卡尔曼滤波简介及算法实现代码

    卡尔曼滤波器（Kalman Filter）是一个最优化自回归数据处理算法（optimal recursive data processing algorithm）。对于解决很大部分的问题，他是最优，效率最高甚至是最有用的。他的广泛应用已经超过30年，包括机器人导航，控制，传感器数据融合甚至在军事方面的雷达系统以及导弹追踪等等。近年来更被应用于计算机图像处理，例如头脸识别，图像分割，图像边缘检测等等。

    最佳线性滤波理论起源于40年代美国科学家Wiener和前苏联科学家Kолмогоров等人的研究工作，后人统称为维纳滤波理论。从理论上说，维纳滤波的最大缺点是必须用到无限过去的数据，不适用于实时处理。为了克服这一缺点，60年代Kalman把状态空间模型引入滤波理论，并导出了一套递推估计算法，后人称之为卡尔曼滤波理论。卡尔曼滤波是以最小均方误差为估计的最佳准则，来寻求一套递推估计的算法，其基本思想是：采用信号与噪声的状态空间模型，利用前一时刻地估计值和现时刻的观测值来更新对状态变量的估计，求出现时刻的估计值。它适合于实时处理和计算机运算。

现设线性时变系统的离散状态方程和观测方程为：

X(k) = F(k,k-1)·X(k-1)+T(k,k-1)·U(k-1)

Y(k) = H(k)·X(k)+N(k)

	其中

	X(k)和Y(k)分别是k时刻的状态矢量和观测矢量

	F(k,k-1)为状态转移矩阵

	U(k)为k时刻动态噪声

	T(k,k-1)为系统控制矩阵

	H(k)为k时刻观测矩阵

	N(k)为k时刻观测噪声

则卡尔曼滤波的算法流程为：

1.预估计

	X(k)^= F(k,k-1)·X(k-1) 

2.计算预估计协方差矩阵

	C(k)^=F(k,k-1)×C(k)×F(k,k-1)'+T(k,k-1)×Q(k)×T(k,k-1)'

	Q(k) = U(k)×U(k)' 

3.计算卡尔曼增益矩阵

	K(k) = C(k)^×H(k)'×[H(k)×C(k)^×H(k)'+R(k)]^(-1)

	R(k) = N(k)×N(k)' 

4.更新估计

	X(k)~=X(k)^+K(k)×[Y(k)-H(k)×X(k)^] 

5.计算更新后估计协防差矩阵

	C(k)~ = [I-K(k)×H(k)]×C(k)^×[I-K(k)×H(k)]'+K(k)×R(k)×K(k)' 

