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学习OpenCV2——卡尔曼滤波(KalmanFilter)详解

2016-03-16 14:27 495 查看

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，

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。

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 <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。

#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

#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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标签： OpenCV2 卡尔曼滤波 KalmanFilter

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