 2011 at 09:07 am - Permalink

Here is an example function that may help you. I have simplified the image to have no baseline or average x, y offsets. These can be added with suitable modification of the user-defined fit function. You may also want to add to the function a way to refine the w_coef initial guesses.
function Ellipse(theta)
    variable theta      //  rotation angle -  radians
    
    Make/O/N=(100,100) w2D  //  make a test wave for the input angle
    setscale x, -4, 4, "" w2D
    setscale y, -4, 4, "" w2D
    variable sigmax =   1
    variable sigmay =   2
    w2D  = (1/(2*pi*sigmax*sigmay))
    w2d*= exp( - (x*cos(theta) + y*sin(theta))^2/(2*sigmax^2)  )
    w2d*= exp( - (-x*sin(theta) + y*cos(theta))^2/(2*sigmay^2) )
    // create a suitable w2D graph to show the fit results and contour lines
    FuncFitMD/NTHR=0/TBOX=768 MyGauss W_coef  w2D /D // and fit it
    
end
 
Function MyGauss(w,x,y) : FitFunc
    Wave w
    Variable x
    Variable y
 
    //CurveFitDialog/ These comments were created by the Curve Fitting dialog. Altering them will
    //CurveFitDialog/ make the function less convenient to work with in the Curve Fitting dialog.
    //CurveFitDialog/ Equation:
    //CurveFitDialog/ f(x,y) = (A/(sigmax*sigmay))*exp(-(x*cos(theta)+y*sin(theta))^2/(2*sigmax^2) -(-x*sin(theta)+y*cos(theta))^2/(2*sigmay^2))
    //CurveFitDialog/ End of Equation
    //CurveFitDialog/ Independent Variables 2
    //CurveFitDialog/ x
    //CurveFitDialog/ y
    //CurveFitDialog/ Coefficients 4
    //CurveFitDialog/ w[0] = A
    //CurveFitDialog/ w[1] = sigmax
    //CurveFitDialog/ w[2] = sigmay
    //CurveFitDialog/ w[3] = theta
 
    return (w[0]/(w[1]*w[2]))*exp(-(x*cos(w[3])+y*sin(w[3]))^2/(2*w[1]^2) -(-x*sin(w[3])+y*cos(w[3]))^2/(2*w[2]^2))
End
Graph0_2.png (213.54 KB) Graph0_2.png (213.54 KB) Graph0_2.png (213.54 KB)

July 28, 2011 at 09:15 am - Permalink