OPENCV中混合高斯背景模型的实现

看高斯混合模型的论文的时候感觉一头雾水，主要是那些概率公式比较难懂，看了几遍论文和代码才有了一点感觉，以下是个人的对混合高斯背景模型代码的理解，后面会通过进一步理解不断对此贴修正.....
注：本人对该代码的理解也有很多不到位的地方，希望以后会慢慢矫正。程序中代码的实现与论文中公式有些出入，不过不影响对全局的把握和理解。

混合高斯背景模型理论的论文：点击下载

////////////////////////cvCreateGaussianBGModel///////////////////////////////////////////
CV_IMPL CvBGStatModel *cvCreateGaussianBGModel( IplImage*first_frame,CvGaussBGStatModelParams* parameters )
{
CvGaussBGModel* bg_model = 0;

CV_FUNCNAME( "cvCreateGaussianBGModel" );

__BEGIN__;

double var_init;
CvGaussBGStatModelParams params;
int i, j, k, m, n;

// init parameters
if( parameters == NULL )
{
params.win_size = CV_BGFG_MOG_WINDOW_SIZE; // 初始化阶段的帧数；用户自定义模型学 习率a=1/win_size;
params.bg_threshold = CV_BGFG_MOG_BACKGROUND_THRESHOLD;

params.std_threshold = CV_BGFG_MOG_STD_THRESHOLD;

params.weight_init = CV_BGFG_MOG_WEIGHT_INIT;

params.variance_init = CV_BGFG_MOG_SIGMA_INIT*CV_BGFG_MOG_SIGMA_INIT; //方差
params.minArea = CV_BGFG_MOG_MINAREA;
params.n_gauss = CV_BGFG_MOG_NGAUSSIANS; //高斯分布函数的个数
}
else
{
params = *parameters; //用户自定义参数
}

if( !CV_IS_IMAGE(first_frame) )
CV_ERROR( CV_StsBadArg, "Invalid or NULL first_frame parameter" );

CV_CALL( bg_model = (CvGaussBGModel*)cvAlloc( sizeof(*bg_model) ));
memset( bg_model, 0, sizeof(*bg_model) );
bg_model->type = CV_BG_MODEL_MOG; //CV_BG_MODEL_MOG为高斯背景模型
bg_model->release = (CvReleaseBGStatModel)icvReleaseGaussianBGModel;
bg_model->update = (CvUpdateBGStatModel)icvUpdateGaussianBGModel;

bg_model->params = params;

//prepare storages
CV_CALL( bg_model->g_point = (CvGaussBGPoint*)cvAlloc(sizeof(CvGaussBGPoint)*
((first_frame->width*first_frame->height) + 256)));

CV_CALL( bg_model->background = cvCreateImage(cvSize(first_frame->width,
first_frame->height), IPL_DEPTH_8U, first_frame->nChannels));
CV_CALL( bg_model->foreground = cvCreateImage(cvSize(first_frame->width,
first_frame->height), IPL_DEPTH_8U, 1));

CV_CALL( bg_model->storage = cvCreateMemStorage());

//initializing
var_init = 2 * params.std_threshold * params.std_threshold; //初始化方差
CV_CALL( bg_model->g_point[0].g_values =
(CvGaussBGValues*)cvAlloc( sizeof(CvGaussBGValues)*params.n_gauss*
(first_frame->width*first_frame->height + 128)));

for( i = 0, n = 0; i < first_frame->height; i++ ) //行
{
for( j = 0; j < first_frame->width; j++, n++ ) //列
{
const int p = i*first_frame->widthStep+j*first_frame->nChannels;
//以下几步是对第一个高斯函数做初始化
bg_model->g_point
.g_values = bg_model->g_point[0].g_values + n*params.n_gauss;
bg_model->g_point
.g_values[0].weight = 1; //权值赋为1
bg_model->g_point
.g_values[0].match_sum = 1; //高斯函数被匹配的次数
for( m = 0; m < first_frame->nChannels; m++)
{
bg_model->g_point
.g_values[0].variance[m] = var_init;

