学习OpenCV2——卡尔曼滤波(KalmanFilter)详解

        本文将简要回顾一下卡尔曼滤波理论,然后详细介绍如何在OpenCV中使用卡尔曼滤波进行跟踪,最后给两个程序实例。

1. 卡尔曼滤波理论回顾

      对于一个动态系统,我们首先定义一组状态空间方程

     状态方程:     

     测量方程:      

        xk是状态向量,zk是测量向量,Ak是状态转移矩阵,uk是控制向量,Bk是控制矩阵,wk是系统误差(噪声),Hk是测量矩阵,vk是测量误差(噪声)。wk和vk都是高斯噪声,即

                             

    整个卡尔曼滤波的过程就是个递推计算的过程,不断的“预测——更新——预测——更新……”

预测

     预测状态值:              

     预测最小均方误差:   

更新

    测量误差:                   

    测量协方差:                

    最优卡尔曼增益:         

    修正状态值:                

    修正最小均方误差:     


2.OpenCV中的KalmanFilter详解

OpenCV中有两个版本的卡尔曼滤波方法KalmanFilter(C++)和CvKalman(C),用法差不太多,这里只介绍KalmanFilter。

C++版本中将KalmanFilter封装到一个类中,其结构如下所示:

class CV_EXPORTS_W KalmanFilter
{
public:    
    CV_WRAP KalmanFilter();                                                                           //构造默认KalmanFilter对象
    CV_WRAP KalmanFilter(int dynamParams, int measureParams, int controlParams=0, int type=CV_32F);  //完整构造KalmanFilter对象方法
    void init(int dynamParams, int measureParams, int controlParams=0, int type=CV_32F);              //初始化KalmanFilter对象,会替换原来的KF对象
  
    CV_WRAP const Mat& predict(const Mat& control=Mat());           //计算预测的状态值    
    CV_WRAP const Mat& correct(const Mat& measurement);             //根据测量值更新状态值

    Mat statePre;            //预测值 (x'(k)): x(k)=A*x(k-1)+B*u(k)
    Mat statePost;           //状态值 (x(k)): x(k)=x'(k)+K(k)*(z(k)-H*x'(k))
    Mat transitionMatrix;    //状态转移矩阵 (A)
    Mat controlMatrix;       //控制矩阵 B 
    Mat measurementMatrix;   //测量矩阵 H
    Mat processNoiseCov;     //系统误差 Q
    Mat measurementNoiseCov; //测量误差 R
    Mat errorCovPre;         //最小均方误差 (P'(k)): P'(k)=A*P(k-1)*At + Q)
    Mat gain;                //卡尔曼增益   (K(k)): K(k)=P'(k)*Ht*inv(H*P'(k)*Ht+R)
    Mat errorCovPost;        //修正的最小均方误差 (P(k)): P(k)=(I-K(k)*H)*P'(k)

    // 临时矩阵
    Mat temp1;
    Mat temp2;
    Mat temp3;
    Mat temp4;
    Mat temp5;
};

enum
{
    OPTFLOW_USE_INITIAL_FLOW = CV_LKFLOW_INITIAL_GUESSES,
    OPTFLOW_LK_GET_MIN_EIGENVALS = CV_LKFLOW_GET_MIN_EIGENVALS,
    OPTFLOW_FARNEBACK_GAUSSIAN = 256
};


       函数原型见:…..\OpenCV2\sources\modules\ocl\src\kalman.cpp

       只有四个方法: 构造KF对象KalmanFilter(DP,MP,CP)、初始化KF对象init(DP,MP,CP)、预测predict( )、更新correct( )。除非你要重新构造KF对象,否则用不到init( )。

KalmanFilter(DP,MP,CP)和init( )就是赋值,没什么好说的。

      注意:KalmanFilter结构体中并没有测量值,测量值需要自己定义,而且一定要定义,因为后面要用。


编程步骤

step1:定义KalmanFilter类并初始化

    //构造KF对象

    KalmanFilter KF(DP, MP, 0);

