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

本文將簡要回顧一下卡爾曼濾波理論，然後詳細介紹如何在OpenCV中使用卡爾曼濾波進行跟蹤，最後給兩個程式例項。

1. 卡爾曼濾波理論回顧

      對於一個動態系統，我們首先定義一組狀態空間方程

     狀態方程：     

     測量方程：      

        x_k是狀態向量，z_k是測量向量，A_k是狀態轉移矩陣，u_k是控制向量，B_k是控制矩陣，w_k是系統誤差（噪聲），H_k是測量矩陣，v_k是測量誤差（噪聲）。w_k和v_k都是高斯噪聲，即

                             

    整個卡爾曼濾波的過程就是個遞推計算的過程，不斷的“預測——更新——預測——更新……”

預測

     預測狀態值：              

     預測最小均方誤差：   

更新

    測量誤差：                   

    測量協方差：                

    最優卡爾曼增益：         

    修正狀態值：                

    修正最小均方誤差：     

2.OpenCV中的KalmanFilter詳解

OpenCV中有兩個版本的卡爾曼濾波方法KalmanFilter(C++)和CvKalman(C)，用法差不太多，這裡只介紹KalmanFilter。

C++版本中將KalmanFilter封裝到一個類中，其結構如下所示：

[cpp] view plain copy print?

1. class CV_EXPORTS_W KalmanFilter  
2. {  
3. public:      
4.     CV_WRAP KalmanFilter();                                                                           //構造預設KalmanFilter物件
5.     CV_WRAP KalmanFilter(int dynamParams, int measureParams, 
   int controlParams=0, int type=CV_32F);  //完整構造KalmanFilter物件方法
6.     void init(int dynamParams, int measureParams, int controlParams=0, int type=CV_32F);              //初始化KalmanFilter物件，會替換原來的KF物件
7.     CV_WRAP const Mat& predict(const Mat& control=Mat());           //計算預測的狀態值    
8.     CV_WRAP const Mat& correct(const Mat& measurement);             //根據測量值更新狀態值
9.     Mat statePre;            //預測值 (x'(k)): x(k)=A*x(k-1)+B*u(k)
10.     Mat statePost;           //狀態值 (x(k)): x(k)=x'(k)+K(k)*(z(k)-H*x'(k))
11.     Mat transitionMatrix;    //狀態轉移矩陣 (A)
12.     Mat controlMatrix;       //控制矩陣 B 
13.     Mat measurementMatrix;   //測量矩陣 H
14.     Mat processNoiseCov;     //系統誤差 Q
15.     Mat measurementNoiseCov; //測量誤差 R
16.     Mat errorCovPre;         //最小均方誤差 (P'(k)): P'(k)=A*P(k-1)*At + Q)
17.     Mat gain;                //卡爾曼增益   (K(k)): K(k)=P'(k)*Ht*inv(H*P'(k)*Ht+R)
18.     Mat errorCovPost;        //修正的最小均方誤差 (P(k)): P(k)=(I-K(k)*H)*P'(k)
19.     // 臨時矩陣
20.     Mat temp1;  
21.     Mat temp2;  
22.     Mat temp3;  
23.     Mat temp4;  
24.     Mat temp5;  
25. };  
26. enum
27. {  
28.     OPTFLOW_USE_INITIAL_FLOW = CV_LKFLOW_INITIAL_GUESSES,  
29.     OPTFLOW_LK_GET_MIN_EIGENVALS = CV_LKFLOW_GET_MIN_EIGENVALS,  
30.     OPTFLOW_FARNEBACK_GAUSSIAN = 256  
31. };  

class CV_EXPORTS_W KalmanFilter
{
public:    
    CV_WRAP KalmanFilter();                                                                           //構造預設KalmanFilter物件
    CV_WRAP KalmanFilter(int dynamParams, int measureParams, int controlParams=0, int type=CV_32F);  //完整構造KalmanFilter物件方法
    void init(int dynamParams, int measureParams, int controlParams=0, int type=CV_32F);              //初始化KalmanFilter物件，會替換原來的KF物件
  
    CV_WRAP const Mat& predict(const Mat& control=Mat());           //計算預測的狀態值    
    CV_WRAP const Mat& correct(const Mat& measurement);             //根據測量值更新狀態值