//均值赋为当前像素的值
bg_model->g_point
.g_values[0].mean[m] = (unsigned char)first_frame->imageData[p + m];
}

//除第一以外的高斯分布函数的初始化(均值、权值和匹配次数都置零)
for( k = 1; k < params.n_gauss; k++)
{
bg_model->g_point
.g_values[k].weight = 0;
bg_model->g_point
.g_values[k].match_sum = 0;
for( m = 0; m < first_frame->nChannels; m++){
bg_model->g_point
.g_values[k].variance[m] = var_init;
bg_model->g_point
.g_values[k].mean[m] = 0;
}
}
}
} //g_point[]:像素，g_values[]:高斯分布函数，mean[]:通道

bg_model->countFrames = 0;

__END__;

if( cvGetErrStatus() < 0 )
{
CvBGStatModel* base_ptr = (CvBGStatModel*)bg_model;

if( bg_model && bg_model->release )
bg_model->release( &base_ptr );
else
cvFree( &bg_model );
bg_model = 0;
}

return (CvBGStatModel*)bg_model;
}

cvUpdateBGStatModel(videoFrame,bgModel);
typedef int (CV_CDECL * CvUpdateBGStatModel)( IplImage* curr_frame, struct CvBGStatModel* bg_model );

/////////////////////////cvUpdateBGStatModel//////////////////////////////////
//函数功能：背景模型的更新，不仅要更新高斯分布函数的参数，还要更新各高斯函数的权重
static int CV_CDECL icvUpdateGaussianBGModel( IplImage* curr_frame, CvGaussBGModel* bg_model )
{
int i, j, k, n;
int region_count = 0;
CvSeq *first_seq = NULL, *prev_seq = NULL, *seq = NULL;

bg_model->countFrames++;

for( i = 0, n = 0; i < curr_frame->height; i++ )
{
for( j = 0; j < curr_frame->width; j++, n++ )
{
int match[CV_BGFG_MOG_MAX_NGAUSSIANS]; //对高斯函数做标记，match[m]=1表示函数m为匹配的高斯分布函数
double sort_key[CV_BGFG_MOG_MAX_NGAUSSIANS]; //此数组存贮每个高斯函数的均值与方差比值
const int nChannels = curr_frame->nChannels;
const int p = curr_frame->widthStep*i+j*nChannels;

CvGaussBGPoint* g_point = &bg_model->g_point
;
const CvGaussBGStatModelParams bg_model_params = bg_model->params;
double pixel[4]; // pixel[]存贮当前像素的各通道RGB值
int no_match;

for( k = 0; k < nChannels; k++ )
pixel[k] = (uchar)curr_frame->imageData[p+k];

no_match = icvMatchTest( pixel, nChannels, match, g_point, &bg_model_params ); //检查是否有与当前像素匹配的高斯函数
if( bg_model->countFrames >= bg_model->params.win_size ) ？？？？？？？？？？？？？
{
icvUpdateFullWindow( pixel, nChannels, match, g_point, &bg_model->params );
if( no_match == -1)
icvUpdateFullNoMatch( curr_frame, p, match, g_point, &bg_model_params );
}
else
{
icvUpdatePartialWindow( pixel, nChannels, match, g_point, &bg_model_params );
if( no_match == -1)
icvUpdatePartialNoMatch( pixel, nChannels, match, g_point, &bg_model_params );
}
icvGetSortKey( nChannels, sort_key, g_point, &bg_model_params );
icvInsertionSortGaussians( g_point, sort_key, (CvGaussBGStatModelParams *)&bg_model_params );
icvBackgroundTest( nChannels, n, i, j, match, bg_model );
}
}

//foreground filtering

//filter small regions
cvClearMemStorage(bg_model->storage);

