基于K-SVD稀疏字典的图像去噪算法

基于K-SVD稀疏字典的图像去噪算法

下面的代码我已经给出了我自己的注释，全部是个人的理解，有误的欢迎指正！

Main.m

%============================================================%               demo2 - denoise an image% this is a run_file the demonstrate how to denoise an image, % using dictionaries. The methods implemented here are the same% one as described in "Image Denoising Via Sparse and Redundant% representations over Learned Dictionaries", (appeared in the % IEEE Trans. on Image Processing, Vol. 15, no. 12, December 2006).% ============================================================ clearbb=8; % block sizeRR=4; % redundancy factor  冗余因素K=RR*bb^2; % number of atoms in the dictionary sigma = 50; %pathForImages ='';%imageName = 'barbara.png';%   [IMin0,pp]=imread('cameraman.tif'); [IMin0,pp]=imread('w.jpg');IMin0=im2double(IMin0);if (length(size(IMin0))>2)    IMin0 = rgb2gray(IMin0);endif (max(IMin0(:))<2)    IMin0 = IMin0*255;end IMin=IMin0+sigma*randn(size(IMin0));%%%%%%此处有随机函数PSNRIn = 20*log10(255/sqrt(mean((IMin(:)-IMin0(:)).^2)));tic%%%基于压缩的那篇论文[IoutAdaptive,output] = denoiseImageKSVD(IMin, sigma,K); PSNROut = 20*log10(255/sqrt(mean((IoutAdaptive(:)-IMin0(:)).^2)));figure;subplot(1,3,1); imshow(IMin0,[]); title('Original clean image');subplot(1,3,2);imshow(IMin,[]); title(strcat(['Noisy image, ',num2str(PSNRIn),'dB']));subplot(1,3,3); imshow(IoutAdaptive,[]); title(strcat(['Clean Image by Adaptive dictionary, ',num2str(PSNROut),'dB'])); figure;I = displayDictionaryElementsAsImage(output.D, floor(sqrt(K)), floor(size(output.D,2)/floor(sqrt(K))),bb,bb);title('The dictionary trained on patches from the noisy image');toc

