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1
2 ω y F() ω y F() 2
3 ω y F() ω y F() 3
4 ω y F() ω y F() 4
5 ω y F() ω y F() 5
6 ω y F() ω y F() 6
7 ω y F() ω y F() 7
8 ω y F() ω y F() 8
9 ωy F() ωy F() 9
10 ωy F() ωy F() 10
11 Gabor Gabor Gabor Gabor
12 Gabor Exp[-x*x/2] Gabor Cos[2P*x] Exp[-x*x/2]*Cos[2P*x] -1 Mathematca Gabor Gabor Plot [ {Exp[-x*x/2]}, {x,-3.0,3.0} ] Plot [ {Cos[2P*x]}, {x,-3.0,3.0} ] Plot [ {Exp[-x*x/2]* Cos[2P*x]}, {x,-3.0,3.0} ] DenstyPlot [ {Exp[-(x*x+y*y)/2]}, {x,-3.0, 3.0}, {y,-3.0,3.0}, PlotRange->{-1,1}, PlotPonts ->100, Mesh -> False] DenstyPlot [ {Cos[2P*x]}, {x,-3.0,3.0},{y,-3.0,3.0}, PlotRange->{-1,1}, PlotPonts ->100, Mesh -> False] DenstyPlot[{Exp[-(x*x+y*y)/2]* Cos[2P*x]},{x,-3.0,3.0}, {y,-3.0,3.0}, PlotRange->{-1,1}, PlotPonts ->100, Mesh -> False] 12
13 13
14 14 Gabor Gabor
15 Mathematca member/yosh/ouec_lecture/mage_recognton/ member/yosh/lecture.html Gabor gabor_xx_yyy.pgm (xx yyy) MAC MAC /Users/w/Desktop/ Desktop/./bar_data0.txt MAC Termnal Mathematca Mathematca Gabor Gabor 15
16 Gabor Gabor g = Import[ d:/presen/oecu_game_lecture/gabor_mages/gabor_ 08_120.pgm ]; gabor08120 = g[[1,1]]-100; LstDenstyPlot[gabor08120, Mesh->False, PlotRange->All]; LstPlot3D[gabor08120, PlotRange->All]; Gabor fgabor08120 = Fourer[gabor08120]; LstDenstyPlot[Abs[fgabor08120], Mesh->False, PlotRange->All]; LstPlot3D[Abs[fgabor08120], PlotRange->All]; Gabor ωy F() 16
17 Gabor ωy F() Gabor ωy F() 17
18 Gabor ωy F() Gabor ωy F() 18
19 Gabor ωy F() Gabor ωy F() Gabor 19
20 ωy F() ωy F() 20
21 ωy F() ωy F() 21
22 Mathematca Gabor gabor_08_000.pgm gabor_08_015.pgm gabor_08_030.pgm gabor_08_045.pgm gabor_08_060.pgm gabor_08_075.pgm gabor_08_090.pgm gabor_08_105.pgm gabor_08_120.pgm gabor_08_135.pgm gabor_08_150.pgm gabor_08_165.pgm cgabor_08 Gabor cgabor08 = gabor gabor gabor gabor gabor08165 LstDenstyPlot[cgabor08, Mesh->False, PlotRange->All]; LstPlot3D[cgabor08, PlotRange->All]; fcgabor08 = Fourer[cgabor08]; LstDenstyPlot[Abs[fcgabor08], Mesh->False, PlotRange->All]; LstPlot3D[Abs[fcgabor08], PlotRange->All]; 22
23 Gabor Gaborpgm g = Import[ / / / /gabor_08_120.pgm ]; gabor08120 = g[[1,1]]100; fgabor08120 = Fourer[gabor08120]; 12 gabor_08_000.pgm(gabor08000), gabor_08_015.pgm(gabor08015), gabor_08_030.pgm(gabor08030),., gabor_08_150.pgm(gabor08150), gabor_08_165.pgm(gabor08165) Gabor cgabor08 = gabor gabor gabor gabor gabor08165 fcgabor08 = Fourer[cgabor08]; LstDenstyPlot[Abs[fcgabor08], Mesh->False, PlotRange->All]; LstPlot3D[Abs[fcgabor08], PlotRange->All]; Gabor Gaborpgm g = Import[ / / / /gabor_08_000.pgm ]; gabor08000 = g[[1,1]]100; g = Import[ / / / /gabor_08_015.pgm ]; gabor08015 = g[[1,1]]100; g = Import[ / / / /gabor_08_030.pgm ]; gabor08030 = g[[1,1]]100; : g = Import[ / / / /gabor_08_150.pgm ]; gabor08150 = g[[1,1]]100; g = Import[ / / / /gabor_08_165.pgm ]; gabor08165 = g[[1,1]]100; cgabor08 = gabor gabor gabor gabor gabor08165 fcgabor08 = Fourer[cgabor08]; LstDenstyPlot[Abs[fcgabor08], Mesh->False, PlotRange->All]; LstPlot3D[Abs[fcgabor08], PlotRange->All]; 23
24 Gabor 04, 08, 16, 32 Gabor cgabor04.pgm cgabor08.pgm cgabor16.pgm cgabor32.pgm Mathematca member/yosh/ouec_lecture/mage_recognton/ member/yosh/lecture.html mglne_xx_yy_zz_p.pgm (xx,yy,zz ) MAC MAC /Users/w/Desktop/ Desktop/./bar_data0.txt MAC Termnal Mathematca 24
25 Mathematca mglne_xx_yy_zz_p.pgm (xx,yy,zz ) 01_02_04 01_04_16 02_04_08 04_08_16 p1 p2 p3 Mathematca mglne_xx_yy_zz_p.pgm (xx,yy,zz ) cgabor08 mglne_01_02_04_p1 cgabor16 cgabor32 25
26 Mathematca mglne_01_02_04_p1.pgm mglne_02_04_08_p1.pgm 04, 08, 16, 32 Gabor cgabor04.pgm cgabor08.pgm cgabor16.pgm cgabor32.pgm Mathematca mglne_01_02_04_p1.pgm g = Import[ /./././mglne_01_02_04_p1.pgm ]; lne = g[[1,1]]; LstDenstyPlot[lne, Mesh->False,PlotRange->All]; flne = Fourer[lne]; LstDenstyPlot[Abs[flne], Mesh->False,PlotRange->All]; 26
27 Mathematca Gabor cgabor08.pgm fcgabor08 = Fourer[cgabor08]; gflne = flne*abs[fcgabor08]]/(8*8); glne = InverseFourer[gflne]; LstDenstyPlot[Abs[gflne], Mesh->False,PlotRange->All]; LstDenstyPlot[Abs[glne], Mesh->False, PlotRange->{40,50}]; Mathematca Gabor g = Import[ /./././mglne_01_02_04_p1.pgm ]; lne = g[[1,1]]; LstDenstyPlot[lne, Mesh->False,PlotRange->All]; flne = Fourer[lne]; LstDenstyPlot[Abs[flne], Mesh->False,PlotRange->All]; fcgabor08 = Fourer[cgabor08]; gflne = flne*abs[fcgabor08]]/(8*8); glne = InverseFourer[gflne]; LstDenstyPlot[Abs[gflne], Mesh->False,PlotRange->All]; LstDenstyPlot[Abs[glne], Mesh->False, PlotRange->{40,50}]; 27
28 Mathematca cgabor08 lne Gabor cgabor08 F(ω) fcgabor08 = Fourer[cgabor08]; flne=fourer[lne] ω Gabor gflne = flne*abs[fcgabor08]]/(8*8) Abs[InverseFourer[gflne]] Mathematca Gabor cgabor16.pgm fcgabor16 = Fourer[cgabor16]; gflne = flne*abs[fcgabor16]]/(16*16); glne = InverseFourer[gflne]; LstDenstyPlot[Abs[gflne], Mesh->False,PlotRange->All]; LstDenstyPlot[Abs[glne], Mesh->False, PlotRange->{40,50}]; 28
29 Mathematca cgabor16 lne Gabor cgabor08 F(ω) fcgabor16 = Fourer[cgabor16]; flne=fourer[lne] ω Gabor gflne = flne*abs[fcgabor16]]/(16*16) Abs[InverseFourer[gflne]] Mathematca Gabor Gabor 29
30 ωy F() Gabor 30
31 ωy F() ωy F() 31
32 ωy F() Gabor ωy F() Gabor 32
33 ωy F() Gabor ωy F() Gabor 33
34 ωy F() Gabor ωy F() Gabor 34