6.X(k+1) = X(k)~	C(k+1) = C(k)~ 

重复以上步骤

其c语言实现代码如下：

#include "stdlib.h"
 #include "rinv.c"
int lman(n,m,k,f,q,r,h,y,x,p,g)
int n,m,k;
double f[],q[],r[],h[],y[],x[],p[],g[];
{ int i,j,kk,ii,l,jj,js;
double*e,*a,*b;
 e=malloc(m*m*sizeof(double));
 l=m;
if (l<n) l=n;
 a=malloc(l*l*sizeof(double));
 b=malloc(l*l*sizeof(double));
for (i=0; i<=n-1; i++)
for (j=0; j<=n-1; j++)
{ ii=i*l+j; a[ii]=0.0;
for (kk=0; kk<=n-1; kk++)
 a[ii]=a[ii]+p[i*n+kk]*f[j*n+kk];
 }
for (i=0; i<=n-1; i++)
for (j=0; j<=n-1; j++)
{ ii=i*n+j; p[ii]=q[ii];
for (kk=0; kk<=n-1; kk++)
 p[ii]=p[ii]+f[i*n+kk]*a[kk*l+j];
 }
for (ii=2; ii<=k; ii++)
{ for (i=0; i<=n-1; i++)
for (j=0; j<=m-1; j++)
{ jj=i*l+j; a[jj]=0.0;
for (kk=0; kk<=n-1; kk++)
 a[jj]=a[jj]+p[i*n+kk]*h[j*n+kk];
 }
for (i=0; i<=m-1; i++)
for (j=0; j<=m-1; j++)
{ jj=i*m+j; e[jj]=r[jj];
for (kk=0; kk<=n-1; kk++)
 e[jj]=e[jj]+h[i*n+kk]*a[kk*l+j];
 }
 js=rinv(e,m);
if (js==0) 
{ free(e); free(a); free(b); return(js);}
for (i=0; i<=n-1; i++)
for (j=0; j<=m-1; j++)
{ jj=i*m+j; g[jj]=0.0;
for (kk=0; kk<=m-1; kk++)
 g[jj]=g[jj]+a[i*l+kk]*e[j*m+kk];
 }
for (i=0; i<=n-1; i++)
{ jj=(ii-1)*n+i; x[jj]=0.0;
for (j=0; j<=n-1; j++)
 x[jj]=x[jj]+f[i*n+j]*x[(ii-2)*n+j];
 }
for (i=0; i<=m-1; i++)
{ jj=i*l; b[jj]=y[(ii-1)*m+i];
for (j=0; j<=n-1; j++)
 b[jj]=b[jj]-h[i*n+j]*x[(ii-1)*n+j];
 }
for (i=0; i<=n-1; i++)
{ jj=(ii-1)*n+i;
for (j=0; j<=m-1; j++)
 x[jj]=x[jj]+g[i*m+j]*b[j*l];
 }
if (ii<k)
{ for (i=0; i<=n-1; i++)
for (j=0; j<=n-1; j++)
{ jj=i*l+j; a[jj]=0.0;
for (kk=0; kk<=m-1; kk++)
 a[jj]=a[jj]-g[i*m+kk]*h[kk*n+j];
if (i==j) a[jj]=1.0+a[jj];
 }
for (i=0; i<=n-1; i++)
for (j=0; j<=n-1; j++)
{ jj=i*l+j; b[jj]=0.0;
for (kk=0; kk<=n-1; kk++)
 b[jj]=b[jj]+a[i*l+kk]*p[kk*n+j];
 }
for (i=0; i<=n-1; i++)
for (j=0; j<=n-1; j++)
{ jj=i*l+j; a[jj]=0.0;
for (kk=0; kk<=n-1; kk++)
 a[jj]=a[jj]+b[i*l+kk]*f[j*n+kk];
 }
for (i=0; i<=n-1; i++)
for (j=0; j<=n-1; j++)
{ jj=i*n+j; p[jj]=q[jj];
for (kk=0; kk<=n-1; kk++)
 p[jj]=p[jj]+f[i*n+kk]*a[j*l+kk];
 }
 }
 }
 free(e); free(a); free(b);
return(js);
 }

C++实现代码如下：

============================kalman.h================================

// kalman.h: interface for the kalman class.
//
//////////////////////////////////////////////////////////////////////

#if !defined(AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_)
#define AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_

#if _MSC_VER > 1000
#pragma once
#endif// _MSC_VER > 1000

#include <math.h>
#include "cv.h"



class kalman 
{
public:
void init_kalman(int x,int xv,int y,int yv);
CvKalman* cvkalman;
CvMat* state; 
CvMat* process_noise;
CvMat* measurement;
const CvMat* prediction;
CvPoint2D32f get_predict(float x, float y);
kalman(int x=0,int xv=0,int y=0,int yv=0);
//virtual ~kalman();


};

#endif// !defined(AFX_KALMAN_H__ED3D740F_01D2_4616_8B74_8BF57636F2C0__INCLUDED_)


============================kalman.cpp================================

#include "kalman.h"
#include <stdio.h>


/* tester de printer toutes les valeurs des vecteurs*/
/* tester de changer les matrices du noises */
/* replace state by cvkalman->state_post ??? */


CvRandState rng;
constdouble T =0.1;
kalman::kalman(int x,int xv,int y,int yv)
{ 
cvkalman = cvCreateKalman( 4, 4, 0 );
state = cvCreateMat( 4, 1, CV_32FC1 );
process_noise = cvCreateMat( 4, 1, CV_32FC1 );
measurement = cvCreateMat( 4, 1, CV_32FC1 );
int code =-1;

