{VERSION 5 0 "APPLE_PPC_MAC" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warni ng" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 380 "# This Maple Worksh eet has been designed and tested with Maple 7.0 for Mac OS9\n#\n# It i mplements the analytical formula for hex-splines as described in the p aper\n# \"Hex-splines: A novel spline family for hexagonal lattices\", by\n# D. Van De Ville, T. Blu, M. Unser, W. Philips, I. Lemahieu, R. \+ Van de Walle\n# submitted and accepted for publication in IEEE Tr. on \+ Image Processing\n#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "resta rt: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "with(linalg): with( inttrans): with(combinat): with(plottools):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the name h ilbert has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 47 "Warning, t he name fibonacci has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 67 "Warning, the protected name Chi has been redefined and unprotected\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 921 "hexlocalization:=proc( e1,e2,e3,k1,k2) \nlocal x,x1,x2,grid1,grid2,grid,gridl,L,iter,element, elementv,s; \ndescription \"compute the localization operators for the hex-splines\";\nx:=vector(2,[x1,x2]);\ngrid1:=exp(-I*2*Pi*innerprod(x ,-e1))+exp(-I*2*Pi*innerprod(x,e1))-exp(-I*2*Pi*innerprod(x,e2))-exp(- I*2*Pi*innerprod(x,-e2));\ngrid2:=exp(-I*2*Pi*innerprod(x,-e1))+exp(-I *2*Pi*innerprod(x,e1))-exp(-I*2*Pi*innerprod(x,e3))-exp(-I*2*Pi*innerp rod(x,-e3));\ngrid:=combine(expand(grid1^k1*grid2^k2));\ngridl:=conver t(grid,list);\nL:=vector(nops(gridl));\nfor iter from 1 to nops(gridl) do\n element:=gridl[iter];\n if type(element,constant)=true then \n \+ elementv:=element;\n L[iter]:=[0,0,elementv];\n else\n if nops(el ement)=1 then elementv:=1; \n else elementv:=op(1,element); element: =op(2,element); end if;\n s:=op(1,element)/(-I*2*Pi); \n L[iter]:= [coeff(s,x1),coeff(s,x2),elementv];\n end if;\nend do;\nreturn eval(L) ;\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1457 "hexbuil ding:=proc(u1,u2,u3,pow,x1,x2,xc1,xc2) \nlocal x,xc,rpow,R,rR,pr1,pr2; \ndescription \"compute the building functions for the hex-splines\"; \nx:=vector(2,[x1,x2]);\nxc:=vector(2,[xc1,xc2]);\nrpow:=matadd(pow,ve ctor(3,[-1,-1,-1]));\nif pow[2]<0 then\n R:=matrix(concat(u1,u2));\n \+ 2*innerprod(u1,u2)*hexbasefunc(u1,u2,u3,matadd(pow,vector(3,[-1,1,0]) ),x1,x2,xc1,xc2)-hexbuilding(u1,u2,u3,matadd(pow,vector(3,[0,1,-1])),x 1,x2,xc1,xc2);\nelif pow[3]<0 then\n R:=matrix(concat(u1,u2)); \n \+ 2*innerprod(u1,u2)*hexbasefunc(u1,u2,u3,matadd(pow,vector(3,[-1,0,1]) ),x1,x2,xc1,xc2)-hexbuilding(u1,u2,u3,matadd(pow,vector(3,[0,-1,1])),x 1,x2,xc1,xc2);\nelif pow[3]=0 then\n R:=matrix(concat(u1,u2)); \n rR :=transpose(inverse(R)); \n pr1:=innerprod(x,col(rR,1)); \n pr2:=inn erprod(x,col(rR,2)); \n Heaviside(innerprod(xc,col(rR,1)))*pr1^rpow[1 ]*Heaviside(innerprod(xc,col(rR,2)))*pr2^rpow[2]/(factorial(rpow[1])*f actorial(rpow[2]));\nelif pow[2]=0 then\n R:=matrix(concat(u1,u3));\n rR:=transpose(inverse(R));\n pr1:=innerprod(x,col(rR,1));\n pr2:=i nnerprod(x,col(rR,2));\n Heaviside(innerprod(xc,col(rR,1)))*pr1^rpow[ 1]*Heaviside(innerprod(xc,col(rR,2)))*pr2^rpow[3]/(factorial(rpow[1])* factorial(rpow[3]));\nelse\n R:=matrix(concat(u1,u2)); \n1/(2*innerpr od(u1,u2)*(rpow[1]/rpow[1]))*((rpow[3]/rpow[3])*hexbuilding(u1,u2,u3,m atadd(pow,vector(3,[1,0,-1])),x1,x2,xc1,xc2)+(rpow[2]/rpow[2])*hexbuil ding(u1,u2,u3,matadd(pow,vector(3,[1,-1,0])),x1,x2,xc1,xc2));\nend if; \nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 990 "diffhexspli ne:=proc(a,b,order,df,x1,x2,xc1,xc2) \nlocal c,sp,comb,combl,filter,fi lterl,iter,pow,x,grid1,grid2,e1,e2,e3,u1,u2,u3,gridR;\ndescription \"c ompose the analytical form of the hex-splines\";\nx:=vector(2,[x1,x2]) ;\ngridR:=matrix(2,2,[b,0,a/2,a]);\nc:=(4*b^2-a^2)/(8*b); \ne1:=vector (2,[-(b-c),0]);\ne2:=vector(2,[-c,a/2]); \ne3:=vector(2,[-c,-a/2]); \n u1:=vector(2,[1,0]);\nu2:=matadd(e2,scalarmul(e1,-1)); u2:=scalarmul(u 2,1/abs(u2[2])); \nu3:=matadd(e3,scalarmul(e1,-1)); u3:=scalarmul(u3,1 /abs(u3[2])); \nsp:=(x1,x2)->0;\ncombl:=composition(order+2,2);\nfor c omb in combl do\n filterl:=hexlocalization(e1,e2,e3,comb[1]-1,comb[2] -1);\n pow:=matadd((comb[1]-1)*vector(3,[1,1,0]),(comb[2]-1)*vector(3 ,[1,0,1]));\n pow:=matadd(pow,-df);\n for iter from 1 to vectdim(fil terl) do\n filter:=filterl[iter];\n sp:=sp+binomial(order,comb[1 ]-1)*filter[3]*hexbuilding(u1,u2,u3,pow,x1-filter[1],x2-filter[2],xc1- filter[1],xc2-filter[2]);\n end do;\nend do;\nreturn eval(sp/det(grid R)^(order-1));\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "hexspline:=proc(a,b,order,x1,x2,xc1,xc2)\ndescription \"compose t he analytical form of the hex-splines\";\ndiffhexspline(a,b,order,vect or(3,[0,0,0]),x1,x2,xc1,xc2);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 175 "diffrhexspline:=proc(order,df,x1,x2,xc1,xc2)\ndescri ption \"compose the analytical form of the regular hex-splines\";\ndif fhexspline(1,sqrt(3)/2,order,df,x1,x2,xc1,xc2);\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "rhexspline:=proc(order,x1,x2,xc1,x c2) \ndescription \"compose the analytical form of the regular hex-spl ines\";\nhexspline(1,sqrt(3)/2,order,x1,x2,xc1,xc2);\nend proc:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "# Examples" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "# REGULAR HEX-SPLINE, FIRST ORDER" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot3d(rhexspline(1,x1,x2,x1 ,x2),x1=-2.5..2.5,x2=-2.5..2.5,grid=[60,60]);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6$-%%GRIDG6%;$!#D!\"\"$\"#DF)F&X,%) anythingG6\"6\"[gl'!%\"!!#[\\y\"gn\"gn00000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 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