35 ωy F() Gabor ωy F() Gabor 35
36 ωy F() Gabor ωy F() Gabor Gabor Gabor 36
37 Gabor Gabor f(x,y) g(x,y) Gabor f * g F(,ω y ) G(,ω y ) FG Gabor Gabor f(x,y) g(x,y) Gabor f * g F(,ω y ) G(,ω y ) FG 37
38 Gabor Gabor f(x,y) g(x,y) Gabor f * g F(,ω y ) G(,ω y ) FG Gabor Gabor 38
39 Gabor Gabor Mathematca 1512Gabor 0, 15, 30, 45,.., 135, 150, 165 Gabor Gabor Gabor 39
40 Mathematca Gabor 0, 15, 30, 45,.., 135, 150, 165 Gabor g = Import[ d: :/././. /gabor_08_120.pgm ]; gabor08120 = g[[1,1]]-100; fgabor08120 = Fourer[gabor08120]; LstDenstyPlot[gabor08120, Mesh->False, PlotRange->All]; LstPlot3D[gabor08120, PlotRange->All]; LstDenstyPlot[Abs[fgabor08120], Mesh->False, PlotRange->All]; LstPlot3D[Abs[fgabor08120], PlotRange->All]; Mathematca Gabor Gabor g = Import[ d:/./././glne_02_120.pgm ]; lne = g[[1,1]]; flne = Fourer[lne]; gflne = InverseFourer[flne*Abs[fgabor08120]]/(8*8); LstDenstyPlot[lne, Mesh->False,PlotRange->All]; LstDenstyPlot[Abs[gflne], Mesh->False,PlotRange->All]; glne_02_110 glne_02_100 glne_02_090 glne_02_080 40
41 Mathematca Gabor g = Import[ d:/./././mglne_02_p2.pgm ]; lne = g[[1,1]]; flne = Fourer[lne]; gflne = InverseFourer[flne*Abs[fgabor08120]]/(8*8); LstDenstyPlot[lne, Mesh->False,PlotRange->{0,20}]; LstDenstyPlot[Abs[gflne], Mesh->False,PlotRange->{0.20}]; fgabor08000, fgabor08015, fgabor08030, fgabor08045,, fgabor08150, gfabor08165 mglne_02_p1 mglne_02_p3 Mathematca Gabor Gabor g = Import[ d:/./././glne_02_110.pgm ]; lne = g[[1,1]]; flne = Fourer[lne]; gflne = flne*abs[fgabor08120] /(8*8); a120 = Norm[Flatten[gflne]]; Prnt[a120]; LstPlot[{a000,a015,a030,.,a165},PlotJoned->True}] fgabor08000, fgabor08015, fgabor08030, fgabor08045,, fgabor08150, gfabor08165 a000, a015, a030, a045,., a150, a165 41
42 Mathematca Gabor 110Gabor ω y Mathematca Gabor 110Gabor ω y 42
43 Mathematca Gabor 110Gabor ω y Mathematca ω y xx M M n = 1 n xx xy M = w x M = w x xy = 1 n M = w yy = 1 2 y y 2 M M xy yy 43
44 Mathematca 0 a000 ω y x y ω Cos(180 o ) Sn(180 o x x Cos(0 o y ) ) Sn(0 o ) Mathematca 15 a015 ω y x y Cos(15 o ) Sn(195 o ) x Cos(15 o y ) Sn(15 o ) 44
45 Mathematca 30 a030 x y Cos(210 o ) Sn(210 o ) ω y x Cos(30 o y ) Sn(30 o ) Mathematca 60 a060 ω y Cos(60 o ) Sn(60 o ) Cos(240 o ) Sn(240 o ) 45
46 Mathematca Cos(120 o )Sn(120 o ) ω y 120 a120 Cos(300 o ) Sn(300 o ) Mathematca 150 a150 Cos(150 o )Sn(150 o ) ω y Cos(330 o ) Sn(330 o ) 46
47 Mathematca 165 a165 ω y Cos(165 o ) Sn(165 o ) Cos(345 o ) Sn(345 o ) Mathematca w :, x : x, y : y =1 a000 Cos(0 o ) Sn(0 o ) =2 a015 Cos(15 o ) Sn(15 o ) =3 a030 Cos(30 o ) Sn(30 o ) : : : : =11 a150 Cos(150 o ) Sn(150 o ) =12 a165 Cos(165 o ) Sn(165 o ) =13 a000 Cos(180 o ) Sn(180 o ) =14 a015 Cos(195 o ) Sn(195 o ) : : : : =23 a150 Cos(330 o ) Sn(330 o ) =24 a165 Cos(345 o ) Sn(345 o ) 47
