基於K-SVD稀疏字典的圖像去噪算法
源代碼下載鏈接：Code
下面的代碼我已經給出了我自己的註釋，全部是個人的理解，有誤的歡迎指正！
Main.m
%============================================================
% demo2 - denoise an image
% this is a run_file the demonstrate how to denoise an image,
% using dictionaries. The methods implemented here are the same
% one as described in "Image Denoising Via Sparse and Redundant
% representations over Learned Dictionaries", (appeared in the
% IEEE Trans. on Image Processing, Vol. 15, no. 12, December 2006).
% ============================================================
clear
bb=8; % block size
RR=4; % redundancy factor 冗餘因素
K=RR*bb^2; % number of atoms in the dictionary
sigma = 50;
%pathForImages ='';
%imageName = 'barbara.png';
% [IMin0,pp]=imread('cameraman.tif');
[IMin0,pp]=imread('w.jpg');
IMin0=im2double(IMin0);
if (length(size(IMin0))>2)
IMin0 = rgb2gray(IMin0);
end
if (max(IMin0(:))<2)
IMin0 = IMin0*255;
end
IMin=IMin0+sigma*randn(size(IMin0));%%%%%%此處有隨機函數
PSNRIn = 20*log10(255/sqrt(mean((IMin(:)-IMin0(:)).^2)));
tic
%%%基於壓縮的那篇論文
[IoutAdaptive,output] = denoiseImageKSVD(IMin, sigma,K);
PSNROut = 20*log10(255/sqrt(mean((IoutAdaptive(:)-IMin0(:)).^2)));
figure;
subplot(1,3,1); imshow(IMin0,[]); title('Original clean image');
subplot(1,3,2);
imshow(IMin,[]); title(strcat(['Noisy image, ',num2str(PSNRIn),'dB']));
subplot(1,3,3);
imshow(IoutAdaptive,[]); title(strcat(['Clean Image by Adaptive dictionary, ',num2str(PSNROut),'dB']));
figure;
I = displayDictionaryElementsAsImage(output.D, floor(sqrt(K)), floor(size(output.D,2)/floor(sqrt(K))),bb,bb);
title('The dictionary trained on patches from the noisy image');
toc
denoiseImageKSVD.m
function [IOut,output] = denoiseImageKSVD(Image,sigma,K,varargin)
%==========================================================================
% P E R F O R M D E N O I S I N G U S I N G A D I C T I O N A R Y
% T R A I N E D O N N O I S Y I M A G E
%==========================================================================
% function IOut = denoiseImageKSVD(Image,sigma,K,varargin)
% denoise an image by sparsely representing each block with the
% already overcomplete trained Dictionary, and averaging the represented parts.
% Detailed description can be found in "Image Denoising Via Sparse and Redundant
% representations over Learned Dictionaries", (appeared in the
% IEEE Trans. on Image Processing, Vol. 15, no. 12, December 2006).
% This function may take some time to process. Possible factor that effect
% the processing time are:
% 1. number of KSVD iterations - the default number of iterations is 10.
% However, fewer iterations may, in most cases, result an acceleration in
% the process, without effecting the result too much. Therefore, when
% required, this parameter may be re-set.
% 2. maxBlocksToConsider - The maximal number of blocks to train on. If this
% number is larger the number of blocks in the image, random blocks
% from the image will be selected for training.
% ===================================================================
% INPUT ARGUMENTS : Image - the noisy image (gray-level scale)
% sigma - the s.d. of the noise (assume to be white Gaussian).
% K - the number of atoms in the trained dictionary.
% Optional arguments:
% 'blockSize' - the size of the blocks the algorithm
% works. All blocks are squares, therefore the given
% parameter should be one number (width or height).
% Default value: 8.
% 'errorFactor' - a factor that multiplies sigma in order
% to set the allowed representation error. In the
% experiments presented in the paper, it was set to 1.15
% (which is also the default value here).
% 'maxBlocksToConsider' - maximal number of blocks that
% can be processed. This number is dependent on the memory
% capabilities of the machine, and performances?
% considerations. If the number of available blocks in the
% image is larger than 'maxBlocksToConsider', the sliding
% distance between the blocks increases. The default value
% is: 250000.
% 'slidingFactor' - the sliding distance between processed
% blocks. Default value is 1. However, if the image is
% large, this number increases automatically (because of
% memory requirements). Larger values result faster
% performances (because of fewer processed blocks).
