##Kalman滤波算法 在视频跟踪处理中，预测目标运动轨迹是一项基本任务。目标运动状态估计的目的有三个：一是对目标过去的状态进行平滑；二是对目标现在的运动状态进行滤波；三是对目标未来的运动状态进行预测。物体的运动状态一般包括目标位置、速度、加速度等。著名的Kalman滤波技术就是其中一种，这是一种线性系统估计技术。 OpenCV中自带了kalman滤波的代码和例程，可参照kalman.cpp，它存在于类KalmanFilter中。kalman滤波算法的调用比较方便，主要的难点是了解多个参数和矩阵计算公式。一个总体的思路是，需要了解前一时刻的状态估计值x和当前的观测值y，然后建立状态方程和观测方程。经过一些运算后即可预测下一步的状态。 **一、离散时间线性动态系统的状态方程** Kalman滤波利用线性系统状态方程，通过系统输入输出观测数据，对系统状态进行最优估计的算法。由于观测数据中包括系统中的噪声和干扰的影响，所以最优估计也可看作是滤波过程。一个**线性系统**是采用状态方程、观测方程及其初始条件来描述。线性离散时间系统的一般状态方程可描述为：   其中， 是状态转移矩阵， 是过程噪声增益矩阵。是k时刻目标的状态向量， 是过程噪声，它是具有均值为零、方差矩阵为Q(k)的高斯噪声向量，即：   **二、传感器的观测方程** 传感器的通用观测方程为：   这里， 是传感器在 时刻的观测向量，观测噪声 是具有零均值和正定协方差矩阵R(k)的高斯分布测量噪声向量，即：   **三、初始状态的描述**  初始状态 是高斯的，具有均值 和协方差 ，即：   以上的描述比较抽象，因此记录一个例子加以说明： 例：目标沿x轴作匀速直线运动，过程噪声为速度噪声，试写出目标的状态方程。 解：由题意知，目标的状态为：  用T表示时间间隔，ux表速度噪声，则有：  写成矩阵形式为：  令：  则有：  其中：  为均值等于0，方差为q的高斯噪声。 在OpenCV中自带的例程里面描述了一个一维的运动跟踪，该点在一个圆弧上运动，只有一个自由度即角度。因此只需建立匀速运动模型即可。 例程的路径：C:\opencv\sources\samples\cpp\kalman.cpp ~~~ 在代码中各变量的对应情况如下： 状态估计值X对应：state 当前观测值Z对应：measurement KalmanFilter类内成员变量transitionMatrix即为状态转移方程中的矩阵A KalmanFilter类内成员变量measurementMatrix即为量测方程中矩阵C Mat statePre; //!< predicted state (x'(k)): x(k)=A*x(k-1)+B*u(k) Mat statePost; //!< corrected state (x(k)): x(k)=x'(k)+K(k)*(z(k)-H*x'(k)) Mat transitionMatrix; //!< state transition matrix (A) Mat controlMatrix; //!< control matrix (B) (not used if there is no control) Mat measurementMatrix; //!< measurement matrix (H) Mat processNoiseCov; //!< process noise covariance matrix (Q) Mat measurementNoiseCov;//!< measurement noise covariance matrix (R) Mat errorCovPre; //!< priori error estimate covariance matrix (P'(k)): P'(k)=A*P(k-1)*At + Q)*/ Mat gain; //!< Kalman gain matrix (K(k)): K(k)=P'(k)*Ht*inv(H*P'(k)*Ht+R) Mat errorCovPost; //!< posteriori error estimate covariance matrix (P(k)): P(k)=(I-K(k)*H)*P'(k) ~~~ 以下是OpenCV/modules/video/src/Kalman.cpp的源代码，后续需继续分析这些代码： ~~~ /*M/////////////////////////////////////////////////////////////////////////////////////// // // IMPORTANT: READ BEFORE DOWNLOADING, COPYING, INSTALLING OR USING. // // By downloading, copying, installing or using the software you agree to this license. // If you do not agree to this license, do not download, install, // copy or use the software. // // // Intel License Agreement // For Open Source Computer Vision Library // // Copyright (C) 2000, Intel Corporation, all rights reserved. // Third party copyrights are property of their respective owners. // // Redistribution and use in source and binary forms, with or without modification, // are permitted provided that the following conditions are met: // // * Redistribution's of source code must retain the above copyright notice, // this list of conditions and the following disclaimer. // // * Redistribution's in binary form must reproduce the above copyright notice, // this list of conditions and the following disclaimer in the documentation // and/or other materials provided with the distribution. // // * The name of Intel Corporation may not be used to endorse or promote products // derived from this software without specific prior written permission. // // This software is provided by the copyright holders and contributors "as is" and // any express or implied warranties, including, but not limited to, the implied // warranties of merchantability and fitness for a particular purpose are disclaimed. // In no event shall the Intel Corporation or contributors be liable for any direct, // indirect, incidental, special, exemplary, or consequential damages // (including, but not limited to, procurement of substitute goods or services; // loss of use, data, or