48 Mathematca ω y xy M M n = 1 xx xy yy M = w x M M xy yy n = 1 M = w y y 2 w :, x : x, y : y =1 a000 Cos(0 o ) Sn(0 o ) =2 a015 Cos(15 o ) Sn(15 o ) =3 a030 Cos(30 o ) Sn(30 o ) =24 xx : a165 n M = w x = 1 2 Cos(345 o ) Sn(345 o ) Mathematca M M xx xy M M xy yy n M = w x xy n = 1 yy M = w x n M = w mxx = N[a000*Cos[0 o ] *Cos[0 o ] + a015*cos[15 o ] *Cos[15 o ] +. + a165* Cos[165 o ] *Cos[165 o ] + a000* Cos[180 o ] *Cos[180 o ] + a015* Cos[195 o ] *Cos[195 o ] +. + a165* Cos[345 o ] *Cos[345 o ]] ; xx = 1 2 y = 1 w :, =1 a000 x : x, Cos(0 o ) y : y Sn(0 o ) =2 =3 a015 a030 : Cos(15 o ) Cos(30 o ) Sn(15 o ) Sn(30 o ) =24 a165 Cos(345 o ) Sn(345 o ) y 2 48
49 Mathematca M M xx xy M M xy yy xx n M = w x = 1 xy 2 n = 1 yy M = w x n M = w mxy = N[a000*Cos[0 o ] *Sn[0 o ] + a015*cos[15 o ] *Sn[15 o ] +. + a165* Cos[165 o ] *Sn[165 o ] + a000* Cos[180 o ] *Sn[180 o ] + a015* Cos[195 o ] *Sn[195 o ] +. + a165* Cos[345 o ] *Sn[345 o ] ]; y = 1 w :, =1 a000 x : x, Cos(0 o ) y : y Sn(0 o ) =2 =3 a015 a030 : Cos(15 o ) Cos(30 o ) Sn(15 o ) Sn(30 o ) =24 a165 Cos(345 o ) Sn(345 o ) y 2 Mathematca M M xx xy M M xy yy n M = w x xy n = 1 yy M = w x n M = w myy = N[a000*Sn[0 o ] *Sn[0 o ] + a015*sn[15 o ] *Sn[15 o ] +. + a165* Sn[165 o ] *Sn[165 o ] + a000* Sn[180 o ] *Sn[180 o ] + a015* Sn[195 o ] *Sn[195 o ] +. + a165*sn[345 o ] *Sn[345 o ] ]; xx = 1 2 y = 1 w :, =1 a000 x : x, Cos(0 o ) y : y Sn(0 o ) =2 =3 a015 a030 : Cos(15 o ) Cos(30 o ) Sn(15 o ) Sn(30 o ) =24 a165 Cos(345 o ) Sn(345 o ) y 2 49
50 Mathematca M M xx xy M M xy yy n M = w x xx = 1 xy 2 n = 1 yy M = w x n M = w y = 1 Prnt[mxx]; Prnt[mxy]; Prnt[myy]; mmtx ={{mxx,mxy},{mxy,myy}}; evec = Egenvectors[mmtx]; Prnt[180.0*ArcTan[evec[[1,1]],evec[[1,2]]]/π]; y 2 50
51 strpe1.jpg strpe2.jpg strpe3.jpg strpe4.jpg strpe5.jpg dot1.jpg dot2.jpg dot3.jpg dot4.jpg 51
52 52
53 53
54 ( g = Import[ /././strpe1.jpg ]; m = g[[1,1]]; LstDenstyPlot[m[[All,All,2]], Mesh->False,PlotRange->All]; fm = Fourer[m[[All,All,2]]; LstDenstyPlot[Abs[fm], Mesh->False,PlotRange->{0,200}]; tfm1 = Jon[Take[fm,-128,128],Take[fm,128,128]]; tfm2 = Jon[Take[fm,-128,-128],Take[fm,128,-128]]; tfm = Transpose[Jon[Transpose[tfm2],Transpose[tfm1]]]; LstDenstyPlot[Abs[tfm], Mesh->False,PlotRange->{0,200}]; Gabor Mathematca yosh@mage.med.osaka-u.ac.jp Subject 54
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The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers
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