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