K-SVD算法

K-SVD算法的基本思想：

Y为训练样本，D为字典，X为稀疏系数。一般分为Sparse Coding和DictionaryUpdate两个步骤：

1：Sparse Coding：固定字典D通过下面的目标函数采用一种追踪算法找到样本的最佳稀疏矩阵。

2：Dictionary Update：按列更新字典，一句可使MSE减少的准则，通过SVD（奇异值分解）循序的更新每一列和该列对应的稀疏矩阵的值。

E_K为字典的第k列的残差，物理意义：没有d_k时表示的误差，也就是字典的第k列在表示Y的过程中究竟起到了多大的作用。

根据上面的E_K的解释可以知道，我们的目的就是找到一个合适的d_k来最大化减小E_K。

为了得到d_k就需要对E_K 进行SVD（奇异值分解），E_k=UΔV^T令矩阵U的第一列作为字典第K列更新后的d_k，同时令Δ（1,1）乘以V的第一列作为更新后的稀疏系数。

下面是一个简单的利用KSVD和OMP算法的演示代码

代码流程：

Step1：读入的一张lena图片img

Step2: 随机生成一个测量矩阵phi

Step3：y=phi*img得到观测值

Step4：利用[Dictionary,]=KSVD[img,para]得到dictionary

Step5：利用A=OMP[phi*Dictionary,y,L]得到稀疏系数矩阵

Step6：img_rec=Dictionary*A得到重建的图像。

Demo_Code_1.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% the K-SVD basis is selected as the sparse representation dictionary% the OMP  algorithm is used to recover the image% Author: zhang ben, [e-mail]%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%***************************** read in the image **************************img=imread('lena.bmp');     % read in the image "lena.bmp"img=double(img);[N,n]=size(img); img0 = img;  % keep an original copy of the input signal%****************form the measurement matrix and Dictionary ***************%form the measurement matrix PhiPhi=randn(N,n);   Phi = Phi./repmat(sqrt(sum(Phi.^2,1)),[N,1]); % normalize each column%fix the parametersparam.L =20;   % number of elements in each linear combination.param.K =150; %number of dictionary elementsparam.numIteration = 50; % number of iteration to execute the K-SVD algorithm.param.errorFlag = 0; % decompose signals until a certain error is reached.                      %do not use fix number of coefficients. %param.errorGoal = sigma;param.preserveDCAtom = 0;param.InitializationMethod ='DataElements';%initialization by the signals themselvesparam.displayProgress = 1; % progress information is displyed.[Dictionary,output]= KSVD(img,param);%Dictionary is N*param.K %************************ projection **************************************y=Phi*img;          % treat each column as a independent signaly0=y;  % keep an original copy of the measurements%********************* recover using OMP *********************************D=Phi*Dictionary;A=OMP(D,y,20);imgr=Dictionary*A;  %***********************  show the results  ******************************** figure(1)subplot(2,2,1),imagesc(img0),title('original image')subplot(2,2,2),imagesc(y0),title('measurement image')subplot(2,2,3),imagesc(Dictionary),title('Dictionary')psnr=20*log10(255/sqrt(mean((img(:)-imgr(:)).^2)));subplot(2,2,4),imagesc(imgr),title(strcat('recover image （',num2str(psnr),'dB）'))disp('over')

OMP.m(这是网友写好的代码)

function [A]=OMP(D,X,L); %=============================================% Sparse coding of a group of signals based on a given % dictionary and specified number of atoms to use. % input arguments: %       D - the dictionary (its columns MUST be normalized).%       X - the signals to represent%       L - the max. number of coefficients for each signal.% output arguments: %       A - sparse coefficient matrix.%=============================================[n,K]=size(D);[n,P]=size(X);for k=1:1:P,    a=[];    x=X(:,k);%令向量x等于矩阵X的第K列的元素长度为n*1    residual=x;%n*1    indx=zeros(L,1);%L*1的0矩阵    for j=1:1:L,        proj=D'*residual;%K*n n*1 变成K*1        [maxVal,pos]=max(abs(proj));%  最大投影系数对应的位置        pos=pos(1);        indx(j)=pos;         a=pinv(D(:,indx(1:j)))*x;        residual=x-D(:,indx(1:j))*a;        if sum(residual.^2) < 1e-6            break;        end    end;    temp=zeros(K,1);    temp(indx(1:j))=a;    A(:,k)=sparse(temp);%A为返回为K*P的矩阵end;return;

