📄 dfilters.m
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function [h0, h1] = dfilters(fname, type)
% DFILTERS Generate directional 2D filters
%
% [h0, h1] = dfilters(fname, type)
%
% Input:
% fname: Filter name. Available 'fname' are:
% 'haar': the "Haar" filters
% 'vk': McClellan transformed of the filter from
% the VK book
% 'ko': orthogonal filter in the Kovacevic's paper
% 'kos': smooth 'ko' filter
% 'lax': 17 x 17 by Lu, Antoniou and Xu
% 'sk': 9 x 9 by Shah and Kalker
% 'cd': 7 and 9 McClellan transformed by
% Cohen and Daubechies
% 'pkva': ladder filters by Phong et al.
% 'oqf_362': regular 3 x 6 filter
% 'dvmlp' : regular linear phase biorthogonal filter with 3 dvm
% 'sinc': ideal filter (*NO perfect recontruction*)
% 'dmaxflat': diamond maxflat filters obtained from a three stage ladder
%
% type: 'd' or 'r' for decomposition or reconstruction filters
%
% Output:
% h0, h1: diamond filter pair (lowpass and highpass)
% To test those filters (for the PR condition for the FIR case), verify that:
% conv2(h0, modulate2(h1, 'b')) + conv2(modulate2(h0, 'b'), h1) = 2
% (replace + with - for even size filters)
%
% To test for orthogonal filter
% conv2(h, reverse2(h)) + modulate2(conv2(h, reverse2(h)), 'b') = 2
% The diamond-shaped filter pair
switch fname
case {'haar'}
if lower(type(1)) == 'd'
h0 = [1, 1] / sqrt(2);
h1 = [-1, 1] / sqrt(2);
else
h0 = [1, 1] / sqrt(2);
h1 = [1, -1] / sqrt(2);
end
case 'vk' % in Vetterli and Kovacevic book
if lower(type(1)) == 'd'
h0 = [1, 2, 1] / 4;
h1 = [-1, -2, 6, -2, -1] / 4;
else
h0 = [-1, 2, 6, 2, -1] / 4;
h1 = [-1, 2, -1] / 4;
end
% McClellan transfrom to obtain 2D diamond filters
t = [0, 1, 0; 1, 0, 1; 0, 1, 0] / 4; % diamond kernel
h0 = ftrans2(h0, t);
h1 = ftrans2(h1, t);
case 'ko' % orthogonal filters in Kovacevic's thesis
a0 = 2; a1 = 0.5; a2 = 1;
h0 = [0, -a1, -a0*a1, 0;
-a2, -a0*a2, -a0, 1;
0, a0*a1*a2, -a1*a2, 0];
% h1 = qmf2(h0);
h1 = [0, -a1*a2, -a0*a1*a2, 0;
1, a0, -a0*a2, a2;
0, -a0*a1, a1, 0];
% Normalize filter sum and norm;
norm = sqrt(2) / sum(h0(:));
h0 = h0 * norm;
h1 = h1 * norm;
if type == 'r'
% Reverse filters for reconstruction
h0 = h0(end:-1:1, end:-1:1);
h1 = h1(end:-1:1, end:-1:1);
end
case 'kos' % Smooth orthogonal filters in Kovacevic's thesis
a0 = -sqrt(3); a1 = -sqrt(3); a2 = 2+sqrt(3);
h0 = [0, -a1, -a0*a1, 0;
-a2, -a0*a2, -a0, 1;
0, a0*a1*a2, -a1*a2, 0];
% h1 = qmf2(h0);
h1 = [0, -a1*a2, -a0*a1*a2, 0;
1, a0, -a0*a2, a2;
0, -a0*a1, a1, 0];
% Normalize filter sum and norm;
norm = sqrt(2) / sum(h0(:));
h0 = h0 * norm;
h1 = h1 * norm;
if type == 'r'
% Reverse filters for reconstruction
h0 = h0(end:-1:1, end:-1:1);
h1 = h1(end:-1:1, end:-1:1);
end
case 'lax' % by Lu, Antoniou and Xu
h = [-1.2972901e-5, 1.2316237e-4, -7.5212207e-5, 6.3686104e-5, ...
9.4800610e-5, -7.5862919e-5, 2.9586164e-4, -1.8430337e-4; ...
1.2355540e-4, -1.2780882e-4, -1.9663685e-5, -4.5956538e-5, ...
-6.5195193e-4, -2.4722942e-4, -2.1538331e-5, -7.0882131e-4; ...