    Mat statePre;            //預測值 (x'(k)): x(k)=A*x(k-1)+B*u(k)
    Mat statePost;           //狀態值 (x(k)): x(k)=x'(k)+K(k)*(z(k)-H*x'(k))
    Mat transitionMatrix;    //狀態轉移矩陣 (A)
    Mat controlMatrix;       //控制矩陣 B 
    Mat measurementMatrix;   //測量矩陣 H
    Mat processNoiseCov;     //系統誤差 Q
    Mat measurementNoiseCov; //測量誤差 R
    Mat errorCovPre;         //最小均方誤差 (P'(k)): P'(k)=A*P(k-1)*At + Q)
    Mat gain;                //卡爾曼增益   (K(k)): K(k)=P'(k)*Ht*inv(H*P'(k)*Ht+R)
    Mat errorCovPost;        //修正的最小均方誤差 (P(k)): P(k)=(I-K(k)*H)*P'(k)

    // 臨時矩陣
    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
};

       函式原型見：…..\OpenCV2\sources\modules\ocl\src\kalman.cpp

       只有四個方法: 構造KF物件KalmanFilter(DP,MP,CP)、初始化KF物件init(DP,MP,CP)、預測predict( )、更新correct( )。除非你要重新構造KF物件，否則用不到init( )。

KalmanFilter(DP,MP,CP)和init( )就是賦值，沒什麼好說的。

      注意：KalmanFilter結構體中並沒有測量值，測量值需要自己定義，而且一定要定義，因為後面要用。

程式設計步驟

step1：定義KalmanFilter類並初始化

    //構造KF物件

    KalmanFilter KF(DP, MP, 0);

    //初始化相關引數

    KF.transitionMatrix                         轉移矩陣 A

    KF.measurementMatrix                  測量矩陣    H

    KF.processNoiseCov                     過程噪聲 Q

    KF.measurementNoiseCov            測量噪聲        R

    KF.errorCovPost                            最小均方誤差 P

    KF.statePost                                系統初始狀態 x(0) 

    Mat measurement                          定義初始測量值 z(0) 

step2：預測

    KF.predict( )                                                 //返回的是下一時刻的狀態值KF.statePost (k+1) 

step3：更新

    更新measurement;                                     //注意measurement不能通過觀測方程進行計算得到，要自己定義！

    更新KF   KF.correct(measurement)

最終的結果應該是更新後的statePost.

相關引數的確定

    對於系統狀態方程，簡記為Y=AX+B，X和Y是表示系統狀態的列向量，A是轉移矩陣，B是其他項。

    狀態值（向量）只要能表示系統的狀態即可，狀態值的維數決定了轉移矩陣A的維數，比如X和Y是N×1的，則A是N×N的。

    A的確定跟X有關，只要保證方程中不相干項的係數為0即可，看下面例子

      X和Y是二維的，

       X和Y是三維的，

          X和Y是三維的，但c和△ c是相關項

      上面的1也可以是其他值。

下面對predict( ) 和correct( )函式介紹下，可以不用看，不影響程式設計。

[cpp] view plain copy print?

1. CV_EXPORTS const oclMat& KalmanFilter::predict(const oclMat& control)  
2. {  
3.     gemm(transitionMatrix, statePost, 1, oclMat(), 0, statePre);  
4.     oclMat temp;  
5.     if(control.data)  
6.         gemm(controlMatrix, control, 1, statePre, 1, statePre);  
7.     gemm(transitionMatrix, errorCovPost, 1, oclMat(), 0, temp1);  
8.     gemm(temp1, transitionMatrix, 1, processNoiseCov, 1, errorCovPre, GEMM_2_T);  
9.     statePre.copyTo(statePost);  
10.     return statePre;  
11. }  

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( )是矩陣的廣義乘法

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()其實是u_k，只不過預設為0，所以沒賦值。整個過程就計算了x'和P’。（用x'代表x的預測值，用P'代表P的預測值）。GEMM_2_T表示對第2個引數轉置。

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·A^T + 1· Q_k= A·P(k-1)·A^T + 1· Q_k

       可見，和第一部分的理論介紹完全一致。 [cpp] view plain copy print?

