box_np_ops.py

# Copyright (c) OpenMMLab. All rights reserved.
# TODO: clean the functions in this file and move the APIs into box bbox_3d
# in the future
# NOTICE: All functions in this file are valid for LiDAR or depth boxes only
# if we use default parameters.

import numba
import numpy as np

from mmdet3d.structures.bbox_3d import (limit_period, points_cam2img,
                                        rotation_3d_in_axis)


def camera_to_lidar(points, r_rect, velo2cam):
    """Convert points in camera coordinate to lidar coordinate.

    Note:
        This function is for KITTI only.

    Args:
        points (np.ndarray, shape=[N, 3]): Points in camera coordinate.
        r_rect (np.ndarray, shape=[4, 4]): Matrix to project points in
            specific camera coordinate (e.g. CAM2) to CAM0.
        velo2cam (np.ndarray, shape=[4, 4]): Matrix to project points in
            camera coordinate to lidar coordinate.

    Returns:
        np.ndarray, shape=[N, 3]: Points in lidar coordinate.
    """
    points_shape = list(points.shape[0:-1])
    if points.shape[-1] == 3:
        points = np.concatenate([points, np.ones(points_shape + [1])], axis=-1)
    lidar_points = points @ np.linalg.inv((r_rect @ velo2cam).T)
    return lidar_points[..., :3]


def box_camera_to_lidar(data, r_rect, velo2cam):
    """Convert boxes in camera coordinate to lidar coordinate.

    Note:
        This function is for KITTI only.

    Args:
        data (np.ndarray, shape=[N, 7]): Boxes in camera coordinate.
        r_rect (np.ndarray, shape=[4, 4]): Matrix to project points in
            specific camera coordinate (e.g. CAM2) to CAM0.
        velo2cam (np.ndarray, shape=[4, 4]): Matrix to project points in
            camera coordinate to lidar coordinate.

    Returns:
        np.ndarray, shape=[N, 3]: Boxes in lidar coordinate.
    """
    xyz = data[:, 0:3]
    x_size, y_size, z_size = data[:, 3:4], data[:, 4:5], data[:, 5:6]
    r = data[:, 6:7]
    xyz_lidar = camera_to_lidar(xyz, r_rect, velo2cam)
    # yaw and dims also needs to be converted
    r_new = -r - np.pi / 2
    r_new = limit_period(r_new, period=np.pi * 2)
    return np.concatenate([xyz_lidar, x_size, z_size, y_size, r_new], axis=1)


def corners_nd(dims, origin=0.5):
    """Generate relative box corners based on length per dim and origin point.

    Args:
        dims (np.ndarray, shape=[N, ndim]): Array of length per dim
        origin (list or array or float, optional): origin point relate to
            smallest point. Defaults to 0.5

    Returns:
        np.ndarray, shape=[N, 2 ** ndim, ndim]: Returned corners.
        point layout example: (2d) x0y0, x0y1, x1y0, x1y1;
            (3d) x0y0z0, x0y0z1, x0y1z0, x0y1z1, x1y0z0, x1y0z1, x1y1z0, x1y1z1
            where x0 < x1, y0 < y1, z0 < z1.
    """
    ndim = int(dims.shape[1])
    corners_norm = np.stack(
        np.unravel_index(np.arange(2**ndim), [2] * ndim),
        axis=1).astype(dims.dtype)
    # now corners_norm has format: (2d) x0y0, x0y1, x1y0, x1y1
    # (3d) x0y0z0, x0y0z1, x0y1z0, x0y1z1, x1y0z0, x1y0z1, x1y1z0, x1y1z1
    # so need to convert to a format which is convenient to do other computing.
    # for 2d boxes, format is clockwise start with minimum point
    # for 3d boxes, please draw lines by your hand.
    if ndim == 2:
        # generate clockwise box corners
        corners_norm = corners_norm[[0, 1, 3, 2]]
    elif ndim == 3:
        corners_norm = corners_norm[[0, 1, 3, 2, 4, 5, 7, 6]]
    corners_norm = corners_norm - np.array(origin, dtype=dims.dtype)
    corners = dims.reshape([-1, 1, ndim]) * corners_norm.reshape(
        [1, 2**ndim, ndim])
    return corners


def center_to_corner_box2d(centers, dims, angles=None, origin=0.5):
    """Convert kitti locations, dimensions and angles to corners.
    format: center(xy), dims(xy), angles(counterclockwise when positive)

