完善以下代码：def em_for_alignment(xs: np.ndarray, ys: np.ndarray, num_iter: int = 10) -> Tuple[np.ndarray, np.ndarray]: """ The em algorithm for aligning two point clouds based on affine transformation :param xs: a set of points with size (N, D), N is the number of samples, D is the dimension of points :param ys: a set of points with size (M, D), M is the number of samples, D is the dimension of points :param num_iter: the number of EM iterations :return: ys_new: the aligned points: ys_new = ys @ affine + translation responsibility: the responsibility matrix P=[p(y_m | x_n)] with size (N, M), whose elements indicating the correspondence between the points """ # TODO: implement the EM algorithm of GMM below for point cloud alignment return

Here is the implementation of the EM algorithm for point cloud alignment:

from typing import Tuple
import numpy as np

def em_for_alignment(xs: np.ndarray, ys: np.ndarray, num_iter: int = 10) -> Tuple[np.ndarray, np.ndarray]:
    """
    The em algorithm for aligning two point clouds based on affine transformation
    :param xs: a set of points with size (N, D), N is the number of samples, D is the dimension of points
    :param ys: a set of points with size (M, D), M is the number of samples, D is the dimension of points
    :param num_iter: the number of EM iterations
    :return:
        ys_new: the aligned points: ys_new = ys @ affine + translation
        responsibility: the responsibility matrix P=[p(y_m | x_n)] with size (N, M),
        whose elements indicating the correspondence between the points
    """
    # initialize the affine matrix and translation vector
    affine = np.eye(xs.shape[1])
    translation = np.zeros(xs.shape[1])

    # initialize the responsibility matrix
    responsibility = np.zeros((xs.shape[0], ys.shape[0]))

    for i in range(num_iter):
        # E-step: compute the responsibility matrix
        for n in range(xs.shape[0]):
            for m in range(ys.shape[0]):
                responsibility[n, m] = 1 / (2 * np.pi) ** (xs.shape[1] / 2) * np.exp(
                    -0.5 * np.linalg.norm(xs[n] - ys[m] @ affine - translation) ** 2)

        responsibility /= np.sum(responsibility, axis=1, keepdims=True)

        # M-step: update the affine matrix and translation vector
        xs_weighted = responsibility.T @ xs
        ys_weighted = responsibility.T @ ys
        affine, _, _, _ = np.linalg.lstsq(xs_weighted, ys_weighted, rcond=None)
        translation = np.mean(ys, axis=0) - np.mean(xs @ affine, axis=0)

    # compute the aligned points
    ys_new = ys @ affine + translation

    return ys_new, responsibility

The EM algorithm is used to estimate the affine matrix and translation vector that aligns the two point clouds. In each iteration, the algorithm computes the responsibility matrix that defines the correspondence between the points in the two clouds, and then updates the affine matrix and translation vector based on the weighted least squares solution. Finally, the algorithm computes the aligned points by applying the affine transformation to the original points and adding the translation vector.