推导基础矩阵的方法有很多种，下面介绍一种。 在双相机的拍摄场景中建立一个空间直角坐标系，称为世界坐标系（图1中的蓝色坐标系）。物体点是场景中物体表面的点，如世界坐标系中点P的坐标 相机的光心是相机镜头组的光学中心。以光心为原点，主光轴为Z轴建立空间直角坐标系，称为相机坐标系（图1中的绿色和红色坐标系）。相机坐标系中平面的方程是z=f，像点是像平面上物点的投影，是透视投影。 用一句话来概括相机的拍摄模型，即物点、像点、光心三点一线，称为针孔相机模型。在这个模型中，从世界坐标系到左右相机坐标系的转换是刚性转换，即只包括旋转和平移，所以我们分别使用增广矩阵[R|t]和[R’|t其中，R和R’是3*旋转矩阵3，t和t为平移向量。对于P的齐次化坐标，左右相机坐标系下物点P的坐标分别为 以相机为例，如图2所示，C为相机光心，Z轴为主轴。相机坐标系下的物点坐标与照片左下角的像点坐标p有以下关系： 类型是相机坐标系下的坐标，主点为图像。 设两相机内参数矩阵同为： 物点与像点的关系如下： 并令代入上式，得到： 由于物点、像点和光心三点一线，物点、一对同名点和两个光心必须在同一平面上。我们称这个平面为平面。平面和像平面的交叉线称为极线l。显然，左片上的每个像点p对应于右片上的极线l’，且p’一定在l上面。两个相机光心的连接和右图像平面的交点称为极点，使用e’表示。 极线在右图像平面内 l方程可以表示为Ax By C=0.平面直线方程的一般公式可视为： 因此，我们可以使用三维向量 (A,B,C)来表示极线l’，并且 l方程可以简单地理由e坐标向量与 p得到坐标向量做向量积，即 ，其中： 令[e’]x表示向量积矩阵形式，然后替换同名点之间的变换关系，得到极线的方程为： 因为p’在 l所以显然有： 即得到：

(1)解决基础矩阵

from PIL import Image from numpy import * from pylab import * import numpy as np  import PCV.geometry.camera as camera import PCV.geometry.homography as homography import PCV.geometry.sfm as sfm import PCV.localdescriptors.sift as sift  im1 = array(Image.open('F:/python/pic/7/1.jpg')) sift.process_image('F:/python/pic/7/1.jpg', 'F:/python/pic/7/im1.sift')  im2 = array(Image.open('F:/python/pic/7/2.jpg')) sift.process_image('F:/python/pic/7/2.jpg', 'F:/python/pic/7/im2.sift')  l1, d1 = sift.read_features_from_file('F:/python/pic/7/im1.sift') l2, d2 = sift.read_features_from_file('F:/python/pic/7/im2.sift')  matches = sift.match_twosided(d1, d2)  ndx = matches.nonzero()[0] x1 = homography.make_homog(l1[ndx, :2].T) ndx2 = [int(matches[i]) for i in ndx] x2 = homography.make_homog(l2[ndx2, :2].T)  d1n = d1[ndx] d2n = d2[ndx2] x1n = x1.copy() x2n = x2.copy()  figure(figsize=(16,16)) sift.plot_matches(im1, im2, l1, l2, matches, True) show()  def F_from_ransac(x1, x2, model, maxiter=5000, match_threshold=1e-3):     """ Robust estimation of a fundamental matrix F from point     correspondences using RANSAC (ransac.py from     http://www.scipy.org/Cookbook/RANSAC).      input: x1, x2 (3*n arrays) points in hom. coordinates. """      import PCV.tools.ransac as ransac     data = np.vstack((x1, x2))     d = 10 # 20 is the original     F, ransac_data = ransac.ransac(data.T, model,                                    7, maxiter, match_threshold, d, return_all=True)     return F, ransac_data['inliers']  model = sfm.RansacModel() F, inliers = F_from_ransac(x1n, x2n, model, maxiter=5000, match_threshold=1e-3) print (F)  P1 = array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]]) P2 = sfm.compute_P_from_fundamental(F)  X = sfm.triangulate(x1n[:, inliers], x2n[:, inliers], P1, P2)   cam1 = camera.Camera(P1) cam2 = camera.Camera(P2) x1p = cam1.project(X) x2p = cam2.project(X)  figure(figsize=(16, 16)) imj = sift.appendimages(im1, im2) imj = vstack((imj, imj))  imshow(imj)  cols1 = im1.shape[1] rows1 = im1.shape[0] for i in range(len(x1p[0])):     if (0<= x1p[0][i]<cols1) and (0<= x2p[0][i]<cols1) and (0<=x1p[1][i]<rows1) and (0<=x2p[1][i]<rows1):         plot([x1p[0][i], x2p[0][i] cols1],[x1p[1][i], x2p[1][i]],'c') axis('off') show()  d1p = d1n[inliers] d2p = d2n[inliers]   print (P1) print (P2) 