/* create matrix data */
constfloat A[] = { 
1, T, 0, 0,
0, 1, 0, 0,
0, 0, 1, T,
0, 0, 0, 1
};

constfloat H[] = { 
1, 0, 0, 0,
0, 0, 0, 0,
0, 0, 1, 0,
0, 0, 0, 0
};

constfloat P[] = {
pow(320,2), pow(320,2)/T, 0, 0,
pow(320,2)/T, pow(320,2)/pow(T,2), 0, 0,
0, 0, pow(240,2), pow(240,2)/T,
0, 0, pow(240,2)/T, pow(240,2)/pow(T,2)
};

constfloat Q[] = {
pow(T,3)/3, pow(T,2)/2, 0, 0,
pow(T,2)/2, T, 0, 0,
0, 0, pow(T,3)/3, pow(T,2)/2,
0, 0, pow(T,2)/2, T
};

constfloat R[] = {
1, 0, 0, 0,
0, 0, 0, 0,
0, 0, 1, 0,
0, 0, 0, 0
};


cvRandInit( &rng, 0, 1, -1, CV_RAND_UNI );

cvZero( measurement );

cvRandSetRange( &rng, 0, 0.1, 0 );
rng.disttype = CV_RAND_NORMAL;

cvRand( &rng, state );

memcpy( cvkalman->transition_matrix->data.fl, A, sizeof(A));
memcpy( cvkalman->measurement_matrix->data.fl, H, sizeof(H));
memcpy( cvkalman->process_noise_cov->data.fl, Q, sizeof(Q));
memcpy( cvkalman->error_cov_post->data.fl, P, sizeof(P));
memcpy( cvkalman->measurement_noise_cov->data.fl, R, sizeof(R));
//cvSetIdentity( cvkalman->process_noise_cov, cvRealScalar(1e-5) ); 
//cvSetIdentity( cvkalman->error_cov_post, cvRealScalar(1));
//cvSetIdentity( cvkalman->measurement_noise_cov, cvRealScalar(1e-1) );

/* choose initial state */

state->data.fl[0]=x;
state->data.fl[1]=xv;
state->data.fl[2]=y;
state->data.fl[3]=yv;
cvkalman->state_post->data.fl[0]=x;
cvkalman->state_post->data.fl[1]=xv;
cvkalman->state_post->data.fl[2]=y;
cvkalman->state_post->data.fl[3]=yv;

cvRandSetRange( &rng, 0, sqrt(cvkalman->process_noise_cov->data.fl[0]), 0 );
cvRand( &rng, process_noise );


}


CvPoint2D32f kalman::get_predict(float x, float y){


/* update state with current position */
state->data.fl[0]=x;
state->data.fl[2]=y;


/* predict point position */
/* x'k=A鈥k+B鈥k
P'k=A鈥k-1*AT + Q */
cvRandSetRange( &rng, 0, sqrt(cvkalman->measurement_noise_cov->data.fl[0]), 0 );
cvRand( &rng, measurement );

/* xk=A?xk-1+B?uk+wk */
cvMatMulAdd( cvkalman->transition_matrix, state, process_noise, cvkalman->state_post );

/* zk=H?xk+vk */
cvMatMulAdd( cvkalman->measurement_matrix, cvkalman->state_post, measurement, measurement );

/* adjust Kalman filter state */
/* Kk=P'k鈥T鈥?H鈥'k鈥T+R)-1
xk=x'k+Kk鈥?zk-H鈥'k)
Pk=(I-Kk鈥)鈥'k */
cvKalmanCorrect( cvkalman, measurement );
float measured_value_x = measurement->data.fl[0];
float measured_value_y = measurement->data.fl[2];


const CvMat* prediction = cvKalmanPredict( cvkalman, 0 );
float predict_value_x = prediction->data.fl[0];
float predict_value_y = prediction->data.fl[2];

return(cvPoint2D32f(predict_value_x,predict_value_y));
}

void kalman::init_kalman(int x,int xv,int y,int yv)
{
state->data.fl[0]=x;
state->data.fl[1]=xv;
state->data.fl[2]=y;
state->data.fl[3]=yv;
cvkalman->state_post->data.fl[0]=x;
cvkalman->state_post->data.fl[1]=xv;
cvkalman->state_post->data.fl[2]=y;
cvkalman->state_post->data.fl[3]=yv;
}