1. CV_EXPORTS const oclMat& KalmanFilter::correct(const oclMat& measurement)  
2. {  
3.     CV_Assert(measurement.empty() == false);  
4.     gemm(measurementMatrix, errorCovPre, 1, oclMat(), 0, temp2);  
5.     gemm(temp2, measurementMatrix, 1, measurementNoiseCov, 1, temp3, GEMM_2_T);  
6.     Mat temp;  
7.     solve(Mat(temp3), Mat(temp2), temp, DECOMP_SVD);  
8.     temp4.upload(temp);  
9.     gain = temp4.t();  
10.     gemm(measurementMatrix, statePre, -1, measurement, 1, temp5);  
11.     gemm(gain, temp5, 1, statePre, 1, statePost);  
12.     gemm(gain, temp2, -1, errorCovPre, 1, errorCovPost);  
13.     return statePost;  
14. }  

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)

求解線型最小二乘估計

temp2 = 1· H·P’ + 0·u(k)

temp3 = 1· temp2·H^T + 1·R = H·P’·H^T+ 1· R   也就是上面的S_k

temp = argmin||tem2- temp3||

K=temp

temp5 = -1· H·x’ + 1·z_k        就是上面的y’。

x = 1·K·temp5 + 1·x’ = K^T·y’ +x’

P =-1·K·temp2 + 1·P’ = -K·H·P’+P’ = (I- K·H) P’

也和第一部分的理論完全一致。

通過深入函式內部，學到了兩個實用的函式哦。矩陣廣義乘法gemm( )、最小二乘估計solve( )

補充：

1）以例2為例，為什麼狀態值一般都設定成（x,y,△x,△y）？我們不妨設定成(x,y,△x)，對應的轉移矩陣也改成3×3的。可以看到仍能跟上，不過在x方向跟蹤速度快，在y方向跟蹤速度慢。進一步設定成(x,y)和2×2的轉移矩陣，程式的跟蹤速度簡直是龜速。所以，簡單理解，△x和△y嚴重影響對應方向上的跟蹤速度。

3.例項

例1 OpenCV自帶的示例程式

[cpp] view plain copy print?

1. #include "opencv2/video/tracking.hpp"
2. #include "opencv2/highgui/highgui.hpp"
3. #include <iostream>
4. #include <stdio.h>
5. usingnamespace std;  
6. usingnamespace cv;  
7. //計算相對視窗的座標值，因為座標原點在左上角，所以sin前有個負號
8. staticinline Point calcPoint(Point2f center, double R, double angle)  
9. {  
10.     return center + Point2f((float)cos(angle), (float)-sin(angle))*(float)R;  
11. }  
12. staticvoid help()  
13. {  
14.     printf( "\nExamle of c calls to OpenCV's Kalman filter.\n"
15. "   Tracking of rotating point.\n"
16. "   Rotation speed is constant.\n"
17. "   Both state and measurements vectors are 1D (a point angle),\n"
18. "   Measurement is the real point angle + gaussian noise.\n"
19. "   The real and the estimated points are connected with yellow line segment,\n"
20. "   the real and the measured points are connected with red line segment.\n"
21. "   (if Kalman filter works correctly,\n"
22. "    the yellow segment should be shorter than the red one).\n"
23.             "\n"
24. "   Pressing any key (except ESC) will reset the tracking with a different speed.\n"
25. "   Pressing ESC will stop the program.\n"
26.             );  
27. }  
28. int main(int, char**)  
29. {  
30.     help();  
31.     Mat img(500, 500, CV_8UC3);  
32.     KalmanFilter KF(2, 1, 0);                                    //建立卡爾曼濾波器物件KF
33.     Mat state(2, 1, CV_32F);                                     //state(角度，△角度)
34.     Mat processNoise(2, 1, CV_32F);  
35.     Mat measurement = Mat::zeros(1, 1, CV_32F);                 //定義測量值
36.     char code = (char)-1;  
37.     for(;;)  
38.     {  
39.         //1.初始化
40.         randn( state, Scalar::all(0), Scalar::all(0.1) );          //
41.         KF.transitionMatrix = *(Mat_<float>(2, 2) << 1, 1, 0, 1);  //轉移矩陣A[1,1;0,1]    
42.         //將下面幾個矩陣設定為對角陣
43.         setIdentity(KF.measurementMatrix);                             //測量矩陣H
44.         setIdentity(KF.processNoiseCov, Scalar::all(1e-5));            //系統噪聲方差矩陣Q
45.         setIdentity(KF.measurementNoiseCov, Scalar::all(1e-1));        //測量噪聲方差矩陣R
46.         setIdentity(KF.errorCovPost, Scalar::all(1));                  //後驗錯誤估計協方差矩陣P
47.         randn(KF.statePost, Scalar::all(0), Scalar::all(0.1));          //x(0)初始化
48.         for(;;)  
49.         {  
50.             Point2f center(img.cols*0.5f, img.rows*0.5f);          //center影象中心點
51.             float R = img.cols/3.f;                                //半徑
52.             double stateAngle = state.at<float>(0);                //跟蹤點角度
53.             Point statePt = calcPoint(center, R, stateAngle);     //跟蹤點座標statePt
54.             //2. 預測
55.             Mat prediction = KF.predict();                       //計算預測值，返回x'
56.             double predictAngle = prediction.at<float>(0);          //預測點的角度
57.             Point predictPt = calcPoint(center, R, predictAngle);   //預測點座標predictPt
58.             //3.更新
59.             //measurement是測量值
60.             randn( measurement, Scalar::all(0), Scalar::all(KF.measurementNoiseCov.at<float>(0)));     //給measurement賦值N(0,R)的隨機值
61.             // generate measurement
62.             measurement += KF.measurementMatrix*state;  //z = z + H*x;
63.             double measAngle = measurement.at<float>(0);  
64.             Point measPt = calcPoint(center, R, measAngle);  
65.             // plot points
66.             //定義了畫十字的方法，值得學習下
67.             #define drawCross( center, color, d )                                 \
68.                 line( img, Point( center.x - d, center.y - d ),                \  
69.                              Point( center.x + d, center.y + d ), color, 1, CV_AA, 0); \  
70.                 line( img, Point( center.x + d, center.y - d ),                \  
71.                              Point( center.x - d, center.y + d ), color, 1, CV_AA, 0 )  
72.             img = Scalar::all(0);  
73.             drawCross( statePt, Scalar(255,255,255), 3 );  
74.             drawCross( measPt, Scalar(0,0,255), 3 );  
75.             drawCross( predictPt, Scalar(0,255,0), 3 );  
76.             line( img, statePt, measPt, Scalar(0,0,255), 3, CV_AA, 0 );  
77.             line( img, statePt, predictPt, Scalar(0,255,255), 3, CV_AA, 0 );  
78.             //呼叫kalman這個類的correct方法得到加入觀察值校正後的狀態變數值矩陣
79.             if(theRNG().uniform(0,4) != 0)  
80.                 KF.correct(measurement);  
81.             //不加噪聲的話就是勻速圓周運動，加了點噪聲類似勻速圓周運動，因為噪聲的原因，運動方向可能會改變
82.             randn( processNoise, Scalar(0), Scalar::all(sqrt(KF.processNoiseCov.at<float>(0, 0))));   //vk
83.             state = KF.transitionMatrix*state + processNoise;     
84.             imshow( "Kalman", img );  
85.             code = (char)waitKey(100);  
86.             if( code > 0 )  
87.                 break;  
88.         }  
89.         if( code == 27 || code == 'q' || code == 'Q' )  
90.             break;  
91.     }  
92.     return 0;  
93. }  