(2)画出极点和极线

import numpy as np import cv2 as cv from matplotlib import pyplot as plt   def drawlines(img1, img2, lines, pts1, pts2):     ''' img1 - image on which we draw the epilines for the points in img2         lines - corresponding epilines '''     r, c = img1.shape     img1 = cv.cvtColor(img1, cv.COLOR_GRAY2BGR)     img2 = cv.cvtColor(img2, cv.COLOR_GRAY2BGR)     for r, pt1, p2 in zip(lines, pts1, pts2):
        color = tuple(np.random.randint(0, 255, 3).tolist())
        x0, y0 = map(int, [0, -r[2] / r[1]])
        x1, y1 = map(int, [c, -(r[2] + r[0] * c) / r[1]])
        img1 = cv.line(img1, (x0, y0), (x1, y1), color, 1)
        img1 = cv.circle(img1, tuple(pt1), 5, color, -1)
        img2 = cv.circle(img2, tuple(pt2), 5, color, -1)
    return img1, img2


img1 = cv.imread('F:/python/pic/7/1.jpg', 0)  # queryimage # left image
img2 = cv.imread('F:/python/pic/7/2.jpg', 0)  # trainimage # right image
sift = cv.xfeatures2d.SIFT_create()
# find the keypoints and descriptors with SIFT
kp1, des1 = sift.detectAndCompute(img1, None)
kp2, des2 = sift.detectAndCompute(img2, None)
# FLANN parameters
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)
good = []
pts1 = []
pts2 = []
# ratio test as per Lowe's paper
for i, (m, n) in enumerate(matches):
    if m.distance < 0.8 * n.distance:
        good.append(m)
        pts2.append(kp2[m.trainIdx].pt)
        pts1.append(kp1[m.queryIdx].pt)

pts1 = np.int32(pts1)
pts2 = np.int32(pts2)
F, mask = cv.findFundamentalMat(pts1, pts2, cv.FM_LMEDS)
# We select only inlier points
pts1 = pts1[mask.ravel() == 1]
pts2 = pts2[mask.ravel() == 1]

# Find epilines corresponding to points in right image (second image) and
# drawing its lines on left image
lines1 = cv.computeCorrespondEpilines(pts2.reshape(-1, 1, 2), 2, F)
lines1 = lines1.reshape(-1, 3)
img5, img6 = drawlines(img1, img2, lines1, pts1, pts2)
# Find epilines corresponding to points in left image (first image) and
# drawing its lines on right image
lines2 = cv.computeCorrespondEpilines(pts1.reshape(-1, 1, 2), 1, F)
lines2 = lines2.reshape(-1, 3)
img3, img4 = drawlines(img2, img1, lines2, pts2, pts1)
plt.subplot(121), plt.imshow(img5)
plt.subplot(122), plt.imshow(img3)
plt.show()

实验图集： （1）求解基础矩阵 （2）画出极点和极线

实验中出现了ValueError: did not meet fit acceptance criteria，经过反复试验，得出是图片匹配点过少，算法无法继续运行，通用的加大阈值也无法解决该问题，只有重新拍摄照片，才得到能够运行出结果的图片。

百度百科：基础矩阵