% 'numKSVDIters' - the number of KSVD iterations processed
% blocks from the noisy image. If the number of
% blocks in the image is larger than this number,
% random blocks from all available blocks will be
% selected. The default value for this parameter is:
% 10 if sigma > 5, and 5 otherwise.
% 'maxNumBlocksToTrainOn' - the maximal number of blocks
% to train on. The default value for this parameter is
% 65000. However, it might not be enough for very large
% images
% 'displayFlag' - if this flag is switched on,
% announcement after finishing each iteration will appear,
% as also a measure concerning the progress of the
% algorithm (the average number of required coefficients
% for representation). The default value is 1 (on).
% 'waitBarOn' - can be set to either 1 or 0. If
% waitBarOn==1 a waitbar, presenting the progress of the
% algorithm will be displayed.
% OUTPUT ARGUMENTS : Iout - a 2-dimensional array in the same size of the
% input image, that contains the cleaned image.
% output.D - the trained dictionary.
% =========================================================================
% first, train a dictionary on the noisy image
reduceDC = 1;
[NN1,NN2] = size(Image);
waitBarOn = 1;
if (sigma > 5)%%%sigma=50 numIterOfKsvd = 10;
numIterOfKsvd = 10;
else
numIterOfKsvd = 5;
end
C = 1.15;
maxBlocksToConsider = 260000;
slidingDis = 1;
bb = 8;
maxNumBlocksToTrainOn = 65000;
displayFlag = 1;
hh=length(varargin)%%%%%%%%%%%測試一下能不能進入下面的for循環中去。
% for argI = 1:2:length(varargin)
% if (strcmp(varargin{argI}, 'slidingFactor'))
% slidingDis = varargin{argI+1};
% end
% if (strcmp(varargin{argI}, 'errorFactor'))
% C = varargin{argI+1};
% end
% if (strcmp(varargin{argI}, 'maxBlocksToConsider'))
% maxBlocksToConsider = varargin{argI+1};
% end
% if (strcmp(varargin{argI}, 'numKSVDIters'))
% numIterOfKsvd = varargin{argI+1};
% end
% if (strcmp(varargin{argI}, 'blockSize'))
% bb = varargin{argI+1};
% end
% if (strcmp(varargin{argI}, 'maxNumBlocksToTrainOn'))
% maxNumBlocksToTrainOn = varargin{argI+1};
% end
% if (strcmp(varargin{argI}, 'displayFlag'))
% displayFlag = varargin{argI+1};
% end
% if (strcmp(varargin{argI}, 'waitBarOn'))
% waitBarOn = varargin{argI+1};
% end
% end
if (sigma <= 5)
numIterOfKsvd = 5;
end
% first, train a dictionary on blocks from the noisy image
if(prod([NN1,NN2]-bb+1)> maxNumBlocksToTrainOn)
randPermutation = randperm(prod([NN1,NN2]-bb+1));
selectedBlocks = randPermutation(1:maxNumBlocksToTrainOn);
blkMatrix = zeros(bb^2,maxNumBlocksToTrainOn);
for i = 1:maxNumBlocksToTrainOn
[row,col] = ind2sub(size(Image)-bb+1,selectedBlocks(i));
currBlock = Image(row:row+bb-1,col:col+bb-1);
blkMatrix(:,i) = currBlock(:);
end
else
blkMatrix = im2col(Image,[bb,bb],'sliding');%%%%%%%8*8=64 所以blkMatrix矩陣大小爲：64*[（NN1-bb+1）*(NN2-bb+1)]
end
param.K = K;%%%K=256 4*8*8=256
param.numIteration = numIterOfKsvd ;%sigma=50 所以numIterOfKsvd = 10;
param.errorFlag = 1; % decompose signals until a certain error is reached. do not use fix number of coefficients.
