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CVBM3D.m
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function [Xdenoised] = CVBM3D(Xnoisy, sigma, Xorig)
% CVBM3D denoising of RGB videos corrupted with AWGN.
%
%
% [Xdenoised] = CVBM3D(Xnoisy, sigma, Xorig)
%
% INPUTS:
%
% 1) Xnoisy --> Either a filename of a noisy AVI RGB uncompressed video (e.g. 'SMg20.avi')
% or a 4-D matrix of dimensions (M x N x 3 x NumberOfFrames)
% The intensity range is [0,255]!
% 2) Sigma --> Noise standard deviation (assumed intensity range is [0,255])
%
% 3) Xorig (optional parameter) --> Filename of the original video
%
% OUTPUT: .avi files are written to the current matlab folder
%
% 1) Xdenoised --> A 4-D matrix with the denoised RGB-video
%
% USAGE EXAMPLES:
% 1) To denoise a video:
% CVBM3D('SMg20.avi', 20)
%
% 2) To denoise a video and print PSNR:
% CVBM3D('SMg20.avi', 20, 'SM.avi')
%
% 1) To denoise a 4-D matrix representing a noisy RGB video:
% CVBM3D(X_4D_matrix, 20)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Copyright © 2009 Tampere University of Technology. All rights reserved.
% This work should only be used for nonprofit purposes.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% If no input argument is provided, then use the internal ones from below:
if exist('sigma', 'var') ~= 1,
Xnoisy = 'SMg20.avi'; sigma = 20; Xorig = 'SM.avi';
end
% Whether or not to print information to the screen
dump_information = 1;
% If the input is a 4-D matrix, then save it as AVI file that is used as
% input to the denoising
if ischar(Xnoisy) == 0;
NumberOfFrames = size(Xnoisy,4);
if NumberOfFrames <= 1
error('The input RGB video should be a 4-D matrix (M x N x 3 x NumberOfFrames)');
end
avi_filename = sprintf('ExternalMatrix_%.6d.avi', round(rand*50000));
if exist(avi_filename, 'file') == 2,
delete(avi_filename);
end
mov = avifile(avi_filename, 'Colormap', gray(256), 'compression', 'None', 'fps', 30);
if mean2(Xnoisy) <= 1
fprintf('Possible error: the input RGB-videos should be in range [0,255] and not in [0,1]!\n');
else
for ii = [1:NumberOfFrames],
mov = addframe(mov, uint8(Xnoisy(:,:,:,ii)));
end
end
mov = close(mov);
fprintf('The input 4-D matrix was written to: %s.\n', avi_filename);
clear Xnoisy
Xnoisy = avi_filename;
end
% Read some properties of the noisy RGB video
noi_avi_file_info = aviinfo(Xnoisy);
NumberOfFrames = noi_avi_file_info.NumFrames;
%%% Read Xorig video --- needed if one wants to compute PSNR and ISNR
if exist('Xorig', 'var') == 1,
if ischar(Xorig) == 1;
org_avi_file_info = aviinfo(Xorig);
mo = aviread(Xorig);
Xorig = zeros([size(mo(1).cdata), NumberOfFrames], 'single');
for cf = 1:NumberOfFrames
Xorig(:,:,:,cf) = single(mo(cf).cdata(:,:,:))/255;
end
clear mo;
if (org_avi_file_info.NumFrames == noi_avi_file_info.NumFrames && org_avi_file_info.FramesPerSecond == noi_avi_file_info.FramesPerSecond && ...
org_avi_file_info.Width == noi_avi_file_info.Width && org_avi_file_info.Height == noi_avi_file_info.Height)
dump_information = 1;
end
else
Xorig = single(Xorig)/255;
if mean2(Xorig) <= 1
fprintf('Possible error: the input RGB-videos should be in range [0,255] and not in [0,1]!\n');
end
end
end
denoiseFrames = min(9, NumberOfFrames);
denoiseFramesW = min(9, NumberOfFrames);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Quality/complexity trade-off
%%%%
%%%% 'np' --> Normal Profile (balanced quality)
%%%% 'lc' --> Low Complexity Profile (fast, lower quality)
%%%%
if (exist('bm3dProfile') ~= 1)
bm3dProfile = 'np';
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Following are the parameters for the Normal Profile.
%%%%
%%%% Select transforms ('dct', 'dst', 'hadamard', or anything that is listed by 'help wfilters'):
transform_2D_HT_name = 'bior1.5'; %% transform used for the HT filt. of size N1 x N1
transform_2D_Wiener_name = 'dct'; %% transform used for the Wiener filt. of size N1_wiener x N1_wiener
transform_3rd_dim_name = 'haar'; %% tranform used in the 3-rd dim, the same for HT and Wiener filt.