    //初始化相关参数

    KF.transitionMatrix                         转移矩阵 A

    KF.measurementMatrix                  测量矩阵    H

    KF.processNoiseCov                     过程噪声 Q

    KF.measurementNoiseCov            测量噪声        R

    KF.errorCovPost                            最小均方误差 P

    KF.statePost                                系统初始状态 x(0) 

    Mat measurement                          定义初始测量值 z(0) 

step2:预测

    KF.predict( )                                                 //返回的是下一时刻的状态值KF.statePost (k+1) 

step3:更新

    更新measurement;                                     //注意measurement不能通过观测方程进行计算得到,要自己定义!

    更新KF   KF.correct(measurement)

最终的结果应该是更新后的statePost.


相关参数的确定

    对于系统状态方程,简记为Y=AX+B,X和Y是表示系统状态的列向量,A是转移矩阵,B是其他项。

    状态值(向量)只要能表示系统的状态即可,状态值的维数决定了转移矩阵A的维数,比如X和Y是N×1的,则A是N×N的。

    A的确定跟X有关,只要保证方程中不相干项的系数为0即可,看下面例子

      X和Y是二维的,

       X和Y是三维的,

学习OpenCV2——卡尔曼滤波(KalmanFilter)详解_第1张图片

          X和Y是三维的,但c和△ c是相关项

学习OpenCV2——卡尔曼滤波(KalmanFilter)详解_第2张图片

学习OpenCV2——卡尔曼滤波(KalmanFilter)详解_第3张图片

      上面的1也可以是其他值。



下面对predict( ) 和correct( )函数介绍下,可以不用看,不影响编程。

CV_EXPORTS const oclMat& KalmanFilter::predict(const oclMat& control)
{
    gemm(transitionMatrix, statePost, 1, oclMat(), 0, statePre);
    oclMat temp;

    if(control.data)
        gemm(controlMatrix, control, 1, statePre, 1, statePre);
    gemm(transitionMatrix, errorCovPost, 1, oclMat(), 0, temp1);
    gemm(temp1, transitionMatrix, 1, processNoiseCov, 1, errorCovPre, GEMM_2_T);
    statePre.copyTo(statePost);
    return statePre;
}

gemm( )是矩阵的广义乘法

void gemm(const GpuMat& src1, constGpuMat& src2, double alpha, const GpuMat& src3, double beta,GpuMat& dst, int flags=0, Stream& stream=Stream::Null())

    dst = alpha · src1 · src2 +beta· src3

   上面,oclMat()其实是uk,只不过默认为0,所以没赋值。整个过程就计算了x'和P’。(用x'代表x的预测值,用P'代表P的预测值)。GEMM_2_T表示对第2个参数转置。

x’(k)=1·A·x(k-1)

如果B非空, x'(k) = 1·B·u + 1·x'(k-1)

temp1 = 1·A·P(k-1) + 0·u(k)

P’(k) = 1· temp1·AT + 1· Qk= A·P(k-1)·AT + 1· Qk

       可见,和第一部分的理论介绍完全一致。

CV_EXPORTS const oclMat& KalmanFilter::correct(const oclMat& measurement)
{
    CV_Assert(measurement.empty() == false);
    gemm(measurementMatrix, errorCovPre, 1, oclMat(), 0, temp2);
    gemm(temp2, measurementMatrix, 1, measurementNoiseCov, 1, temp3, GEMM_2_T);
    Mat temp;
    solve(Mat(temp3), Mat(temp2), temp, DECOMP_SVD);
    temp4.upload(temp);
    gain = temp4.t();
    gemm(measurementMatrix, statePre, -1, measurement, 1, temp5);
    gemm(gain, temp5, 1, statePre, 1, statePost);
    gemm(gain, temp2, -1, errorCovPre, 1, errorCovPost);
    return statePost;
}
bool solve(InputArray src1, InputArray src2, OutputArray dst, int flags=DECOMP_LU)