profits; or business interruption) however caused // and on any theory of liability, whether in contract, strict liability, // or tort (including negligence or otherwise) arising in any way out of // the use of this software, even if advised of the possibility of such damage. // //M*/ #include "precomp.hpp" CV_IMPL CvKalman* cvCreateKalman( int DP, int MP, int CP ) { CvKalman *kalman = 0; if( DP <= 0 || MP <= 0 ) CV_Error( CV_StsOutOfRange, "state and measurement vectors must have positive number of dimensions" ); if( CP < 0 ) CP = DP; /* allocating memory for the structure */ kalman = (CvKalman *)cvAlloc( sizeof( CvKalman )); memset( kalman, 0, sizeof(*kalman)); kalman->DP = DP; kalman->MP = MP; kalman->CP = CP; kalman->state_pre = cvCreateMat( DP, 1, CV_32FC1 ); cvZero( kalman->state_pre ); kalman->state_post = cvCreateMat( DP, 1, CV_32FC1 ); cvZero( kalman->state_post ); kalman->transition_matrix = cvCreateMat( DP, DP, CV_32FC1 ); cvSetIdentity( kalman->transition_matrix ); kalman->process_noise_cov = cvCreateMat( DP, DP, CV_32FC1 ); cvSetIdentity( kalman->process_noise_cov ); kalman->measurement_matrix = cvCreateMat( MP, DP, CV_32FC1 ); cvZero( kalman->measurement_matrix ); kalman->measurement_noise_cov = cvCreateMat( MP, MP, CV_32FC1 ); cvSetIdentity( kalman->measurement_noise_cov ); kalman->error_cov_pre = cvCreateMat( DP, DP, CV_32FC1 ); kalman->error_cov_post = cvCreateMat( DP, DP, CV_32FC1 ); cvZero( kalman->error_cov_post ); kalman->gain = cvCreateMat( DP, MP, CV_32FC1 ); if( CP > 0 ) { kalman->control_matrix = cvCreateMat( DP, CP, CV_32FC1 ); cvZero( kalman->control_matrix ); } kalman->temp1 = cvCreateMat( DP, DP, CV_32FC1 ); kalman->temp2 = cvCreateMat( MP, DP, CV_32FC1 ); kalman->temp3 = cvCreateMat( MP, MP, CV_32FC1 ); kalman->temp4 = cvCreateMat( MP, DP, CV_32FC1 ); kalman->temp5 = cvCreateMat( MP, 1, CV_32FC1 ); #if 1 kalman->PosterState = kalman->state_pre->data.fl; kalman->PriorState = kalman->state_post->data.fl; kalman->DynamMatr = kalman->transition_matrix->data.fl; kalman->MeasurementMatr = kalman->measurement_matrix->data.fl; kalman->MNCovariance = kalman->measurement_noise_cov->data.fl; kalman->PNCovariance = kalman->process_noise_cov->data.fl; kalman->KalmGainMatr = kalman->gain->data.fl; kalman->PriorErrorCovariance = kalman->error_cov_pre->data.fl; kalman->PosterErrorCovariance = kalman->error_cov_post->data.fl; #endif return kalman; } CV_IMPL void cvReleaseKalman( CvKalman** _kalman ) { CvKalman *kalman; if( !_kalman ) CV_Error( CV_StsNullPtr, "" ); kalman = *_kalman; if( !kalman ) return; /* freeing the memory */ cvReleaseMat( &kalman->state_pre ); cvReleaseMat( &kalman->state_post ); cvReleaseMat( &kalman->transition_matrix ); cvReleaseMat( &kalman->control_matrix ); cvReleaseMat( &kalman->measurement_matrix ); cvReleaseMat( &kalman->process_noise_cov ); cvReleaseMat( &kalman->measurement_noise_cov ); cvReleaseMat( &kalman->error_cov_pre ); cvReleaseMat( &kalman->gain ); cvReleaseMat( &kalman->error_cov_post ); cvReleaseMat( &kalman->temp1 ); cvReleaseMat( &kalman->temp2 ); cvReleaseMat( &kalman->temp3 ); cvReleaseMat( &kalman->temp4 ); cvReleaseMat( &kalman->temp5 ); memset( kalman, 0, sizeof(*kalman)); /* deallocating the structure */ cvFree( _kalman ); } CV_IMPL const CvMat* cvKalmanPredict( CvKalman* kalman, const CvMat* control ) { if( !kalman ) CV_Error( CV_StsNullPtr, "" ); /* update the state */ /* x'(k) = A*x(k) */ cvMatMulAdd( kalman->transition_matrix, kalman->state_post, 0, kalman->state_pre ); if( control && kalman->CP > 0 ) /* x'(k) = x'(k) + B*u(k) */ cvMatMulAdd( kalman->control_matrix, control, kalman->state_pre, kalman->state_pre ); /* update error covariance matrices */ /* temp1 = A*P(k) */ cvMatMulAdd( kalman->transition_matrix, kalman->error_cov_post, 0, kalman->temp1 ); /* P'(k) = temp1*At + Q */ cvGEMM( kalman->temp1, kalman->transition_matrix, 1, kalman->process_noise_cov, 1, kalman->error_cov_pre, CV_GEMM_B_T ); /* handle the case when there will be measurement before the next predict */ cvCopy(kalman->state_pre, kalman->state_post); return kalman->state_pre; } CV_IMPL const CvMat* cvKalmanCorrect( CvKalman* kalman, const CvMat* measurement ) { if( !kalman || !measurement ) CV_Error( CV_StsNullPtr, "" ); /* temp2 = H*P'(k) */ cvMatMulAdd( kalman->measurement_matrix, kalman->error_cov_pre, 0, kalman->temp2 ); /* temp3 = temp2*Ht + R */ cvGEMM( kalman->temp2, kalman->measurement_matrix, 1, kalman->measurement_noise_cov, 1, kalman->temp3, CV_GEMM_B_T ); /* temp4 = inv(temp3)*temp2 = Kt(k) */ cvSolve( kalman->temp3, kalman->temp2, kalman->temp4, CV_SVD ); /* K(k) */ cvTranspose( kalman->temp4, kalman->gain ); /* temp5 = z(k) - H*x'(k) */ cvGEMM( kalman->measurement_matrix, kalman->state_pre, -1, measurement, 1, kalman->temp5 ); /* x(k) = x'(k) + K(k)*temp5 */ cvMatMulAdd( kalman->gain, kalman->temp5, kalman->state_pre, kalman->state_post ); /* P(k) = P'(k) - K(k)*temp2 */ cvGEMM( kalman->gain, kalman->temp2, -1, kalman->error_cov_pre, 1, kalman->error_cov_post, 0 ); return kalman->state_post; } namespace cv { KalmanFilter::KalmanFilter() {} KalmanFilter::KalmanFilter(int dynamParams, int measureParams, int controlParams, int type) { init(dynamParams, measureParams, controlParams, type); } void KalmanFilter::init(int DP, int MP, int CP, int type) { CV_Assert( DP > 0 && MP > 0 ); CV_Assert( type == CV_32F || type == CV_64F ); CP = std::max(CP, 0); statePre = Mat::zeros(DP, 1, type); statePost = Mat::zeros(DP, 1, type); transitionMatrix = Mat::eye(DP, DP, type); processNoiseCov = Mat::eye(DP, DP, type); measurementMatrix = Mat::zeros(MP, DP, type); measurementNoiseCov = Mat::eye(MP, MP, type); errorCovPre = Mat::zeros(DP, DP, type); errorCovPost = Mat::zeros(DP, DP, type); gain = Mat::zeros(DP, MP, type); if( CP > 0 ) controlMatrix = Mat::zeros(DP, CP, type); else controlMatrix.release(); temp1.create(DP, DP, type); temp2.create(MP, DP, type); temp3.create(MP, MP, type); temp4.create(MP, DP, type); temp5.create(MP, 1, type); } const Mat& KalmanFilter::predict(const Mat& control) { // update the state: x'(k) = A*x(k) statePre = transitionMatrix*statePost; if( control.data ) // x'(k) = x'(k) + B*u(k) statePre += controlMatrix*control; // update error covariance matrices: temp1 = A*P(k) temp1 = transitionMatrix*errorCovPost; // P'(k) = temp1*At + Q gemm(temp1, transitionMatrix, 1, processNoiseCov, 1, errorCovPre, GEMM_2_T); // handle the case when there will be measurement before the next predict. statePre.copyTo(statePost); errorCovPre.copyTo(errorCovPost); return statePre; } const Mat& KalmanFilter::correct(const Mat& measurement) { // temp2 = H*P'(k) temp2 = measurementMatrix * errorCovPre; // temp3 = temp2*Ht + R gemm(temp2, measurementMatrix, 1, measurementNoiseCov, 1, temp3, GEMM_2_T); // temp4 = inv(temp3)*temp2 = Kt(k) solve(temp3, temp2, temp4, DECOMP_SVD); // K(k) gain = temp4.t(); // temp5 = z(k) - H*x'(k) temp5 = measurement - measurementMatrix*statePre; // x(k) = x'(k) + K(k)*temp5 statePost = statePre + gain*temp5; // P(k) = P'(k) - K(k)*temp2 errorCovPost = errorCovPre - gain*temp2; return statePost; } } ~~~ 参考链接：[http://blog.csdn.net/yang_xian521/article/details/7050398](http://blog.csdn.net/yang_xian521/article/details/7050398)