denoiseImageKSVD.m

function [IOut,output] = denoiseImageKSVD(Image,sigma,K,varargin)%==========================================================================%   P E R F O R M   D E N O I S I N G   U S I N G   A  D I C T  I O N A R Y%                  T R A I N E D   O N   N O I S Y   I M A G E%==========================================================================% function IOut = denoiseImageKSVD(Image,sigma,K,varargin)% denoise an image by sparsely representing each block with the% already overcomplete trained Dictionary, and averaging the represented parts.% Detailed description can be found in "Image Denoising Via Sparse and Redundant% representations over Learned Dictionaries", (appeared in the % IEEE Trans. on Image Processing, Vol. 15, no. 12, December 2006).% This function may take some time to process. Possible factor that effect% the processing time are:%  1. number of KSVD iterations - the default number of iterations is 10.%  However, fewer iterations may, in most cases, result an acceleration in%  the process, without effecting  the result too much. Therefore, when%  required, this parameter may be re-set.%  2. maxBlocksToConsider - The maximal number of blocks to train on. If this %  number is larger the number of blocks in the image, random blocks%  from the image will be selected for training. % ===================================================================% INPUT ARGUMENTS : Image - the noisy image (gray-level scale)%                   sigma - the s.d. of the noise (assume to be white Gaussian).%                   K - the number of atoms in the trained dictionary.%    Optional arguments:              %                  'blockSize' - the size of the blocks the algorithm%                       works. All blocks are squares, therefore the given%                       parameter should be one number (width or height).%                       Default value: 8.%                       'errorFactor' - a factor that multiplies sigma in order%                       to set the allowed representation error. In the%                       experiments presented in the paper, it was set to 1.15%                       (which is also the default  value here).%                  'maxBlocksToConsider' - maximal number of blocks that%                       can be processed. This number is dependent on the memory%                       capabilities of the machine, and performances?%                       considerations. If the number of available blocks in the%                       image is larger than 'maxBlocksToConsider', the sliding%                       distance between the blocks increases. The default value%                       is: 250000.%                  'slidingFactor' - the sliding distance between processed%                       blocks. Default value is 1. However, if the image is%                       large, this number increases automatically (because of%                       memory requirements). Larger values result faster%                       performances (because of fewer processed blocks).%                  'numKSVDIters' - the number of KSVD iterations processed%                       blocks from the noisy image. If the number of%                       blocks in the image is larger than this number,%                       random blocks from all available blocks will be%                       selected. The default value for this parameter is:%                       10 if sigma > 5, and 5 otherwise.%                  'maxNumBlocksToTrainOn' - the maximal number of blocks%                       to train on. The default value for this parameter is%                       65000. However, it might not be enough for very large%                       images%                  'displayFlag' - if this flag is switched on,%                       announcement after finishing each iteration will appear,%                       as also a measure concerning the progress of the%                       algorithm (the average number of required coefficients%                       for representation). The default value is 1 (on).%                  'waitBarOn' - can be set to either 1 or 0. If%                       waitBarOn==1 a waitbar, presenting the progress of the%                       algorithm will be displayed.% OUTPUT ARGUMENTS : Iout - a 2-dimensional array in the same size of the%                       input image, that contains the cleaned image.%                    output.D - the trained dictionary.