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blurry ellipse

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92033] blurry ellipse
  • From: "Solomon, Joshua" <J.A.Solomon at city.ac.uk>
  • Date: Thu, 18 Sep 2008 06:11:07 -0400 (EDT)
  • Organization: Posted via ULCC Internet Services

NB: This is a mathematical problem, not necessarily a Mathematica problem.

I need a rasterized, blurry ellipse, i.e. an ellipse convolved with a
Gaussian. I already know how to make a blurry circle. You take the Fourier
transform of a circle (i.e. a Bessel function), the Fourier transform of a
Guassian, multiply them, and transform them back. No problem. Since an
ellipse is just a circle that has been squashed in one dimension, I figured
I could make a blurry one the same way as I make blurry circles. I just
needed to squash the Bessel function first. And I was right. The code below
works just fine, *except* the intensity varies as you go around the elipse.
If anyone could tell me how to fix that problem, I'd be very grateful!

BlurryEllipse[sizePix_, radiusPix_, sigmaPix_,stretchFactors_: {1, 1}] := 

    Module[{half = sizePix/2, x, y, f, r, a, pr2}, 

        f = 2*Pi*radiusPix/sizePix;
        pr2 = -2 (Pi*sigmaPix/sizePix)^2; 

        RotateRight[

            Abs[
                Chop[

                    InverseFourier[

                        RotateRight[

                            Table[r1=
                                    Abs[stretchFactors[[1]]*x+
                                        stretchFactors[[2]]*I*y];
                                 r2 = Abs[x + I*y];
                                 BesselJ[0, f*r1] Exp[pr2 (r2^2)],
                                {y, -half, half-1} , {x, -half, half-1}],
                            {half, half}]]]], {half,half}]]

size = 64; rad = 24; scale = 3;
tmp = BlurryEllipse[size, rad, scale, {1, .5}];
Show[Graphics[Raster[tmp/Max[Max[tmp]]]], ImageSize -> 400]

Intensity varies around the elipse. We can compare amplitude (or power)
ratios between major and minor axes:

In[]:=Max[tmp[[33]]]/Max[Transpose[tmp][[33]]]


Out[]=1.93009


In[]:= Sqrt[Total[tmp[[33]]^2]/Total[Transpose[tmp][[33]]^2]]

Out[]= 1.98887



I was surprised these numbers weren't closer to 2. How can we make intensity
invariant around the ellipse?



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