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OpenCV2——卡尔曼滤波(KalmanFilter)详解

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# -KalmanFilter-
OpenCV2——卡尔曼滤波(KalmanFilter)详解

http://blog.csdn.net/gdfsg/article/details/50904811

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：
[cpp] view plain copy
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，

X和Y是三维的，但c和△ c是相关项

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。
[cpp] view plain copy
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。
[cpp] view plain copy
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
[cpp] view plain copy
#include "opencv2/video/tracking.hpp"  
#include "opencv2/highgui/highgui.hpp"  
#include <iostream>  
#include <stdio.h>  
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_<float>(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<float>(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<float>(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<float>(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<float>(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<float>(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

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。
[cpp] view plain copy
#include "opencv2/video/tracking.hpp"  
#include "opencv2/highgui/highgui.hpp"  
#include <stdio.h>  
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_<float>(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<float>(0),prediction.at<float>(1) );   //预测值(x',y')  
  
        //3.update measurement  
        measurement.at<float>(0) = (float)mousePosition.x;  
        measurement.at<float>(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

例3 
[cpp] view plain copy
#include "opencv2/video/tracking.hpp"   
#include <opencv2/legacy/legacy.hpp>    //#include "cvAux.h"  
#include <opencv2/highgui/highgui.hpp>  
#include <opencv2/core/core.hpp>  
#include <stdio.h>  
  
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_<float>(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_<float>(3, 1) <<A[0][0],A[0][1],A[0][2]);   
    cout<<"state0="<<KF.statePost<<endl;  
  
    for(int i=1;i<=9;i++)  
    {  
        //预测  
        Mat prediction = KF.predict();            
            //计算测量值  
        measurement.at<float>(0) = (float)A[i][0];    
        measurement.at<float>(1) = (float)A[i][1];   
        measurement.at<float>(2) = (float)A[i][2];   
        //更新  
        KF.correct(measurement);    
            //输出结果  
        cout<<"predict ="<<"\t"<<prediction.at<float>(0)<<"\t"<<prediction.at<float>(1)<<"\t"<<prediction.at<float>(2)<<endl;  
        cout<<"measurement="<<"\t"<<measurement.at<float>(0)<<"\t"<<measurement.at<float>(1)<<"\t"<<measurement.at<float>(2)<<endl;  
        cout<<"correct ="<<"\t"<<KF.statePost.at<float>(0)<<"\t"<<KF.statePost.at<float>(1)<<"\t"<<KF.statePost.at<float>(2)<<endl;  
    }  
    system("pause");  
}   
结果如下

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。

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