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。
/****************************************************************************************\ * 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