% ========================================================================= % first, train a dictionary on the noisy image reduceDC = 1;[NN1,NN2] = size(Image);waitBarOn = 1;if (sigma > 5)%%%sigma=50   numIterOfKsvd = 10;    numIterOfKsvd = 10;else    numIterOfKsvd = 5;endC = 1.15;maxBlocksToConsider = 260000;slidingDis = 1;bb = 8;maxNumBlocksToTrainOn = 65000;displayFlag = 1;hh=length(varargin)%%%%%%%%%%%测试一下能不能进入下面的for循环中去。% for argI = 1:2:length(varargin)%     if (strcmp(varargin{argI}, 'slidingFactor'))%         slidingDis = varargin{argI+1};%     end%     if (strcmp(varargin{argI}, 'errorFactor'))%         C = varargin{argI+1};%     end%     if (strcmp(varargin{argI}, 'maxBlocksToConsider'))%         maxBlocksToConsider = varargin{argI+1};%     end%     if (strcmp(varargin{argI}, 'numKSVDIters'))%         numIterOfKsvd = varargin{argI+1};%     end%     if (strcmp(varargin{argI}, 'blockSize'))%         bb = varargin{argI+1};%     end%     if (strcmp(varargin{argI}, 'maxNumBlocksToTrainOn'))%         maxNumBlocksToTrainOn = varargin{argI+1};%     end%     if (strcmp(varargin{argI}, 'displayFlag'))%         displayFlag = varargin{argI+1};%     end%     if (strcmp(varargin{argI}, 'waitBarOn'))%         waitBarOn = varargin{argI+1};%     end% end if (sigma <= 5)    numIterOfKsvd = 5;end % first, train a dictionary on blocks from the noisy image if(prod([NN1,NN2]-bb+1)> maxNumBlocksToTrainOn)    randPermutation =  randperm(prod([NN1,NN2]-bb+1));    selectedBlocks = randPermutation(1:maxNumBlocksToTrainOn);     blkMatrix = zeros(bb^2,maxNumBlocksToTrainOn);    for i = 1:maxNumBlocksToTrainOn        [row,col] = ind2sub(size(Image)-bb+1,selectedBlocks(i));        currBlock = Image(row:row+bb-1,col:col+bb-1);        blkMatrix(:,i) = currBlock(:);    endelse    blkMatrix = im2col(Image,[bb,bb],'sliding');%%%%%%%8*8=64   所以blkMatrix矩阵大小为：64*[（NN1-bb+1）*(NN2-bb+1)]end param.K = K;%%%K=256  4*8*8=256param.numIteration = numIterOfKsvd ;%sigma=50   所以numIterOfKsvd = 10; param.errorFlag = 1; % decompose signals until a certain error is reached. do not use fix number of coefficients.param.errorGoal = sigma*C;param.preserveDCAtom = 0; Pn=ceil(sqrt(K));%%Pn=16DCT=zeros(bb,Pn);%%bb=8for k=0:1:Pn-1,    V=cos([0:1:bb-1]'*k*pi/Pn);    if k>0, V=V-mean(V); end;    DCT(:,k+1)=V/norm(V);end;DCT=kron(DCT,DCT);%%%%%跟DCT中的代码一样的     64*256的矩阵 param.initialDictionary = DCT(:,1:param.K );%%%% 取了256列。也就是全部都取了param.InitializationMethod =  'GivenMatrix'; if (reduceDC)%%reduceDC=1    vecOfMeans = mean(blkMatrix);    blkMatrix = blkMatrix-ones(size(blkMatrix,1),1)*vecOfMeans;%%%减去平均数  blkMatrix矩阵大小为：64*[（NN1-bb+1）*(NN2-bb+1)]end if (waitBarOn)%waitBarOn=1    counterForWaitBar = param.numIteration+1;%param.numIteration = numIterOfKsvd ;  =10    h = waitbar(0,'Denoising In Process ...');    param.waitBarHandle = h;    param.counterForWaitBar = counterForWaitBar;end  param.displayProgress = displayFlag;%displayFlag = 1;[Dictionary,output] = KSVD(blkMatrix,param);%%%%%%%最核心的函数%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%output.D = Dictionary; if (displayFlag)%displayFlag = 1;    disp('finished Trainning dictionary');end   % denoise the image using the resulted dictionaryerrT = sigma*C;IMout=zeros(NN1,NN2);Weight=zeros(NN1,NN2);%blocks = im2col(Image,[NN1,NN2],[bb,bb],'sliding');while (prod(floor((size(Image)-bb)/slidingDis)+1)>maxBlocksToConsider)    slidingDis = slidingDis+1;end[blocks,idx] = my_im2col(Image,[bb,bb],slidingDis); if (waitBarOn)    newCounterForWaitBar = (param.numIteration+1)*size(blocks,2);end  % go with jumps of 30000for jj = 1:30000:size(blocks,2)    if (waitBarOn)        waitbar(((param.numIteration*size(blocks,2))+jj)/newCounterForWaitBar);    end    jumpSize = min(jj+30000-1,size(blocks,2));    if (reduceDC)        vecOfMeans = mean(blocks(:,jj:jumpSize));        blocks(:,jj:jumpSize) = blocks(:,jj:jumpSize) - repmat(vecOfMeans,size(blocks,1),1);    end        %Coefs = mexOMPerrIterative(blocks(:,jj:jumpSize),Dictionary,errT);    Coefs = OMPerr(Dictionary,blocks(:,jj:jumpSize),errT);    if (reduceDC)        blocks(:,jj:jumpSize)= Dictionary*Coefs + ones(size(blocks,1),1) * vecOfMeans;    else        blocks(:,jj:jumpSize)= Dictionary*Coefs ;    endend count = 1;Weight = zeros(NN1,NN2);IMout = zeros(NN1,NN2);[rows,cols] = ind2sub(size(Image)-bb+1,idx);for i  = 1:length(cols)    col = cols(i); row = rows(i);            block =reshape(blocks(:,count),[bb,bb]);    IMout(row:row+bb-1,col:col+bb-1)=IMout(row:row+bb-1,col:col+bb-1)+block;    Weight(row:row+bb-1,col:col+bb-1)=Weight(row:row+bb-1,col:col+bb-1)+ones(bb);    count = count+1;end; if (waitBarOn)    close(h);endIOut = (Image+0.034*sigma*IMout)./(1+0.034*sigma*Weight); 

