混合高斯背景建模原理及实现

转自：http://blog.csdn.net/jinshengtao/article/details/26278725

一、理论

混合高斯背景建模是基于像素样本统计信息的背景表示方法，利用像素在较长时间内大量样本值的概率密度等统计信息(如模式数量、每个模式的均值和标准差)表示背景，然后使用统计差分(如3σ原则)进行目标像素判断，可以对复杂动态背景进行建模，计算量较大。

在混合高斯背景模型中，认为像素之间的颜色信息互不相关，对各像素点的处理都是相互独立的。对于视频图像中的每一个像素点，其值在序列图像中的变化可看作是不断产生像素值的随机过程，即用高斯分布来描述每个像素点的颜色呈现规律【单模态(单峰)，多模态(多峰)】。

对于多峰高斯分布模型，图像的每一个像素点按不同权值的多个高斯分布的叠加来建模，每种高斯分布对应一个可能产生像素点所呈现颜色的状态，各个高斯分布的权值和分布参数随时间更新。当处理彩色图像时，假定图像像素点R、G、B三色通道相互独立并具有相同的方差。对于随机变量X的观测数据集{x₁,x₂,…,x_N}，x_t=(r_t,g_t,b_t)为t时刻像素的样本，则单个采样点x_t其服从的混合高斯分布概率密度函数：

其中k为分布模式总数，η(x_t,μ_i,t, τ_i,t)为t时刻第i个高斯分布，μ_i,t为其均值，τ_i,t为其协方差矩阵，δ_i,t为方差，I为三维单位矩阵，ω_i,t为t时刻第i个高斯分布的权重。

详细算法流程：

二、代码实现（我还没验证，不知效果）

// my_mixgaussians.cpp : 定义控制台应用程序的入口点。  
//  
 
#include "stdafx.h"  
#include "cv.h"  
#include "highgui.h"  
  
int _tmain(int argc, _TCHAR* argv[])  
{  
    CvCapture *capture=cvCreateFileCapture("test.avi");  
    IplImage *mframe,*current,*frg,*test;  
    int *fg,*bg_bw,*rank_ind;  
    double *w,*mean,*sd,*u_diff,*rank;  
    int C,M,sd_init,i,j,k,m,rand_temp=0,rank_ind_temp=0,min_index=0,x=0,y=0,counter_frame=0;  
    double D,alph,thresh,p,temp;  
    CvRNG state;  
    int match,height,width;  
    mframe=cvQueryFrame(capture);  
  
    frg = cvCreateImage(cvSize(mframe->width,mframe->height),IPL_DEPTH_8U,1);  
    current = cvCreateImage(cvSize(mframe->width,mframe->height),IPL_DEPTH_8U,1);  
    test = cvCreateImage(cvSize(mframe->width,mframe->height),IPL_DEPTH_8U,1);  
      
    C = 4;                      //number of gaussian components (typically 3-5)  
    M = 4;                      //number of background components  
    sd_init = 6;                //initial standard deviation (for new components) var = 36 in paper  
    alph = 0.01;                //learning rate (between 0 and 1) (from paper 0.01)  
    D = 2.5;                    //positive deviation threshold  
    thresh = 0.25;              //foreground threshold (0.25 or 0.75 in paper)  
    p = alph/(1/C);         //initial p variable (used to update mean and sd)  
  
    height=current->height;width=current->widthStep;  
      
    fg = (int *)malloc(sizeof(int)*width*height);                   //foreground array  
    bg_bw = (int *)malloc(sizeof(int)*width*height);                //background array  
    rank = (double *)malloc(sizeof(double)*1*C);                    //rank of components (w/sd)  
    w = (double *)malloc(sizeof(double)*width*height*C);            //weights array  
    mean = (double *)malloc(sizeof(double)*width*height*C);         //pixel means  
    sd = (double *)malloc(sizeof(double)*width*height*C);           //pixel standard deviations  
    u_diff = (double *)malloc(sizeof(double)*width*height*C);       //difference of each pixel from mean  
      
    for (i=0;i<height;i++)  
    {  
        for (j=0;j<width;j++)  
        {  
            for(k=0;k<C;k++)  
            {  
                mean[i*width*C+j*C+k] = cvRandReal(&state)*255;  
                w[i*width*C+j*C+k] = (double)1/C;  
                sd[i*width*C+j*C+k] = sd_init;  
            }  
        }  
    }  
  