param.errorGoal = sigma*C;
param.preserveDCAtom = 0;
Pn=ceil(sqrt(K));%%Pn=16
DCT=zeros(bb,Pn);%%bb=8
for k=0:1:Pn-1,
V=cos([0:1:bb-1]'*k*pi/Pn);
if k>0, V=V-mean(V); end;
DCT(:,k+1)=V/norm(V);
end;
DCT=kron(DCT,DCT);%%%%%跟DCT中的代碼一樣的 64*256的矩陣
param.initialDictionary = DCT(:,1:param.K );%%%% 取了256列。也就是全部都取了
param.InitializationMethod = 'GivenMatrix';
if (reduceDC)%%reduceDC=1
vecOfMeans = mean(blkMatrix);
blkMatrix = blkMatrix-ones(size(blkMatrix,1),1)*vecOfMeans;%%%減去平均數 blkMatrix矩陣大小爲：64*[（NN1-bb+1）*(NN2-bb+1)]
end
if (waitBarOn)%waitBarOn=1
counterForWaitBar = param.numIteration+1;%param.numIteration = numIterOfKsvd ; =10
h = waitbar(0,'Denoising In Process ...');
param.waitBarHandle = h;
param.counterForWaitBar = counterForWaitBar;
end
param.displayProgress = displayFlag;%displayFlag = 1;
[Dictionary,output] = KSVD(blkMatrix,param);%%%%%%%最核心的函數%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
output.D = Dictionary;
if (displayFlag)%displayFlag = 1;
disp('finished Trainning dictionary');
end
% denoise the image using the resulted dictionary
errT = sigma*C;
IMout=zeros(NN1,NN2);
Weight=zeros(NN1,NN2);
%blocks = im2col(Image,[NN1,NN2],[bb,bb],'sliding');
while (prod(floor((size(Image)-bb)/slidingDis)+1)>maxBlocksToConsider)
slidingDis = slidingDis+1;
end
[blocks,idx] = my_im2col(Image,[bb,bb],slidingDis);
if (waitBarOn)
newCounterForWaitBar = (param.numIteration+1)*size(blocks,2);
end
% go with jumps of 30000
for jj = 1:30000:size(blocks,2)
if (waitBarOn)
waitbar(((param.numIteration*size(blocks,2))+jj)/newCounterForWaitBar);
end
jumpSize = min(jj+30000-1,size(blocks,2));
if (reduceDC)
vecOfMeans = mean(blocks(:,jj:jumpSize));
blocks(:,jj:jumpSize) = blocks(:,jj:jumpSize) - repmat(vecOfMeans,size(blocks,1),1);
end
%Coefs = mexOMPerrIterative(blocks(:,jj:jumpSize),Dictionary,errT);
Coefs = OMPerr(Dictionary,blocks(:,jj:jumpSize),errT);
if (reduceDC)
blocks(:,jj:jumpSize)= Dictionary*Coefs + ones(size(blocks,1),1) * vecOfMeans;
else
blocks(:,jj:jumpSize)= Dictionary*Coefs ;
end
end
count = 1;
Weight = zeros(NN1,NN2);
IMout = zeros(NN1,NN2);
[rows,cols] = ind2sub(size(Image)-bb+1,idx);
for i = 1:length(cols)
col = cols(i); row = rows(i);
block =reshape(blocks(:,count),[bb,bb]);
IMout(row:row+bb-1,col:col+bb-1)=IMout(row:row+bb-1,col:col+bb-1)+block;
Weight(row:row+bb-1,col:col+bb-1)=Weight(row:row+bb-1,col:col+bb-1)+ones(bb);
count = count+1;
end;
if (waitBarOn)
close(h);
end
IOut = (Image+0.034*sigma*IMout)./(1+0.034*sigma*Weight);
my_im2col.m
function [blocks,idx] = my_im2col(I,blkSize,slidingDis);
if (slidingDis==1)
blocks = im2col(I,blkSize,'sliding');%行爲blksize元素的總個數，列爲(m-bb+1) x (n-bb+1)=62001
% http://fuda641.blog.163.com/blog/static/20751421620135483846711/
idx = [1:size(blocks,2)];
return
end
idxMat = zeros(size(I)-blkSize+1);
idxMat([[1:slidingDis:end-1],end],[[1:slidingDis:end-1],end]) = 1; % take blocks in distances of 'slidingDix', but always take the first and last one (in each row and column).