求解线型最小二乘估计



temp2 = 1· H·P’ + 0·u(k)

temp3 = 1· temp2·HT + 1·R = H·P’·HT+ 1· R   也就是上面的Sk

temp = argmin||tem2- temp3||

K=temp

temp5 = -1· H·x’ + 1·zk        就是上面的y’。

x = 1·K·temp5 + 1·x’ = KT·y’ +x’

P =-1·K·temp2 + 1·P’ = -K·H·P’+P’ = (I- K·H) P’

也和第一部分的理论完全一致。


通过深入函数内部,学到了两个实用的函数哦。矩阵广义乘法gemm( )、最小二乘估计solve( )


补充

1)以例2为例,为什么状态值一般都设置成(x,y,△x,△y)?我们不妨设置成(x,y,△x),对应的转移矩阵也改成3×3的。可以看到仍能跟上,不过在x方向跟踪速度快,在y方向跟踪速度慢。进一步设置成(x,y)和2×2的转移矩阵,程序的跟踪速度简直是龟速。所以,简单理解,△x和△y严重影响对应方向上的跟踪速度。



3.实例

例1 OpenCV自带的示例程序

#include "opencv2/video/tracking.hpp"
#include "opencv2/highgui/highgui.hpp"
#include 
#include 
using namespace std;
using namespace cv;

//计算相对窗口的坐标值,因为坐标原点在左上角,所以sin前有个负号
static inline Point calcPoint(Point2f center, double R, double angle)
{
    return center + Point2f((float)cos(angle), (float)-sin(angle))*(float)R;
}

static void help()
{
    printf( "\nExamle of c calls to OpenCV's Kalman filter.\n"
"   Tracking of rotating point.\n"
"   Rotation speed is constant.\n"
"   Both state and measurements vectors are 1D (a point angle),\n"
"   Measurement is the real point angle + gaussian noise.\n"
"   The real and the estimated points are connected with yellow line segment,\n"
"   the real and the measured points are connected with red line segment.\n"
"   (if Kalman filter works correctly,\n"
"    the yellow segment should be shorter than the red one).\n"
            "\n"
"   Pressing any key (except ESC) will reset the tracking with a different speed.\n"
"   Pressing ESC will stop the program.\n"
            );
}

int main(int, char**)
{
    help();
    Mat img(500, 500, CV_8UC3);
    KalmanFilter KF(2, 1, 0);                                    //创建卡尔曼滤波器对象KF
    Mat state(2, 1, CV_32F);                                     //state(角度,△角度)
    Mat processNoise(2, 1, CV_32F);
    Mat measurement = Mat::zeros(1, 1, CV_32F);                 //定义测量值
    char code = (char)-1;

    for(;;)
    {
		//1.初始化
        randn( state, Scalar::all(0), Scalar::all(0.1) );          //
        KF.transitionMatrix = *(Mat_(2, 2) << 1, 1, 0, 1);  //转移矩阵A[1,1;0,1]    
		

		//将下面几个矩阵设置为对角阵
        setIdentity(KF.measurementMatrix);                             //测量矩阵H
        setIdentity(KF.processNoiseCov, Scalar::all(1e-5));            //系统噪声方差矩阵Q
        setIdentity(KF.measurementNoiseCov, Scalar::all(1e-1));        //测量噪声方差矩阵R
        setIdentity(KF.errorCovPost, Scalar::all(1));                  //后验错误估计协方差矩阵P

        randn(KF.statePost, Scalar::all(0), Scalar::all(0.1));          //x(0)初始化
		
        for(;;)
        {
            Point2f center(img.cols*0.5f, img.rows*0.5f);          //center图像中心点
            float R = img.cols/3.f;                                //半径
            double stateAngle = state.at(0);                //跟踪点角度
            Point statePt = calcPoint(center, R, stateAngle);     //跟踪点坐标statePt

