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。
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)
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。
可见，和第一部分的理论介绍完全一致。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
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( )
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