#include "opencv2/video/tracking.hpp"
#include "opencv2/highgui/highgui.hpp"
#include <iostream>
#include <stdio.h>
using namespace std;
using namespace cv;

//計算相對視窗的座標值，因為座標原點在左上角，所以sin前有個負號
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);                                    //建立卡爾曼濾波器物件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);                 //定義測量值
    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);  //轉移矩陣A[1,1;0,1]    
		

		//將下面幾個矩陣設定為對角陣
        setIdentity(KF.measurementMatrix);                             //測量矩陣H
        setIdentity(KF.processNoiseCov, Scalar::all(1e-5));            //系統噪聲方差矩陣Q
        setIdentity(KF.measurementNoiseCov, Scalar::all(1e-1));        //測量噪聲方差矩陣R
        setIdentity(KF.errorCovPost, Scalar::all(1));                  //後驗錯誤估計協方差矩陣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影象中心點
            float R = img.cols/3.f;                                //半徑
            double stateAngle = state.at<float>(0);                //跟蹤點角度
            Point statePt = calcPoint(center, R, stateAngle);     //跟蹤點座標statePt

			//2. 預測
            Mat prediction = KF.predict();                       //計算預測值，返回x'
            double predictAngle = prediction.at<float>(0);          //預測點的角度
            Point predictPt = calcPoint(center, R, predictAngle);   //預測點座標predictPt


			//3.更新
			//measurement是測量值
            randn( measurement, Scalar::all(0), Scalar::all(KF.measurementNoiseCov.at<float>(0)));     //給measurement賦值N(0,R)的隨機值

            // generate measurement
            measurement += KF.measurementMatrix*state;  //z = z + H*x;
			
            double measAngle = measurement.at<float>(0);
            Point measPt = calcPoint(center, R, measAngle);

            // plot points
			//定義了畫十字的方法，值得學習下
            #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 );


			//呼叫kalman這個類的correct方法得到加入觀察值校正後的狀態變數值矩陣
			if(theRNG().uniform(0,4) != 0)
                KF.correct(measurement);

			//不加噪聲的話就是勻速圓周運動，加了點噪聲類似勻速圓周運動，因為噪聲的原因，運動方向可能會改變
            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;
}

程式結果

例2  跟蹤滑鼠位置

[cpp] view plain copy print?

