从变换矩阵提取无旋转平移信息：SIFT刚性2D变换参数求解

我使用SIFT特征检测器处理两幅经过刚性2D变换的图像，需要直接从变换矩阵中提取旋转角度以及中心平移量。附上最小可运行示例，思路是利用图像配准得到的角度构建旋转矩阵，再通过矩阵乘法反向提取平移矩阵，但结果存在细微误差。该示例的目标是从图像配准结果中还原初始设定的参数（tx、ty、angle）。

import numpy as np
import cv2 as cv, cv2
import matplotlib.pyplot as plt
import math

img=cv2.imread(r"\lena_std.tif",cv2.IMREAD_GRAYSCALE)

tx=0 #parameters to play which should be totally compensated by the code
ty=0
angle=60

rows,cols= img.shape

rot_mat = cv2.getRotationMatrix2D((cols/2,rows/2),angle,1)
rot_mat=np.vstack([rot_mat, np.array([0,0,1])])
trans_mat = np.array([
    [1, 0, tx],
    [0, 1, ty],
    [0, 0, 1]
], dtype=np.float32)
M=trans_mat @ rot_mat

reference_image = img
test_image = cv2.warpAffine(img,M[:2,:3],(cols,rows))

def similarity_sift_flann_affine2D (reference_image, test_image):

    img1=test_image.astype(np.uint8)
    img2=reference_image.astype(np.uint8)

    # Initiate SIFT detector
    sift = cv.SIFT_create()
    # find the keypoints and descriptors with SIFT
    kp1, des1 = sift.detectAndCompute(img1,None)
    kp2, des2 = sift.detectAndCompute(img2,None)
    FLANN_INDEX_KDTREE = 1
    index_params = dict(algorithm = FLANN_INDEX_KDTREE, trees = 5)
    search_params = dict(checks = 50)
    flann = cv.FlannBasedMatcher(index_params, search_params)
    matches = flann.knnMatch(des1,des2,k=2)
    # store all the good matches as per Lowe's ratio test.
    good = []
    for m,n in matches:
        if m.distance < 0.7*n.distance:     #Lowes ratio test
            good.append(m)
        
    MIN_MATCH_COUNT=10
        
    if len(good)>=MIN_MATCH_COUNT:
        src_pts = np.float32([ kp1[m.queryIdx].pt for m in good ]).reshape(-1,1,2)
        dst_pts = np.float32([ kp2[m.trainIdx].pt for m in good ]).reshape(-1,1,2)
        M, mask = cv2.estimateAffinePartial2D(src_pts, dst_pts)
        #M,mask = cv.estimateAffine2D(src_pts, dst_pts)
    
    else:
        print( "Not enough matches are found - {}/{}".format(len(good), MIN_MATCH_COUNT) )
        matchesMask = None
        
    return M

M1=similarity_sift_flann_affine2D (reference_image, test_image)

sizeImg1=img.shape
u, _, vh = np.linalg.svd(M[0:2, 0:2])   
R = u @ vh
angle2 = math.atan2(R[1,0], R[0,0])
angle2 =np.rad2deg(angle2)

cor_rot_mat = cv2.getRotationMatrix2D((cols/2,rows/2),-angle2,1)
cor_rot_mat=np.vstack([cor_rot_mat, np.array([0,0,1])])

M1=np.vstack([M1, np.array([0,0,1])])
#cor_trans_mat= np.linalg.inv(cor_rot_mat) @ M1
cor_trans_mat= M1 @ np.linalg.inv(cor_rot_mat)

print("angle diff", str(np.abs(angle2)-np.abs(angle)))
print("tx diff", str(np.abs(tx)-np.abs(cor_trans_mat[0][2])))
print("ty diff", str(np.abs(ty)-np.abs(cor_trans_mat[1][2])))

恳请各位帮忙解决，非常感谢！

内容的提问来源于stack exchange，提问作者TMul