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.
% 选择不同的原子数代表信号，在程序中可以通过参数errorFlag来设置选择不同的模型。
% 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.如果指定显示差异，在字典和训练字典之间
% 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)
% =========================================================================
% isfield函数功能： 判断输入是否是结构体数组的域（成员）。
% 调用格式：tf = isfield(S, 'fieldname')
% 检查结构体S是否包含由fieldname指定的域， 如果包含， 返回逻辑1； 如果S不包含fieldname域或者S不是结构体类型的， 返回逻辑0。
if (~isfield(param,'displayProgress'))
param.displayProgress = 0;
end
totalerr(1) = 99999;%代表的累积误差
if (isfield(param,'errorFlag')==0)
param.errorFlag = 0;
end
if (isfield(param,'TrueDictionary'))%显示差异
displayErrorWithTrueDictionary = 1;
ErrorBetweenDictionaries = zeros(param.numIteration+1,1);
ratio = zeros(param.numIteration+1,1);
else
displayErrorWithTrueDictionary = 0;
ratio = 0; %参数包含一个长度为param.numIteration的矢量，包含每个中的检测比率
end
if (param.preserveDCAtom>0)%字典中的第一个原子被设定为恒定的，并且不会改变。这个可能对于处理自然图像有用（在这种情况下，只有param.K-1原子被训练）。
FixedDictionaryElement(1:size(Data,1),1) = 1/sqrt(size(Data,1));
else
FixedDictionaryElement = [];
end
% coefficient calculation method is OMP with fixed number of coefficients
%%初始化字典为DCT字典
if (size(Data,2) < param.K)%r=size(A,1)该语句返回的是矩阵A的行数， c=size(A,2) 该语句返回的是矩阵A的列数。判断原始数据的列数是否小于字典的原子数
disp('Size of data is smaller than the dictionary size. Trivial solution...');
Dictionary = Data(:,1:size(Data,2)); %data(:,[y1:y2]) 就是指y1到y2列的对应的所有行的值组成的一个矩阵，此处即为扩大字典
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'))%比较两个字符串是否相等
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.归一化字典
Dictionary = Dictionary*diag(1./sqrt(sum(Dictionary.*Dictionary)));%把D的每一列数据除以该列数据的平方和，从而进行归一化
%diag生成一个对角矩阵 ，sqrt开平方根
Dictionary = Dictionary.*repmat(sign(Dictionary(1,:)),size(Dictionary,1),1);
%B = repmat(A,m,n)，将矩阵 A 复制 m×n 块，即把 A 作为 B 的元素，B 由 m×n 个 A 平铺而成。
%sign 取符号，size返回行数
%字典集中的每个元素的化为正数
totalErr = zeros(1,param.numIteration);%返回m*n的零矩阵
% the K-SVD algorithm starts here.
for iterNum = 1:param.numIteration%迭代次数
% find the coefficients
%得到稀疏矩阵
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);%[A]=OMP(D,X,L)生成稀疏矩阵
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;%信号最大数设为1？？？？
end
replacedVectorCounter = 0;
rPerm = randperm(size(Dictionary,2));%生成一个1到size(Dictionary,2)的随机的向量 y = randperm(n)，y是把1到n这些数随机打乱得到的一个数字序列。
%随机更新某一列，即K-means思想
for j = rPerm%随机更新某一列
[betterDictionaryElement,CoefMatrix,addedNewVector] = I_findBetterDictionaryElement(Data,...%基于块结构化字典学习方法
[FixedDictionaryElement,Dictionary],j+size(FixedDictionaryElement,2),...
CoefMatrix ,param.L);
Dictionary(:,j) = betterDictionaryElement;%更新字典集
if (param.preserveDCAtom)%不执行
tmpCoef = FixedDictionaryElement\betterDictionaryElement;
Dictionary(:,j) = betterDictionaryElement - FixedDictionaryElement*tmpCoef;
Dictionary(:,j) = Dictionary(:,j)./sqrt(Dictionary(:,j)'*Dictionary(:,j));%去掉一列以后重新归一化
end
replacedVectorCounter = replacedVectorCounter+addedNewVector;%实验证明（针对w.jpg图像），值累加了一次
end
if (iterNum>1 & param.displayProgress)
if (param.errorFlag==0)%不执行
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
%显示迭代过程
output.numCoef(iterNum-1) = length(find(CoefMatrix))/size(Data,2);%%CoefMatrix中所有非0元素的长度/数据的列数=平均每列非零系数的个数
disp(['Iteration ',num2str(iterNum),' Average number of coefficients: ',num2str(output.numCoef(iterNum-1))]);% disp将内容输出在MATLAB命令行窗口中
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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%得到非零元在第j行的系数矩阵中的位置
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.
%某一列全为0时怎么做？？？？
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
%执行除掉某一行后得到的误差,只计算非零项
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，记录非零项的位置
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.
%SVD计算得到特征值
[betterDictionaryElement,singularValue,betaVector] = svds(errors,1);%取出第一主分量
%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（删除相同的原子） 列求和 CoefMatrix(j,relevantDataIndices)的大小为256*relevantDataIndices
G=Dictionary'*Dictionary; G = G-diag(diag(G)); %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
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);
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;
来源：CSDN
作者：鲁青的青未
链接：https://blog.csdn.net/w1378236582/article/details/80455587