-7.5319075e-5, -1.9350810e-5, -7.1947086e-4, 1.2295412e-3, ...
5.7411214e-4, 4.4705422e-4, 1.9623554e-3, 3.3596717e-4; ...
6.3400249e-5, -2.4947178e-4, 4.4905711e-4, -4.1053629e-3, ...
-2.8588307e-3, 4.3782726e-3, -3.1690509e-3, -3.4371484e-3; ...
9.6404973e-5, -4.6116254e-5, 1.2371871e-3, -1.1675575e-2, ...
1.6173911e-2, -4.1197559e-3, 4.4911165e-3, 1.1635130e-2; ...
-7.6955555e-5, -6.5618379e-4, 5.7752252e-4, 1.6211426e-2, ...
2.1310378e-2, -2.8712621e-3, -4.8422645e-2, -5.9246338e-3; ...
2.9802986e-4, -2.1365364e-5, 1.9701350e-3, 4.5047673e-3, ...
-4.8489158e-2, -3.1809526e-3, -2.9406153e-2, 1.8993868e-1; ...
-1.8556637e-4, -7.1279432e-4, 3.3839195e-4, 1.1662001e-2, ...
-5.9398223e-3, -3.4467920e-3, 1.9006499e-1, 5.7235228e-1];
h0 = sqrt(2) * [h, h(:, end-1:-1:1); ...
h(end-1:-1:1, :), h(end-1:-1:1, end-1:-1:1)];
h1 = modulate2(h0, 'b');
case 'sk' % by Shah and Kalker
h = [ 0.621729, 0.161889, -0.0126949, -0.00542504, 0.00124838; ...
0.161889, -0.0353769, -0.0162751, -0.00499353, 0; ...
-0.0126949, -0.0162751, 0.00749029, 0, 0; ...
-0.00542504, 0.00499353, 0, 0, 0; ...
0.00124838, 0, 0, 0, 0];
h0 = sqrt(2) * [h(end:-1:2, end:-1:2), h(end:-1:2, :); ...
h(:, end:-1:2), h];
h1 = modulate2(h0, 'b');
case 'dvmlp'
q= sqrt(2); b=.02; b1 = b*b;
h = [b/q 0 -2*q*b 0 3*q*b 0 -2*q*b 0 b/q;
0 -1/(16*q) 0 9/(16*q) 1/q 9/(16*q) 0 -1/(16*q) 0;
b/q 0 -2*q*b 0 3*q*b 0 -2*q*b 0 b/q];
g0 = [-b1/q 0 4*b1*q 0 -14*q*b1 0 28*q*b1 0 -35*q*b1 0 28*q*b1 0 -14*q*b1 0 4*b1*q 0 -b1/q;
0 b/(8*q) 0 -13*b/(8*q) b/q 33*b/(8*q) -2*q*b -21*b/(8*q) 3*q*b -21*b/(8*q) -2*q*b 33*b/(8*q) b/q -13*b/(8*q) 0 b/(8*q) 0;
-q*b1 0 -1/(256*q) + 8*q*b1 0 9/(128*q) - 28*q*b1 -1/(q*16) -63/(256*q) + 56*q*b1 9/(16*q) 87/(64*q)-70*q*b1 9/(16*q) -63/(256*q) + 56*q*b1 -1/(q*16) 9/(128*q) - 28*q*b1 0 -1/(256*q) + 8*q*b1 0 -q*b1;
0 b/(8*q) 0 -13*b/(8*q) b/q 33*b/(8*q) -2*q*b -21*b/(8*q) 3*q*b -21*b/(8*q) -2*q*b 33*b/(8*q) b/q -13*b/(8*q) 0 b/(8*q) 0;
-b1/q 0 4*b1*q 0 -14*q*b1 0 28*q*b1 0 -35*q*b1 0 28*q*b1 0 -14*q*b1 0 4*b1*q 0 -b1/q];
h1 = modulate2(g0,'b' );
h0 = h;
if lower(type(1)) == 'r'
h1 = modulate2(h ,'b');
h0 = g0;
end
case {'cd', '7-9'} % by Cohen and Daubechies
% 1D prototype filters: the '7-9' pair
h0 = [0.026748757411, -0.016864118443, -0.078223266529, ...
0.266864118443, 0.602949018236, 0.266864118443, ...
-0.078223266529, -0.016864118443, 0.026748757411];
g0 = [-0.045635881557, -0.028771763114, 0.295635881557, ...
0.557543526229, 0.295635881557, -0.028771763114, ...