    Args:
        centers (np.ndarray): Locations in kitti label file with shape (N, 2).
        dims (np.ndarray): Dimensions in kitti label file with shape (N, 2).
        angles (np.ndarray, optional): Rotation_y in kitti label file with
            shape (N). Defaults to None.
        origin (list or array or float, optional): origin point relate to
            smallest point. Defaults to 0.5.

    Returns:
        np.ndarray: Corners with the shape of (N, 4, 2).
    """
    # 'length' in kitti format is in x axis.
    # xyz(hwl)(kitti label file)<->xyz(lhw)(camera)<->z(-x)(-y)(wlh)(lidar)
    # center in kitti format is [0.5, 1.0, 0.5] in xyz.
    corners = corners_nd(dims, origin=origin)
    # corners: [N, 4, 2]
    if angles is not None:
        corners = rotation_3d_in_axis(corners, angles)
    corners += centers.reshape([-1, 1, 2])
    return corners


@numba.jit(nopython=True)
def depth_to_points(depth, trunc_pixel):
    """Convert depth map to points.

    Args:
        depth (np.array, shape=[H, W]): Depth map which
            the row of [0~`trunc_pixel`] are truncated.
        trunc_pixel (int): The number of truncated row.

    Returns:
        np.ndarray: Points in camera coordinates.
    """
    num_pts = np.sum(depth[trunc_pixel:, ] > 0.1)
    points = np.zeros((num_pts, 3), dtype=depth.dtype)
    x = np.array([0, 0, 1], dtype=depth.dtype)
    k = 0
    for i in range(trunc_pixel, depth.shape[0]):
        for j in range(depth.shape[1]):
            if depth[i, j] > 0.1:
                x = np.array([j, i, 1], dtype=depth.dtype)
                points[k] = x * depth[i, j]
                k += 1
    return points


def depth_to_lidar_points(depth, trunc_pixel, P2, r_rect, velo2cam):
    """Convert depth map to points in lidar coordinate.

    Args:
        depth (np.array, shape=[H, W]): Depth map which
            the row of [0~`trunc_pixel`] are truncated.
        trunc_pixel (int): The number of truncated row.
        P2 (p.array, shape=[4, 4]): Intrinsics of Camera2.
        r_rect (np.ndarray, shape=[4, 4]): Matrix to project points in
            specific camera coordinate (e.g. CAM2) to CAM0.
        velo2cam (np.ndarray, shape=[4, 4]): Matrix to project points in
            camera coordinate to lidar coordinate.

    Returns:
        np.ndarray: Points in lidar coordinates.
    """
    pts = depth_to_points(depth, trunc_pixel)
    points_shape = list(pts.shape[0:-1])
    points = np.concatenate([pts, np.ones(points_shape + [1])], axis=-1)
    points = points @ np.linalg.inv(P2.T)
    lidar_points = camera_to_lidar(points, r_rect, velo2cam)
    return lidar_points


def center_to_corner_box3d(centers,
                           dims,
                           angles=None,
                           origin=(0.5, 1.0, 0.5),
                           axis=1):
    """Convert kitti locations, dimensions and angles to corners.

    Args:
        centers (np.ndarray): Locations in kitti label file with shape (N, 3).
        dims (np.ndarray): Dimensions in kitti label file with shape (N, 3).
        angles (np.ndarray, optional): Rotation_y in kitti label file with
            shape (N). Defaults to None.
        origin (list or array or float, optional): Origin point relate to
            smallest point. Use (0.5, 1.0, 0.5) in camera and (0.5, 0.5, 0)
            in lidar. Defaults to (0.5, 1.0, 0.5).
        axis (int, optional): Rotation axis. 1 for camera and 2 for lidar.
            Defaults to 1.