%%%% Step 1: Hard-thresholding (HT) parameters:
N1 = 8; %% N1 x N1 is the block size used for the hard-thresholding (HT) filtering
Nstep = 5; %% sliding step to process every next refernece block
N2 = 8; %% maximum number of similar blocks (maximum size of the 3rd dimension of the 3D groups)
Ns = 7; %% length of the side of the search neighborhood for full-search block-matching (BM)
Npr = 3; %% length of the side of the motion-adaptive search neighborhood, use din the predictive-search BM
tau_match = 3000; %% threshold for the block distance (d-distance)
lambda_thr3D = 2.7; %% threshold parameter for the hard-thresholding in 3D DFT domain
dsub = 13; %% a small value subtracted from the distnce of blocks with the same spatial coordinate as the reference one
Nb = 2; %% number of blocks to follow in each next frame, used in the predictive-search BM
beta = 2.0; %% the beta parameter of the 2D Kaiser window used in the reconstruction
%%%% Step 2: Wiener filtering parameters:
N1_wiener = 7;
Nstep_wiener = 4;
N2_wiener = 8;
Ns_wiener = 7;
Npr_wiener = 3;
tau_match_wiener = 1000;
beta_wiener = 2.0;
dsub_wiener = 1.5;
Nb_wiener = 2;
%%%% Block-matching parameters:
stepFS = 1; %% step that firces to switch to full-search BM, "1" implies always full-search
stepFSW = 1;
thrToIncStep = 8; %% used in the HT filtering to increase the sliding step in uniform regions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Following are the parameters for the Low Complexity Profile.
%%%%
if strcmp(bm3dProfile, 'lc') == 1,
lambda_thr3D = 2.8;
denoiseFrames = min(5, NumberOfFrames);
denoiseFramesW = min(5, NumberOfFrames);
N2_wiener = 4;
N2 = 4;
Ns = 3;
Ns_wiener = 3;
Nb = 1;
Nb_wiener = 1;
end
if strcmp(bm3dProfile, 'hi') == 1,
Nstep = 3;
Nstep_wiener = 3;
end
if sigma > 30,
N1_wiener = 8;
tau_match = 4500;
tau_match_wiener = 3000;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Note: touch below this point only if you know what you are doing!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Create transform matrices, etc.
%%%%
decLevel = 0; %% dec. levels of the dyadic wavelet 2D transform for blocks (0 means full decomposition, higher values decrease the dec. number)
decLevel3 = 0; %% dec. level for the wavelet transform in the 3rd dimension
[Tfor, Tinv] = getTransfMatrix(N1, transform_2D_HT_name, decLevel); %% get (normalized) forward and inverse transform matrices
[TforW, TinvW] = getTransfMatrix(N1_wiener, transform_2D_Wiener_name); %% get (normalized) forward and inverse transform matrices
if (strcmp(transform_3rd_dim_name, 'haar') == 1 || strcmp(transform_3rd_dim_name(end-2:end), '1.1') == 1),
%%% Fast internal transform is used, no need to generate transform
%%% matrices.
