MOD 之"Hello World"

首先声明，MOD不是取模函数！MOD是字典学习和sparse coding的一种方法… 最近在看KSVD，其简化版就是MOD（method of directions）,这么说吧，KSVD和MOD的优化目标函数是相同的，MOD之所以可以称作KSVD的简化版是因为KSVD在MOD的基础上做了顺序更新列的优化。关于KSVD和MOD的理论知识请见下面我给出的一页note和referenc中的paper。本文主要给出其基本思想及我的代码，已经过测试，如有bug欢迎提出。

Reference

<<From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images>>, Page 68~70

KSVD & MOD's principle & objective function 

Principle:

简单来说，其优化就是一个OMP（orthogonal matching pursuit）与Regression的迭代过程，因此代码包括一个OMP.m, regression.m.

Objective Function & the variation from MOD to KSVD:

Code

CODE1. MOD

运行Main（Main中通过MOD）学习字典和稀疏表示，MOD迭代调用Regression学习字典，调用和OMP获得sparse representation.

Main.m

%% Main.m
clc;
clear;
P = 512;
N = 256;
M = 128;
K = 100;

%% Data Generator Method 1
% sparsity_X = 0.4;
% Y = randi(10,M,P);
% X = floor(sprand(N,P,sparsity_X)*10);

%% Data Generator Method 2
Y = randn(M,P);%Notice that Y should be full rank, that is, rank(Y) = N
X = randn(N,P);% initialization of X

%% Main Iteration
[D,X] = MOD(Y,X,K,1e-4);

MOD.m

%   @Function: Method Of Dirction of 2D signal
%   For dictionary and sparse representation learning
%   @CreateTime: 2013-2-22
%   @Author: Rachel Zhang  @  http://blog.csdn.net/abcjennifer
%   
%   @Reference: From Sparse Solutions of Systems of Equations to 
%   Sparse Modeling of Signals and Images

function [ D , X ] = MOD( Y ,X ,K ,ErrorThreshold )
%MOD Summary of this function goes here
%   Detailed explanation goes here
%   Sample_Data is Y
%   Coefficient is X
%   Dictionary is D
%   sparsity is K

disp('Run Method of directions');
iteration_time = 1;
error = ErrorThreshold+1;


while error>=ErrorThreshold;
    disp(['iteration time = ' num2str(iteration_time)]);
    D = Regression(Y,X);
    X = OMP(Y,D,K);
    iteration_time = iteration_time+1;
    error = sum(sum(abs(Y-D*X)))
end

end

OMP.m

%   @Function: Orthogonal Matching Pursuit of 2D signal
%   Learning Sparse Representation Given Dictionary
%   @CreateTime: 2013-2-21
%   @Author: Rachel Zhang  @  http://blog.csdn.net/abcjennifer
%   
%   @Reference: http://www.eee.hku.hk/~wsha/Freecode/freecode.htm   

function [ X ] = OMP( Y,D,K )
% Y is the sample data to be recovered M*P
% D is the dictionary M*N
% X is the sparse coefficient N*P
% K is the sparsity

if nargin==2
    K = size(D,2);
end;

M = size(D,1);
P = size(Y,2);
N = size(D,2);
m = K*2;  % execute iterations

for idx = 1:P
    % recover the idx-th column sample
    y = Y(:,idx);
    residual = y;
    Aug_D = [];
    D1 = D;
    
    for times = 1:m;
        product = abs(D1'*residual);
        [~,pos] = max(product); %  最大投影系数对应的位置
        Aug_D = [Aug_D, D1(:,pos)];
        D1(:,pos) = zeros(M,1);    %去掉选中的列
        indx(times) = pos;
        Aug_x = (Aug_D'*Aug_D)^-1*Aug_D'*y; %  最小二乘,使残差最小,i.e. x = pinv(Aug_D)*y
        residual = y - Aug_D*Aug_x;
        
        if sum(residual.^2)<1e-6
            break;
        end
    end
    temp = zeros(N,1);
    temp(indx(1:times)) = Aug_x;
    X(:,idx) = sparse(temp);
end
end

Regression.m

%   @Function: Dictionary learning & Regression
%   Learning Dictionary Given Sparse Representation
%   @CreateTime: 2013-2-21
%   @Author: Rachel Zhang  @  http://blog.csdn.net/abcjennifer
%   
function [ D ] = Regression( Y,X )
% Y is the sample data to be recovered M*P
% D is the dictionary M*N
% X is the sparse coefficient N*P
% P>N>M

%由于X是扁矩阵,需要转置求D0 = min(D) ||Y^T-X^TD^T||
%这样就是N个未知数，P个方程去求解；
%每次解得D中的一列，共解M次

Y = Y';
X = X';
P = size(Y,1);
N = size(X,2);
M = size(Y,2);
D = zeros(N,M);

for i = 1:M;
    y = Y(:,i);
    D(:,i) = regress(y,X);
end
D = D';
end

============================================================================

CODE2. KSVD

ksvd函数代码是国外的人写的，很规矩，这里贴过来。

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;
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.
Dictionary = Dictionary*diag(1./sqrt(sum(Dictionary.*Dictionary)));
Dictionary = Dictionary.*repmat(sign(Dictionary(1,:)),size(Dictionary,1),1); % multiply in the sign of the first element.
totalErr = zeros(1,param.numIteration);

% the K-SVD algorithm starts here.

for iterNum = 1:param.numIteration
    % find the coefficients
    if (param.errorFlag==0)
        %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);
        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
            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)
if (length(who('numCoefUsed'))==0)
    numCoefUsed = 1;
end
relevantDataIndices = find(CoefMatrix(j,:)); % the data indices that uses the j'th dictionary element.
if (length(relevantDataIndices)<1) %(length(relevantDataIndices)==0)
    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;
    return;
end

NewVectorAdded = 0;
tmpCoefMatrix = CoefMatrix(:,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
% % 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);
CoefMatrix(j,relevantDataIndices) = singularValue*betaVector';% *signOfFirstElem

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  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;

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