BM3D/IDDBM3D/BM3DDEB_init.m

   1 function [ISNR, y_hat_RI,y_hat_RWI,zRI] = BM3DDEB_init(experiment_number, y, z, v, sigma)
   2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   3 %
   4 % Copyright © 2008 Tampere University of Technology. All rights reserved.
   5 % This work should only be used for nonprofit purposes.
   6 %
   7 % AUTHORS:
   8 %     Kostadin Dabov, email: kostadin.dabov _at_ tut.fi
   9 %     Alessandro Foi
  10 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  11 %
  12 %  This function implements the image deblurring method proposed in:
  13 %
  14 %  [1] K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, "Image
  15 %   restoration  by sparse 3D transform-domain collaborative filtering,"
  16 %   Proc SPIE Electronic Imaging, January 2008.
  17 %
  18 %  FUNCTION INTERFACE:
  19 %
  20 %  [PSNR, y_hat_RWI] = BM3DDEB(experiment_number, test_image_name)
  21 %
  22 %  INPUT:
  23 %   1) experiment_number: 1 -> PSF 1, sigma^2 = 2
  24 %                         2 -> PSF 1, sigma^2 = 8
  25 %                         3 -> PSF 2, sigma^2 = 0.308
  26 %                         4 -> PSF 3, sigma^2 = 49
  27 %                         5 -> PSF 4, sigma^2 = 4
  28 %                         6 -> PSF 5, sigma^2 = 64
  29 %
  30 %   2) test_image_name:   a valid filename of a grayscale test image
  31 %
  32 %  OUTPUT:
  33 %   1) ISNR:              the output improvement in SNR, dB
  34 %   2) y_hat_RWI:         the restored image
  35 %
  36 %  ! The function can work without any of the input arguments,
  37 %   in which case, the internal default ones are used !
  38 %
  39 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  40
  41 %%%% Fixed regularization parameters (obtained empirically after a rough optimization)
  42 Regularization_alpha_RI = 4e-4;
  43 Regularization_alpha_RWI = 5e-3;
  44
  45 %%%% Experiment number (see below for details, e.g. how the blur is generated, etc.)
  46 if (exist('experiment_number') ~= 1)
  47     experiment_number = 3; % 1 -- 6
  48 end
  49
  50 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  51 %%%% Select a single image filename (might contain path)
  52 %%%%
  53 % if (exist('test_image_name') ~= 1)
  54 %     test_image_name = [
  55 % %        'Lena512.png'
  56 %         'Cameraman256.png'
  57 % %        'barbara.png'
  58 % %        'house.png'
  59 %     ];
  60 % end
  61
  62 %%%% Select 2D transforms ('dct', 'dst', 'hadamard', or anything that is listed by 'help wfilters'):
  63 transform_2D_HT_name      = 'dst'; %% 2D transform (of size N1 x N1) used in Step 1
  64 transform_2D_Wiener_name  = 'dct'; %% 2D transform (of size N1_wiener x N1_wiener) used in Step 2
  65 transform_3rd_dimage_name = 'haar'; %% 1D tranform used in the 3-rd dim, the same for both steps
  66
  67 %%%% Step 1 (BM3D with collaborative hard-thresholding) parameters:
  68 N1                  = 8;   %% N1 x N1 is the block size
  69 Nstep               = 3;   %% sliding step to process every next refernece block
  70 N2                  = 16;  %% maximum number of similar blocks (maximum size of the 3rd dimensiona of a 3D array)
  71 Ns                  = 39;  %% length of the side of the search neighborhood for full-search block-matching (BM)
  72 tau_match           = 6000;%% threshold for the block distance (d-distance)
  73 lambda_thr2D        = 0;   %% threshold for the coarse initial denoising used in the d-distance measure
  74 lambda_thr3D        = 2.9; %% threshold for the hard-thresholding
  75 beta                = 0; %% the beta parameter of the 2D Kaiser window used in the reconstruction
  76
  77 %%%% Step 2 (BM3D with collaborative Wiener filtering) parameters:
  78 N1_wiener           = 8;
  79 Nstep_wiener        = 2;
  80 N2_wiener           = 16;
  81 Ns_wiener           = 39;
  82 tau_match_wiener    = 800;
  83 beta_wiener         = 0;
  84
  85 %%%%  Specify whether to print results and display images
  86 print_to_screen     = 0;
  87
  88
  89 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  90 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  91 %%%% Note: touch below this point only if you know what you are doing!
