请写出SVD计算旋转矩阵的Python代码

以下是使用SVD计算旋转矩阵的Python代码：

import numpy as np

def svd_rotation_matrix(A):
    U, _, V = np.linalg.svd(A)
    R = np.dot(U, V)
    if np.linalg.det(R) < 0:
        R[:, -1] *= -1
    return R

其中，`A`是原始矩阵，`U`、`V`是经过SVD分解后的矩阵，`R`是旋转矩阵。如果计算得到的旋转矩阵的行列式小于0，则将最后一列乘以-1以保证其为旋转矩阵。

相关问题

基于NDT算法写出python代码实现：1.将点云数据划分为若干个体素；2.对每个体素内的点云数据进行高斯分布拟合；3.计算每个体素内的点云数据与目标点云数据的匹配度；4.根据匹配度进行点云配准。

感谢您的提问。对于您的问题，我可以回答。基于NDT算法的python代码实现如下：

1.将点云数据划分为若干个体素：

import numpy as np
from sklearn.neighbors import KDTree

def voxel_downsample(points, voxel_size):
    """
    将点云数据划分为若干个体素
    :param points: 点云数据
    :param voxel_size: 体素大小
    :return: 体素化后的点云数据
    """
    min_point = np.min(points, axis=0)
    max_point = np.max(points, axis=0)
    voxel_grid = np.ceil((max_point - min_point) / voxel_size)
    voxel_grid = voxel_grid.astype(np.int32)
    voxel_points, voxel_point_idx = [], []
    for i in range(voxel_grid[0]):
        for j in range(voxel_grid[1]):
            for k in range(voxel_grid[2]):
                voxel_min = min_point + np.array([i, j, k]) * voxel_size
                voxel_max = voxel_min + voxel_size
                idx = np.where((points[:, 0] >= voxel_min[0]) & (points[:, 0] < voxel_max[0]) &
                               (points[:, 1] >= voxel_min[1]) & (points[:, 1] < voxel_max[1]) &
                               (points[:, 2] >= voxel_min[2]) & (points[:, 2] < voxel_max[2]))[0]
                if len(idx) > 0:
                    voxel_points.append(np.mean(points[idx], axis=0))
                    voxel_point_idx.append(idx)
    return np.array(voxel_points), voxel_point_idx

2.对每个体素内的点云数据进行高斯分布拟合：

from scipy.optimize import minimize

def gauss_newton(x0, y, sigma, f, jacobian, max_iter=20):
    """
    高斯牛顿法拟合高斯分布
    :param x0: 初始参数
    :param y: 数据
    :param sigma: 数据权重
    :param f: 模型函数
    :param jacobian: 模型函数的雅可比矩阵
    :param max_iter: 最大迭代次数
    :return: 拟合后的参数
    """
    x = x0
    for i in range(max_iter):
        J = jacobian(x, y)
        r = f(x, y) - sigma
        H = J.T.dot(np.diag(sigma ** 2)).dot(J)
        g = J.T.dot(sigma ** 2).dot(r)
        dx = np.linalg.solve(H, -g)
        x += dx
        if np.linalg.norm(dx) < 1e-6:
            break
    return x

def gaussian_model(x, y):
    """
    高斯分布模型函数
    :param x: 参数
    :param y: 数据
    :return: 模型值
    """
    mu = x[:3]
    sigma = np.diag(x[3:6])
    det = np.linalg.det(sigma)
    if det == 0:
        return np.zeros(len(y))
    else:
        inv_sigma = np.linalg.inv(sigma)
        diff = y - mu
        exponent = np.exp(-0.5 * np.sum(diff.dot(inv_sigma) * diff, axis=1))
        return exponent / np.sqrt((2 * np.pi) ** 3 * det)

def gaussian_jacobian(x, y):
    """
    高斯分布模型函数的雅可比矩阵
    :param x: 参数
    :param y: 数据
    :return: 雅可比矩阵
    """
    mu = x[:3]
    sigma = np.diag(x[3:6])
    inv_sigma = np.linalg.inv(sigma)
    diff = y - mu
    exponent = np.exp(-0.5 * np.sum(diff.dot(inv_sigma) * diff, axis=1))
    dmu = np.sum(diff.dot(inv_sigma), axis=0)
    dsigma = -0.5 * inv_sigma.dot(diff.T).dot(diff).dot(inv_sigma)
    return np.hstack((np.tile(exponent, (3, 1)).T * dmu, np.tile(exponent, (6, 1)).T * dsigma))

def fit_gaussian(points):
    """
    对每个体素内的点云数据进行高斯分布拟合
    :param points: 体素内的点云数据
    :return: 高斯分布的均值和协方差矩阵
    """
    x0 = np.hstack((np.mean(points, axis=0), np.std(points, axis=0)))
    res = minimize(lambda x: np.sum((gaussian_model(x, points) - 1) ** 2), x0, method='trust-krylov',
                   jac=lambda x: 2 * (gaussian_model(x, points) - 1)[:, np.newaxis] * gaussian_jacobian(x, points),
                   options={'maxiter': 100})
    mu = res.x[:3]
    sigma = np.diag(res.x[3:6])
    return mu, sigma

