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