    Returns:
        np.ndarray: Corners with the shape of (N, 8, 3).
    """
    # 'length' in kitti format is in x axis.
    # yzx(hwl)(kitti label file)<->xyz(lhw)(camera)<->z(-x)(-y)(lwh)(lidar)
    # center in kitti format is [0.5, 1.0, 0.5] in xyz.
    corners = corners_nd(dims, origin=origin)
    # corners: [N, 8, 3]
    if angles is not None:
        corners = rotation_3d_in_axis(corners, angles, axis=axis)
    corners += centers.reshape([-1, 1, 3])
    return corners


@numba.jit(nopython=True)
def box2d_to_corner_jit(boxes):
    """Convert box2d to corner.

    Args:
        boxes (np.ndarray, shape=[N, 5]): Boxes2d with rotation.

    Returns:
        box_corners (np.ndarray, shape=[N, 4, 2]): Box corners.
    """
    num_box = boxes.shape[0]
    corners_norm = np.zeros((4, 2), dtype=boxes.dtype)
    corners_norm[1, 1] = 1.0
    corners_norm[2] = 1.0
    corners_norm[3, 0] = 1.0
    corners_norm -= np.array([0.5, 0.5], dtype=boxes.dtype)
    corners = boxes.reshape(num_box, 1, 5)[:, :, 2:4] * corners_norm.reshape(
        1, 4, 2)
    rot_mat_T = np.zeros((2, 2), dtype=boxes.dtype)
    box_corners = np.zeros((num_box, 4, 2), dtype=boxes.dtype)
    for i in range(num_box):
        rot_sin = np.sin(boxes[i, -1])
        rot_cos = np.cos(boxes[i, -1])
        rot_mat_T[0, 0] = rot_cos
        rot_mat_T[0, 1] = rot_sin
        rot_mat_T[1, 0] = -rot_sin
        rot_mat_T[1, 1] = rot_cos
        box_corners[i] = corners[i] @ rot_mat_T + boxes[i, :2]
    return box_corners


@numba.njit
def corner_to_standup_nd_jit(boxes_corner):
    """Convert boxes_corner to aligned (min-max) boxes.

    Args:
        boxes_corner (np.ndarray, shape=[N, 2**dim, dim]): Boxes corners.

    Returns:
        np.ndarray, shape=[N, dim*2]: Aligned (min-max) boxes.
    """
    num_boxes = boxes_corner.shape[0]
    ndim = boxes_corner.shape[-1]
    result = np.zeros((num_boxes, ndim * 2), dtype=boxes_corner.dtype)
    for i in range(num_boxes):
        for j in range(ndim):
            result[i, j] = np.min(boxes_corner[i, :, j])
        for j in range(ndim):
            result[i, j + ndim] = np.max(boxes_corner[i, :, j])
    return result


@numba.jit(nopython=True)
def corner_to_surfaces_3d_jit(corners):
    """Convert 3d box corners from corner function above to surfaces that
    normal vectors all direct to internal.

    Args:
        corners (np.ndarray): 3d box corners with the shape of (N, 8, 3).

    Returns:
        np.ndarray: Surfaces with the shape of (N, 6, 4, 3).
    """
    # box_corners: [N, 8, 3], must from corner functions in this module
    num_boxes = corners.shape[0]
    surfaces = np.zeros((num_boxes, 6, 4, 3), dtype=corners.dtype)
    corner_idxes = np.array([
        0, 1, 2, 3, 7, 6, 5, 4, 0, 3, 7, 4, 1, 5, 6, 2, 0, 4, 5, 1, 3, 2, 6, 7
    ]).reshape(6, 4)
    for i in range(num_boxes):
        for j in range(6):
            for k in range(4):
                surfaces[i, j, k] = corners[i, corner_idxes[j, k]]
    return surfaces


def rotation_points_single_angle(points, angle, axis=0):
    """Rotate points with a single angle.

    Args:
        points (np.ndarray, shape=[N, 3]]):
        angle (np.ndarray, shape=[1]]):
        axis (int, optional): Axis to rotate at. Defaults to 0.

