1.5 Radial Basis Function
1.5 Radial Basis Function
1.5.1 RBF Approximation
1.5.1 RBF Approximation
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data=Table[{i0.3,i0.3Sin[i0.3]},{i,1,10}]
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{{0.3,0.0886561},{0.6,0.338785},{0.9,0.704994},{1.2,1.11845},{1.5,1.49624},{1.8,1.75293},{2.1,1.81274},{2.4,1.62111},{2.7,1.15393},{3.,0.42336}}
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p1=ListPlot[data,FillingAxis]
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TPSpline=Function[{x,xi},If[x≠xi,Log[Abs[x-xi]],{0}]]
2
Abs[{x-xi}]
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Function[{x,xi},If[x≠xi,Log[Abs[x-xi]],{0}]]
2
Abs[{x-xi}]
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Plot[{TPSpline[x,1.5],TPSpline[x,1.8]},{x,0.3,3},PlotRange{-0.5,0.5}]
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np=Length[data];datax=Transpose[data][[1]];
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datay=Transpose[data][[2]];
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M=Table[First[TPSpline[datax[[i]],datax[[j]]]],{i,1,np},{j,1,np}];
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Norm[Inverse[M]]Norm[M]
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308.843
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c=PseudoInverse[M].datay
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{1.94971,-2.26345,-0.412961,-0.419176,-0.282941,-0.22277,-0.224389,-0.178729,-1.23521,1.00129}
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ListPlot[c,FillingAxis]
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g[x_]:=First[c.Map[TPSpline[x,#]&,datax]]
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Show[{Plot[xSin[x],{x,0.5,3},PlotStyleRed],Plot[g[x],{x,0.5,3},PlotRange{0,3}],p1}]
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1.5.2 RBF Image Compression
1.5.2 RBF Image Compression
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image=
;
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img=ColorConvert[image,"Grayscale"];
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M=ImageData[img];M//Dimensions
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{173,174}
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ListPlot[Flatten[M],PlotStylePointSize[Tiny],FrameTrue]
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MR=M;
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SeedRandom[4567]
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Do[MR[[RandomInteger[{1,173}],RandomInteger[{1,174}]]]=1.,{i,1,60000}];
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Image[MR]
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dataz={};dataxy={};
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Do[If[MR[[i,j]]≠1.,AppendTo[dataxy,{i/173.,j/174.}];AppendTo[dataz,MR[[i,j]]]],{i,1,173},{j,1,174}]
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ListPlot[dataxy,FrameTrue,AspectRatio1,PlotStylePointSize[Tiny]]
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