MOD 之 Hello World

首先声明，MOD不是取模函数！MOD是字典学习和sparse coding的一种方法… 最近在看KSVD，其简化版就是MOD（method of directions）,这么说吧，KSVD和MOD的优化目标函数是相同的，MOD之所以可以称作KSVD的简化版是因为KSVD在MOD的基础上做了顺序更新列的优化。关于KSVD和MOD的理论知识请见下面我给出的一页note和referenc中的paper。本文主要给出其基本思想及我的代码，已经过测试，如有bug欢迎提出。

Reference

<<From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images>>, Page 68~70

KSVD & MOD's principle & objective function 

Principle:

简单来说，其优化就是一个OMP（orthogonal matching pursuit）与Regression的迭代过程，因此代码包括一个OMP.m, regression.m.

Objective Function & the variation from MOD to KSVD:

Code

CODE1. MOD

运行Main（Main中通过MOD）学习字典和稀疏表示，MOD迭代调用Regression学习字典，调用和OMP获得sparse representation.

Main.m

%% Main.mclc;clear;P = 512;N = 256;M = 128;K = 100;%% Data Generator Method 1% sparsity_X = 0.4;% Y = randi(10,M,P);% X = floor(sprand(N,P,sparsity_X)*10);%% Data Generator Method 2Y = randn(M,P);%Notice that Y should be full rank, that is, rank(Y) = NX = randn(N,P);% initialization of X%% Main Iteration[D,X] = MOD(Y,X,K,1e-4);

MOD.m

%   @Function: Method Of Dirction of 2D signal%   For dictionary and sparse representation learning%   @CreateTime: 2013-2-22%   @Author: Rachel Zhang  @  http://blog.csdn.net/abcjennifer%   %   @Reference: From Sparse Solutions of Systems of Equations to %   Sparse Modeling of Signals and Imagesfunction [ D , X ] = MOD( Y ,X ,K ,ErrorThreshold )%MOD Summary of this function goes here%   Detailed explanation goes here%   Sample_Data is Y%   Coefficient is X%   Dictionary is D%   sparsity is Kdisp('Run Method of directions');iteration_time = 1;error = ErrorThreshold+1;while error>=ErrorThreshold;    disp(['iteration time = ' num2str(iteration_time)]);    D = Regression(Y,X);    X = OMP(Y,D,K);    iteration_time = iteration_time+1;    error = sum(sum(abs(Y-D*X)))endend

OMP.m

%   @Function: Orthogonal Matching Pursuit of 2D signal%   Learning Sparse Representation Given Dictionary%   @CreateTime: 2013-2-21%   @Author: Rachel Zhang  @  http://blog.csdn.net/abcjennifer%   %   @Reference: http://www.eee.hku.hk/~wsha/Freecode/freecode.htm   function [ X ] = OMP( Y,D,K )% Y is the sample data to be recovered M*P% D is the dictionary M*N% X is the sparse coefficient N*P% K is the sparsityif nargin==2    K = size(D,2);end;M = size(D,1);P = size(Y,2);N = size(D,2);m = K*2;  % execute iterationsfor idx = 1:P    % recover the idx-th column sample    y = Y(:,idx);    residual = y;    Aug_D = [];    D1 = D;        for times = 1:m;        product = abs(D1'*residual);        [~,pos] = max(product); %  最大投影系数对应的位置        Aug_D = [Aug_D, D1(:,pos)];        D1(:,pos) = zeros(M,1);    %去掉选中的列        indx(times) = pos;        Aug_x = (Aug_D'*Aug_D)^-1*Aug_D'*y; %  最小二乘,使残差最小,i.e. x = pinv(Aug_D)*y        residual = y - Aug_D*Aug_x;                if sum(residual.^2)<1e-6            break;        end    end    temp = zeros(N,1);    temp(indx(1:times)) = Aug_x;    X(:,idx) = sparse(temp);endend

Regression.m

%   @Function: Dictionary learning & Regression%   Learning Dictionary Given Sparse Representation%   @CreateTime: 2013-2-21%   @Author: Rachel Zhang  @  http://blog.csdn.net/abcjennifer%   function [ D ] = Regression( Y,X )% Y is the sample data to be recovered M*P% D is the dictionary M*N% X is the sparse coefficient N*P% P>N>M%由于X是扁矩阵,需要转置求D0 = min(D) ||Y^T-X^TD^T||%这样就是N个未知数，P个方程去求解；%每次解得D中的一列，共解M次Y = Y';X = X';P = size(Y,1);N = size(X,2);M = size(Y,2);D = zeros(N,M);for i = 1:M;    y = Y(:,i);    D(:,i) = regress(y,X);endD = D';end

============================================================================

CODE2. KSVD

ksvd函数代码是国外的人写的，很规矩，这里贴过来。

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);    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;

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