卡尔曼滤波之Opencv（二）

今天研究了一下卡尔曼滤波跟踪，同时也看了一下卡尔曼滤波Opencv的源代码，总是看懂了，具体原理可以看看【1】。下面是opencv自带的一个程序，代码如下：

// kalman.cpp : 定义控制台应用程序的入口点。
//

#include "stdafx.h"


#include "opencv2/video/tracking.hpp"
#include "opencv2/highgui/highgui.hpp"

#include <stdio.h>

using namespace cv;

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);
	//[x1,x2]=[角度，角速度]
	/*
	运动模型：x1(k+1) = x1(k) + x2(k)*T
	         x2(k+1) = x2(k)
	状态转移方程：
	x^ = AX + w
	测量方程：
	z = Hx + v
	*/
	//状态估计值x=[x1,x2] --> state
    Mat state(2, 1, CV_32F); /* (phi, delta_phi) */
    Mat processNoise(2, 1, CV_32F);
	//当前观测值Z=Hx + v ---> measurement
    Mat measurement = Mat::zeros(1, 1, CV_32F);
    char code = (char)-1;

    for(;;)
    {
        randn( state, Scalar::all(0), Scalar::all(0.1) );
		//transitionMatrix对应到状态转移方程中的矩阵A
        KF.transitionMatrix = *(Mat_<float>(2, 2) << 1, 1, 0, 1);
		//measurementMatrix对应到测量方程的矩阵H
        setIdentity(KF.measurementMatrix);
		//processNoiseCov对应过程噪声协方差Q
        setIdentity(KF.processNoiseCov, Scalar::all(1e-5));
		//measurementNoiseCov对一个测量噪声协方差R
        setIdentity(KF.measurementNoiseCov, Scalar::all(1e-1));
		//errorCovPost对应最优值对应的偏差P(k|k)
        setIdentity(KF.errorCovPost, Scalar::all(1));
		//statePost对应系统状态最优值x(k|k)
        randn(KF.statePost, Scalar::all(0), Scalar::all(0.1));

        for(;;)
        {
            Point2f center(img.cols*0.5f, img.rows*0.5f);
            float R = img.cols/3.f;
			//角度
            double stateAngle = state.at<float>(0);
            Point statePt = calcPoint(center, R, stateAngle);

			//X(k|k-1) = A*X(k-1|k-1)
            Mat prediction = KF.predict();
			//角度
            double predictAngle = prediction.at<float>(0);
            Point predictPt = calcPoint(center, R, predictAngle);

            randn( measurement, Scalar::all(0), Scalar::all(KF.measurementNoiseCov.at<float>(0)));

            // generate measurement
			//Z(k) = H*X(k)
            measurement += KF.measurementMatrix*state;

			//角度
            double measAngle = measurement.at<float>(0);
            Point measPt = calcPoint(center, R, measAngle);

            // plot points
            #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 );

            if(theRNG().uniform(0,4) != 0)
			//X(k|k) = X(k|k-1) + Kg(k)*[Z(k) - H*X(k|k-1)
                KF.correct(measurement);

            randn( processNoise, Scalar(0), Scalar::all(sqrt(KF.processNoiseCov.at<float>(0, 0))));
            //X(k) = AX(k-1) + W(k)
			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;
}

同时为了更好的理解代码，我们需要知道一下的东西

代码1.

 Mat statePre;           //!< predicted state (x'(k)): x(k)=A*x(k-1)+B*u(k)
    Mat statePost;          //!< corrected state (x(k)): x(k)=x'(k)+K(k)*(z(k)-H*x'(k))
    Mat transitionMatrix;   //!< state transition matrix (A)
    Mat controlMatrix;      //!< control matrix (B) (not used if there is no control)
    Mat measurementMatrix;  //!< measurement matrix (H)
    Mat processNoiseCov;    //!< process noise covariance matrix (Q)
    Mat measurementNoiseCov;//!< measurement noise covariance matrix (R)
    Mat errorCovPre;        //!< priori error estimate covariance matrix (P'(k)): P'(k)=A*P(k-1)*At + Q)*/
    Mat gain;               //!< Kalman gain matrix (K(k)): K(k)=P'(k)*Ht*inv(H*P'(k)*Ht+R)
    Mat errorCovPost;       //!< posteriori error estimate covariance matrix (P(k)): P(k)=(I-K(k)*H)*P'(k)

一看上面的注释大概也明白什么意思了。

此外，还需要看2断代码

const Mat& KalmanFilter::predict(const Mat& control)
{
    // update the state: x'(k) = A*x(k)
    statePre = transitionMatrix*statePost;

    if( control.data )
        // x'(k) = x'(k) + B*u(k)
        statePre += controlMatrix*control;

    // update error covariance matrices: temp1 = A*P(k)
    temp1 = transitionMatrix*errorCovPost;

    // P'(k) = temp1*At + Q
    gemm(temp1, transitionMatrix, 1, processNoiseCov, 1, errorCovPre, GEMM_2_T);

    // handle the case when there will be measurement before the next predict.
    statePre.copyTo(statePost);

    return statePre;
}

这个段代码其实就是：X(k|k-1) = A(k-1|k-1) ，同时得到预测结果X(k|k-1)的偏差P(k|k-1)

const Mat& KalmanFilter::correct(const Mat& measurement)
{
    // temp2 = H*P'(k)
    temp2 = measurementMatrix * errorCovPre;

    // temp3 = temp2*Ht + R
    gemm(temp2, measurementMatrix, 1, measurementNoiseCov, 1, temp3, GEMM_2_T);

    // temp4 = inv(temp3)*temp2 = Kt(k)
    solve(temp3, temp2, temp4, DECOMP_SVD);

    // K(k):卡尔曼增益
    gain = temp4.t();

    // temp5 = z(k) - H*x'(k)
    temp5 = measurement - measurementMatrix*statePre;

    // x(k) = x'(k) + K(k)*temp5
    statePost = statePre + gain*temp5;

    // P(k) = P'(k) - K(k)*temp2
    errorCovPost = errorCovPre - gain*temp2;

    return statePost;
}

上面的代码其实就是：求Kg(k)=P(k|k-1)H' / (HP(k|k-1)H' + R) ，X(k|k) = X(k|k-1) + Kg(k)(Z(k) - HX(k|k-1)，P(k|k) = ( 1 - Kg(k)H)P(k|k-1)

参考

【1】卡尔曼滤波的简单应用