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Re: blurry ellipse
*To*: mathgroup at smc.vnet.net
*Subject*: [mg92072] Re: blurry ellipse
*From*: dh <dh at metrohm.ch>
*Date*: Fri, 19 Sep 2008 05:16:20 -0400 (EDT)
*References*: <gat9g4$ec5$1@smc.vnet.net>
Hi Joshua,
you can blurry any picture by a convolution with a suitable kernel. Here
is an example. We first create a picture of an ellips. Then we
increasingly blurry it three times. Note that we must keep the numbers
between 0..1 because of "Raster":
n=100;
pict=Table[0,{n},{n}];
Scan[(pict[[Round[n/2+n/5 Sin[#]],Round[n/2+2
n/5Cos[#]]]]=1)&,Range[0,2Pi,0.01]];
Graphics[Raster[pict]]
kernel={{1,1},{1,1}}; pict=ListConvolve[kernel,pict];pict=pict/Max[pict];
Graphics[Raster[pict]]
kernel={{1,1,1},{1,1,1},{1,1,1}};
pict=ListConvolve[kernel,pict];pict=pict/Max[pict];
Graphics[Raster[pict]]
kernel={{1,1,1,1},{1,1,1,1},{1,1,1,1},{1,1,1,1}};
pict=ListConvolve[kernel,pict];pict=pict/Max[pict];
Graphics[Raster[pict]]
Note that you may instead of using a larger kernel, repeat the operation.
hope this helps, Daniel
Solomon, Joshua wrote:
> 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?
>
>
--
Daniel Huber
Metrohm Ltd.
Oberdorfstr. 68
CH-9100 Herisau
Tel. +41 71 353 8585, Fax +41 71 353 8907
E-Mail:<mailto:dh at metrohm.com>
Internet:<http://www.metrohm.com>
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