DescriptionDOG vs MHF.png	Comparison of Difference of Gaussians and Mexican Hat Function, intended to illustrate that DoG is a reasonable approximation to MHF.
Date	6 December 2005, 04:58 (UTC)
Source	Originally uploaded to en.wikipedia. Generated using Mathematica 5.0.   This diagram was created with Mathematica.
Author	JustinWick

This work has been released into the public domain by its author, JustinWick at English Wikipedia. This applies worldwide. In some countries this may not be legally possible; if so: JustinWick grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.Public domainPublic domainfalsefalse

To reproduce this plot in mathematica, use the following code:

<< "Statistics`NormalDistribution`";
DOG[mu_, s1_, s2_][x_] := PDF[NormalDistribution[mu, s1], x] - PDF[NormalDistribution[mu, s2], x]; 
MHF[x_] := ((2/(Sqrt[3]*Pi^(1/4)))*(1 - x^2))/E^(x^2/2);
Show[Plot[c*DOG[0, a, b][x] /. {a -> 1.3383326406221043,  b -> 0.7477302449634321, c -> -3.812773094633841}, {x, -5, 5}, 
PlotRange -> {-1,  1}, Frame -> True, Axes -> False, PlotStyle -> {RGBColor[0, 0, 1]}, PlotLabel -> "Difference of Gaussians"], 
Plot[MHF[x], {x, -5, 5}, PlotRange -> {-1, 1}, Frame -> True, Axes -> False, 
PlotStyle -> RGBColor[1, 0, 0], PlotLabel -> "Mexican Hat Function"], PlotLabel -> "Difference of Gaussians and Mexican Hat Function"]

Note, to fit the DoG best to the MHF, I utilized the following code:

NMinimize[{Hold[NIntegrate[Abs[c*DOG[0, a, b][x] - MHF[x]], {x, -5, 5}]], {a > 0, b > 0}}, {a, b, c}]

This uses least squares fitting, which may be slightly suboptimal.

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File versions

• 04:58, 6 December 2005 . . JustinWick . . 486x900 (10415 bytes) (Comparison of Difference of Gaussians Difference of Gaussians and Mexican Hat Function. This file was created at 19:02, 18 March 2006 (UTC) by JustinWick, using Mathematica 5.0.)