my_im2col.m

function [blocks,idx] = my_im2col(I,blkSize,slidingDis); if (slidingDis==1)    blocks = im2col(I,blkSize,'sliding');%行为blksize元素的总个数，列为(m-bb+1) x (n-bb+1)=62001       % http://fuda641.blog.163.com/blog/static/20751421620135483846711/    idx = [1:size(blocks,2)];    returnend idxMat = zeros(size(I)-blkSize+1);idxMat([[1:slidingDis:end-1],end],[[1:slidingDis:end-1],end]) = 1; % take blocks in distances of 'slidingDix', but always take the first and last one (in each row and column).idx = find(idxMat);[rows,cols] = ind2sub(size(idxMat),idx);blocks = zeros(prod(blkSize),length(idx));for i = 1:length(idx)    currBlock = I(rows(i):rows(i)+blkSize(1)-1,cols(i):cols(i)+blkSize(2)-1);    blocks(:,i) = currBlock(:);end

OMPerr.m

function [A]=OMPerr(D,X,errorGoal); 

%=============================================

% Sparse coding of a group of signals based on a given 

% dictionary and specified number of atoms to use. 

% input arguments: D - the dictionary

%                  X - the signals to represent

%                  errorGoal - the maximal allowed representation error for

%                  each siganl.

% output arguments: A - sparse coefficient matrix.

%=============================================

[n,P]=size(X);%n=64  P= 62001=249*249

[n,K]=size(D);%n=64 K=256

E2 = errorGoal^2*n;

maxNumCoef = n/2;%%%%%%32

A = sparse(size(D,2),size(X,2));%参考稀疏矩阵的帮助256*10000

for k=1:1:P,

    a=[];

    x=X(:,k);

    residual=x;

    indx = [];

    a = [];

    currResNorm2 = sum(residual.^2);

    j = 0;

 

    while currResNorm2>E2 & j < maxNumCoef,

        j = j+1;

        proj=D'*residual;%参考pinv函数的帮助 256*1

        pos=find(abs(proj)==max(abs(proj)));%看看D（256列）中哪一列的值最大

        pos=pos(1);

        indx(j)=pos;%%%index的值为1到256

        %c++的opm优化速度的算法     http://blog.csdn.net/pi9nc/article/details/26593003

        a=pinv(D(:,indx(1:j)))*x;%j*64  *64*1=j*1    

        residual=x-D(:,indx(1:j))*a;

        currResNorm2 = sum(residual.^2);

   end;

   if (length(indx)>0)

       A(indx,k)=a;%%%a是j*1的矩阵,其中j=maxNumCoef

   end

end;

return;

