卡爾曼濾波原理

卡爾曼濾波最早可以追溯到Wiener濾波，不同的是卡爾曼采用狀態(tài)空間來(lái)描述它的濾波器，卡爾曼濾波器同時(shí)具有模糊/平滑與預(yù)測(cè)功能，特別是后者在視頻分析與對(duì)象跟蹤應(yīng)用場(chǎng)景中被發(fā)揚(yáng)光大，在離散空間(圖像或者視頻幀)使用卡爾曼濾波器相對(duì)簡(jiǎn)單。假設(shè)我們根據(jù)一個(gè)處理想知道一個(gè)變量值如下：

最終卡爾曼濾波完整的評(píng)估與空間預(yù)測(cè)模型工作流程如下：

OpenCV API

cv::KalmanFilter::KalmanFilter(
    int dynamParams, 
    int measureParams,
    int controlParams = 0,
    int type = CV_32F 
)
# dynamParams表示state的維度
# measureParams表示測(cè)量維度
# controlParams表示控制向量
# type表示創(chuàng)建的matrices

代碼演示

import cv2
from math import cos, sin, sqrt
import numpy as np

if __name__ == "__main__":

    img_height = 500
    img_width = 500
    kalman = cv2.KalmanFilter(2, 1, 0)

    cv2.namedWindow("Kalman", cv2.WINDOW_AUTOSIZE)

    while True:
        state = 0.1 * np.random.randn(2, 1)

        # 初始化
        kalman.transitionMatrix = np.array([[1., 1.], [0., 1.]])
        kalman.measurementMatrix = 1. * np.ones((1, 2))
        kalman.processNoiseCov = 1e-5 * np.eye(2)
        kalman.measurementNoiseCov = 1e-1 * np.ones((1, 1))
        kalman.errorCovPost = 1. * np.ones((2, 2))
        kalman.statePost = 0.1 * np.random.randn(2, 1)

        while True:
            def calc_point(angle):
                return (np.around(img_width/2 + img_width/3*cos(angle), 0).astype(int),
                        np.around(img_height/2 - img_width/3*sin(angle), 1).astype(int))

            state_angle = state[0, 0]
            state_pt = calc_point(state_angle)
            # 預(yù)測(cè)
            prediction = kalman.predict()
            predict_angle = prediction[0, 0]
            predict_pt = calc_point(predict_angle)

            measurement = kalman.measurementNoiseCov * np.random.randn(1, 1)

            # 生成測(cè)量
            measurement = np.dot(kalman.measurementMatrix, state) + measurement
            measurement_angle = measurement[0, 0]
            measurement_pt = calc_point(measurement_angle)

            # plot points
            def draw_cross(center, color, d):
                cv2.line(img,
                         (center[0] - d, center[1] - d), (center[0] + d, center[1] + d),
                         color, 1, cv2.LINE_AA, 0)
                cv2.line(img,
                         (center[0] + d, center[1] - d), (center[0] - d, center[1] + d),
                         color, 1, cv2.LINE_AA, 0)

            img = np.zeros((img_height, img_width, 3), np.uint8)

            cv2.line(img, state_pt, measurement_pt, (0, 0, 255), 3, cv2.LINE_AA, 0)
            cv2.line(img, state_pt, predict_pt, (255, 0, 0), 3, cv2.LINE_AA, 0)

            # 校正預(yù)測(cè)與測(cè)量值差異
            kalman.correct(measurement)

            # 更新noise矩陣與狀態(tài)
            process_noise = sqrt(kalman.processNoiseCov[0,0]) * np.random.randn(2, 1)
            state = np.dot(kalman.transitionMatrix, state) + process_noise

            cv2.imshow("Kalman", img)

            code = cv2.waitKey(100)
            if code != -1:
                break

        if code in [27, ord('q'), ord('Q')]:
            break

    cv2.destroyWindow("Kalman")