			//2. 预测
            Mat prediction = KF.predict();                       //计算预测值,返回x'
            double predictAngle = prediction.at(0);          //预测点的角度
            Point predictPt = calcPoint(center, R, predictAngle);   //预测点坐标predictPt


			//3.更新
			//measurement是测量值
            randn( measurement, Scalar::all(0), Scalar::all(KF.measurementNoiseCov.at(0)));     //给measurement赋值N(0,R)的随机值

            // generate measurement
            measurement += KF.measurementMatrix*state;  //z = z + H*x;
			
            double measAngle = measurement.at(0);
            Point measPt = calcPoint(center, R, measAngle);

            // plot points
			//定义了画十字的方法,值得学习下
            #define drawCross( center, color, d )                                 \
                line( img, Point( center.x - d, center.y - d ),                \
                             Point( center.x + d, center.y + d ), color, 1, CV_AA, 0); \
                line( img, Point( center.x + d, center.y - d ),                \
                             Point( center.x - d, center.y + d ), color, 1, CV_AA, 0 )

            img = Scalar::all(0);
            drawCross( statePt, Scalar(255,255,255), 3 );
            drawCross( measPt, Scalar(0,0,255), 3 );
            drawCross( predictPt, Scalar(0,255,0), 3 );
            line( img, statePt, measPt, Scalar(0,0,255), 3, CV_AA, 0 );
            line( img, statePt, predictPt, Scalar(0,255,255), 3, CV_AA, 0 );


			//调用kalman这个类的correct方法得到加入观察值校正后的状态变量值矩阵
			if(theRNG().uniform(0,4) != 0)
                KF.correct(measurement);

			//不加噪声的话就是匀速圆周运动,加了点噪声类似匀速圆周运动,因为噪声的原因,运动方向可能会改变
            randn( processNoise, Scalar(0), Scalar::all(sqrt(KF.processNoiseCov.at(0, 0))));   //vk
            state = KF.transitionMatrix*state + processNoise;   

            imshow( "Kalman", img );
            code = (char)waitKey(100);

            if( code > 0 )
                break;
        }
        if( code == 27 || code == 'q' || code == 'Q' )
            break;
    }

    return 0;
}
程序结果
学习OpenCV2——卡尔曼滤波(KalmanFilter)详解_第4张图片


例2  跟踪鼠标位置

在我介绍粒子滤波的博文“学习Opencv2——粒子滤波Condensation算法”里,有个例3,是跟踪鼠标位置。现在我们用卡尔曼滤波来实现。

#include "opencv2/video/tracking.hpp"
#include "opencv2/highgui/highgui.hpp"
#include 
using namespace cv;
using namespace std;

const int winHeight=600;
const int winWidth=800;


Point mousePosition= Point(winWidth>>1,winHeight>>1);

//mouse event callback
void mouseEvent(int event, int x, int y, int flags, void *param )
{
	if (event==CV_EVENT_MOUSEMOVE) {
		mousePosition = Point(x,y);
	}
}

int main (void)
{
	RNG rng;
	//1.kalman filter setup
	const int stateNum=4;                                      //状态值4×1向量(x,y,△x,△y)
	const int measureNum=2;                                    //测量值2×1向量(x,y)	
	KalmanFilter KF(stateNum, measureNum, 0);	