  92 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  93 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  94
  95 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  96 %%%% Make parameters compatible with the interface of the mex-functions
  97 %%%%
  98
  99 [Tfor, Tinv]   = getTransfMatrix(N1, transform_2D_HT_name, 0); %% get (normalized) forward and inverse transform matrices
 100 [TforW, TinvW] = getTransfMatrix(N1_wiener, transform_2D_Wiener_name, 0); %% get (normalized) forward and inverse transform matrices
 101
 102 if (strcmp(transform_3rd_dimage_name, 'haar') == 1),
 103     %%% Fast internal transform is used, no need to generate transform
 104     %%% matrices.
 105     hadper_trans_single_den         = {};
 106     inverse_hadper_trans_single_den = {};
 107 else
 108     %%% Create transform matrices. The transforms are later applied by
 109     %%% vector-matrix multiplications
 110     for hpow = 0:ceil(log2(max(N2,N2_wiener))),
 111         h = 2^hpow;
 112         [Tfor3rd, Tinv3rd] = getTransfMatrix(h, transform_3rd_dimage_name, 0);
 113         hadper_trans_single_den{h}         = single(Tfor3rd);
 114         inverse_hadper_trans_single_den{h} = single(Tinv3rd');
 115     end
 116 end
 117
 118 if beta == 0 & beta_wiener == 0
 119     Wwin2D = ones(N1_wiener,N1_wiener);
 120     Wwin2D_wiener = ones(N1,N1);
 121 else
 122     Wwin2D        = kaiser(N1, beta) * kaiser(N1, beta)'; % Kaiser window used in the hard-thresholding part
 123     Wwin2D_wiener = kaiser(N1_wiener, beta_wiener) * kaiser(N1_wiener, beta_wiener)'; % Kaiser window used in the Wiener filtering part
 124 end
 125
 126 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 127 % %%%% Read an image and generate a blurred and noisy image
 128 % %%%%
 129 % y = im2double(imread(test_image_name));
 130 %
 131 % if experiment_number==1
 132 %     sigma=sqrt(2)/255;
 133 %     for x1=-7:7; for x2=-7:7; v(x1+8,x2+8)=1/(x1^2+x2^2+1); end, end; v=v./sum(v(:));
 134 % end
 135 % if experiment_number==2
 136 %     sigma=sqrt(8)/255;
 137 %     s1=0; for a1=-7:7; s1=s1+1; s2=0; for a2=-7:7; s2=s2+1; v(s1,s2)=1/(a1^2+a2^2+1); end, end;  v=v./sum(v(:));
 138 % end
 139 % if experiment_number==3
 140 %     BSNR=40; sigma=-1; % if "sigma=-1", then the value of sigma depends on the BSNR
 141 %     v=ones(9); v=v./sum(v(:));
 142 % end
 143 % if experiment_number==4
 144 %     sigma=7/255;
 145 %     v=[1 4 6 4 1]'*[1 4 6 4 1]; v=v./sum(v(:));  % PSF
 146 % end
 147 % if experiment_number==5
 148 %     sigma=2/255;
 149 %     v=fspecial('gaussian', 25, 1.6);
 150 % end
 151 % if experiment_number==6
 152 %     sigma=8/255;
 153 %     v=fspecial('gaussian', 25, .4);
 154 % end
 155 %
 156 %
 157 [Xv, Xh]  = size(y);
 158 [ghy,ghx] = size(v);
 159 big_v  = zeros(Xv,Xh); big_v(1:ghy,1:ghx)=v; big_v=circshift(big_v,-round([(ghy-1)/2 (ghx-1)/2])); % pad PSF with zeros to whole image domain, and center it
 160 V      = fft2(big_v); % frequency response of the PSF
 161 % y_blur = imfilter(y, v, 'circular'); % performs blurring (by circular convolution)
 162 %
