【EM算法】小波域隐马尔科夫树模型参数的EM算法估计MATLAB仿真

1.软件版本

matlab2013b

2.本算法理论知识

 

        EM算法可以解决由不完全观测数据估计概率模型参数的问题。HMT模型中的EM算法思想是构造一个递推算法，使每一步估计的参数都使似然函数更大，使估计的参数逐渐接近真实值。其中，隐马尔科夫树的观测树的似然函数可以表示为：

 

 

 隐马尔科夫树模型参数的E步算法

隐马尔科夫树模型参数的M步算法

3.部分源码

function [ESN,PSN,MUN,SIN]=emlht(w,ES,PS,MU,SI,zm)

%
% Input : 
% w : data PxP matrix (wavelet transform of an image)   
% ES : state transition matrix (MxMxPxP)
% PS : state probability matrix (MxPxP)
% MU : mean matrix (MxPxP)
% SI : variance matrix (MxPxP)
% zm : type of density functions
%  zm = 1 : zero mean (do not update MU)
%  zm = 0 : nonzero mean (update MU)


M=size(ES,1);
P=size(w,1);
level=log2(P);
BE=zeros(M,P,P);
BEP=zeros(M,P,P);
BER=zeros(M,P,P);
 
AL=zeros(M,P,P);
P1=zeros(M,P,P);
P2=zeros(M,M,P,P);

%UP step
wtmp = repmat(w,[1 1 M]);
wtmp = shiftdim(wtmp,2);
si=1; ei=2^(level-1); sj=2^(level-1)+1; ej=P;
gtmp = gauss(wtmp,MU,SI);
scale = repmat(mean(gtmp,1),[M 1 1]);
BE(:,si:ei,sj:ej) = gtmp(:,si:ei,sj:ej)./scale(:,si:ei,sj:ej);

%clear MUtmp SItmp;

for k=level:-1:2
 J=2^(k-1);J2=J*J; si = 1; ei = J; sj = J+1; ej = 2*J;

 EStmp = reshape(ES(:,:,si:ei,sj:ej),M,M*J2);

 if M == 2
  %%%%%% For M=2 the following is faster
  BEtmp = zeros(M,M*J2);
  BEtmp(:,1:M:(M*J2))=reshape(BE(:,si:ei,sj:ej),M,J2);
  BEtmp(:,2:M:(M*J2))=BEtmp(:,1:M:(M*J2)); 
 else
  % For general M (not equal to 2) use the following
  %
  BEtmp = zeros(M,M*4^(k-1));
  for m=1:M
   BEtmp(:,m:M:(M*4^(k-1)))=reshape(BE(:,si:ei,sj:ej,:),M,4^(k-1));
  end;
 end;

 BEtmp = reshape(EStmp.*BEtmp,[M M J J]);
 BEP(:,si:ei,sj:ej) = squeeze(sum(BEtmp,1));
 
 sni = 1; eni = J/2; snj = J/2+1; enj = J;

%construct betachild matrix here
 BCtmp = BEP(:,si:2:ei,sj:2:ej);
 BCtmp = BCtmp.*BEP(:,si+1:2:ei,sj:2:ej);
 BCtmp = BCtmp.*BEP(:,si:2:ei,sj+1:2:ej);
 BCtmp = BCtmp.*BEP(:,si+1:2:ei,sj+1:2:ej); 
 scaletmp = repmat(mean(BCtmp,1),[M 1 1]);
 scale(:,sni:eni,snj:enj) = scale(:,sni:eni,snj:enj).*scaletmp;
 BE(:,sni:eni,snj:enj)=gtmp(:,sni:eni,snj:enj)./scale(:,sni:eni,snj:enj).*BCtmp;

%construct BE(:,pai(i),paj(j),dindex) matrix
 Btmp=zeros(M,J,J);
 Btmp(:,1:2:J,1:2:J)=BE(:,sni:eni,snj:enj);
 Btmp(:,2:2:J,1:2:J)=BE(:,sni:eni,snj:enj);
 Btmp(:,1:2:J,2:2:J)=BE(:,sni:eni,snj:enj);
 Btmp(:,2:2:J,2:2:J)=BE(:,sni:eni,snj:enj);
 BER(:,si:ei,sj:ej)=Btmp./BEP(:,si:ei,sj:ej);
end;

clear EStmp BEtmp BCtmp Btmp;

%DOWN step
 %initialize
 AL(:,1,2) = PS(:,1,2);

 for k=2:level
  J = 2^(k-1); J2=J*J;
  si=1; ei=J; sj=J+1; ej=2*J;
  sni = 1; eni = J/2; snj = J/2+1; enj = J;
  Atmp=zeros(M,J,J);
  Atmp(:,1:2:J,1:2:J)=AL(:,sni:eni,snj:enj);
  Atmp(:,2:2:J,1:2:J)=AL(:,sni:eni,snj:enj);
  Atmp(:,1:2:J,2:2:J)=AL(:,sni:eni,snj:enj);
  Atmp(:,2:2:J,2:2:J)=AL(:,sni:eni,snj:enj);

