ksvd

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)
% =========================================================================

if (~isfield(param,'displayProgress'))
    param.displayProgress = 0;
end
totalerr(1) = 99999;
if (isfield(param,'errorFlag')==0)
    param.errorFlag = 0;
end

if (isfield(param,'TrueDictionary'))
    displayErrorWithTrueDictionary = 1;
    ErrorBetweenDictionaries = zeros(param.numIteration+1,1);
    ratio = zeros(param.numIteration+1,1);
else
    displayErrorWithTrueDictionary = 0;
	ratio = 0;
end
if (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 coefficients

if (size(Data,2) < param.K)
    disp('Size of data is smaller than the dictionary size. Trivial solution...');
    Dictionary = Data(:,1:size(Data,2));
    return;
    %若参数K大于信号的个数 则将数据集作为字典集
elseif (strcmp(param.InitializationMethod,'DataElements'))
    Dictionary(:,1:param.K-param.preserveDCAtom) = Data(:,1:param.K-param.preserveDCAtom);%将数据集的1到param.K-param.preserveDCAtom数据作为字典集
elseif (strcmp(param.InitializationMethod,'GivenMatrix'))
    Dictionary(:,1:param.K-param.preserveDCAtom) = param.initialDictionary(:,1:param.K-param.preserveDCAtom);%将initialDictionary的1到param.K-param.preserveDCAtom列作为字典集
end
% reduce the components in Dictionary that are spanned by the fixed
% elements
if (param.preserveDCAtom)
    tmpMat = FixedDictionaryElement \ Dictionary;%求minimal norm（FixedDictionaryElement×Dictionary-dictionary）
    Dictionary = Dictionary - FixedDictionaryElement*tmpMat;
end
%normalize the dictionary.
Dictionary = Dictionary*diag(1./sqrt(sum(Dictionary.*Dictionary)));%归一化
Dictionary = Dictionary.*repmat(sign(Dictionary(1,:)),size(Dictionary,1),1); % multiply in the sign of the first element.%字典集中的每个元素的化为正数
totalErr = zeros(1,param.numIteration);

% the K-SVD algorithm starts here.

for iterNum = 1:param.numIteration
    % find the coefficients
    if (param.errorFlag==0)%固定表达系数的个数
        %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);%设定允许的误差
        param.L = 1;
    end
    
    replacedVectorCounter = 0;
	rPerm = randperm(size(Dictionary,2));%生成一个1到size(Dictionary,2)的随机的向量
    for j = rPerm
        [betterDictionaryElement,CoefMatrix,addedNewVector] = I_findBetterDictionaryElement(Data,...
            [FixedDictionaryElement,Dictionary],j+size(FixedDictionaryElement,2),...
            CoefMatrix ,param.L);
        Dictionary(:,j) = betterDictionaryElement;%更新字典集
        if (param.preserveDCAtom)
            tmpCoef = FixedDictionaryElement\betterDictionaryElement;
            Dictionary(:,j) = betterDictionaryElement - FixedDictionaryElement*tmpCoef;
            Dictionary(:,j) = Dictionary(:,j)./sqrt(Dictionary(:,j)'*Dictionary(:,j));%归一化
        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);
    end
end

output.CoefMatrix = CoefMatrix;
Dictionary = [FixedDictionaryElement,Dictionary];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  findBetterDictionaryElement
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [betterDictionaryElement,CoefMatrix,NewVectorAdded] = I_findBetterDictionaryElement(Data,Dictionary,j,CoefMatrix,numCoefUsed)%CoefMatrix为字典最终的系数
if (length(who('numCoefUsed'))==0)
    numCoefUsed = 1;
end
relevantDataIndices = find(CoefMatrix(j,:)); % 非零元在第j行的系数矩阵中的位置the data indices that uses the j'th dictionary element.
if (length(relevantDataIndices)<1) %(length(relevantDataIndices)==0)如果系数矩阵的第j列全为零
    ErrorMat = Data-Dictionary*CoefMatrix;%在已有的字典集下和系数下对data项的估计误差
    ErrorNormVec = sum(ErrorMat.^2);%对误 差每项平方
    [d,i] = max(ErrorNormVec);%d为所有列中最大项，i为其第几列
    betterDictionaryElement = Data(:,i);%ErrorMat(:,i); %数据项的i列赋给betterDictionaryElement
    betterDictionaryElement = betterDictionaryElement./sqrt(betterDictionaryElement'*betterDictionaryElement);%归一化betterDictionaryElement
    betterDictionaryElement = betterDictionaryElement.*sign(betterDictionaryElement(1));%将betterDictionaryElement中负的元素化为正的
    CoefMatrix(j,:) = 0;%将系数矩阵的第j行赋值为0
    NewVectorAdded = 1;
    return;
end

NewVectorAdded = 0;
tmpCoefMatrix = CoefMatrix(:,relevantDataIndices); %将系数矩阵的第j行的非零项所在的列赋给tmpCoefMatrix
tmpCoefMatrix(j,:) = 0;% the coeffitients of the element we now improve are not relevant.将tmpCoefMatrix第j行赋0
errors =(Data(:,relevantDataIndices) - Dictionary*tmpCoefMatrix); %在除去字典中第j个的元素后数据集与预测数据之间的误差% 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);%betterDictionaryElement为右特征向量 singularValue为最大特征值 betaVector左特征向量
CoefMatrix(j,relevantDataIndices) = singularValue*betaVector';% *signOfFirstElem 系数矩阵的第j行的非零元的位置换为singularValue*betaVector的值

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  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);
end
for 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);
end
ratio = 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 atoms
G=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;

KSVD_Demo.m

%KSVD demo
clc;clear;
param.K = 5;
param.numIteration = 100;
param.preserveDCAtom = 1;
param.displayProgress = 1;
param.InitializationMethod = 'DataElements';
param.errorFlag = 1;
param.errorGoal = 0;

data = rand(10,100);
[Dictionary,output] = KSVD(data,param);

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,K]=size(D);
E2 = errorGoal^2*n;
maxNumCoef = n/2;
A = sparse(size(D,2),size(X,2));
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;
        pos=find(abs(proj)==max(abs(proj)));
        pos=pos(1);
        indx(j)=pos;
        a=pinv(D(:,indx(1:j)))*x;
        residual=x-D(:,indx(1:j))*a;
		currResNorm2 = sum(residual.^2);
   end;
   if (length(indx)>0)
       A(indx,k)=a;
   end
end;
return;