KSVD.m

function [Dictionary,output] = KSVD(...    Data,... % an nXN matrix that contins N signals (Y), each of dimension n.    param)% =========================================================================%                          K-SVD algorithm% =========================================================================% The K-SVD algorithm finds a dictionary for linear representation of% signals. Given a set of signals, it searches for the best dictionary that% can sparsely represent each signal. Detailed discussion on the algorithm% and possible applications can be found in "The K-SVD: An Algorithm for % Designing of Overcomplete Dictionaries for Sparse Representation", written% by M. Aharon, M. Elad, and A.M. Bruckstein and appeared in the IEEE Trans. % On Signal Processing, Vol. 54, no. 11, pp. 4311-4322, November 2006. % =========================================================================% INPUT ARGUMENTS:% Data                         an nXN matrix that contins N signals (Y), each of dimension n. % param                        structure that includes all required%                                 parameters for the K-SVD execution.%                                 Required fields are:%    K, ...                    the number of dictionary elements to train%    numIteration,...          number of iterations to perform.%    errorFlag...              if =0, a fix number of coefficients is%                                 used for representation of each signal. If so, param.L must be%                                 specified as the number of representing atom. if =1, arbitrary number%                                 of atoms represent each signal, until a specific representation error%                                 is reached. If so, param.errorGoal must be specified as the allowed%                                 error.%    preserveDCAtom...         if =1 then the first atom in the dictionary%                                 is set to be constant, and does not ever change. This%                                 might be useful for working with natural%                                 images (in this case, only param.K-1%                                 atoms are trained).%    (optional, see errorFlag) L,...                 % maximum coefficients to use in OMP coefficient calculations.%    (optional, see errorFlag) errorGoal, ...        % allowed representation error in representing each signal.%    InitializationMethod,...  mehtod to initialize the dictionary, can%                                 be one of the following arguments: %                                 * 'DataElements' (initialization by the signals themselves), or: %                                 * 'GivenMatrix' (initialization by a given matrix param.initialDictionary).%    (optional, see InitializationMethod) initialDictionary,...      % if the initialization method %                                 is 'GivenMatrix', this is the matrix that will be used.%    (optional) TrueDictionary, ...        % if specified, in each%                                 iteration the difference between this dictionary and the trained one%                                 is measured and displayed.%    displayProgress, ...      if =1 progress information is displyed. If param.errorFlag==0, %                                 the average repersentation error (RMSE) is displayed, while if %                                 param.errorFlag==1, the average number of required coefficients for %                                 representation of each signal is displayed.% =========================================================================% OUTPUT ARGUMENTS:%  Dictionary                  The extracted dictionary of size nX(param.K).%  output                      Struct that contains information about the current run. It may include the following fields:%    CoefMatrix                  The final coefficients matrix (it should hold that Data equals approximately Dictionary*output.CoefMatrix.%    ratio                       If the true dictionary was defined (in%                                synthetic experiments), this parameter holds a vector of length%                                param.numIteration that includes the detection ratios in each%                                iteration).