OpenCV（3）ML库->Expectation - Maximization 算法

最大期望算法（Expectation-maximization algorithm，又译期望最大化算法）在统计中被用于寻找，依赖于不可观察的隐性变量的概率模型中，参数的最大似然估计。

在统计计算中，最大期望（EM）算法是在概率（probabilistic）模型中寻找参数最大似然估计或者最大后验估计的算法，其中概率模型依赖于无法观测的隐藏变量（Latent Variable）。最大期望经常用在机器学习和计算机视觉的数据聚类（Data Clustering）领域。
最大期望算法经过两个步骤交替进行计算：
第一步是计算期望（E），利用对隐藏变量的现有估计值，计算其最大似然估计值；
第二步是最大化（M），最大化在 E 步上求得的最大似然值来计算参数的值。
M 步上找到的参数估计值被用于下一个 E 步计算中，这个过程不断交替进行。
总体来说，EM的算法流程如下：
1.初始化分布参数
2.重复直到收敛：
E步骤：估计未知参数的期望值，给出当前的参数估计。
M步骤：重新估计分布参数，以使得数据的似然性最大，给出未知变量的期望估计。

/****************************************************************************************\
*                              Expectation - Maximization                                *
\****************************************************************************************/

struct CV_EXPORTS CvEMParams   //参数设定 EM算法估计混合高斯模型所需要的参数
{
CvEMParams() : nclusters(10), cov_mat_type(1/*CvEM::COV_MAT_DIAGONAL*/),
start_step(0/*CvEM::START_AUTO_STEP*/), probs(0), weights(0), means(0), covs(0)
{
term_crit=cvTermCriteria( CV_TERMCRIT_ITER+CV_TERMCRIT_EPS, 100, FLT_EPSILON );
}

CvEMParams( int _nclusters, int _cov_mat_type=1/*CvEM::COV_MAT_DIAGONAL*/,
int _start_step=0/*CvEM::START_AUTO_STEP*/,
CvTermCriteria _term_crit=cvTermCriteria(CV_TERMCRIT_ITER+CV_TERMCRIT_EPS, 100, FLT_EPSILON),
const CvMat* _probs=0, const CvMat* _weights=0, const CvMat* _means=0, const CvMat** _covs=0 ) :
nclusters(_nclusters), cov_mat_type(_cov_mat_type), start_step(_start_step),
probs(_probs), weights(_weights), means(_means), covs(_covs), term_crit(_term_crit)
{}

int nclusters;
int cov_mat_type;
int start_step;
const CvMat* probs; //初始的后验概率
const CvMat* weights; //初始的各个成分的概率
const CvMat* means;   //初始的均值
const CvMat** covs;  //初始的协方差矩阵
CvTermCriteria term_crit; //E步和M步 迭代停止的准则。 EM算法会在一定的迭代次数之后（term_crit.num_iter)，或者当模型参数在两次迭代之间的变化小于预定值（term_crit.epsilon）时停止
};

class CV_EXPORTS CvEM : public CvStatModel //模型设置
{
public:
// Type of covariation matrices
enum { COV_MAT_SPHERICAL=0, COV_MAT_DIAGONAL=1, COV_MAT_GENERIC=2 };   //

// The initial step
enum { START_E_STEP=1, START_M_STEP=2, START_AUTO_STEP=0 };

CvEM();
CvEM( const CvMat* samples, const CvMat* sample_idx=0,
CvEMParams params=CvEMParams(), CvMat* labels=0 );
//CvEM (CvEMParams params, CvMat * means, CvMat ** covs, CvMat * weights, CvMat * probs, CvMat * log_weight_div_det, CvMat * inv_eigen_values, CvMat** cov_rotate_mats);

virtual ~CvEM();

virtual bool train( const CvMat* samples, const CvMat* sample_idx=0,
CvEMParams params=CvEMParams(), CvMat* labels=0 );

virtual float predict( const CvMat* sample, CvMat* probs ) const;

#ifndef SWIG
CvEM( const cv::Mat& samples, const cv::Mat& sample_idx=cv::Mat(),
CvEMParams params=CvEMParams(), cv::Mat* labels=0 );

virtual bool train( const cv::Mat& samples, const cv::Mat& sample_idx=cv::Mat(),
CvEMParams params=CvEMParams(), cv::Mat* labels=0 );

virtual float predict( const cv::Mat& sample, cv::Mat* probs ) const;
#endif

virtual void clear();

int           get_nclusters() const;
const CvMat*  get_means()     const;
const CvMat** get_covs()      const;
const CvMat*  get_weights()   const;
const CvMat*  get_probs()     const;

inline double         get_log_likelihood     () const { return log_likelihood;     };

//    inline const CvMat *  get_log_weight_div_det () const { return log_weight_div_det; };
//    inline const CvMat *  get_inv_eigen_values   () const { return inv_eigen_values;   };
//    inline const CvMat ** get_cov_rotate_mats    () const { return cov_rotate_mats;    };

protected:

virtual void set_params( const CvEMParams& params,
const CvVectors& train_data );
virtual void init_em( const CvVectors& train_data );
virtual double run_em( const CvVectors& train_data );
virtual void init_auto( const CvVectors& samples );
virtual void kmeans( const CvVectors& train_data, int nclusters,
CvMat* labels, CvTermCriteria criteria,
const CvMat* means );
CvEMParams params;
double log_likelihood;