3.计算每个体素内的点云数据与目标点云数据的匹配度：

def compute_correspondences(source_points, target_points, source_normals, target_normals, voxel_size, max_distance):
    """
    计算每个体素内的点云数据与目标点云数据的匹配度
    :param source_points: 源点云数据
    :param target_points: 目标点云数据
    :param source_normals: 源点云法向量
    :param target_normals: 目标点云法向量
    :param voxel_size: 体素大小
    :param max_distance: 最大匹配距离
    :return: 匹配点对
    """
    source_voxel_points, source_voxel_point_idx = voxel_downsample(source_points, voxel_size)
    target_voxel_points, target_voxel_point_idx = voxel_downsample(target_points, voxel_size)
    source_voxel_normals = np.zeros_like(source_voxel_points)
    target_voxel_normals = np.zeros_like(target_voxel_points)
    for i in range(len(source_voxel_point_idx)):
        idx = source_voxel_point_idx[i]
        source_voxel_normals[i] = np.mean(source_normals[idx], axis=0)
    for i in range(len(target_voxel_point_idx)):
        idx = target_voxel_point_idx[i]
        target_voxel_normals[i] = np.mean(target_normals[idx], axis=0)
    tree = KDTree(target_voxel_points)
    dist, idx = tree.query(source_voxel_points, k=1, distance_upper_bound=max_distance)
    valid_idx = np.where(np.isfinite(dist))[0]
    source_voxel_points = source_voxel_points[valid_idx]
    source_voxel_normals = source_voxel_normals[valid_idx]
    target_voxel_points = target_voxel_points[idx[valid_idx]]
    target_voxel_normals = target_voxel_normals[idx[valid_idx]]
    source_voxel_features = np.hstack((source_voxel_points, source_voxel_normals))
    target_voxel_features = np.hstack((target_voxel_points, target_voxel_normals))
    tree = KDTree(target_voxel_features)
    dist, idx = tree.query(source_voxel_features, k=1)
    valid_idx = np.where(dist < max_distance)[0]
    source_voxel_points = source_voxel_points[valid_idx]
    target_voxel_points = target_voxel_points[idx[valid_idx]]
    return source_voxel_points, target_voxel_points

4.根据匹配度进行点云配准：

from scipy.spatial.transform import Rotation

def icp(source_points, target_points, source_normals, target_normals, voxel_size, max_distance, max_iter=20):
    """
    点云配准
    :param source_points: 源点云数据
    :param target_points: 目标点云数据
    :param source_normals: 源点云法向量
    :param target_normals: 目标点云法向量
    :param voxel_size: 体素大小
    :param max_distance: 最大匹配距离
    :param max_iter: 最大迭代次数
    :return: 配准后的源点云数据
    """
    source_points = source_points.copy()
    source_normals = source_normals.copy()
    for i in range(max_iter):
        source_voxel_points, target_voxel_points = compute_correspondences(source_points, target_points, source_normals,
                                                                           target_normals, voxel_size, max_distance)
        R, t = align_points(source_voxel_points, target_voxel_points)
        source_points = source_points.dot(R.T) + t
        source_normals = source_normals.dot(R.T)
    return source_points

def align_points(source_points, target_points):
    """
    计算点云配准的旋转矩阵和平移向量
    :param source_points: 源点云数据
    :param target_points: 目标点云数据
    :return: 旋转矩阵和平移向量
    """
    mu_s = np.mean(source_points, axis=0)
    mu_t = np.mean(target_points, axis=0)
    source_points_centered = source_points - mu_s
    target_points_centered = target_points - mu_t
    W = source_points_centered.T.dot(target_points_centered)
    U, _, Vt = np.linalg.svd(W)
    S = np.eye(3)
    if np.linalg.det(U) * np.linalg.det(Vt) < 0:
        S[2, 2] = -1
    R = U.dot(S).dot(Vt)
    t = mu_t - mu_s.dot(R.T)
    return R, t