hadper_trans_single_den = {};
inverse_hadper_trans_single_den = {};
else
%%% Create transform matrices. The transforms are later computed by
%%% matrix multiplication with them
for hh = [1 2 4 8 16 32];
[Tfor3rd, Tinv3rd] = getTransfMatrix(hh, transform_3rd_dim_name, decLevel3);
hadper_trans_single_den{hh} = single(Tfor3rd);
inverse_hadper_trans_single_den{hh} = single(Tinv3rd');
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% 2D Kaiser windows that scale the reconstructed blocks
%%%%
if beta_wiener==2 & beta==2 & N1_wiener==7 & N1==8 % hardcode the window function so that the signal processing toolbox is not needed by default
Wwin2D = [ 0.1924 0.2989 0.3846 0.4325 0.4325 0.3846 0.2989 0.1924;
0.2989 0.4642 0.5974 0.6717 0.6717 0.5974 0.4642 0.2989;
0.3846 0.5974 0.7688 0.8644 0.8644 0.7688 0.5974 0.3846;
0.4325 0.6717 0.8644 0.9718 0.9718 0.8644 0.6717 0.4325;
0.4325 0.6717 0.8644 0.9718 0.9718 0.8644 0.6717 0.4325;
0.3846 0.5974 0.7688 0.8644 0.8644 0.7688 0.5974 0.3846;
0.2989 0.4642 0.5974 0.6717 0.6717 0.5974 0.4642 0.2989;
0.1924 0.2989 0.3846 0.4325 0.4325 0.3846 0.2989 0.1924 ];
Wwin2D_wiener = [ 0.1924 0.3151 0.4055 0.4387 0.4055 0.3151 0.1924;
0.3151 0.5161 0.6640 0.7184 0.6640 0.5161 0.3151;
0.4055 0.6640 0.8544 0.9243 0.8544 0.6640 0.4055;
0.4387 0.7184 0.9243 1.0000 0.9243 0.7184 0.4387;
0.4055 0.6640 0.8544 0.9243 0.8544 0.6640 0.4055;
0.3151 0.5161 0.6640 0.7184 0.6640 0.5161 0.3151;
0.1924 0.3151 0.4055 0.4387 0.4055 0.3151 0.1924 ];
else
Wwin2D = kaiser(N1, beta) * kaiser(N1, beta)'; % Kaiser window used in the aggregation of the HT part
Wwin2D_wiener = kaiser(N1_wiener, beta_wiener) * kaiser(N1_wiener, beta_wiener)'; % Kaiser window used in the aggregation of the Wiener filt. part
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Read an image, generate noise and add it to the image
%%%%
if dump_information == 1
fprintf('Input video: %s, sigma: %.1f\n', Xnoisy, sigma);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Determine unique filenames of intermediate avi files
%%%%
HT_avi_file = sprintf('%s_cvbm3d_step1_0.avi', Xnoisy(1:end-4));
Denoised_avi_file = sprintf('%s_cvbm3d_0.avi', Xnoisy(1:end-4));
i = 1;
while (exist(['./' HT_avi_file], 'file') ~= 0) | (exist(['./' Denoised_avi_file],'file') ~= 0)
HT_avi_file = sprintf('%s_cvbm3d_step1_%d.avi', Xnoisy(1:end-4),i);
Denoised_avi_file = sprintf('%s_cvbm3d_%d.avi', Xnoisy(1:end-4),i);
i = i + 1;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Initial estimate by hard-thresholding filtering
HT_IO = {which(Xnoisy), HT_avi_file};
tic;
bm3d_thr_video_c(HT_IO, hadper_trans_single_den, Nstep, N1, N2, 0,...
lambda_thr3D, tau_match*N1*N1/(255*255), (Ns-1)/2, sigma/255, thrToIncStep,...
single(Tfor), single(Tinv)', inverse_hadper_trans_single_den, single(ones(N1)),...
'unused arg', dsub*dsub/255 * (sigma^2 / 255), ones(NumberOfFrames,1), Wwin2D,...
(Npr-1)/2, stepFS, denoiseFrames, Nb, 0 );
estimate_elapsed_time = toc;
if dump_information == 1
% mo = aviread(HT_avi_file);
% y_hat = zeros([size(mo(1).cdata(:,:,1)), 3, NumberOfFrames], 'single');
% for cf = 1:NumberOfFrames
% y_hat(:,:,:,cf) = single(mo(cf).cdata(:,:,:))/255;
% end
% clear mo
%
% PSNR_HT_ESTIMATE = 10*log10(1/mean2((Xorig-y_hat).^2));
% fprintf('HT ESTIMATE, PSNR: %.3f dB\n', PSNR_HT_ESTIMATE);
% clear y_hat;
fprintf('STEP1 completed!\n');
end
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%% Final estimate by Wiener filtering (using the hard-thresholding
% initial estimate)
lut_ic = ClipComp16b(sigma/255);
WIE_IO = {which(Xnoisy), HT_avi_file, Denoised_avi_file};
tic;
bm3d_wiener_video_c(WIE_IO, 'unused', hadper_trans_single_den, Nstep_wiener, N1_wiener, N2_wiener, ...