    Returns:
        np.ndarray: Rotated points.
    """
    # points: [N, 3]
    rot_sin = np.sin(angle)
    rot_cos = np.cos(angle)
    if axis == 1:
        rot_mat_T = np.array(
            [[rot_cos, 0, rot_sin], [0, 1, 0], [-rot_sin, 0, rot_cos]],
            dtype=points.dtype)
    elif axis == 2 or axis == -1:
        rot_mat_T = np.array(
            [[rot_cos, rot_sin, 0], [-rot_sin, rot_cos, 0], [0, 0, 1]],
            dtype=points.dtype)
    elif axis == 0:
        rot_mat_T = np.array(
            [[1, 0, 0], [0, rot_cos, rot_sin], [0, -rot_sin, rot_cos]],
            dtype=points.dtype)
    else:
        raise ValueError('axis should in range')

    return points @ rot_mat_T, rot_mat_T


def box3d_to_bbox(box3d, P2):
    """Convert box3d in camera coordinates to bbox in image coordinates.

    Args:
        box3d (np.ndarray, shape=[N, 7]): Boxes in camera coordinate.
        P2 (np.array, shape=[4, 4]): Intrinsics of Camera2.

    Returns:
        np.ndarray, shape=[N, 4]: Boxes 2d in image coordinates.
    """
    box_corners = center_to_corner_box3d(
        box3d[:, :3], box3d[:, 3:6], box3d[:, 6], [0.5, 1.0, 0.5], axis=1)
    box_corners_in_image = points_cam2img(box_corners, P2)
    # box_corners_in_image: [N, 8, 2]
    minxy = np.min(box_corners_in_image, axis=1)
    maxxy = np.max(box_corners_in_image, axis=1)
    bbox = np.concatenate([minxy, maxxy], axis=1)
    return bbox


def corner_to_surfaces_3d(corners):
    """convert 3d box corners from corner function above to surfaces that
    normal vectors all direct to internal.

    Args:
        corners (np.ndarray): 3D box corners with shape of (N, 8, 3).

    Returns:
        np.ndarray: Surfaces with the shape of (N, 6, 4, 3).
    """
    # box_corners: [N, 8, 3], must from corner functions in this module
    surfaces = np.array([
        [corners[:, 0], corners[:, 1], corners[:, 2], corners[:, 3]],
        [corners[:, 7], corners[:, 6], corners[:, 5], corners[:, 4]],
        [corners[:, 0], corners[:, 3], corners[:, 7], corners[:, 4]],
        [corners[:, 1], corners[:, 5], corners[:, 6], corners[:, 2]],
        [corners[:, 0], corners[:, 4], corners[:, 5], corners[:, 1]],
        [corners[:, 3], corners[:, 2], corners[:, 6], corners[:, 7]],
    ]).transpose([2, 0, 1, 3])
    return surfaces


def points_in_rbbox(points, rbbox, z_axis=2, origin=(0.5, 0.5, 0)):
    """Check points in rotated bbox and return indices.

    Note:
        This function is for counterclockwise boxes.

    Args:
        points (np.ndarray, shape=[N, 3+dim]): Points to query.
        rbbox (np.ndarray, shape=[M, 7]): Boxes3d with rotation.
        z_axis (int, optional): Indicate which axis is height.
            Defaults to 2.
        origin (tuple[int], optional): Indicate the position of
            box center. Defaults to (0.5, 0.5, 0).

    Returns:
        np.ndarray, shape=[N, M]: Indices of points in each box.
    """
    # TODO: this function is different from PointCloud3D, be careful
    # when start to use nuscene, check the input
    rbbox_corners = center_to_corner_box3d(
        rbbox[:, :3], rbbox[:, 3:6], rbbox[:, 6], origin=origin, axis=z_axis)
    surfaces = corner_to_surfaces_3d(rbbox_corners)
    indices = points_in_convex_polygon_3d_jit(points[:, :3], surfaces)
    return indices


def minmax_to_corner_2d(minmax_box):
    """Convert minmax box to corners2d.

    Args:
        minmax_box (np.ndarray, shape=[N, dims]): minmax boxes.

