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;
