计算机视觉学习 多视图几何
对极几何
参考:https://blog.csdn.net/my88site/article/details/53967141
原理
对极几何（Epipolar Geometry）描述的是两幅视图之间的内在射影关系，与外部场景无关，只依赖于摄像机内参数和这两幅试图之间的的相对姿态。假设两个相机的内部参数一致，比如焦距、镜头等，为了数学描述的方便，需引入坐标，由于坐标是人为引入的，因此客观世界中的事物可以处于不同的坐标系中。假设两个相机的X轴方向一致，像平面重叠，坐标系以左相机为准，右相机相对于左相机是简单的平移，用坐标表示为(Tx,0,0)。如下所示
实际模型
（1）本质矩阵E
推导过程
左像平面上的一点乘以本质矩阵E，结果为一条直线，该直线就是的对极线，且过在右像平面上的对应点。本质矩阵E的基本性质：秩为2，且仅依赖于外部参数R和T。其中，P表示物点矢量，p表示像点矢量。
（2）基础矩阵
基本矩阵F求解(8点算法)
基本矩阵方程
八点法求解基本矩阵可参考 https://blog.csdn.net/a6333230/article/details/83413835
实验代码：
from PIL import Image
from numpy import *
from pylab import *
import numpy as np
from PCV.geometry import camera
from PCV.geometry import homography
from PCV.geometry import sfm
from PCV.localdescriptors import sift
# Read features
# 载入图像，并计算特征
im1 = array(Image.open('img3.jpg'))
sift.process_image('img3.jpg', 'im1.sift')
l1, d1 = sift.read_features_from_file('im1.sift')
im2 = array(Image.open('img4.jpg'))
sift.process_image('img4.jpg', 'im2.sift')
l2, d2 = sift.read_features_from_file('im2.sift')
# 匹配特征
matches = sift.match_twosided(d1, d2)
ndx = matches.nonzero()[0]
# 使用齐次坐标表示，并使用 inv(K) 归一化
x1 = homography.make_homog(l1[ndx, :2].T)
ndx2 = [int(matches[i]) for i in ndx]
x2 = homography.make_homog(l2[ndx2, :2].T)
x1n = x1.copy()
x2n = x2.copy()
print(len(ndx))
figure(figsize=(16,16))
sift.plot_matches(im1, im2, l1, l2, matches, True)
show()
# Don't use K1, and K2
#def F_from_ransac(x1, x2, model, maxiter=5000, match_threshold=1e-6):
def F_from_ransac(x1, x2, model, maxiter=5000, match_threshold=1e-6):
""" 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. """
from PCV.tools import ransac
data = np.vstack((x1, x2))
d = 20 # 20 is the original
# compute F and return with inlier index
F, ransac_data = ransac.ransac(data.T, model,
8, maxiter, match_threshold, d, return_all=True)
return F, ransac_data['inliers']
# find E through RANSAC
# 使用 RANSAC 方法估计 E
model = sfm.RansacModel()
F, inliers = F_from_ransac(x1n, x2n, model, maxiter=5000, match_threshold=1e-4)
print(len(x1n[0]))
print(len(inliers))
# 计算照相机矩阵（P2 是 4 个解的列表）
P1 = array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]])
P2 = sfm.compute_P_from_fundamental(F)
# triangulate inliers and remove points not in front of both cameras
X = sfm.triangulate(x1n[:, inliers], x2n[:, inliers], P1, P2)
# plot the projection of X
cam1 = camera.Camera(P1)
cam2 = camera.Camera(P2)
x1p = cam1.project(X)
x2p = cam2.project(X)
figure()
imshow(im1)
gray()
plot(x1p[0], x1p[1], 'o')
#plot(x1[0], x1[1], 'r.')
axis('off')
figure()
imshow(im2)
gray()
plot(x2p[0], x2p[1], 'o')
#plot(x2[0], x2[1], 'r.')
axis('off')
show()
figure(figsize=(16, 16))
im3 = sift.appendimages(im1, im2)
im3 = vstack((im3, im3))
imshow(im3)
cols1 = im1.shape[1]
rows1 = im1.shape[0]
for i in range(len(x1p[0])):
if (0<= x1p[0][i]0: #plot([locs1[i][0],locs2[m][0]+cols1],[locs1[i][1],locs2[m][1]],'c')
x1=int(l1[i][0])
y1=int(l1[i][1])
x2=int(l2[int(m)][0])
y2=int(l2[int(m)][1])
# p1 = array([l1[i][0], l1[i][1], 1])
# p2 = array([l2[m][0], l2[m][1], 1])
p1 = array([x1, y1, 1])
p2 = array([x2, y2, 1])
# Use Sampson distance as error
Fx1 = dot(F, p1)
Fx2 = dot(F, p2)
denom = Fx1[0]**2 + Fx1[1]**2 + Fx2[0]**2 + Fx2[1]**2
e = (dot(p1.T, dot(F, p2)))**2 / denom
x1e.append([p1[0], p1[1]])
x2e.append([p2[0], p2[1]])
ers.append(e)
x1e = array(x1e)
x2e = array(x2e)
ers = array(ers)
indices = np.argsort(ers)
x1s = x1e[indices]
x2s = x2e[indices]
ers = ers[indices]
x1s = x1s[:20]
x2s = x2s[:20]
figure(figsize=(16, 16))
im3 = sift.appendimages(im1, im2)
im3 = vstack((im3, im3))
imshow(im3)
cols1 = im1.shape[1]
rows1 = im1.shape[0]
for i in range(len(x1s)):
if (0<= x1s[i][0] 运行结果：
1.室外案例
sift匹配
蓝色为投影特征点，红色为原始特征点
基础矩阵：
投影矩阵：
2.室内案例
sift匹配：
基础矩阵：
投影矩阵·：
