KSVD程序中要求信号Y的列数要大于字典D的列数,不知道为什么有这个要求。

用KSVD.m训练字典,程序中有一段要求信号Y的列数要大于字典D的列数,否则就会报错,但是理论上并不需要这个要求,帮忙解释下为什么会有这个要求?(程序第91行)如果Y是n*1的向量,就不能训练出字典了吗?

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
%    K  原子个数
%    numIteration,...          number of iterations to perform.
%    numIteration    迭代次数
%    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)%  问题在这里,Data的列小于K就不能运行
    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
% elements
if (param.preserveDCAtom)
    tmpMat = FixedDictionaryElement \ Dictionary;
    Dictionary = Dictionary - FixedDictionaryElement*tmpMat;
end
%normalize the dictionary.   对字典进行归一化
Dictionary = Dictionary*diag(1./sqrt(sum(Dictionary.*Dictionary)));
Dictionary = Dictionary.*repmat(sign(Dictionary(1,:)),size(Dictionary,1),1); 
totalErr = zeros(1,param.numIteration);

%%
% 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));
    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 %执行此句
            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];

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。
end
relevantDataIndices = 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 。 如果relevantDataIndices为0,说明没有patch表达粗腰用到第j个原子
    ErrorMat = Data-Dictionary*CoefMatrix;
    ErrorNormVec = sum(ErrorMat.^2);
    [d,i] = max(ErrorNormVec);
    betterDictionaryElement = Data(:,i);%ErrorMat(:,i); %
    betterDictionaryElement = betterDictionaryElement./sqrt(betterDictionaryElement'*betterDictionaryElement);%归一化
    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.
%%在这里使用SVD就可以达到|| errors - beta*element ||_F^2误差最小的效果
[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);
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); %%K=256
Er=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*relevantDataIndices
G=Dictionary'*Dictionary; %256*256  G表示不同的原子求内积,可以认为是计算相似性 G 的大小是 K*K 
G = G-diag(diag(G));%例如:G=magic(3)     diag(diag(G))   也就是将对角的元素赋值为0
for jj=1:1:K,
    if max(G(jj,:))>T2 | length(find(abs(CoefMatrix(jj,:))>1e-7))<=T1 ,  %G(jj,:))>T2 表示两个原子间相似性很高,大于0.99 
        %length(find(abs(CoefMatrix(jj,:))>1e-7) 表示这使用到第jj个原子的patch少于3个
        [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   此整句相当于把误差最大的列归一化
        G=Dictionary'*Dictionary;
        G = G-diag(diag(G));
    end;
end;

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