'unused_arg', tau_match_wiener*N1_wiener*N1_wiener/(255*255), (Ns_wiener-1)/2, sigma/255, 'unused arg',...
single(TforW), single(TinvW)', inverse_hadper_trans_single_den, 'unused arg', dsub_wiener*dsub_wiener/255*(sigma^2 / 255),...
ones(NumberOfFrames,1), Wwin2D_wiener, (Npr_wiener-1)/2, stepFSW, denoiseFramesW, Nb_wiener, 0, lut_ic);
wiener_elapsed_time = toc;
if nargout == 1
mo = aviread(Denoised_avi_file);
Xdenoised = zeros([size(mo(1).cdata(:,:,1)), 3, NumberOfFrames], 'single');
for cf = 1:NumberOfFrames
Xdenoised(:,:,:,cf) = single(mo(cf).cdata(:,:,:))/255;
end
clear mo
end
if dump_information == 1
if nargout ~= 1
mo = aviread(Denoised_avi_file);
Xdenoised = zeros([size(mo(1).cdata(:,:,1)), 3, NumberOfFrames], 'single');
for cf = 1:NumberOfFrames
Xdenoised(:,:,:,cf) = single(mo(cf).cdata(:,:,:))/255;
end
clear mo
end
PSNR_TEXT='';
if exist('Xorig', 'var') == 1
PSNR = 10*log10(1/mean2((Xorig-Xdenoised).^2));
PSNR_TEXT=sprintf(' PSNR: %.3f dB,', PSNR);
New_Denoised_avi_file = sprintf('%s_PSNR%.2f.avi',Denoised_avi_file(1:end-4),PSNR);
movefile(Denoised_avi_file, New_Denoised_avi_file);
Denoised_avi_file = New_Denoised_avi_file;
end
% PSNRs = zeros(NumberOfFrames,1);
% for ii = 1:NumberOfFrames,
% PSNRs(ii) = 10*log10(1/mean2( (Xorig(:,:,:,ii)-Xdenoised(:,:,:,ii)).^2));
% fprintf('Frame: %d, PSNR: %.2f\n', ii, PSNRs(ii));
% end
if nargout == 0
clear Xdenoised
end
fprintf('FILTERING COMPLETED (frames/sec: %.2f,%s denoised video saved as %s)\n', ...
NumberOfFrames/(wiener_elapsed_time + estimate_elapsed_time), PSNR_TEXT, Denoised_avi_file);
end
return;
function [Tforward, Tinverse] = getTransfMatrix (N, transform_type, dec_levels)
%
% Create forward and inverse transform matrices, which allow for perfect
% reconstruction. The forward transform matrix is normalized so that the
% l2-norm of each basis element is 1.
%
% [Tforward, Tinverse] = getTransfMatrix (N, transform_type, dec_levels)
%
% INPUTS:
%
% N --> Size of the transform (for wavelets, must be 2^K)
%
% transform_type --> 'dct', 'dst', 'hadamard', or anything that is
% listed by 'help wfilters' (bi-orthogonal wavelets)
% 'DCrand' -- an orthonormal transform with a DC and all
% the other basis elements of random nature
%
% dec_levels --> If a wavelet transform is generated, this is the
% desired decomposition level. Must be in the
% range [0, log2(N)-1], where "0" implies
% full decomposition.
%
% OUTPUTS:
%
% Tforward --> (N x N) Forward transform matrix
%
% Tinverse --> (N x N) Inverse transform matrix
%
if exist('dec_levels') ~= 1,
dec_levels = 0;
end
if N == 1,
Tforward = 1;
elseif strcmp(transform_type, 'hadamard') == 1,
Tforward = hadamard(N);
elseif (N == 8) & strcmp(transform_type, 'bior1.5')==1 % hardcoded transform so that the wavelet toolbox is not needed to generate it
Tforward = [ 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274;
0.219417649252501 0.449283757993216 0.449283757993216 0.219417649252501 -0.219417649252501 -0.449283757993216 -0.449283757993216 -0.219417649252501;
0.569359398342846 0.402347308162278 -0.402347308162278 -0.569359398342846 -0.083506045090284 0.083506045090284 -0.083506045090284 0.083506045090284;
-0.083506045090284 0.083506045090284 -0.083506045090284 0.083506045090284 0.569359398342846 0.402347308162278 -0.402347308162278 -0.569359398342846;
0.707106781186547 -0.707106781186547 0 0 0 0 0 0;
0 0 0.707106781186547 -0.707106781186547 0 0 0 0;
0 0 0 0 0.707106781186547 -0.707106781186547 0 0;
0 0 0 0 0 0 0.707106781186547 -0.707106781186547];
elseif (N == 8) & strcmp(transform_type, 'dct')==1 % hardcoded transform so that the signal processing toolbox is not needed to generate it
Tforward = [ 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274 0.353553390593274;
0.490392640201615 0.415734806151273 0.277785116509801 0.097545161008064 -0.097545161008064 -0.277785116509801 -0.415734806151273 -0.490392640201615;