//cvMorphologyEx( bg_model->foreground, bg_model->foreground, 0, 0, CV_MOP_OPEN, 1 );
//cvMorphologyEx( bg_model->foreground, bg_model->foreground, 0, 0, CV_MOP_CLOSE, 1 );

cvFindContours( bg_model->foreground, bg_model->storage, &first_seq, sizeof(CvContour), CV_RETR_LIST );
for( seq = first_seq; seq; seq = seq->h_next )
{
CvContour* cnt = (CvContour*)seq;
if( cnt->rect.width * cnt->rect.height < bg_model->params.minArea )
{
//delete small contour
prev_seq = seq->h_prev;
if( prev_seq )
{
prev_seq->h_next = seq->h_next;
if( seq->h_next ) seq->h_next->h_prev = prev_seq;
}
else
{
first_seq = seq->h_next;
if( seq->h_next ) seq->h_next->h_prev = NULL;
}
}
else
{
region_count++;
}
}
bg_model->foreground_regions = first_seq;
cvZero(bg_model->foreground);
cvDrawContours(bg_model->foreground, first_seq, CV_RGB(0, 0, 255), CV_RGB(0, 0, 255), 10, -1);

return region_count;
}

/////////////////////////////////////icvMatchTest//////////////////////////////////////////
//函数功能：拿当前像素的值与已存在的高斯分布函数比较，查找是否存在匹配的的高斯分布函数，如果有则返回 k值(高斯分布函数的序号)
static int icvMatchTest( double* src_pixel, int nChannels, int* match,
const CvGaussBGPoint* g_point,
const CvGaussBGStatModelParams *bg_model_params )
{ //参数的传递：src_pixel为piexl[]:即当前像素的各通道值
int k;
int matchPosition=-1;
for ( k = 0; k < bg_model_params->n_gauss; k++)
match[k]=0;

for ( k = 0; k < bg_model_params->n_gauss; k++)
if (g_point->g_values[k].match_sum > 0)
{
double sum_d2 = 0.0;
double var_threshold = 0.0;
for(int m = 0; m < nChannels; m++)
{
double d = g_point->g_values[k].mean[m]- src_pixel[m]; //通道m的原始模型值与当前像素的值之差
sum_d2 += (d*d);
var_threshold += g_point->g_values[k].variance[m];
}

//当前sum_d2为d0,d1,d2的平方和，var_threshold的值为像素各通道方差之和
var_threshold = bg_model_params->std_threshold*

bg_model_params- >std_threshold*var_threshold;
if(sum_d2 < var_threshold) //查看是否可以与某高斯分布匹配 ？？？？？？？？？？？？？？？？
{
match[k] = 1;
matchPosition = k;
break; //如果和第k个高斯函数匹配，则终止与后续函数的匹配
}
}

return matchPosition;
}

///////////////////////icvUpdateFullWindow////////////////////////////////////////
//函数功能：更新每个高斯分布的权值(对匹配的高斯函数k加大权值，其余的则减小权值)，如果前面的结果中存在匹配的高斯分布函数k,则需要再对第k个高斯分布函数的均值mean和方差variance做修正
static void icvUpdateFullWindow( double* src_pixel, int nChannels, int* match,
CvGaussBGPoint* g_point,
const CvGaussBGStatModelParams *bg_model_params )
{ //参数的传递：src_pixel为piexl[]:即当前帧中该像素的RGB值
const double learning_rate_weight = (1.0/(double)bg_model_params->win_size); //用户自定义模型学习率a

for(int k = 0; k < bg_model_params->n_gauss; k++)
{
//对每个高斯分布的权值做修正：w=(1-a)w+a*m (a:模型学习率，m是匹配，匹配就是1，不匹配就是0)
g_point->g_values[k].weight = g_point->g_values[k].weight +
(learning_rate_weight*((double)match[k] -g_point->g_values[k].weight));

if(match[k]) //如果存在匹配的高斯分布函数k（当前像素为背景像素）,则需要再对第k个高斯分布函数的均值mean和方差variance更新
{
double learning_rate_gaussian = (double)match[k]/(g_point->g_values[k].weight*
(double)bg_model_params->win_size); //参数学习率p(p=a/w)
for(int m = 0; m < nChannels; m++)
{ //参数更新公式：u=(1-p)*u0+p*x; o*o=(1-p)*o*o+p*tmpDiff*tmpDiff
const double tmpDiff = src_pixel[m] - g_point->g_values[k].mean[m]; //当前像素的通道m的值与原始模型值之差
g_point->g_values[k].mean[m] = g_point->g_values[k].mean[m] + (learning_rate_gaussian * tmpDiff);
g_point->g_values[k].variance[m] = g_point->g_values[k].variance[m]+
(learning_rate_gaussian*((tmpDiff*tmpDiff) - g_point->g_values[k].variance[m]));
}
}
}
}

