function x = iQDLifting(L, H, dinfo, pinfo, lsi)
% LIFTING wavelet decomposition with directional lifting on
% qincux lattice
% 
% USAGE:
%    x = iQDLifting(L, H, dinfo, pinfo, lsi)
% INPUT:   
%    L    - Lower band
%    H    - Higher band
%  dinfo  - directional information 
%  pinfo  - partition information
%    lsi  - Lifting lattice
% OUTPUT:
%    x    - reconstructed signal
% 
% REFERENCE:
%    W. Ding, F. Wu, and S. Li, ¡°Lifting-based wavelet transform
%    with directionally spatial prediction,¡± in Proc. Picture 
%    Coding Symposium 2004, San Francisco, CA, USA, Dec. 2004.

y{1} = L;
y{2} = H;

N = size(lsi, 1);
y = Normalize(y, lsi{N,1}, lsi{N,2});
y = iQExtract(y);
for i = (N-1):-1:1
     f  = lsi{i,2}; % filter taps
     fd = lsi{i,3}; % filter delay: z(-fd)
     cd = lsi{i,4}; % component delay which corresponds non-one diagonal components: z(-cd)    
    if lsi{i,1}=='p'
        y{2} = Predict(y, f, dinfo, pinfo, fd);
    elseif lsi{i,1}=='u'
%         y{1} = Update(y, f, dinfo, pinfo, fd);
    end
end
x = QMerge(y{1}, y{2}); 


function [h] = Predict(x, f, dinfo, pinfo, fd)
% primal lift: predicting along conlumns
s = x{1};
d = x{2};
[r c]= size(d);
dset = [0 1; 1 2; 0.5 0.5; 2 1; 1 0; 2 -1; 0.5 -0.5; 1 -2];
ldset = length(dset);

h_all = zeros(r, c, ldset);
h = zeros(r, c);
bw = 8;
bh = 8;
br = r/bh;
bc = c/bw;
hsad = zeros(br, bc, ldset);
minsad = zeros(br,bc);
dinfo = zeros(br, bc);
pinfo = zeros(br, bc);
rextn = 2;
cextn = 2;

% symmetric extension
extns  = extn2D(s',1:(rextn+1), 'symodd')'; % odd symmtric extension with rextn samples for each row
extns  = extn2D(extns, 1:(cextn+1), 'symodd'); % odd symmtric extension with cextn samples for each row 
% generate 
% Predict for all spatial directions except directiona 
% 3 and 7 since they lie in half pixel position

for i = 1:ldset;
    if (i~=3&i~=7)
    h_all(1:r, :, i) = d(1:r, :) + 0.5*extns( (1:r)+cextn + dset(i,1), (1:c)+rextn + dset(i,2) ) ...
                                 + 0.5*extns( (1:r)+cextn - dset(i,1), (1:c)+rextn - dset(i,2) );
    end
end

% generate half pixel value for direction 3
halfh3 = zeros(1+r, 1+c);
for i = 1:(1+r)
    for j = 1:(1+c)
        half3(i, j) = (-extns(cextn + i + 1, rextn + j -2) + 3*extns(cextn + i, rextn + j -1) + ...
            3*extns(cextn + i - 1, rextn + j) - extns(cextn + i - 2, rextn + j + 1) )/4;
    end
end
% predict for direction 3
    h_all(1:r, 1:c, 3) = d(1:r, 1:c) + 0.5*half3(1:r, 1:c) + 0.5*half3((1:r)+1, (1:c)+1);
    
% generate half pixel value for direction 7
halfh7 = zeros(1+r, 1+c);
for i = 1:(1+r)
    for j = 1:(1+c)
        half7(i, j) = (-extns(cextn + i - 2 , rextn + j - 1 -1) + 3*extns(cextn + i -1, rextn + j -1) + ...
            3*extns(cextn + i , rextn + j + 1 - 1) - extns(cextn + i + 1, rextn + j + 2 -1) )/4;
    end
end
% predict for direction 7
    h_all(1:r, 1:c, 7) = d(1:r, 1:c) + 0.5*half7((1:r), (1:c)+1) + 0.5*half7((1:r)+1, (1:c));

for i=1:r
    for j=1:c
        if mod(i+j,2)==0
            h_all(i,j, :) = 0;          
        end
    end
end    

for i = 1:br
    for j = 1:bc
        roff = (i-1)*bh;
        coff = (j-1)*bw;
        hsad(i, j, 1:ldset) = sum(sum(abs(h_all((roff+1):(roff+bh), (coff+1):(coff+bw), 1:ldset)),1),2);
        [minsad(i,j) dinfo(i,j)] = min(hsad(i, j, :));
        h((roff+1):(roff+bh), (coff+1):(coff+bw)) = h_all((roff+1):(roff+bh), (coff+1):(coff+bw), dinfo(i,j));
    end
end

function [l] = Update(x, f, dinfo, pinfo, fd)
% dual lift: updating along conlumns
s = x{1};
d = x{2};
dset = -3:3;
ldset = length(dset);
[r c] = size(s);
bw = 4;
bh = 4;
br = r/bh;
bc = c/bw;
l = zeros(r, c);
l_all  = zeros(r, c, ldset);

rextn = 3;
cextn = 1;
extnd  = extn2D(d',1:(rextn+1), 'symodd')'; % odd symmtric extension with 3 samples for each row
extnd  = extn2D(extnd, 1:(cextn+1), 'symodd'); % odd symmtric extension with 1 samples for each row

% Update for all spatial directions
for i = 1:ldset
    bos1 = rextn+1 + dset(i);
    bos2 = rextn+1 - dset(i);
    l_all(1:r, :, i) = s(1:r, :) - 0.25*extnd((1:r)+cextn - 1, bos1:(bos1+c-1)) - 0.25*extnd((1:r)+cextn , bos2:(bos2+c-1));
end

% Pick the right one with dinfo
for i = 1:br
    for j = 1:bc
        roff = (i-1)*bh;
        coff = (j-1)*bw;
        l((roff+1):(roff+bh), (coff+1):(coff+bw)) = l_all((roff+1):(roff+bh), (coff+1):(coff+bw), dinfo(i,j));
    end
end


function y = Normalize(x, Lscale, Hscale)

y{1} = (1/Lscale)*x{1};
y{2} = (1/Hscale)*x{2};

function x  = QMerge(L, H)
% Spliting along conlumns
%    Input:     L - Lower band
%               H - Lower band
%   Output:      x - recontructed signal
%   Notes: the indices of the signal is supposed to
%          be 0, 1, 2, ..., M-1 and  0, 1, 2, ..., N-1
%          for the size of MxN. This is not consistent
%          with the convention of MATLAB                                     

[h, w] = size(L);
for i=1:h
    for j=1:w
        if mod(i+j,2)==0
            x(i,j) = L(i,j);
        else
            x(i,j) = H(i,j);            
        end
    end
end
 
function y = iQExtract(x)
[h, w] = size(x{1});
H = x{2};
He = H(:, 1:2:w);
Ho = H(:, 2:2:w);
Het = upsample(He, 2, 1);
Hot = upsample(Ho, 2);
y{2}(:, 1:2:w) = Het(:,:);
y{2}(:, 2:2:w) = Hot(:,:);
y{1} = x{1};