%    totalerr                    The total representation error after each%                                iteration (defined only if%                                param.displayProgress=1 and%                                param.errorFlag = 0)%    numCoef                     A vector of length param.numIteration that%                                include the average number of coefficients required for representation%                                of each signal (in each iteration) (defined only if%                                param.displayProgress=1 and%                                param.errorFlag = 1)% =========================================================================  %isfield(param,'displayProgress'):表示的是param中是否含有displayPrograess，如果含有则返回1，没有则返回0if (~isfield(param,'displayProgress'))%%%原来的程序中含有param.displayProgress = displayFlag;%displayFlag = 1;  所以此句也不会执行    param.displayProgress = 0;endtotalerr(1) = 99999;%代表的累积误差if (isfield(param,'errorFlag')==0)%%%param.errorFlag = 1;   此句也不会执行    param.errorFlag = 0;end if (isfield(param,'TrueDictionary'))%%%param中没有TrueDictionary    displayErrorWithTrueDictionary = 1;    ErrorBetweenDictionaries = zeros(param.numIteration+1,1);    ratio = zeros(param.numIteration+1,1);else    displayErrorWithTrueDictionary = 0;%%执行此句    ratio = 0;%看开头的说明endif (param.preserveDCAtom>0)  %param.preserveDCAtom = 0;    FixedDictionaryElement(1:size(Data,1),1) = 1/sqrt(size(Data,1));else    FixedDictionaryElement = [];%执行此句end% coefficient calculation method is OMP with fixed number of coefficientsif (size(Data,2) < param.K)%K=256    size(Data,2)=249*249  此句不满足if条件    disp('Size of data is smaller than the dictionary size. Trivial solution...');    Dictionary = Data(:,1:size(Data,2));    return;elseif (strcmp(param.InitializationMethod,'DataElements'))%%比较两个字符串是否相等    param.InitializationMethod =  'GivenMatrix';    Dictionary(:,1:param.K-param.preserveDCAtom) = Data(:,1:param.K-param.preserveDCAtom);elseif (strcmp(param.InitializationMethod,'GivenMatrix'))%% param.InitializationMethod =  'GivenMatrix';   执行此句    Dictionary(:,1:param.K-param.preserveDCAtom) = param.initialDictionary(:,1:param.K-param.preserveDCAtom);%param.initialDictionary = DCT(:,1:param.K );%%%% 取了256列。也就是全部都取了%param.preserveDCAtom=0   param.K-param.preserveDCAtom=K=256   初始化字典就是DCT字典end% reduce the components in Dictionary that are spanned by the fixed% elementsif (param.preserveDCAtom)%param.preserveDCAtom = 0;   此句不执行    tmpMat = FixedDictionaryElement \ Dictionary;    Dictionary = Dictionary - FixedDictionaryElement*tmpMat;end    %%进入正题了！！！！！%normalize the dictionary.   对字典进行归一化Dictionary = Dictionary*diag(1./sqrt(sum(Dictionary.*Dictionary)));%64*256    *256*256（可以借助帮助文档）：diag(1./sqrt(sum(Dictionary.*Dictionary)))  将sum(Dictionary.*Dictionary)作为对角线生成一个对角的矩阵Dictionary = Dictionary.*repmat(sign(Dictionary(1,:)),size(Dictionary,1),1); % multiply in the sign of the first element.  64*256  64*256totalErr = zeros(1,param.numIteration);%param.numIteration = numIterOfKsvd=10 ;    %sigma=50   所以numIterOfKsvd = 10;   % the K-SVD algorithm starts here.for iterNum = 1:param.numIteration  %param.numIteration = numIterOfKsvd=10    % find the coefficients    if (param.errorFlag==0)  %param.errorFlag = 1;           %CoefMatrix = mexOMPIterative2(Data, [FixedDictionaryElement,Dictionary],param.L);        CoefMatrix = OMP([FixedDictionaryElement,Dictionary],Data, param.L); %size(Data,2)=249*249    else          %CoefMatrix = mexOMPerrIterative(Data, [FixedDictionaryElement,Dictionary],param.errorGoal);        CoefMatrix = OMPerr([FixedDictionaryElement,Dictionary],Data, param.errorGoal);%%%%%%%%%%param.errorGoal = sigma*C;   稀疏矩阵        param.L = 1;    end        replacedVectorCounter = 0;    rPerm = randperm(size(Dictionary,2));%size(Dictionary,2)=256  测试一下就知道该函数的用法了(产生1到256的随机的整数，没有重合的整数)    for j = rPerm  %j的值为从1到256的随机整数值（没有重复的）        [betterDictionaryElement,CoefMatrix,addedNewVector] = I_findBetterDictionaryElement(Data,...%%%%%%%%参考基于块结构化字典学习                [FixedDictionaryElement,Dictionary],j+size(FixedDictionaryElement,2),...            CoefMatrix,param.L);        Dictionary(:,j) = betterDictionaryElement;%%%%%已看懂        if (param.preserveDCAtom)%param.preserveDCAtom = 0;   此句不执行            tmpCoef = FixedDictionaryElement\betterDictionaryElement;            Dictionary(:,j) = betterDictionaryElement - FixedDictionaryElement*tmpCoef;            Dictionary(:,j) = Dictionary(:,j)./sqrt(Dictionary(:,j)'*Dictionary(:,j));        end        replacedVectorCounter = replacedVectorCounter+addedNewVector;%%%%实验证明（针对w.jpg图像），值累加了一次    end         if (iterNum>1 & param.displayProgress)%param.displayProgress = 1        if (param.errorFlag==0)%param.errorFlag = 1;            output.totalerr(iterNum-1) = sqrt(sum(sum((Data-[FixedDictionaryElement,Dictionary]*CoefMatrix).