希望这些代码能够帮助到您。

写一段深度相机和机械臂手眼标定的程序

好的，以下是一段深度相机和机械臂手眼标定的程序：

import numpy as np
import cv2
import glob

# 初始化机械臂和深度相机的变量
robot_pose = []
depth_pose = []
robot_pose_inv = []
depth_pose_inv = []

# 读取机械臂和深度相机的标定图像
robot_images = sorted(glob.glob('robot_calib_images/*.jpg'))
depth_images = sorted(glob.glob('depth_calib_images/*.png'))

# 从标定图像中提取角点
robot_criteria = (cv2.TERM_CRITERIA_EPS + cv2.TERM_CRITERIA_MAX_ITER, 30, 0.001)
depth_criteria = (cv2.TERM_CRITERIA_EPS + cv2.TERM_CRITERIA_MAX_ITER, 30, 0.001)
robot_objp = np.zeros((6*7,3), np.float32)
robot_objp[:,:2] = np.mgrid[0:7,0:6].T.reshape(-1,2)
depth_objp = np.zeros((6*7,3), np.float32)
depth_objp[:,:2] = np.mgrid[0:7,0:6].T.reshape(-1,2)
robot_objpoints = []
depth_objpoints = []
robot_imgpoints = []
depth_imgpoints = []
for robot_img, depth_img in zip(robot_images, depth_images):
    robot_img = cv2.imread(robot_img)
    depth_img = cv2.imread(depth_img, cv2.IMREAD_UNCHANGED)
    gray_robot = cv2.cvtColor(robot_img, cv2.COLOR_BGR2GRAY)
    gray_depth = cv2.cvtColor(depth_img, cv2.COLOR_BGR2GRAY)
    ret_robot, corners_robot = cv2.findChessboardCorners(gray_robot, (7,6), None)
    ret_depth, corners_depth = cv2.findChessboardCorners(gray_depth, (7,6), None)
    if ret_robot and ret_depth:
        robot_objpoints.append(robot_objp)
        depth_objpoints.append(depth_objp)
        corners_robot = cv2.cornerSubPix(gray_robot,corners_robot,(11,11),(-1,-1),robot_criteria)
        corners_depth = cv2.cornerSubPix(gray_depth,corners_depth,(11,11),(-1,-1),depth_criteria)
        robot_imgpoints.append(corners_robot)
        depth_imgpoints.append(corners_depth)

# 进行机械臂和深度相机的手眼标定
retval, robot_matrix, robot_distCoeffs, robot_rvecs, robot_tvecs = cv2.calibrateCamera(robot_objpoints, robot_imgpoints, gray_robot.shape[::-1], None, None)
retval, depth_matrix, depth_distCoeffs, depth_rvecs, depth_tvecs = cv2.calibrateCamera(depth_objpoints, depth_imgpoints, gray_depth.shape[::-1], None, None)
for robot_rvec, robot_tvec, depth_rvec, depth_tvec in zip(robot_rvecs, robot_tvecs, depth_rvecs, depth_tvecs):
    robot_rmat, _ = cv2.Rodrigues(robot_rvec)
    depth_rmat, _ = cv2.Rodrigues(depth_rvec)
    robot_pose.append(np.hstack((robot_rmat, robot_tvec)))
    depth_pose.append(np.hstack((depth_rmat, depth_tvec)))
    robot_pose_inv.append(np.hstack((robot_rmat.T, -robot_rmat.T.dot(robot_tvec))))
    depth_pose_inv.append(np.hstack((depth_rmat.T, -depth_rmat.T.dot(depth_tvec))))
robot_pose = np.array(robot_pose)
depth_pose = np.array(depth_pose)
robot_pose_inv = np.array(robot_pose_inv)
depth_pose_inv = np.array(depth_pose_inv)
H = np.zeros((3,3))
for i in range(len(robot_pose)):
    H += np.dot(robot_pose[i], depth_pose_inv[i])
U, S, V = np.linalg.svd(H)
R = np.dot(U, V)
T = np.dot(depth_pose.mean(axis=0)[:3], R.T) - np.dot(robot_pose.mean(axis=0)[:3], R.T)

# 输出标定结果
print('Robot camera matrix:\n', robot_matrix)
print('Robot distortion coefficients:\n', robot_distCoeffs)
print('Depth camera matrix:\n', depth_matrix)
print('Depth distortion coefficients:\n', depth_distCoeffs)
print('Rotation matrix:\n', R)
print('Translation vector:\n', T)

这段程序首先读取了机械臂和深度相机的标定图像，并从中提取出角点。然后，使用这些角点进行机械臂和深度相机的相机内参数标定，得到了相机矩阵和畸变系数。接着，使用这些角点进行机械臂和深度相机的手眼标定，得到了机械臂和深度相机之间的转换矩阵。最后，使用SVD方法求解旋转矩阵和平移向量，输出标定结果。