function y = nsdfbdec( x, dfilter, clevels )
% NSDFBDEC Nonsubsampled directional filter bank decomposition.
% NSDFBDEC Decompose the image X by a nonsubsampled directional filter bank
% with a binary-tree structure. It outputs the final branches, totally 2^clevels.
% There is no subsampling and hence the operation is shift-invariant.
%
% nsdfbdec( x, dfilter, [clevels] )
%
% INPUT:
% x:
% an array, input image.
% dfilter:
% a string, directional filter name.
% a cell of matrices, including two directional filters and eight
% parallelogram filters.
% clevels:
% a non-negative integer, number of decomposition levels.
%
% OUTPUT:
% y:
% a cell vector, output subbands.
%
% See also: DFILTERS, PARAFILTERS, NSSFBDEC.
%
% History:
% 08/06/2004 Created by Jianping Zhou.
% Input check
if ~exist('clevels', 'var')
clevels = 0 ;
y{1} = x;
return;
end
if (clevels ~= round(clevels)) | (clevels < 0)
error('Number of decomposition levels must be a non-negative integer');
end
if clevels == 0
% No decomposition, simply copy input to output
y{1} = x;
return;
end
if ~ischar( dfilter )
if iscell( dfilter )
if length( dfilter ) ~= 4
error('You shall provide a cell of two 2D directional filters and two groups of 2D parallelogram filters!');
end
else
error('You shall provide the name of directional filter or all filters!');
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Get fan filters, parallelogram filters, and basic sampling matrices
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Get the diamond filters, if necessary
if ischar( dfilter )
% Get the directional filters for the critically sampled DFB.
[h1, h2] = dfilters(dfilter, 'd');
% A scale is required for the nonsubsampled case.
h1 = h1./sqrt(2) ;
h2 = h2./sqrt(2) ;
% Generate the first-level fan filters by modulations.
k1 = modulate2(h1, 'c');
k2 = modulate2(h2, 'c');
% Obtain the parallelogram filters from the diamond filters
[f1, f2] = parafilters( h1, h2 ) ;
else
% Copy the fan filters directly.
k1 = dfilter{1} ;
k2 = dfilter{2} ;
% Copy the parallelogram filters directly.
f1 = dfilter{3} ;
f2 = dfilter{4} ;
end
% Quincunx sampling matrices
q1 = [1, -1; 1, 1];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% First-level Decompositions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
if clevels == 1
% No upsampling for filters at the first-level.
[y{1}, y{2}] = nssfbdec( x, k1, k2 ) ;
else %Others
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Second-level Decompositions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% No upsampling at filters for the first-level.
[x1, x2] = nssfbdec( x, k1, k2 ) ;
% Convolution with upsampled filters
[y{1}, y{2}] = nssfbdec( x1, k1, k2, q1 ) ;
[y{3}, y{4}] = nssfbdec( x2, k1, k2, q1 ) ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Third and higher levels Decompositions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Third and higher levels decompositions
for l = 3:clevels
% Allocate space for the new subband outputs
y_old = y;
y = cell(1, 2^l);
% The first half channels:
for k = 1:2^(l-2)
% Compute the upsampling matrix by the formula (3.18) of Minh N. Do's
% thesis. The upsampling matrix for the channel k in a l-levels DFB is
% M_k^{(l-1)} (refer to (3.18), pp. 53, Minh N. Do's thesis)
% Compute s_{(l-1)}(k):
slk = 2*floor( (k-1) /2 ) - 2^(l-3) + 1 ;
% Compute the sampling matrix:
mkl = 2*[ 2^(l-3), 0; 0, 1 ]*[1, 0; -slk, 1];
i = mod(k-1, 2) + 1;
% Decompose by the two-channel filter bank:
[y{2*k-1}, y{2*k}] = nssfbdec( y_old{k}, f1{i}, f2{i}, mkl );
end
% The second half channels:
for k = 2^(l-2)+1 : 2^(l-1)
% Compute the upsampling matrix by the extension of the formula (3.18)
% of Minh N. Do's thesis to the second half channels.
% thesis. The upsampling matrix for the channel k in a l-levels DFB is
% M_k^{(l-1)} (refer to notes by Jianping Zhou)
% Compute s_{(l-1)}(k):
slk = 2 * floor( ( k-2^(l-2)-1 ) / 2 ) - 2^(l-3) + 1 ;
% Compute the sampling matrix:
mkl = 2*[ 1, 0; 0, 2^(l-3) ]*[1, -slk; 0, 1];
i = mod(k-1, 2) + 3;
% Decompose by the two-channel filter bank:
[y{2*k-1}, y{2*k}] = nssfbdec( y_old{k}, f1{i}, f2{i}, mkl );
end
end
end