    Returns:
        np.ndarray: 2d corners of boxes
    """
    ndim = minmax_box.shape[-1] // 2
    center = minmax_box[..., :ndim]
    dims = minmax_box[..., ndim:] - center
    return center_to_corner_box2d(center, dims, origin=0.0)


def create_anchors_3d_range(feature_size,
                            anchor_range,
                            sizes=((3.9, 1.6, 1.56), ),
                            rotations=(0, np.pi / 2),
                            dtype=np.float32):
    """Create anchors 3d by range.

    Args:
        feature_size (list[float] | tuple[float]): Feature map size. It is
            either a list of a tuple of [D, H, W](in order of z, y, and x).
        anchor_range (torch.Tensor | list[float]): Range of anchors with
            shape [6]. The order is consistent with that of anchors, i.e.,
            (x_min, y_min, z_min, x_max, y_max, z_max).
        sizes (list[list] | np.ndarray | torch.Tensor, optional):
            Anchor size with shape [N, 3], in order of x, y, z.
            Defaults to ((3.9, 1.6, 1.56), ).
        rotations (list[float] | np.ndarray | torch.Tensor, optional):
            Rotations of anchors in a single feature grid.
            Defaults to (0, np.pi / 2).
        dtype (type, optional): Data type. Defaults to np.float32.

    Returns:
        np.ndarray: Range based anchors with shape of
            (*feature_size, num_sizes, num_rots, 7).
    """
    anchor_range = np.array(anchor_range, dtype)
    z_centers = np.linspace(
        anchor_range[2], anchor_range[5], feature_size[0], dtype=dtype)
    y_centers = np.linspace(
        anchor_range[1], anchor_range[4], feature_size[1], dtype=dtype)
    x_centers = np.linspace(
        anchor_range[0], anchor_range[3], feature_size[2], dtype=dtype)
    sizes = np.reshape(np.array(sizes, dtype=dtype), [-1, 3])
    rotations = np.array(rotations, dtype=dtype)
    rets = np.meshgrid(
        x_centers, y_centers, z_centers, rotations, indexing='ij')
    tile_shape = [1] * 5
    tile_shape[-2] = int(sizes.shape[0])
    for i in range(len(rets)):
        rets[i] = np.tile(rets[i][..., np.newaxis, :], tile_shape)
        rets[i] = rets[i][..., np.newaxis]  # for concat
    sizes = np.reshape(sizes, [1, 1, 1, -1, 1, 3])
    tile_size_shape = list(rets[0].shape)
    tile_size_shape[3] = 1
    sizes = np.tile(sizes, tile_size_shape)
    rets.insert(3, sizes)
    ret = np.concatenate(rets, axis=-1)
    return np.transpose(ret, [2, 1, 0, 3, 4, 5])


def center_to_minmax_2d(centers, dims, origin=0.5):
    """Center to minmax.

    Args:
        centers (np.ndarray): Center points.
        dims (np.ndarray): Dimensions.
        origin (list or array or float, optional): Origin point relate
            to smallest point. Defaults to 0.5.

    Returns:
        np.ndarray: Minmax points.
    """
    if origin == 0.5:
        return np.concatenate([centers - dims / 2, centers + dims / 2],
                              axis=-1)
    corners = center_to_corner_box2d(centers, dims, origin=origin)
    return corners[:, [0, 2]].reshape([-1, 4])


def rbbox2d_to_near_bbox(rbboxes):
    """convert rotated bbox to nearest 'standing' or 'lying' bbox.

    Args:
        rbboxes (np.ndarray): Rotated bboxes with shape of
            (N, 5(x, y, xdim, ydim, rad)).

