python计算机视觉—多视图几何

今天博主带大家领略多视图几何的魅力

基础矩阵

基本矩阵体现了两视图几何（对极几何，epipolar geometry）的内在射影几何（projective geometry）关系，基本矩阵只依赖于摄像机的内参KK和外参R,tR,t。

上图是一个两视图的几何描述，其中OO、O′O′是两个相机的光心，两点连线OO′OO′称为基线，基线与图像平面的交点ee、e′e′称为对极点，其中ll、l′l′分别是图像点x′x′、xx对应的对极线。

上图的左侧相机的图像平面上的一个点xx，反向投影得到射线OXOX。由于点的深度未知，图像平面上的点xx可能是射线上某一深度的3D点XX。而射线OXOX在第二个相机的图像平面上的投影为l′l′。也就是说，给定一对图像，第一幅图像上的每个点xx，在另外一幅图像上存在一条直线l′l′与之对应。换言之，第二幅图像上与点xx对应的点x′x′必定在线l′l′上。

我们可以看到这里存在一个从一副图像上的点到另外一幅图像与之对应的对极线的映射x→l′x→l′。而基本矩阵就表示了这种从点到直线的射影映射关系。

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('1.jpg'))
sift.process_image('1.jpg', 'im1.sift')
l1, d1 = sift.read_features_from_file('im1.sift')

im2 = array(Image.open('2.jpg'))
sift.process_image('2.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.geometry import ransac
    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]<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()

print(F)



x1e = []
x2e = []
ers = []
for i,m in enumerate(matches):
    if m>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]<cols1) and (0<= x2s[i][0]<cols1) and (0<=x1s[i][1]<rows1) and (0<=x2s[i][1]<rows1):
        plot([x1s[i][0], x2s[i][0]+cols1],[x1s[i][1], x2s[i][1]],'c')
axis('off')
show()

实验结果：

室内图像对

在这里插入图片描述


基础矩阵：

相机矩阵：

谢谢阅读！