File:DOG vs MHF.png

描述DOG vs MHF.png	Comparison of Difference of Gaussians and Mexican Hat Function, intended to illustrate that DoG is a reasonable approximation to MHF.
日期	2005年十二月6日, 04:58 (UTC)
來源	Originally uploaded to en.wikipedia. Generated using Mathematica 5.0.   本PNG 點陣圖使用Mathematica創作。
作者	JustinWick

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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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• 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.)