-0.045635881557];
if lower(type(1)) == 'd'
h1 = modulate2(g0, 'c');
else
h1 = modulate2(h0, 'c');
h0 = g0;
end
% Use McClellan to obtain 2D filters
t = [0, 1, 0; 1, 0, 1; 0, 1, 0] / 4; % diamond kernel
h0 = sqrt(2) * ftrans2(h0, t);
h1 = sqrt(2) * ftrans2(h1, t);
case {'pkva', 'ldtest'} % Filters from the ladder structure
% Allpass filter for the ladder structure network
beta = ldfilter(fname);
% Analysis filters
[h0, h1] = ld2quin(beta);
% Normalize norm
h0 = sqrt(2) * h0;
h1 = sqrt(2) * h1;
% Synthesis filters
if lower(type(1)) == 'r'
f0 = modulate2(h1, 'b');
f1 = modulate2(h0, 'b');
h0 = f0;
h1 = f1;
end
case {'pkva-half4'} % Filters from the ladder structure
% Allpass filter for the ladder structure network
beta = ldfilterhalf( 4);
% Analysis filters
[h0, h1] = ld2quin(beta);
% Normalize norm
h0 = sqrt(2) * h0;
h1 = sqrt(2) * h1;
% Synthesis filters
if lower(type(1)) == 'r'
f0 = modulate2(h1, 'b');
f1 = modulate2(h0, 'b');
h0 = f0;
h1 = f1;
end
case {'pkva-half6'} % Filters from the ladder structure
% Allpass filter for the ladder structure network
beta = ldfilterhalf( 6);
% Analysis filters
[h0, h1] = ld2quin(beta);
% Normalize norm
h0 = sqrt(2) * h0;
h1 = sqrt(2) * h1;
% Synthesis filters
if lower(type(1)) == 'r'
f0 = modulate2(h1, 'b');
f1 = modulate2(h0, 'b');
h0 = f0;
h1 = f1;
end
case {'pkva-half8'} % Filters from the ladder structure
% Allpass filter for the ladder structure network
beta = ldfilterhalf( 8);
% Analysis filters
[h0, h1] = ld2quin(beta);
% Normalize norm
h0 = sqrt(2) * h0;
h1 = sqrt(2) * h1;
% Synthesis filters
if lower(type(1)) == 'r'
f0 = modulate2(h1, 'b');
f1 = modulate2(h0, 'b');
h0 = f0;
h1 = f1;
end
case 'sinc' % The "sinc" case, NO Perfect Reconstruction
% Ideal low and high pass filters
flength = 30;
h0 = fir1(flength, 0.5);
h1 = modulate2(h0, 'c');
% Use McClellan to obtain 2D filters
t = [0, 1, 0; 1, 0, 1; 0, 1, 0] / 4; % diamond kernel
h0 = sqrt(2) * ftrans2(h0, t);
h1 = sqrt(2) * ftrans2(h1, t);
case {'oqf_362'} % Some "home-made" filters!
h0 = sqrt(2) / 64 * ...
[ sqrt(15), -3, 0; ...
0, 5, -sqrt(15); ...
-2*sqrt(15), 30, 0; ...
0, 30, 2*sqrt(15); ...
sqrt(15), 5, 0; ...
0, -3, -sqrt(15)]';
h1 = -reverse2(modulate2(h0, 'b'));
if type == 'r'
% Reverse filters for reconstruction
h0 = h0(end:-1:1, end:-1:1);
h1 = h1(end:-1:1, end:-1:1);
end
case {'test'} % Only for the shape, not for PR
h0 = [0, 1, 0; 1, 4, 1; 0, 1, 0];
h1 = [0, -1, 0; -1, 4, -1; 0, -1, 0];
case {'testDVM'} % Only for directional vanishing moment
h0 = [1, 1; 1, 1] / sqrt(2);
h1 = [-1, 1; 1, -1] / sqrt(2);
case 'qmf' % by Lu, Antoniou and Xu
% ideal response
% window
m=2;
n=2;
w=[];
w1d=kaiser(4*m+1,2.6)
for n1=-m:m
for n2 = -n:n