  Atmp = repmat(reshape(Atmp.*BER(:,si:ei,sj:ej),1,M*J2),[M 1]);
  EStmp = reshape(ES(:,:,si:ei,sj:ej),M,M*J2);
  ALtmp = reshape(EStmp.*Atmp,[M M J J]);
  AL(:,si:ei,sj:ej) = squeeze(sum(ALtmp,2));
 end;

 clear Atmp EStmp ALtmp;

%compute probabilities
for k=2:level
 J=2^(k-1); J2=J*J;
 si=1; ei=J; sj=J+1; ej=2*J;
 sni = 1; eni = J/2; snj = J/2+1; enj = J;

 temp = repmat(sum(AL(:,si:ei,sj:ej).*BE(:,si:ei,sj:ej), 1),[M 1]);
 P1(:,si:ei,sj:ej) = AL(:,si:ei,sj:ej).*BE(:,si:ei,sj:ej)./temp;
  
 %compute P2
 
 if M == 2 
  % For M=2 the following may be faster
  BEtmp = zeros(M,M*J2);
  BEtmp(:,1:M:(M*J2))=reshape(BE(:,si:ei,sj:ej),M,J2);
  BEtmp(:,2:M:(M*J2))=BEtmp(:,1:M:(M*J2));
 else
  % For general M (not equal to 2) use the following
   BEtmp = zeros(M,M*J2);
  for m=1:M
   BEtmp(:,m:M:(M*J2))=reshape(BE(:,si:ei,sj:ej,:),M,J2);
  end;
 end;
 
 BEtmp = reshape(BEtmp,[M M J J]);

 EStmp = ES(:,:,si:ei,sj:ej);
 Atmp=zeros(M,J,J);
 Atmp(:,1:2:J,1:2:J)=AL(:,sni:eni,snj:enj);
 Atmp(:,2:2:J,1:2:J)=AL(:,sni:eni,snj:enj);
 Atmp(:,1:2:J,2:2:J)=AL(:,sni:eni,snj:enj);
 Atmp(:,2:2:J,2:2:J)=AL(:,sni:eni,snj:enj);
 Atmp = repmat(reshape(Atmp,1,M*J2),[M 1]);
 Atmp = reshape(Atmp,[M M J J]);
 BERtmp = repmat(reshape(BER(:,si:ei,sj:ej),1,M*J2),[M 1]);
 BERtmp = reshape(BERtmp,[M M J J]);
 temp = repmat(reshape(temp,1,M*J2),[M 1]);
 temp = reshape(temp, [M M J J]);
 P2(:,:,si:ei,sj:ej)=BEtmp.*EStmp.*Atmp.*BERtmp./temp;
end;
P1(:,1,2)=AL(:,1,2).*BE(:,1,2)./repmat(sum(AL(:,1,2).*BE(:,1,2),1),[M 1 1]);

clear temp BEtmp EStmp Atmp BERtmp;

%PSP=PS; ESP=ES; MUP=MU; SIP=SI;

%M step
PS(:,1,2)=P1(:,1,2);

for k=2:level
 J=2^(k-1); J2=J*J;
 si=1; ei=J; sj=J+1; ej=2*J;
 sni = 1; eni = J/2; snj = J/2+1; enj = J;

 pstmp = sum(sum(P1(:,si:ei,sj:ej),3),2)/J2;
 pstmp = pstmp.*(pstmp>1e-4)+1e-4*(pstmp<=1e-4);
 PS(:,si:ei,sj:ej) = repmat(pstmp,[1 J J]);

 if zm == 0
  % do not update MU if zero mean densities
  mutmp = sum(sum(wtmp(:,si:ei,sj:ej).*P1(:,si:ei,sj:ej),3),2)/J2;
  MU(:,si:ei,sj:ej) = repmat(mutmp,[1 J J])./PS(:,si:ei,sj:ej);
 end;

 sitmp = sum(sum((wtmp(:,si:ei,sj:ej)-MU(:,si:ei,sj:ej)).^2.*P1(:,si:ei,sj:ej),3),2)/J2;

 SI(:,si:ei,sj:ej) = repmat(sitmp,[1 J J])./PS(:,si:ei,sj:ej);

 estmp =sum(sum(P2(:,:,si:ei,sj:ej),4),3)/J2;
 ptmp = [PS(:,sni,snj)'; PS(:,sni,snj)'];
 ES(:,:,si:ei,sj:ej)= repmat(estmp,[1 1 J J])./repmat(ptmp,[1 1 J J]);
end; %k

ESN=ES; PSN=PS; MUN=MU; SIN=SI;

4.仿真结论

5.参考文献

[1]陈静, 范文兵, 甄姬娜. 基于小波变换及隐式马尔科夫树模型的图像信号去噪[J]. 现代电子技术, 2006.A09-06