0.461939766255643 0.191341716182545 -0.191341716182545 -0.461939766255643 -0.461939766255643 -0.191341716182545 0.191341716182545 0.461939766255643;
0.415734806151273 -0.097545161008064 -0.490392640201615 -0.277785116509801 0.277785116509801 0.490392640201615 0.097545161008064 -0.415734806151273;
0.353553390593274 -0.353553390593274 -0.353553390593274 0.353553390593274 0.353553390593274 -0.353553390593274 -0.353553390593274 0.353553390593274;
0.277785116509801 -0.490392640201615 0.097545161008064 0.415734806151273 -0.415734806151273 -0.097545161008064 0.490392640201615 -0.277785116509801;
0.191341716182545 -0.461939766255643 0.461939766255643 -0.191341716182545 -0.191341716182545 0.461939766255643 -0.461939766255643 0.191341716182545;
0.097545161008064 -0.277785116509801 0.415734806151273 -0.490392640201615 0.490392640201615 -0.415734806151273 0.277785116509801 -0.097545161008064];
elseif (N == 8) & strcmp(transform_type, 'dst')==1 % hardcoded transform so that the PDE toolbox is not needed to generate it
Tforward = [ 0.161229841765317 0.303012985114696 0.408248290463863 0.464242826880013 0.464242826880013 0.408248290463863 0.303012985114696 0.161229841765317;
0.303012985114696 0.464242826880013 0.408248290463863 0.161229841765317 -0.161229841765317 -0.408248290463863 -0.464242826880013 -0.303012985114696;
0.408248290463863 0.408248290463863 0 -0.408248290463863 -0.408248290463863 0 0.408248290463863 0.408248290463863;
0.464242826880013 0.161229841765317 -0.408248290463863 -0.303012985114696 0.303012985114696 0.408248290463863 -0.161229841765317 -0.464242826880013;
0.464242826880013 -0.161229841765317 -0.408248290463863 0.303012985114696 0.303012985114696 -0.408248290463863 -0.161229841765317 0.464242826880013;
0.408248290463863 -0.408248290463863 0 0.408248290463863 -0.408248290463863 0 0.408248290463863 -0.408248290463863;
0.303012985114696 -0.464242826880013 0.408248290463863 -0.161229841765317 -0.161229841765317 0.408248290463863 -0.464242826880013 0.303012985114696;
0.161229841765317 -0.303012985114696 0.408248290463863 -0.464242826880013 0.464242826880013 -0.408248290463863 0.303012985114696 -0.161229841765317];
elseif (N == 7) & strcmp(transform_type, 'dct')==1 % hardcoded transform so that the signal processing toolbox is not needed to generate it
Tforward =[ 0.377964473009227 0.377964473009227 0.377964473009227 0.377964473009227 0.377964473009227 0.377964473009227 0.377964473009227;
0.521120889169602 0.417906505941275 0.231920613924330 0 -0.231920613924330 -0.417906505941275 -0.521120889169602;
0.481588117120063 0.118942442321354 -0.333269317528993 -0.534522483824849 -0.333269317528993 0.118942442321354 0.481588117120063;
0.417906505941275 -0.231920613924330 -0.521120889169602 0 0.521120889169602 0.231920613924330 -0.417906505941275;
0.333269317528993 -0.481588117120063 -0.118942442321354 0.534522483824849 -0.118942442321354 -0.481588117120063 0.333269317528993;
0.231920613924330 -0.521120889169602 0.417906505941275 0 -0.417906505941275 0.521120889169602 -0.231920613924330;
0.118942442321354 -0.333269317528993 0.481588117120063 -0.534522483824849 0.481588117120063 -0.333269317528993 0.118942442321354];
elseif strcmp(transform_type, 'dct') == 1,
Tforward = dct(eye(N));
elseif strcmp(transform_type, 'dst') == 1,
Tforward = dst(eye(N));
elseif strcmp(transform_type, 'DCrand') == 1,
x = randn(N); x(1:end,1) = 1; [Q,R] = qr(x);
if (Q(1) < 0),
Q = -Q;
end;
Tforward = Q';
else %% a wavelet decomposition supported by 'wavedec'
%%% Set periodic boundary conditions, to preserve bi-orthogonality
dwtmode('per','nodisp');
Tforward = zeros(N,N);
for i = 1:N
Tforward(:,i)=wavedec(circshift([1 zeros(1,N-1)],[dec_levels i-1]), log2(N), transform_type); %% construct transform matrix
end
end
%%% Normalize the basis elements
Tforward = (Tforward' * diag(sqrt(1./sum(Tforward.^2,2))))';
%%% Compute the inverse transform matrix
Tinverse = inv(Tforward);
return;