CvMat* means;
CvMat** covs;
CvMat* weights;
CvMat* probs;

CvMat* log_weight_div_det;
CvMat* inv_eigen_values;
CvMat** cov_rotate_mats;
};

循环重复直到收敛 {

（E步）对于每一个i，计算





（M步）计算

#include "stdafx.h"
#include <ml.h>
#include <iostream>
#include <highgui.h>
#include <cv.h>
#include <cxcore.h>
using namespace cv;
using namespace std;

int main( int argc, char** argv )
{
const int N = 4;
const int N1 = (int)sqrt((double)N);
const CvScalar colors[] = {{{0,0,255}},{{0,255,0}},{{0,255,255}},{{255,255,0}}};
int i, j;
int nsamples = 100;
CvRNG rng_state = cvRNG(-1);
CvMat* samples = cvCreateMat( nsamples, 2, CV_32FC1 );
CvMat* labels = cvCreateMat( nsamples, 1, CV_32SC1 );
IplImage* img = cvCreateImage( cvSize( 500, 500 ), 8, 3 );
float _sample[2];
CvMat sample = cvMat( 1, 2, CV_32FC1, _sample );

//EM算法初始化
CvEM em_model;
CvEMParams params;

CvMat samples_part;

cvReshape( samples, samples, 2, 0 );
for( i = 0; i < N; i++ )
{
CvScalar mean, sigma;

// form the training samples
cvGetRows( samples, &samples_part, i*nsamples/N, (i+1)*nsamples/N );
mean = cvScalar(((i%N1)+1.)*img->width/(N1+1), ((i/N1)+1.)*img->height/(N1+1));
sigma = cvScalar(30,30);
cvRandArr( &rng_state, &samples_part, CV_RAND_NORMAL, mean, sigma );
}
cvReshape( samples, samples, 1, 0 );

// initialize model's parameters
params.covs      = NULL;
params.means     = NULL;
params.weights   = NULL;
params.probs     = NULL;
params.nclusters = N;
params.cov_mat_type       = CvEM::COV_MAT_SPHERICAL;
params.start_step         = CvEM::START_AUTO_STEP;
params.term_crit.max_iter = 10;
params.term_crit.epsilon  = 0.1;
params.term_crit.type     = CV_TERMCRIT_ITER|CV_TERMCRIT_EPS;

// cluster the data
em_model.train( samples, 0, params, labels );

#if 0
// the piece of code shows how to repeatedly optimize the model
// with less-constrained parameters (COV_MAT_DIAGONAL instead of COV_MAT_SPHERICAL)
// when the output of the first stage is used as input for the second.
CvEM em_model2;
params.cov_mat_type = CvEM::COV_MAT_DIAGONAL;
params.start_step = CvEM::START_E_STEP;
params.means = em_model.get_means();
params.covs = (const CvMat**)em_model.get_covs();
params.weights = em_model.get_weights();

em_model2.train( samples, 0, params, labels );
// to use em_model2, replace em_model.predict() with em_model2.predict() below
#endif
// classify every image pixel
cvZero( img );
for( i = 0; i < img->height; i++ )
{
for( j = 0; j < img->width; j++ )
{
CvPoint pt = cvPoint(j, i);
sample.data.fl[0] = (float)j;
sample.data.fl[1] = (float)i;
int response = cvRound(em_model.predict( &sample, NULL ));
CvScalar c = colors[response];

cvCircle( img, pt, 1, cvScalar(c.val[0]*0.75,c.val[1]*0.75,c.val[2]*0.75), CV_FILLED );
}
}

//draw the clustered samples
for( i = 0; i < nsamples; i++ )
{
CvPoint pt;
pt.x = cvRound(samples->data.fl[i*2]);
pt.y = cvRound(samples->data.fl[i*2+1]);
cvCircle( img, pt, 1, colors[labels->data.i[i]], CV_FILLED );
}

cvNamedWindow( "EM-clustering result", 1 );
cvShowImage( "EM-clustering result", img );
cvWaitKey(0);

cvReleaseMat( &samples );
cvReleaseMat( &labels );
return 0;
}

参考：http://www.cnblogs.com/jerrylead/archive/2011/04/06/2006936.html

http://zhidao.baidu.com/link?url=12xrCFpWm1U-bYb4V8uxf3uu2ZDTFlwpDzbWe7HjOrNWXdsCQTlA466N78ZUDWP-jFAcVsTQo9JyKW28o86ng_

http://www.360doc.com/content/13/0624/13/10942270_295158557.shtml

http://fuliang.iteye.com/blog/1621633

http://wiki.opencv.org.cn/index.php/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0%E4%B8%AD%E6%96%87%E5%8F%82%E8%80%83%E6%89%8B%E5%86%8C#CvEMParams

http://www.seas.upenn.edu/~bensapp/opencvdocs/ref/opencvref_ml.htm#ch_em

http://hi.baidu.com/darkhorse/item/cc58043eb19800159dc65e70