 163 % randn('seed',0);  %%% fix seed for the random number generator
 164 % if sigma == -1;   %% check whether to use BSNR in order to define value of sigma
 165 %     sigma=sqrt(norm(y_blur(:)-mean(y_blur(:)),2)^2 /(Xh*Xv*10^(BSNR/10))); % compute sigma from the desired BSNR
 166 % end
 167 %
 168 % %%%% Create a blurred and noisy observation
 169 % z = y_blur + sigma*randn(Xv,Xh);
 170
 171
 172 tic;
 173 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 174 %%%% Step 1: Final estimate by Regularized Inversion (RI) followed by
 175 %%%% BM3D with collaborative hard-thresholding
 176 %%%%
 177
 178 %%%% Step 1.1. Regularized Inversion
 179 RI= conj(V)./( (abs(V).^2) + Regularization_alpha_RI * Xv*Xh*sigma^2); % Transfer Matrix for RI    %% Standard Tikhonov Regularization
 180 zRI=real(ifft2( fft2(z).* RI ));   % Regularized Inverse Estimate (RI OBSERVATION)
 181
 182 stdRI = zeros(N1, N1);
 183 for ii = 1:N1,
 184     for jj = 1:N1,
 185         UnitMatrix = zeros(N1,N1); UnitMatrix(ii,jj)=1;
 186         BasisElementPadded = zeros(Xv, Xh); BasisElementPadded(1:N1,1:N1) = Tinv*UnitMatrix*Tinv';
 187         TransfBasisElementPadded = fft2(BasisElementPadded);
 188         stdRI(ii,jj) = sqrt( (1/(Xv*Xh)) * sum(sum(abs(TransfBasisElementPadded.*RI).^2)) )*sigma;
 189     end,
 190 end
 191
 192 %%%% Step 1.2. Colored noise suppression by BM3D with collaborative hard-
 193 %%%% thresholding
 194
 195 y_hat_RI = bm3d_thr_colored_noise(zRI, hadper_trans_single_den, Nstep, N1, N2, lambda_thr2D,...
 196     lambda_thr3D, tau_match*N1*N1/(255*255), (Ns-1)/2, sigma, 0, single(Tfor), single(Tinv)',...
 197     inverse_hadper_trans_single_den, single(stdRI'), Wwin2D, 0, 1 );
 198
 199 PSNR_INITIAL_ESTIMATE = 10*log10(1/mean((y(:)-y_hat_RI(:)).^2));
 200 ISNR_INITIAL_ESTIMATE = PSNR_INITIAL_ESTIMATE - 10*log10(1/mean((y(:)-z(:)).^2));
 201
 202 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 203 %%%% Step 2: Final estimate by Regularized Wiener Inversion (RWI) followed
 204 %%%% by BM3D with collaborative Wiener filtering
 205 %%%%
 206
 207 %%%% Step 2.1. Regularized Wiener Inversion
 208 Wiener_Pilot = abs(fft2(double(y_hat_RI)));   %%% Wiener reference estimate
 209 RWI  = conj(V).*Wiener_Pilot.^2./(Wiener_Pilot.^2.*(abs(V).^2) + Regularization_alpha_RWI*Xv*Xh*sigma^2);   % Transfer Matrix for RWI (uses standard regularization 'a-la-Tikhonov')
 210 zRWI = real(ifft2(fft2(z).*RWI));   % RWI OBSERVATION
 211
 212 stdRWI = zeros(N1_wiener, N1_wiener);
 213 for ii = 1:N1,
 214     for jj = 1:N1,
 215         UnitMatrix = zeros(N1,N1); UnitMatrix(ii,jj)=1;
 216         BasisElementPadded = zeros(Xv, Xh); BasisElementPadded(1:N1,1:N1) = idct2(UnitMatrix);
 217         TransfBasisElementPadded = fft2(BasisElementPadded);
 218         stdRWI(ii,jj) = sqrt( (1/(Xv*Xh)) * sum(sum(abs(TransfBasisElementPadded.*RWI).^2)) )*sigma;
 219     end,
 220 end
 221
 222 %%%% Step 2.2. Colored noise suppression by BM3D with collaborative Wiener
 223 %%%% filtering
 224 y_hat_RWI = bm3d_wiener_colored_noise(zRWI, y_hat_RI, hadper_trans_single_den, Nstep_wiener, N1_wiener, N2_wiener, ...