KSVD算法实现代码：

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)% =========================================================================if (~isfield(param,'displayProgress'))    param.displayProgress = 0;endtotalerr(1) = 99999;if (isfield(param,'errorFlag')==0)    param.errorFlag = 0;endif (isfield(param,'TrueDictionary'))    displayErrorWithTrueDictionary = 1;    ErrorBetweenDictionaries = zeros(param.numIteration+1,1); %产生零矩阵    ratio = zeros(param.numIteration+1,1);else    displayErrorWithTrueDictionary = 0;ratio = 0;endif (param.preserveDCAtom>0)    FixedDictionaryElement(1:size(Data,1),1) = 1/sqrt(size(Data,1));else    FixedDictionaryElement = [];end% coefficient calculation method is OMP with fixed number of coefficientsif (size(Data,2) < param.K)    disp('Size of data is smaller than the dictionary size. Trivial solution...');    Dictionary = Data(:,1:size(Data,2));    return;elseif (strcmp(param.InitializationMethod,'DataElements'))    Dictionary(:,1:param.K-param.preserveDCAtom) = Data(:,1:param.K-param.preserveDCAtom);elseif (strcmp(param.InitializationMethod,'GivenMatrix'))    Dictionary(:,1:param.K-param.preserveDCAtom) = param.initialDictionary(:,1:param.K-param.preserveDCAtom);end% reduce the components in Dictionary that are spanned by the fixed% elementsif (param.preserveDCAtom)    tmpMat = FixedDictionaryElement \ Dictionary;    Dictionary = Dictionary - FixedDictionaryElement*tmpMat;end%normalize the dictionary.Dictionary = Dictionary*diag(1./sqrt(sum(Dictionary.*Dictionary)));Dictionary = Dictionary.*repmat(sign(Dictionary(1,:)),size(Dictionary,1),1); % multiply in the sign of the first element.totalErr = zeros(1,param.numIteration);% the K-SVD algorithm starts here.for iterNum = 1:param.numIteration    % find the coefficients    if (param.errorFlag==0)        %CoefMatrix = mexOMPIterative2(Data, [FixedDictionaryElement,Dictionary],param.L);        CoefMatrix = OMP([FixedDictionaryElement,Dictionary],Data, param.L);    else         %CoefMatrix = mexOMPerrIterative(Data, [FixedDictionaryElement,Dictionary],param.errorGoal);        CoefMatrix = OMPerr([FixedDictionaryElement,Dictionary],Data, param.errorGoal);        param.L = 1;    end        replacedVectorCounter = 0;rPerm = randperm(size(Dictionary,2));    for j = rPerm        [betterDictionaryElement,CoefMatrix,addedNewVector] = I_findBetterDictionaryElement(Data,...            [FixedDictionaryElement,Dictionary],j+size(FixedDictionaryElement,2),...            CoefMatrix ,param.L);        Dictionary(:,j) = betterDictionaryElement;        if (param.preserveDCAtom)            tmpCoef = FixedDictionaryElement\betterDictionaryElement;            Dictionary(:,j) = betterDictionaryElement - FixedDictionaryElement*tmpCoef;            Dictionary(:,j) = Dictionary(:,j)./sqrt(Dictionary(:,j)'*Dictionary(:,j));        end        replacedVectorCounter = replacedVectorCounter+addedNewVector;    end    if (iterNum>1 & param.displayProgress)        if (param.errorFlag==0)            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            output.numCoef(iterNum-1) = length(find(CoefMatrix))/size(Data,2);            disp(['Iteration   ',num2str(iterNum),'   Average number of coefficients: ',num2str(output.numCoef(iterNum-1))]);        end    end    if (displayErrorWithTrueDictionary )         [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);        if (isfield(param,'waitBarHandle'))        waitbar(iterNum/param.counterForWaitBar);    endendoutput.CoefMatrix = CoefMatrix;Dictionary = [FixedDictionaryElement,Dictionary];%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  findBetterDictionaryElement%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%function [betterDictionaryElement,CoefMatrix,NewVectorAdded] = I_findBetterDictionaryElement(Data,Dictionary,j,CoefMatrix,numCoefUsed)if (length(who('numCoefUsed'))==0)    numCoefUsed = 1;endrelevantDataIndices = find(CoefMatrix(j,:)); % the data indices that uses the j'th dictionary element.if (length(relevantDataIndices)<1) %(length(relevantDataIndices)==0)    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;    return;endNewVectorAdded = 0;tmpCoefMatrix = CoefMatrix(:,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% % 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);CoefMatrix(j,relevantDataIndices) = singularValue*betaVector';% *signOfFirstElem%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  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);Er=sum((Data-Dictionary*CoefMatrix).^2,1); % remove identical atomsG=Dictionary'*Dictionary; G = G-diag(diag(G));for jj=1:1:K,    if max(G(jj,:))>T2 | length(find(abs(CoefMatrix(jj,:))>1e-7))<=T1 ,        [val,pos]=max(Er);        Er(pos(1))=0;        Dictionary(:,jj)=Data(:,pos(1))/norm(Data(:,pos(1)));        G=Dictionary'*Dictionary; G = G-diag(diag(G));    end;end;

这是运行代码之后的结果：