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防城和观测方程为:
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:
- 预估计X(k)^= F(k,k-1)·X(k-1)
- 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)'
- 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)'
- 更新估计
X(k)~=X(k)^+K(k)×[Y(k)-H(k)×X(k)^]
- 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)'
- 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:
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============================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;
const double 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 */
const float A[] = {
1, T, 0, 0,
0, 1, 0, 0,
0, 0, 1, T,
0, 0, 0, 1
};
const float H[] = {
1, 0, 0, 0,
0, 0, 0, 0,
0, 0, 1, 0,
0, 0, 0, 0
};
const float 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)
};
const float 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
};
const float 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;
}
// 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
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/* tester de changer les matrices du noises */
/* replace state by cvkalman->state_post ??? */
CvRandState rng;
const double 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 */
const float A[] = {
1, T, 0, 0,
0, 1, 0, 0,
0, 0, 1, T,
0, 0, 0, 1
};
const float H[] = {
1, 0, 0, 0,
0, 0, 0, 0,
0, 0, 1, 0,
0, 0, 0, 0
};
const float 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)
};
const float 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
};
const float 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;
}