Gibbs Sampling for Ising model

20171208更新：

之前理解有错，没有体现出Gibbs抽样的“抽样”操作。重贴主程序代码如下：

close all; clear all; rng(0);

%% Gibbs sampling method for Ising model
% chapter 24.2.2 in MLaPP
% by forseerwang, http://blog.csdn.net/foreseerwang
% QQ: 50834
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Image generation
a=double(imread('LetterB.bmp'))*2-1;
figure(1); subplot(2,2,1);
imagesc(a);
colorbar;
title('original image', 'fontsize', 20);
axis('square');
colormap gray;
axis off;

% add noise
sigma  = 2; % noise level
y = a + sigma*randn(size(a)); %y = noisy signal
figure(1); subplot(2,2,2);
imagesc(y);
colorbar;
title('noisy image', 'fontsize', 20);
axis('square');
colormap gray;
axis off;

%%
% 迭代参数
maxIter = 30;
% 最初需要舍弃的抽样数
Nburnin = 10;

psi_p = log(normpdf(y(:), 1, sigma)); 
psi_m = log(normpdf(y(:), -1, sigma));
logodds = psi_p - psi_m;
J = 1;

xpred = y(:);
xrec = zeros(length(xpred), maxIter - Nburnin);

[M,N] = size(y);
for iter = 1:maxIter
    for ii = 1:N,
        for jj = 1:M,
            pos = jj + M*(ii-1);
            nbr = pos + [-1,1,-M,M];
            nbr(([jj==1,jj==M,ii==1,ii==N])) = [];
            eta = sum(xpred(nbr));
            
            % MLaPP中的公式（24.6），书中有typo
            prob = fun_sigm(2*J*eta + logodds(pos));
            
            % 抽样
            if rand>prob,
                xpred(pos) = -1;
            else
                xpred(pos) = +1;
            end;
        end;
    end;
    
    if iter>Nburnin,
        xrec(:,iter-Nburnin) = xpred;
    end;
    
end;

xmean = mean(xrec,2);
figure(1); subplot(2,2,3);
imagesc(reshape(xmean,M,N));
colorbar;
title(sprintf('Mean image over last %d iter', maxIter-Nburnin), 'fontsize', 20);
axis('square');
colormap gray;
axis off;

xpred=reshape(xpred,M,N);

figure(1); subplot(2,2,4);
imagesc(xpred);
colorbar;
title(sprintf('image after %d iter', maxIter), 'fontsize', 20);
axis('square');
colormap gray;
axis off;

这个代码的运行结果如下：

%% 旧的版本，理解和代码有误。

Gibbs SamplingGibbs Sampling是MCMC（蒙特卡洛马尔科夫链：Markov chain Monte Carlo）中最流行的一个算法。抛开数学推导不说，这个算法的形式非常简单。详见MLaPP第24.2节。

作为一个入门示例，此处给出了Gibbs sampling用于Ising模型的实现，即MLaPP第24.2.2节的实现。

算法很简单，核心公式就是（24.1）和（24.6），只是需要注意的是公式(24.6)里有typo，括号中的减号应该是加号，这个自己稍微推导一下就出来了，很简单。

还是我之前画的那个“B”，降噪结果如下（可以和之前Mean Field算法对比一下，http://blog.csdn.net/foreseerwang/article/details/78343238）：

%% 20171208更新：以下主程序代码有误。

代码如下：

close all; clear all; rng(0);

%% Gibbs sampling method for Ising model
% chapter 24.2.2 in MLaPP
% by forseerwang, http://blog.csdn.net/foreseerwang
% QQ: 50834
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Image generation
a=double(imread('LetterB.bmp'))*2-1;
figure(1); subplot(2,2,1);
imagesc(a);
colorbar;
title('original image', 'fontsize', 20);
axis('square');
colormap gray;
axis off;

% add noise
sigma  = 2; % noise level
y = a + sigma*randn(size(a)); %y = noisy signal
figure(1); subplot(2,2,2);
imagesc(y);
colorbar;
title('noisy image', 'fontsize', 20);
axis('square');
colormap gray;
axis off;

%%
% 迭代参数
maxIter = 10;

psi_p = log(normpdf(y(:), 1, sigma)); 
psi_m = log(normpdf(y(:), -1, sigma));
logodds = psi_p - psi_m;
J = 1;

xpred = y(:);

[M,N] = size(y);
for iter = 1:maxIter
    for ii = 1:N,
        for jj = 1:M,
            pos = jj + M*(ii-1);
            nbr = pos + [-1,1,-M,M];
            nbr(([jj==1,jj==M,ii==1,ii==N])) = [];
            eta = sum(xpred(nbr));
            
            % MLaPP中的公式（24.6），书中有typo
            xpred(pos) = fun_sigm(2*J*eta + logodds(pos))*2-1;
        end;
    end;
    
    if iter == round(maxIter/2),
        xpred=reshape(xpred,M,N);
        figure(1); subplot(2,2,3);
        imagesc(xpred);
        colorbar;
        title(sprintf('image after %d iter', iter), 'fontsize', 20);
        axis('square');
        colormap gray;
        axis off;
    end;

end;

xpred=reshape(xpred,M,N);

figure(1); subplot(2,2,4);
imagesc(xpred);
colorbar;
title(sprintf('image after %d iter', maxIter), 'fontsize', 20);
axis('square');
colormap gray;
axis off;

Gibbs Sampling for Ising model

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