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：
#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; 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 */ 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 */ 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; }