/////////////////////////icvUpdatePartialWindow/////////////////////////////
//函数功能：对所有的高斯分布函数做更新.至少每个高斯分布的权值必须修正，如果前面的结果中存在匹配的高斯分布函数k,则需要再对第k个高斯分布函数的match_sum修改，最终对那些匹配的高斯分布函数k的参数match_sum>0的做均值mean和方差variance修正
static void icvUpdatePartialWindow( double* src_pixel, int nChannels, int* match, CvGaussBGPoint* g_point, const CvGaussBGStatModelParams *bg_model_params )
{
int k, m;
int window_current = 0;

for( k = 0; k < bg_model_params->n_gauss; k++ )
window_current += g_point->g_values[k].match_sum; //window_current为k个高斯分布函数的match_sum值之和

for( k = 0; k < bg_model_params->n_gauss; k++ )
{
g_point->g_values[k].match_sum += match[k]; //修正匹配的高斯分布函数k的match_sum值
double learning_rate_weight = (1.0/((double)window_current + 1.0)); //increased by one since sum
//修正每个高斯分布的权值
g_point->g_values[k].weight = g_point->g_values[k].weight +
(learning_rate_weight*((double)match[k] - g_point->g_values[k].weight));

if( g_point->g_values[k].match_sum > 0 && match[k] )
{
double learning_rate_gaussian = (double)match[k]/((double)g_point->g_values[k].match_sum);
for( m = 0; m < nChannels; m++ )
{
const double tmpDiff = src_pixel[m] - g_point->g_values[k].mean[m];
g_point->g_values[k].mean[m] = g_point->g_values[k].mean[m] +
(learning_rate_gaussian*tmpDiff);
g_point->g_values[k].variance[m] = g_point->g_values[k].variance[m]+
(learning_rate_gaussian*((tmpDiff*tmpDiff) - g_point->g_values[k].variance[m]));
}
}
}
}

//////////////////////////icvUpdateFullNoMatch//////////////////////////
//函数功能：当所有的高斯函数均不匹配时，说明有新的分布出现，需要将原高斯函数中sort_key最小的替换为新的高斯函数(权值小，方差大),其余的高斯函数对应的只需更新权值
static void icvUpdateFullNoMatch( IplImage* gm_image, int p, int* match,
CvGaussBGPoint* g_point,
const CvGaussBGStatModelParams *bg_model_params)
{ //参数的传递：gm_image为当前帧curr_frame
int k, m;
double alpha;
int match_sum_total = 0;

g_point->g_values[bg_model_params->n_gauss - 1].match_sum = 1; //将新的高斯分布函数的match_sum置为1

for( k = 0; k < bg_model_params->n_gauss ; k++ )
match_sum_total += g_point->g_values[k].match_sum;

g_point->g_values[bg_model_params->n_gauss - 1].weight = 1./(double)match_sum_total; //要给新的高斯分布函数赋一个较小的权值

//将新的高斯分布函数的variance[m]全部置为variance_init；mean[m]的值置为当前像素各通道的值
for( m = 0; m < gm_image->nChannels ; m++ )
{
g_point->g_values[bg_model_params->n_gauss - 1].variance[m] = bg_model_params->variance_init;
g_point->g_values[bg_model_params->n_gauss - 1].mean[m] = (unsigned char)gm_image->imageData[p + m];
}

//对其他的高斯分布函数做权值更新：w=(1-a)*w+a*m (a:模型学习率，m是匹配，匹配就是1，不匹配就是0)
alpha = 1.0 - (1.0/bg_model_params->win_size); //alpha=1-a;
for( k = 0; k < bg_model_params->n_gauss - 1; k++ )
{
g_point->g_values[k].weight *= alpha;
if( match[k] )
g_point->g_values[k].weight += alpha;
}
}