idx = find(idxMat);
[rows,cols] = ind2sub(size(idxMat),idx);
blocks = zeros(prod(blkSize),length(idx));
for i = 1:length(idx)
currBlock = I(rows(i):rows(i)+blkSize(1)-1,cols(i):cols(i)+blkSize(2)-1);
blocks(:,i) = currBlock(:);
end
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=64 P= 62001=249*249
[n,K]=size(D);%n=64 K=256
E2 = errorGoal^2*n;
maxNumCoef = n/2;%%%%%%32
A = sparse(size(D,2),size(X,2));%參考稀疏矩陣的幫助256*10000
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;%參考pinv函數的幫助 256*1
pos=find(abs(proj)==max(abs(proj)));%看看D（256列）中哪一列的值最大
pos=pos(1);
indx(j)=pos;%%%index的值爲1到256
%c++的opm優化速度的算法 http://blog.csdn.net/pi9nc/article/details/26593003
a=pinv(D(:,indx(1:j)))*x;%j*64 *64*1=j*1
residual=x-D(:,indx(1:j))*a;
currResNorm2 = sum(residual.^2);
end;
if (length(indx)>0)
A(indx,k)=a;%%%a是j*1的矩陣,其中j=maxNumCoef
end
end;
return;
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)
% =========================================================================
%isfield(param,'displayProgress'):表示的是param中是否含有displayPrograess，如果含有則返回1，沒有則返回0
if (~isfield(param,'displayProgress'))%%%原來的程序中含有param.displayProgress = displayFlag;%displayFlag = 1; 所以此句也不會執行
param.displayProgress = 0;
end
totalerr(1) = 99999;%代表的累積誤差
if (isfield(param,'errorFlag')==0)%%%param.errorFlag = 1; 此句也不會執行
param.errorFlag = 0;
end
if (isfield(param,'TrueDictionary'))%%%param中沒有TrueDictionary
displayErrorWithTrueDictionary = 1;
ErrorBetweenDictionaries = zeros(param.numIteration+1,1);
ratio = zeros(param.numIteration+1,1);
else
displayErrorWithTrueDictionary = 0;%%執行此句
ratio = 0;%看開頭的說明
end
if (param.preserveDCAtom>0) %param.preserveDCAtom = 0;
FixedDictionaryElement(1:size(Data,1),1) = 1/sqrt(size(Data,1));
else
FixedDictionaryElement = [];%執行此句
end
% coefficient calculation method is OMP with fixed number of coefficients
if (size(Data,2) < param.K)%K=256 size(Data,2)=249*249 此句不滿足if條件
disp('Size of data is smaller than the dictionary size. Trivial solution...');
Dictionary = Data(:,1:size(Data,2));
return;
elseif (strcmp(param.InitializationMethod,'DataElements'))%%比較兩個字符串是否相等 param.InitializationMethod = 'GivenMatrix';
Dictionary(:,1:param.K-param.preserveDCAtom) = Data(:,1:param.K-param.preserveDCAtom);
elseif (strcmp(param.InitializationMethod,'GivenMatrix'))%% param.InitializationMethod = 'GivenMatrix'; 執行此句
Dictionary(:,1:param.K-param.preserveDCAtom) = param.initialDictionary(:,1:param.K-param.preserveDCAtom);%param.initialDictionary = DCT(:,1:param.K );%%%% 取了256列。也就是全部都取了
%param.preserveDCAtom=0 param.K-param.preserveDCAtom=K=256 初始化字典就是DCT字典
end
% reduce the components in Dictionary that are spanned by the fixed
% elements
if (param.preserveDCAtom)%param.preserveDCAtom = 0; 此句不執行
tmpMat = FixedDictionaryElement \ Dictionary;
Dictionary = Dictionary - FixedDictionaryElement*tmpMat;
end
%%進入正題了！！！！！
%normalize the dictionary. 對字典進行歸一化
Dictionary = Dictionary*diag(1./sqrt(sum(Dictionary.*Dictionary)));%64*256 *256*256（可以藉助幫助文檔）：diag(1./sqrt(sum(Dictionary.*Dictionary))) 將sum(Dictionary.*Dictionary)作爲對角線生成一個對角的矩陣
Dictionary = Dictionary.*repmat(sign(Dictionary(1,:)),size(Dictionary,1),1); % multiply in the sign of the first element. 64*256 64*256
totalErr = zeros(1,param.numIteration);%param.numIteration = numIterOfKsvd=10 ; %sigma=50 所以numIterOfKsvd = 10;
% 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; 稀疏矩陣
param.L = 1;
end
replacedVectorCounter = 0;
rPerm = randperm(size(Dictionary,2));%size(Dictionary,2)=256 測試一下就知道該函數的用法了(產生1到256的隨機的整數，沒有重合的整數)
for j = rPerm %j的值爲從1到256的隨機整數值（沒有重複的）
[betterDictionaryElement,CoefMatrix,addedNewVector] = I_findBetterDictionaryElement(Data,...%%%%%%%%參考基於塊結構化字典學習
[FixedDictionaryElement,Dictionary],j+size(FixedDictionaryElement,2),...