1. #include "opencv2/video/tracking.hpp"
2. #include "opencv2/highgui/highgui.hpp"
3. #include <stdio.h>
4. usingnamespace cv;  
5. usingnamespace std;  
6. constint winHeight=600;  
7. constint winWidth=800;  
8. Point mousePosition= Point(winWidth>>1,winHeight>>1);  
9. //mouse event callback
10. void mouseEvent(int event, int x, int y, int flags, void *param )  
11. {  
12.     if (event==CV_EVENT_MOUSEMOVE) {  
13.         mousePosition = Point(x,y);  
14.     }  
15. }  
16. int main (void)  
17. {  
18.     RNG rng;  
19.     //1.kalman filter setup
20.     constint stateNum=4;                                      //狀態值4×1向量(x,y,△x,△y)
21.     constint measureNum=2;                                    //測量值2×1向量(x,y)  
22.     KalmanFilter KF(stateNum, measureNum, 0);     
23.     KF.transitionMatrix = *(Mat_<float>(4, 4) <<1,0,1,0,0,1,0,1,0,0,1,0,0,0,0,1);  //轉移矩陣A
24.     setIdentity(KF.measurementMatrix);                                             //測量矩陣H
25.     setIdentity(KF.processNoiseCov, Scalar::all(1e-5));                            //系統噪聲方差矩陣Q
26.     setIdentity(KF.measurementNoiseCov, Scalar::all(1e-1));                        //測量噪聲方差矩陣R
27.     setIdentity(KF.errorCovPost, Scalar::all(1));                                  //後驗錯誤估計協方差矩陣P
28.     rng.fill(KF.statePost,RNG::UNIFORM,0,winHeight>winWidth?winWidth:winHeight);   //初始狀態值x(0)
29.     Mat measurement = Mat::zeros(measureNum, 1, CV_32F);                           //初始測量值x'(0)，因為後面要更新這個值，所以必須先定義
30.     namedWindow("kalman");  
31.     setMouseCallback("kalman",mouseEvent);  
32.     Mat image(winHeight,winWidth,CV_8UC3,Scalar(0));  
33.     while (1)  
34.     {  
35.         //2.kalman prediction
36.         Mat prediction = KF.predict();  
37.         Point predict_pt = Point(prediction.at<float>(0),prediction.at<float>(1) );   //預測值(x',y')
38.         //3.update measurement
39.         measurement.at<float>(0) = (float)mousePosition.x;  
40.         measurement.at<float>(1) = (float)mousePosition.y;          
41.         //4.update
42.         KF.correct(measurement);  
43.         //draw 
44.         image.setTo(Scalar(255,255,255,0));  
45.         circle(image,predict_pt,5,Scalar(0,255,0),3);    //predicted point with green
46.         circle(image,mousePosition,5,Scalar(255,0,0),3); //current position with red        
47.         char buf[256];  
48.         sprintf_s(buf,256,"predicted position:(%3d,%3d)",predict_pt.x,predict_pt.y);  
49.         putText(image,buf,Point(10,30),CV_FONT_HERSHEY_SCRIPT_COMPLEX,1,Scalar(0,0,0),1,8);  
50.         sprintf_s(buf,256,"current position :(%3d,%3d)",mousePosition.x,mousePosition.y);  
51.         putText(image,buf,cvPoint(10,60),CV_FONT_HERSHEY_SCRIPT_COMPLEX,1,Scalar(0,0,0),1,8);  
52.         imshow("kalman", image);  
53.         int key=waitKey(3);  
54.         if (key==27){//esc   
55.             break;     
56.         }         
57.     }  
58. }  