 225      0, tau_match_wiener*N1_wiener*N1_wiener/(255*255), (Ns_wiener-1)/2, 0, single(stdRWI'), single(TforW), single(TinvW)',...
 226      inverse_hadper_trans_single_den, Wwin2D_wiener, 0, 1, single(ones(N1_wiener)) );
 227
 228 elapsed_time = toc;
 229
 230
 231 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 232 %%%% Calculate the final estimate's PSNR and ISNR, print them, and show the
 233 %%%% restored image
 234 %%%%
 235 PSNR = 10*log10(1/mean((y(:)-y_hat_RWI(:)).^2));
 236 ISNR = PSNR - 10*log10(1/mean((y(:)-z(:)).^2));
 237
 238 if print_to_screen == 1
 239 fprintf('Image: %s, Exp %d, Time: %.1f sec, PSNR-RI: %.2f dB, PSNR-RWI: %.2f, ISNR-RWI: %.2f dB\n', ...
 240     test_image_name, experiment_number, elapsed_time, PSNR_INITIAL_ESTIMATE, PSNR, ISNR);
 241     figure,imshow(z);
 242     figure,imshow(double(y_hat_RWI));
 243 end
 244
 245 return;
 246
 247
 248 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 249 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 250 % Some auxiliary functions
 251 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 252 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 253
 254
 255
 256 function [Tforward, Tinverse] = getTransfMatrix (N, transform_type, dec_levels)
 257 %
 258 % Create forward and inverse transform matrices, which allow for perfect
 259 % reconstruction. The forward transform matrix is normalized so that the
 260 % l2-norm of each basis element is 1.
 261 %
 262 % [Tforward, Tinverse] = getTransfMatrix (N, transform_type, dec_levels)
 263 %
 264 %  INPUTS:
 265 %
 266 %   N               --> Size of the transform (for wavelets, must be 2^K)
 267 %
 268 %   transform_type  --> 'dct', 'dst', 'hadamard', or anything that is
 269 %                       listed by 'help wfilters' (bi-orthogonal wavelets)
 270 %                       'DCrand' -- an orthonormal transform with a DC and all
 271 %                       the other basis elements of random nature
 272 %
 273 %   dec_levels      --> If a wavelet transform is generated, this is the
 274 %                       desired decomposition level. Must be in the
 275 %                       range [0, log2(N)-1], where "0" implies
 276 %                       full decomposition.
 277 %
 278 %  OUTPUTS:
 279 %
 280 %   Tforward        --> (N x N) Forward transform matrix
 281 %
 282 %   Tinverse        --> (N x N) Inverse transform matrix
 283 %
 284
 285 if exist('dec_levels') ~= 1,
 286     dec_levels = 0;
 287 end
 288
 289 if N == 1,
 290     Tforward = 1;
 291 elseif strcmp(transform_type, 'hadamard') == 1,
 292     Tforward    = hadamard(N);
 293 elseif (N == 8) & strcmp(transform_type, 'bior1.5')==1 % hardcoded transform so that the wavelet toolbox is not needed to generate it
 294     Tforward =  [ 0.353553390593274   0.353553390593274   0.353553390593274   0.353553390593274   0.353553390593274   0.353553390593274   0.353553390593274   0.353553390593274;
 295        0.219417649252501   0.449283757993216   0.449283757993216   0.219417649252501  -0.219417649252501  -0.449283757993216  -0.449283757993216  -0.219417649252501;
 296        0.569359398342846   0.402347308162278  -0.402347308162278  -0.569359398342846  -0.083506045090284   0.083506045090284  -0.083506045090284   0.083506045090284;
 297       -0.083506045090284   0.083506045090284  -0.083506045090284   0.083506045090284   0.569359398342846   0.402347308162278  -0.402347308162278  -0.569359398342846;
 298        0.707106781186547  -0.707106781186547                   0                   0                   0                   0                   0                   0;
 299                        0                   0   0.707106781186547  -0.707106781186547                   0                   0                   0                   0;
 300                        0                   0                   0                   0   0.707106781186547  -0.707106781186547                   0                   0;
 301                        0                   0                   0                   0                   0                   0   0.707106781186547  -0.707106781186547];