CoefMatrix,param.L);
Dictionary(:,j) = betterDictionaryElement;%%%%%已看懂
if (param.preserveDCAtom)%param.preserveDCAtom = 0; 此句不執行
tmpCoef = FixedDictionaryElement\betterDictionaryElement;
Dictionary(:,j) = betterDictionaryElement - FixedDictionaryElement*tmpCoef;
Dictionary(:,j) = Dictionary(:,j)./sqrt(Dictionary(:,j)'*Dictionary(:,j));
end
replacedVectorCounter = replacedVectorCounter+addedNewVector;%%%%實驗證明（針對w.jpg圖像），值累加了一次
end
if (iterNum>1 & param.displayProgress)%param.displayProgress = 1
if (param.errorFlag==0)%param.errorFlag = 1;
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 %執行此句
output.numCoef(iterNum-1) = length(find(CoefMatrix))/size(Data,2);%%CoefMatrix中所有非0元素的長度/DATE的列數
disp(['Iteration ',num2str(iterNum),' Average number of coefficients: ',num2str(output.numCoef(iterNum-1))]);
end
end
if (displayErrorWithTrueDictionary ) %displayErrorWithTrueDictionary = 0;
[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);%%%%%%%%%%size(FixedDictionaryElement,2)=0 CoefMatrix有256行
% h = waitbar(0,'Denoising In Process ...');
% param.waitBarHandle = h;
if (isfield(param,'waitBarHandle'))
waitbar(iterNum/param.counterForWaitBar);
end
end
output.CoefMatrix = CoefMatrix;
Dictionary = [FixedDictionaryElement,Dictionary];%% FixedDictionaryElement = [];%執行此句
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% findBetterDictionaryElement
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%將字典原子D的解定義爲U中的第一列，將係數向量CoefMatrix的解定義爲V的第一列與S(1，1)的乘積
function [betterDictionaryElement,CoefMatrix,NewVectorAdded] = I_findBetterDictionaryElement(Data,Dictionary,j,CoefMatrix,numCoefUsed)
if (length(who('numCoefUsed'))==0)
numCoefUsed = 1;
% liu=1%%%%沒有進行此句，說明if條件不滿足。
end
relevantDataIndices = find(CoefMatrix(j,:)); % the data indices that uses the j'th dictionary element. 查找出係數矩陣中每一行中非0元素的序號 參考DCT字典的程序：relevantDataIndices = find(Coefs(3,:));
if (length(relevantDataIndices)<1) %(length(relevantDataIndices)==0) 如果係數矩陣爲空，則進行如下的語句
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%%%%%實驗證明（針對w.jpg圖像），值累加了一次
% liuzhe=1 沒進行此句，說明稀疏矩陣的每一行都有非零的元素
return;
end
NewVectorAdded = 0;
tmpCoefMatrix = CoefMatrix(:,relevantDataIndices); %將稀疏矩陣中非0 的取出來 tmpCoefMatrix尺寸爲：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尺寸爲：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.
[betterDictionaryElement,singularValue,betaVector] = svds(errors,1);%%%%%%%僅僅取出了第一主分量 errors的大小爲;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
%對於svds函數：a爲M*N的矩陣，那麼u:M*M S:M*N(簡寫成M*M) V=N*M V'=M*N
%對於svd函數：a爲M*N的矩陣， 那麼u:M*M S:M*N V=N*N V'=N*N
%將字典原子D的解定義爲U中的第一列，將係數向量CoefMatrix的解定義爲V的第一列與S(1，1)的乘積 這個是核心 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 ，所以此時s*v'矩陣大小爲 1*N，即CoefMatrix(j,relevantDataIndices)也爲：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（刪除相同的原子） 列求和 CoefMatrix(j,relevantDataIndices)的大小爲256*relevantDataIndices
G=Dictionary'*Dictionary; %256*256
G = G-diag(diag(G));%例如：G=magic(3) diag(diag(G)) 也就是將對角的元素賦值爲0
for jj=1:1:K,
if max(G(jj,:))>T2 | length(find(abs(CoefMatrix(jj,:))>1e-7))<=T1 ,
[val,pos]=max(Er);
clearDictionary=1%%%%%%%%%%%%%%%%%%%%%%%%測試滿足if條件的有多少次
Er(pos(1))=0;%將最大誤差處的值賦值爲0
Dictionary(:,jj)=Data(:,pos(1))/norm(Data(:,pos(1)));%%norm(Data(:,pos(1))：求向量的模 此整句相當於歸一化
G=Dictionary'*Dictionary;
G = G-diag(diag(G));
end;
end;