////////////////////////////icvUpdatePartialNoMatch////////////////////////////////
static void
icvUpdatePartialNoMatch(double *pixel,
int nChannels,
int* /*match*/,
CvGaussBGPoint* g_point,
const CvGaussBGStatModelParams *bg_model_params)
{
int k, m;
//new value of last one
g_point->g_values[bg_model_params->n_gauss - 1].match_sum = 1;

//get sum of all but last value of match_sum
int match_sum_total = 0;
for(k = 0; k < bg_model_params->n_gauss ; k++)
match_sum_total += g_point->g_values[k].match_sum;

for(m = 0; m < nChannels; m++)
{
//first pass mean is image value
g_point->g_values[bg_model_params->n_gauss - 1].variance[m] = bg_model_params->variance_init;
g_point->g_values[bg_model_params->n_gauss - 1].mean[m] = pixel[m];
}
for(k = 0; k < bg_model_params->n_gauss; k++)
{ //更新所有高斯分布的权值
g_point->g_values[k].weight = (double)g_point->g_values[k].match_sum /
(double)match_sum_total;
}
}

/////////////////////////////////icvGetSortKey///////////////////////////////////
//函数功能：计算各个高斯分布weight/sqrt(variance_sum)的值，后面将对该值进行排序（该值越大则表示背景的可能性就越大）
static void icvGetSortKey( const int nChannels, double* sort_key, const CvGaussBGPoint* g_point,
const CvGaussBGStatModelParams *bg_model_params )
{
int k, m;
for( k = 0; k < bg_model_params->n_gauss; k++ )
{
// Avoid division by zero
if( g_point->g_values[k].match_sum > 0 )
{
// Independence assumption between components
double variance_sum = 0.0;
for( m = 0; m < nChannels; m++ )
variance_sum += g_point->g_values[k].variance[m];

sort_key[k] = g_point->g_values[k].weight/sqrt(variance_sum); //sort_key=w/(o*o)
}
else
sort_key[k]= 0.0;
}
}

//////////////////////////////icvInsertionSortGaussians////////////////////////////
static void icvInsertionSortGaussians( CvGaussBGPoint* g_point, double* sort_key, CvGaussBGStatModelParams *bg_model_params )
{
int i, j;
for( i = 1; i < bg_model_params->n_gauss; i++ )
{
double index = sort_key[i];
for( j = i; j > 0 && sort_key[j-1] < index; j-- ) //对sort_key[]按降序排序
{
double temp_sort_key = sort_key[j];
sort_key[j] = sort_key[j-1];
sort_key[j-1] = temp_sort_key;

CvGaussBGValues temp_gauss_values = g_point->g_values[j];
g_point->g_values[j] = g_point->g_values[j-1];
g_point->g_values[j-1] = temp_gauss_values;
}
// sort_key[j] = index;
}
}

///////////////////////////////////icvBackgroundTest/////////////////////////
static void icvBackgroundTest( const int nChannels, int n, int i, int j, int *match, CvGaussBGModel* bg_model )
{
int m, b;
uchar pixelValue = (uchar)255; // 像素默认都为前景
double weight_sum = 0.0;
CvGaussBGPoint* g_point = bg_model->g_point;

for( m = 0; m < nChannels; m++)？？？？？？？？？？？？？
bg_model->background->imageData[ bg_model->background->widthStep*i + j*nChannels + m] = (unsigned char)(g_point
.g_values[0].mean[m]+0.5);

for( b = 0; b < bg_model->params.n_gauss; b++)
{
weight_sum += g_point
.g_values[b].weight;
if( match[b] )
pixelValue = 0; //if为真，说明该像素已与某高斯函数匹配，该像素为背景
if( weight_sum > bg_model->params.bg_threshold )
break; //如果if语句为真，则前b个高斯分布被选为描述背景的函数
}

bg_model->foreground->imageData[ bg_model->foreground->widthStep*i + j] = pixelValue;
}