	KF.transitionMatrix = *(Mat_(4, 4) <<1,0,1,0,0,1,0,1,0,0,1,0,0,0,0,1);  //转移矩阵A
	setIdentity(KF.measurementMatrix);                                             //测量矩阵H
	setIdentity(KF.processNoiseCov, Scalar::all(1e-5));                            //系统噪声方差矩阵Q
	setIdentity(KF.measurementNoiseCov, Scalar::all(1e-1));                        //测量噪声方差矩阵R
	setIdentity(KF.errorCovPost, Scalar::all(1));                                  //后验错误估计协方差矩阵P
	rng.fill(KF.statePost,RNG::UNIFORM,0,winHeight>winWidth?winWidth:winHeight);   //初始状态值x(0)
	Mat measurement = Mat::zeros(measureNum, 1, CV_32F);                           //初始测量值x'(0),因为后面要更新这个值,所以必须先定义
	
	namedWindow("kalman");
	setMouseCallback("kalman",mouseEvent);
		
	Mat image(winHeight,winWidth,CV_8UC3,Scalar(0));

	while (1)
	{
		//2.kalman prediction
		Mat prediction = KF.predict();
		Point predict_pt = Point(prediction.at(0),prediction.at(1) );   //预测值(x',y')

		//3.update measurement
		measurement.at(0) = (float)mousePosition.x;
		measurement.at(1) = (float)mousePosition.y;		

		//4.update
		KF.correct(measurement);

		//draw 
		image.setTo(Scalar(255,255,255,0));
		circle(image,predict_pt,5,Scalar(0,255,0),3);    //predicted point with green
		circle(image,mousePosition,5,Scalar(255,0,0),3); //current position with red		
		
		char buf[256];
		sprintf_s(buf,256,"predicted position:(%3d,%3d)",predict_pt.x,predict_pt.y);
		putText(image,buf,Point(10,30),CV_FONT_HERSHEY_SCRIPT_COMPLEX,1,Scalar(0,0,0),1,8);
		sprintf_s(buf,256,"current position :(%3d,%3d)",mousePosition.x,mousePosition.y);
		putText(image,buf,cvPoint(10,60),CV_FONT_HERSHEY_SCRIPT_COMPLEX,1,Scalar(0,0,0),1,8);
		
		imshow("kalman", image);
		int key=waitKey(3);
		if (key==27){//esc   
			break;   
		}		
	}
}

结果

学习OpenCV2——卡尔曼滤波(KalmanFilter)详解_第5张图片


例3 

#include "opencv2/video/tracking.hpp" 
#include     //#include "cvAux.h"
#include 
#include 
#include 

using namespace cv;
using namespace std;

int main( )  
{  
	float A[10][3] = 
	{
		10,    50,     15.6,
		12,    49,     16,
		11,    52,     15.8,
		13,    52.2,   15.8,
		12.9,  50,     17,
		14,    48,     16.6,
		13.7,  49,     16.5,
		13.6,  47.8,   16.4,
		12.3,  46,     15.9,
		13.1,  45,     16.2
	};	

	const int stateNum=3;
	const int measureNum=3;
	KalmanFilter KF(stateNum, measureNum, 0); 
	KF.transitionMatrix = *(Mat_(3, 3) <<1,0,0,0,1,0,0,0,1);  //转移矩阵A  
	setIdentity(KF.measurementMatrix);                                             //测量矩阵H  
	setIdentity(KF.processNoiseCov, Scalar::all(1e-5));                            //系统噪声方差矩阵Q  
	setIdentity(KF.measurementNoiseCov, Scalar::all(1e-1));                        //测量噪声方差矩阵R  
	setIdentity(KF.errorCovPost, Scalar::all(1)); 
	Mat measurement = Mat::zeros(measureNum, 1, CV_32F); 
	
	//初始状态值
	KF.statePost = *(Mat_(3, 1) <(0)<<"\t"<(1)<<"\t"<(2)<(0)<<"\t"<(1)<<"\t"<(2)<(0)<<"\t"<(1)<<"\t"<(2)<
结果如下
学习OpenCV2——卡尔曼滤波(KalmanFilter)详解_第6张图片

这里预测值和上一个状态值一样,原因是转移矩阵A是单位阵,如果改成非单位阵,结果就不一样了。


你可能感兴趣的:(OpenCV2,目标跟踪)