bundle adjustment 玩具程序

Bundle Adjustment 是一种用于同时估计多个相机位置，姿态和三维点云的优化算法。下面是一个用Python编写的简单的Bundle Adjustment玩具程序：

import numpy as np
import scipy.sparse as sp
import scipy.sparse.linalg as splinalg

# 生成测试数据
num_cameras = 3
num_points = 5
num_observations = 10
noise_scale = 0.1

# 相机的位置和姿态
camera_positions = np.random.randn(num_cameras, 3)
camera_orientations = np.random.randn(num_cameras, 3)

# 三维点云
points_3d = np.random.randn(num_points, 3)

# 观测到的2D点
observed_points = np.zeros((num_observations, 2))
camera_indices = np.zeros(num_observations, dtype=int)
point_indices = np.zeros(num_observations, dtype=int)

for i in range(num_observations):
    camera_idx = np.random.randint(num_cameras)
    point_idx = np.random.randint(num_points)
    observed_points[i] = np.random.randn(2) + \
        np.dot(camera_orientations[camera_idx], points_3d[point_idx]) + \
        camera_positions[camera_idx]
    observed_points[i] += np.random.randn(2) * noise_scale
    camera_indices[i] = camera_idx
    point_indices[i] = point_idx

# 构建稀疏矩阵
A = sp.lil_matrix((num_observations * 2, num_cameras * 6 + num_points * 3))
b = np.zeros(num_observations * 2)

for i in range(num_observations):
    camera_idx = camera_indices[i]
    point_idx = point_indices[i]
    point_3d = points_3d[point_idx]
    camera_pos = camera_positions[camera_idx]
    camera_orient = camera_orientations[camera_idx]

    # 计算重投影误差
    projected_point = np.dot(camera_orient, point_3d) + camera_pos
    projected_point /= projected_point[2]
    projected_point = projected_point[:2]

    error = observed_points[i] - projected_point

    # 构建雅可比矩阵
    J_camera = np.hstack((point_3d, np.zeros(3), np.eye(3)))
    J_point = np.hstack((np.zeros(2), camera_orient[:2].reshape(2, 1) * point_3d.reshape(1, 3)))
    J = np.vstack((J_camera, J_point))

    A[i * 2: (i+1) * 2, camera_idx * 6: (camera_idx+1) * 6] = J[:, :6]
    A[i * 2: (i+1) * 2, num_cameras * 6 + point_idx * 3: num_cameras * 6 + (point_idx+1) * 3] = J[:, 6:]

    b[i * 2: (i+1) * 2] = error

# 优化
x0 = np.hstack((camera_positions.ravel(), camera_orientations.ravel(), points_3d.ravel()))
res = splinalg.lsmr(A, b, x0=x0)

optimized_positions = res[0][:num_cameras * 3].reshape(num_cameras, 3)
optimized_orientations = res[0][num_cameras * 3: num_cameras * 6].reshape(num_cameras, 3)
optimized_points_3d = res[0][num_cameras * 6:].reshape(num_points, 3)

print("Original Camera Positions:\n", camera_positions)
print("Original Camera Orientations:\n", camera_orientations)
print("Original 3D Points:\n", points_3d)

print("Optimized Camera Positions:\n", optimized_positions)
print("Optimized Camera Orientations:\n", optimized_orientations)
print("Optimized 3D Points:\n", optimized_points_3d)

这个程序生成了一些随机的相机位置、姿态和三维点云，然后随机生成了一些观测到的2D点，并添加了一些高斯噪声。程序使用Bundle Adjustment来估计相机和3D点云的位置，并输出优化后的结果。