    while(1){  
        rank_ind = (int *)malloc(sizeof(int)*C);  
        cvCvtColor(mframe,current,CV_BGR2GRAY);  
        // calculate difference of pixel values from mean  
        for (i=0;i<height;i++)  
        {  
            for (j=0;j<width;j++)  
            {  
                for (m=0;m<C;m++)  
                {  
                    u_diff[i*width*C+j*C+m] = abs((uchar)current->imageData[i*width+j]-mean[i*width*C+j*C+m]);  
                }  
            }  
        }  
        //update gaussian components for each pixel  
        for (i=0;i<height;i++)  
        {  
            for (j=0;j<width;j++)  
            {  
                match = 0;  
                temp = 0;  
                for(k=0;k<C;k++)  
                {  
                    if (abs(u_diff[i*width*C+j*C+k]) <= D*sd[i*width*C+j*C+k])      //pixel matches component  
                    {  
                         match = 1;                                                 // variable to signal component match  
                           
                         //update weights, mean, sd, p  
                         w[i*width*C+j*C+k] = (1-alph)*w[i*width*C+j*C+k] + alph;  
                         p = alph/w[i*width*C+j*C+k];                    
                         mean[i*width*C+j*C+k] = (1-p)*mean[i*width*C+j*C+k] + p*(uchar)current->imageData[i*width+j];  
                         sd[i*width*C+j*C+k] =sqrt((1-p)*(sd[i*width*C+j*C+k]*sd[i*width*C+j*C+k]) + p*(pow((uchar)current->imageData[i*width+j] - mean[i*width*C+j*C+k],2)));  
                    }else{  
                        w[i*width*C+j*C+k] = (1-alph)*w[i*width*C+j*C+k];           // weight slighly decreases  
                    }  
                    temp += w[i*width*C+j*C+k];  
                }  
                  
                for(k=0;k<C;k++)  
                {  
                    w[i*width*C+j*C+k] = w[i*width*C+j*C+k]/temp;  
                }  
              
                temp = w[i*width*C+j*C];  
                bg_bw[i*width+j] = 0;  
                for (k=0;k<C;k++)  
                {  
                    bg_bw[i*width+j] = bg_bw[i*width+j] + mean[i*width*C+j*C+k]*w[i*width*C+j*C+k];  
                    if (w[i*width*C+j*C+k]<=temp)  
                    {  
                        min_index = k;  
                        temp = w[i*width*C+j*C+k];  
                    }  
                    rank_ind[k] = k;  
                }  
  
                test->imageData[i*width+j] = (uchar)bg_bw[i*width+j];  
  
                //if no components match, create new component  
                if (match == 0)  
                {  
                    mean[i*width*C+j*C+min_index] = (uchar)current->imageData[i*width+j];  
                    //printf("%d ",(uchar)bg->imageData[i*width+j]);  
                    sd[i*width*C+j*C+min_index] = sd_init;  
                }  
                for (k=0;k<C;k++)  
                {  
                    rank[k] = w[i*width*C+j*C+k]/sd[i*width*C+j*C+k];  
                    //printf("%f ",w[i*width*C+j*C+k]);  
                }  
  
                //sort rank values  
                for (k=1;k<C;k++)  
                {  
                    for (m=0;m<k;m++)  
                    {  
                        if (rank[k] > rank[m])  
                        {  
                            //swap max values  
                            rand_temp = rank[m];  
                            rank[m] = rank[k];  
                            rank[k] = rand_temp;  
  
                            //swap max index values  
                            rank_ind_temp = rank_ind[m];  
                            rank_ind[m] = rank_ind[k];  
                            rank_ind[k] = rank_ind_temp;  
                        }  
                    }  
                }  
  
                //calculate foreground  
                match = 0;k = 0;  
                //frg->imageData[i*width+j]=0;  
                while ((match == 0)&&(k<M)){  
                    if (w[i*width*C+j*C+rank_ind[k]] >= thresh)  
                        if (abs(u_diff[i*width*C+j*C+rank_ind[k]]) <= D*sd[i*width*C+j*C+rank_ind[k]]){  
                            frg->imageData[i*width+j] = 0;  
                            match = 1;  
                        }  
                        else  
                            frg->imageData[i*width+j] = (uchar)current->imageData[i*width+j];       
                    k = k+1;  
                }  
            }  
        }         
  
        mframe = cvQueryFrame(capture);  
        cvShowImage("fore",frg);  
        cvShowImage("back",test);  
        char s=cvWaitKey(33);  
        if(s==27) break;  
        free(rank_ind);  
    }  
      
    free(fg);free(w);free(mean);free(sd);free(u_diff);free(rank);  
    cvNamedWindow("back",0);  
    cvNamedWindow("fore",0);  
    cvReleaseCapture(&capture);  
    cvDestroyWindow("fore");  
    cvDestroyWindow("back");  
    return 0;  
}  

（原作者）实验结果：

前景：

背景：