w(n1+m+1,n2+n+1) = w1d(2*m+n1+n2+1)*w1d(2*m+n1-n2+1)
end
end
h=[];
for n1=-m:m
for n2 = -n:n
h(n1+m+1,n2+n+1) = .5*sinc((n1+n2)/2)*.5*sinc((n1-n2)/2);
end
end
c=sum(sum(h));
h = sqrt(2)*h/c;
h0=h.*w;
h1 = modulate2(h0,'b');
%h0 = modulate2(h,'r');
%h1 = modulate2(h,'b');
case 'qmf2' % by Lu, Antoniou and Xu
% ideal response
% window
h=[-.001104 .002494 -0.001744 0.004895 -0.000048 -.000311;
0.008918 -0.002844 -0.025197 -0.017135 0.003905 -0.000081;
-0.007587 -0.065904 00.100431 -0.055878 0.007023 0.001504;
0.001725 0.184162 0.632115 0.099414 -0.027006 -0.001110;
-0.017935 -0.000491 0.191397 -0.001787 -0.010587 0.002060;
.001353 0.005635 -0.001231 -0.009052 -0.002668 0.000596];
h0 = h./sum(sum(h));
h1 = modulate2(h0,'b');
%h0 = modulate2(h,'r');
%h1 = modulate2(h,'b');
case 'dmaxflat4'
M1 = 1/sqrt(2); M2=M1;
k1=1-sqrt(2);k3=k1;
k2 = M1;
h = [.25*k2*k3 .5*k2 1+.5*k2*k3]*M1; h = [h fliplr(h(1:end-1))];
g = [-.125*k1*k2*k3 0.25*k1*k2 (-0.5*k1-0.5*k3-0.375*k1*k2*k3) 1+ .5*k1*k2]*M2;
g = [g fliplr(g(1:end-1))];
B = dmaxflat(4,0);
h0 = mctrans(h,B);
g0 = mctrans(g,B);
h0 = sqrt(2)*(h0./sum(h0(:)));
g0 = sqrt(2)*(g0./sum(g0(:)));
h1 = modulate2(g0,'b' );
if lower(type(1)) == 'r'
h1 = modulate2(h0 ,'b');
h0 = g0;
end
case 'dmaxflat5'
M1 = 1/sqrt(2); M2=M1;
k1=1-sqrt(2);k3=k1;
k2 = M1;
h = [.25*k2*k3 .5*k2 1+.5*k2*k3]*M1; h = [h fliplr(h(1:end-1))];
g = [-.125*k1*k2*k3 0.25*k1*k2 (-0.5*k1-0.5*k3-0.375*k1*k2*k3) 1+ .5*k1*k2]*M2;
g = [g fliplr(g(1:end-1))];
B = dmaxflat(5,0);
h0 = mctrans(h,B);
g0 = mctrans(g,B);
h0 = sqrt(2)*(h0./sum(h0(:)));
g0 = sqrt(2)*(g0./sum(g0(:)));
h1 = modulate2(g0,'b' );
if lower(type(1)) == 'r'
h1 = modulate2(h0 ,'b');
h0 = g0;
end
case 'dmaxflat6'
M1 = 1/sqrt(2); M2=M1;
k1=1-sqrt(2);k3=k1;
k2 = M1;
h = [.25*k2*k3 .5*k2 1+.5*k2*k3]*M1; h = [h fliplr(h(1:end-1))];
g = [-.125*k1*k2*k3 0.25*k1*k2 (-0.5*k1-0.5*k3-0.375*k1*k2*k3) 1+ .5*k1*k2]*M2;
g = [g fliplr(g(1:end-1))];
B = dmaxflat(6,0);
h0 = mctrans(h,B);
g0 = mctrans(g,B);
h0 = sqrt(2)*(h0./sum(h0(:)));
g0 = sqrt(2)*(g0./sum(g0(:)));
h1 = modulate2(g0,'b' );
if lower(type(1)) == 'r'
h1 = modulate2(h0 ,'b');
h0 = g0;
end
case 'dmaxflat7'
M1 = 1/sqrt(2); M2=M1;
k1=1-sqrt(2);k3=k1;
k2 = M1;
h = [.25*k2*k3 .5*k2 1+.5*k2*k3]*M1; h = [h fliplr(h(1:end-1))];
g = [-.125*k1*k2*k3 0.25*k1*k2 (-0.5*k1-0.5*k3-0.375*k1*k2*k3) 1+ .5*k1*k2]*M2;
g = [g fliplr(g(1:end-1))];
B = dmaxflat(7,0);
h0 = mctrans(h,B);
g0 = mctrans(g,B);
h0 = sqrt(2)*(h0./sum(h0(:)));
g0 = sqrt(2)*(g0./sum(g0(:)));
h1 = modulate2(g0,'b' );
if lower(type(1)) == 'r'
h1 = modulate2(h0 ,'b');
h0 = g0;
end
otherwise
% Assume the "degenerated" case: 1D wavelet filters
[h0, h1] = wfilters(fname, type);
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