卡尔曼 javascript 卡尔曼滤波器

卡尔曼滤波器是一种由卡尔曼（Kalman）提出的用于时变线性系统的递归滤波器。这个系统可用包含正交状态变量的微分方程模型来描述，这种滤波器是将过去的测量估计误差合并到新的测量误差中来估计将来的误差。

一. 卡尔曼滤波理论回顾

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：

二. 编程步骤

step1：定义KalmanFilter类并初始化

cv::KalmanFilter KF;//定义KF对象

//初始化相关参数

KF(DP, MP, 0); //初始化KF

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，对于目标检测而言，它是由网络输出给的（x,y,w,h）

KF.correct(measurement) //更新KF

KF.statePost.at(i); 得到更新后的statePost

三、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;
}