 302 elseif (N == 8) & strcmp(transform_type, 'dct')==1 % hardcoded transform so that the signal processing toolbox is not needed to generate it
 303     Tforward = [ 0.353553390593274   0.353553390593274   0.353553390593274   0.353553390593274   0.353553390593274   0.353553390593274   0.353553390593274   0.353553390593274;
 304        0.490392640201615   0.415734806151273   0.277785116509801   0.097545161008064  -0.097545161008064  -0.277785116509801  -0.415734806151273  -0.490392640201615;
 305        0.461939766255643   0.191341716182545  -0.191341716182545  -0.461939766255643  -0.461939766255643  -0.191341716182545   0.191341716182545   0.461939766255643;
 306        0.415734806151273  -0.097545161008064  -0.490392640201615  -0.277785116509801   0.277785116509801   0.490392640201615   0.097545161008064  -0.415734806151273;
 307        0.353553390593274  -0.353553390593274  -0.353553390593274   0.353553390593274   0.353553390593274  -0.353553390593274  -0.353553390593274   0.353553390593274;
 308        0.277785116509801  -0.490392640201615   0.097545161008064   0.415734806151273  -0.415734806151273  -0.097545161008064   0.490392640201615  -0.277785116509801;
 309        0.191341716182545  -0.461939766255643   0.461939766255643  -0.191341716182545  -0.191341716182545   0.461939766255643  -0.461939766255643   0.191341716182545;
 310        0.097545161008064  -0.277785116509801   0.415734806151273  -0.490392640201615   0.490392640201615  -0.415734806151273   0.277785116509801  -0.097545161008064];
 311 elseif (N == 8) & strcmp(transform_type, 'dst')==1 % hardcoded transform so that the PDE toolbox is not needed to generate it
 312     Tforward = [ 0.161229841765317   0.303012985114696   0.408248290463863   0.464242826880013   0.464242826880013   0.408248290463863   0.303012985114696   0.161229841765317;
 313        0.303012985114696   0.464242826880013   0.408248290463863   0.161229841765317  -0.161229841765317  -0.408248290463863  -0.464242826880013  -0.303012985114696;
 314        0.408248290463863   0.408248290463863                   0  -0.408248290463863  -0.408248290463863                   0   0.408248290463863   0.408248290463863;
 315        0.464242826880013   0.161229841765317  -0.408248290463863  -0.303012985114696   0.303012985114696   0.408248290463863  -0.161229841765317  -0.464242826880013;
 316        0.464242826880013  -0.161229841765317  -0.408248290463863   0.303012985114696   0.303012985114696  -0.408248290463863  -0.161229841765317   0.464242826880013;
 317        0.408248290463863  -0.408248290463863                   0   0.408248290463863  -0.408248290463863                   0   0.408248290463863  -0.408248290463863;
 318        0.303012985114696  -0.464242826880013   0.408248290463863  -0.161229841765317  -0.161229841765317   0.408248290463863  -0.464242826880013   0.303012985114696;
 319        0.161229841765317  -0.303012985114696   0.408248290463863  -0.464242826880013   0.464242826880013  -0.408248290463863   0.303012985114696  -0.161229841765317];
 320 elseif strcmp(transform_type, 'dct') == 1,
 321     Tforward    = dct(eye(N));
 322 elseif strcmp(transform_type, 'dst') == 1,
 323     Tforward    = dst(eye(N));
 324 elseif strcmp(transform_type, 'DCrand') == 1,
 325     x = randn(N); x(1:end,1) = 1; [Q,R] = qr(x);
 326     if (Q(1) < 0),
 327         Q = -Q;
 328     end;
 329     Tforward = Q';
 330 else %% a wavelet decomposition supported by 'wavedec'
 331     %%% Set periodic boundary conditions, to preserve bi-orthogonality
 332     dwtmode('per','nodisp');
 333
 334     Tforward = zeros(N,N);
 335     for i = 1:N
 336         Tforward(:,i)=wavedec(circshift([1 zeros(1,N-1)],[dec_levels i-1]), log2(N), transform_type);  %% construct transform matrix
 337     end
 338 end
 339
 340 %%% Normalize the basis elements
 341 Tforward = (Tforward' * diag(sqrt(1./sum(Tforward.^2,2))))';
 342
 343 %%% Compute the inverse transform matrix
 344 Tinverse = inv(Tforward);
 345
 346 return;