^2))/prod(size(Data)));            disp(['Iteration   ',num2str(iterNum),'   Total error is: ',num2str(output.totalerr(iterNum-1))]);        else %执行此句            output.numCoef(iterNum-1) = length(find(CoefMatrix))/size(Data,2);%%CoefMatrix中所有非0元素的长度/DATE的列数            disp(['Iteration   ',num2str(iterNum),'   Average number of coefficients: ',num2str(output.numCoef(iterNum-1))]);        end    end    if (displayErrorWithTrueDictionary ) %displayErrorWithTrueDictionary = 0;        [ratio(iterNum+1),ErrorBetweenDictionaries(iterNum+1)] = I_findDistanseBetweenDictionaries(param.TrueDictionary,Dictionary);%%%%%%        disp(strcat(['Iteration  ', num2str(iterNum),' ratio of restored elements: ',num2str(ratio(iterNum+1))]));        output.ratio = ratio;    end       Dictionary = I_clearDictionary(Dictionary,CoefMatrix(size(FixedDictionaryElement,2)+1:end,:),Data);%%%%%%%%%%size(FixedDictionaryElement,2)=0  CoefMatrix有256行       %     h = waitbar(0,'Denoising In Process ...');%     param.waitBarHandle = h;    if (isfield(param,'waitBarHandle'))        waitbar(iterNum/param.counterForWaitBar);    endend output.CoefMatrix = CoefMatrix;Dictionary = [FixedDictionaryElement,Dictionary];%% FixedDictionaryElement = [];%执行此句   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  findBetterDictionaryElement%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%将字典原子D的解定义为U中的第一列，将系数向量CoefMatrix的解定义为V的第一列与S(1，1)的乘积function [betterDictionaryElement,CoefMatrix,NewVectorAdded] = I_findBetterDictionaryElement(Data,Dictionary,j,CoefMatrix,numCoefUsed)if (length(who('numCoefUsed'))==0)    numCoefUsed = 1;%     liu=1%%%%没有进行此句，说明if条件不满足。endrelevantDataIndices = find(CoefMatrix(j,:)); % the data indices that uses the j'th dictionary element.    查找出系数矩阵中每一行中非0元素的序号  参考DCT字典的程序：relevantDataIndices = find(Coefs(3,:));if (length(relevantDataIndices)<1) %(length(relevantDataIndices)==0)  如果系数矩阵为空，则进行如下的语句    ErrorMat = Data-Dictionary*CoefMatrix;    ErrorNormVec = sum(ErrorMat.^2);    [d,i] = max(ErrorNormVec);    betterDictionaryElement = Data(:,i);%ErrorMat(:,i); %    betterDictionaryElement = betterDictionaryElement./sqrt(betterDictionaryElement'*betterDictionaryElement);    betterDictionaryElement = betterDictionaryElement.*sign(betterDictionaryElement(1));    CoefMatrix(j,:) = 0;    NewVectorAdded = 1%%%%%实验证明（针对w.jpg图像），值累加了一次%     liuzhe=1  没进行此句，说明稀疏矩阵的每一行都有非零的元素    return;end NewVectorAdded = 0;tmpCoefMatrix = CoefMatrix(:,relevantDataIndices); %将稀疏矩阵中非0 的取出来  tmpCoefMatrix尺寸为：256*length(relevantDataIndices)tmpCoefMatrix(j,:) = 0;% the coeffitients of the element we now improve are not relevant.errors =(Data(:,relevantDataIndices) - Dictionary*tmpCoefMatrix); % vector of errors that we want to minimize with the new element    D:64*256     tmpCoefMatrix尺寸为：256*length(relevantDataIndices)  Data(:,relevantDataIndices):64*relevantDataIndices% % the better dictionary element and the values of beta are found using svd.% % This is because we would like to minimize || errors - beta*element ||_F^2. % % that is, to approximate the matrix 'errors' with a one-rank matrix. This% % is done using the largest singular value.[betterDictionaryElement,singularValue,betaVector] = svds(errors,1);%%%%%%%仅仅取出了第一主分量     errors的大小为;64*relevantDataIndices   M=64  N=relevantDataIndices     betterDictionaryElement*singularValue*betaVector'近似的可以表示errors%a=[1 2 3 4;5 6 7 8;9 10 11 12;2 4 6 7.99999]; [u,s,v]=svds(a)   u*s*v'    [u,s,v]=svds(a,1):取出的第一主成分 %对于svds函数：a为M*N的矩阵，那么u:M*M   S:M*N(简写成M*M)   V=N*M    V'=M*N%对于svd函数：a为M*N的矩阵， 那么u:M*M   S:M*N             V=N*N    V'=N*N%将字典原子D的解定义为U中的第一列，将系数向量CoefMatrix的解定义为V的第一列与S(1，1)的乘积    这个是核心  核心 核心！！！！！！！！！！！！！！！CoefMatrix(j,relevantDataIndices) = singularValue*betaVector';% *signOfFirstElem  s*v'    [u,s,v]=svds(a,1):取出的第一主成分 ，所以此时s*v'矩阵大小为 1*N，即CoefMatrix(j,relevantDataIndices)也为：1*N     betterDictionaryElement:M*1,即64*1的向量    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  findDistanseBetweenDictionaries%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [ratio,totalDistances] = I_findDistanseBetweenDictionaries(original,new)% first, all the column in oiginal starts with positive values.catchCounter = 0;totalDistances = 0;for i = 1:size(new,2)    new(:,i) = sign(new(1,i))*new(:,i);endfor i = 1:size(original,2)    d = sign(original(1,i))*original(:,i);    distances =sum ( (new-repmat(d,1,size(new,2))).^2);    [minValue,index] = min(distances);    errorOfElement = 1-abs(new(:,index)'*d);    totalDistances = totalDistances+errorOfElement;    catchCounter = catchCounter+(errorOfElement<0.01);endratio = 100*catchCounter/size(original,2);    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  I_clearDictionary%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function Dictionary = I_clearDictionary(Dictionary,CoefMatrix,Data)T2 = 0.99;T1 = 3;K=size(Dictionary,2); %%K=256Er=sum((Data-Dictionary*CoefMatrix).^2,1); % remove identical atoms（删除相同的原子）  列求和   CoefMatrix(j,relevantDataIndices)的大小为256*relevantDataIndicesG=Dictionary'*Dictionary; %256*256G = G-diag(diag(G));%例如：G=magic(3)     diag(diag(G))   也就是将对角的元素赋值为0for jj=1:1:K,    if max(G(jj,:))>T2 | length(find(abs(CoefMatrix(jj,:))>1e-7))<=T1 ,        [val,pos]=max(Er);        clearDictionary=1%%%%%%%%%%%%%%%%%%%%%%%%测试满足if条件的有多少次        Er(pos(1))=0;%将最大误差处的值赋值为0        Dictionary(:,jj)=Data(:,pos(1))/norm(Data(:,pos(1)));%%norm(Data(:,pos(1))：求向量的模   此整句相当于归一化        G=Dictionary'*Dictionary;        G = G-diag(diag(G));    end;end;