#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;                                      //狀態值4×1向量(x,y,△x,△y)
	const int measureNum=2;                                    //測量值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);  //轉移矩陣A
	setIdentity(KF.measurementMatrix);                                             //測量矩陣H
	setIdentity(KF.processNoiseCov, Scalar::all(1e-5));                            //系統噪聲方差矩陣Q
	setIdentity(KF.measurementNoiseCov, Scalar::all(1e-1));                        //測量噪聲方差矩陣R
	setIdentity(KF.errorCovPost, Scalar::all(1));                                  //後驗錯誤估計協方差矩陣P
	rng.fill(KF.statePost,RNG::UNIFORM,0,winHeight>winWidth?winWidth:winHeight);   //初始狀態值x(0)
	Mat measurement = Mat::zeros(measureNum, 1, CV_32F);                           //初始測量值x'(0)，因為後面要更新這個值，所以必須先定義
	
	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) );   //預測值(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;   
		}		
	}
}

結果

例3 

[cpp] view plain copy print?

1. #include "opencv2/video/tracking.hpp" 
2. #include <opencv2/legacy/legacy.hpp>    //#include "cvAux.h"
3. #include <opencv2/highgui/highgui.hpp>
4. #include <opencv2/core/core.hpp>
5. #include <stdio.h>
6. usingnamespace cv;  
7. usingnamespace std;  
8. int main( )    
9. {    
10.     float A[10][3] =   
11.     {  
12.         10,    50,     15.6,  
13.         12,    49,     16,  
14.         11,    52,     15.8,  
15.         13,    52.2,   15.8,  
16.         12.9,  50,     17,  
17.         14,    48,     16.6,  
18.         13.7,  49,     16.5,  
19.         13.6,  47.8,   16.4,  
20.         12.3,  46,     15.9,  
21.         13.1,  45,     16.2  
22.     };    
23.     constint stateNum=3;  
24.     constint measureNum=3;  
25.     KalmanFilter KF(stateNum, measureNum, 0);   
26.     KF.transitionMatrix = *(Mat_<float>(3, 3) <<1,0,0,0,1,0,0,0,1);  //轉移矩陣A  
27.     setIdentity(KF.measurementMatrix);                                             //測量矩陣H  
28.     setIdentity(KF.processNoiseCov, Scalar::all(1e-5));                            //系統噪聲方差矩陣Q  
29.     setIdentity(KF.measurementNoiseCov, Scalar::all(1e-1));                        //測量噪聲方差矩陣R  
30.     setIdentity(KF.errorCovPost, Scalar::all(1));   
31.     Mat measurement = Mat::zeros(measureNum, 1, CV_32F);   
32.     //初始狀態值
33.     KF.statePost = *(Mat_<float>(3, 1) <<A[0][0],A[0][1],A[0][2]);   
34.     cout<<"state0="<<KF.statePost<<endl;  
35.     for(int i=1;i<=9;i++)  
36.     {  
37.         //預測
38.         Mat prediction = KF.predict();            
39.             //計算測量值
40.         measurement.at<float>(0) = (float)A[i][0];    
41.         measurement.at<float>(1) = (float)A[i][1];   
42.         measurement.at<float>(2) = (float)A[i][2];   
43.         //更新
44.         KF.correct(measurement);    
45.             //輸出結果
46.         cout<<"predict ="<<"\t"<<prediction.at<float>(0)<<"\t"<<prediction.at<float>(1)<<"\t"<<prediction.at<float>(2)<<endl;  
47.         cout<<"measurement="<<"\t"<<measurement.at<float>(0)<<"\t"<<measurement.at<float>(1)<<"\t"<<measurement.at<float>(2)<<endl;  
48.         cout<<"correct ="<<"\t"<<KF.statePost.at<float>(0)<<"\t"<<KF.statePost.at<float>(1)<<"\t"<<KF.statePost.at<float>(2)<<endl;  
49.     }  
50.     system("pause");  
51. }   

#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);  //轉移矩陣A  
	setIdentity(KF.measurementMatrix);                                             //測量矩陣H  
	setIdentity(KF.processNoiseCov, Scalar::all(1e-5));                            //系統噪聲方差矩陣Q  
	setIdentity(KF.measurementNoiseCov, Scalar::all(1e-1));                        //測量噪聲方差矩陣R  
	setIdentity(KF.errorCovPost, Scalar::all(1)); 
	Mat measurement = Mat::zeros(measureNum, 1, CV_32F); 
	
	//初始狀態值
	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++)
	{
	    //預測
	    Mat prediction = KF.predict(); 			
            //計算測量值
	    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);  
            //輸出結果
	    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");
} 

結果如下

這裡預測值和上一個狀態值一樣，原因是轉移矩陣A是單位陣，如果改成非單位陣，結果就不一樣了。