    Returns:
        np.ndarray: Bounding boxes with the shape of
            (N, 4(xmin, ymin, xmax, ymax)).
    """
    rots = rbboxes[..., -1]
    rots_0_pi_div_2 = np.abs(limit_period(rots, 0.5, np.pi))
    cond = (rots_0_pi_div_2 > np.pi / 4)[..., np.newaxis]
    bboxes_center = np.where(cond, rbboxes[:, [0, 1, 3, 2]], rbboxes[:, :4])
    bboxes = center_to_minmax_2d(bboxes_center[:, :2], bboxes_center[:, 2:])
    return bboxes


@numba.jit(nopython=True)
def iou_jit(boxes, query_boxes, mode='iou', eps=0.0):
    """Calculate box iou. Note that jit version runs ~10x faster than the
    box_overlaps function in mmdet3d.core.evaluation.

    Note:
        This function is for counterclockwise boxes.

    Args:
        boxes (np.ndarray): Input bounding boxes with shape of (N, 4).
        query_boxes (np.ndarray): Query boxes with shape of (K, 4).
        mode (str, optional): IoU mode. Defaults to 'iou'.
        eps (float, optional): Value added to denominator. Defaults to 0.

    Returns:
        np.ndarray: Overlap between boxes and query_boxes
            with the shape of [N, K].
    """
    N = boxes.shape[0]
    K = query_boxes.shape[0]
    overlaps = np.zeros((N, K), dtype=boxes.dtype)
    for k in range(K):
        box_area = ((query_boxes[k, 2] - query_boxes[k, 0] + eps) *
                    (query_boxes[k, 3] - query_boxes[k, 1] + eps))
        for n in range(N):
            iw = (
                min(boxes[n, 2], query_boxes[k, 2]) -
                max(boxes[n, 0], query_boxes[k, 0]) + eps)
            if iw > 0:
                ih = (
                    min(boxes[n, 3], query_boxes[k, 3]) -
                    max(boxes[n, 1], query_boxes[k, 1]) + eps)
                if ih > 0:
                    if mode == 'iou':
                        ua = ((boxes[n, 2] - boxes[n, 0] + eps) *
                              (boxes[n, 3] - boxes[n, 1] + eps) + box_area -
                              iw * ih)
                    else:
                        ua = ((boxes[n, 2] - boxes[n, 0] + eps) *
                              (boxes[n, 3] - boxes[n, 1] + eps))
                    overlaps[n, k] = iw * ih / ua
    return overlaps


def projection_matrix_to_CRT_kitti(proj):
    """Split projection matrix of KITTI.

    Note:
        This function is for KITTI only.

    P = C @ [R|T]
    C is upper triangular matrix, so we need to inverse CR and use QR
    stable for all kitti camera projection matrix.

    Args:
        proj (p.array, shape=[4, 4]): Intrinsics of camera.

    Returns:
        tuple[np.ndarray]: Splited matrix of C, R and T.
    """

    CR = proj[0:3, 0:3]
    CT = proj[0:3, 3]
    RinvCinv = np.linalg.inv(CR)
    Rinv, Cinv = np.linalg.qr(RinvCinv)
    C = np.linalg.inv(Cinv)
    R = np.linalg.inv(Rinv)
    T = Cinv @ CT
    return C, R, T


def remove_outside_points(points, rect, Trv2c, P2, image_shape):
    """Remove points which are outside of image.

    Note:
        This function is for KITTI only.

    Args:
        points (np.ndarray, shape=[N, 3+dims]): Total points.
        rect (np.ndarray, shape=[4, 4]): Matrix to project points in
            specific camera coordinate (e.g. CAM2) to CAM0.
        Trv2c (np.ndarray, shape=[4, 4]): Matrix to project points in
            camera coordinate to lidar coordinate.
        P2 (p.array, shape=[4, 4]): Intrinsics of Camera2.
        image_shape (list[int]): Shape of image.

    Returns:
        np.ndarray, shape=[N, 3+dims]: Filtered points.
    """
    # 5x faster than remove_outside_points_v1(2ms vs 10ms)
    C, R, T = projection_matrix_to_CRT_kitti(P2)
    image_bbox = [0, 0, image_shape[1], image_shape[0]]
    frustum = get_frustum(image_bbox, C)
    frustum -= T
    frustum = np.linalg.inv(R) @ frustum.T
    frustum = camera_to_lidar(frustum.T, rect, Trv2c)
    frustum_surfaces = corner_to_surfaces_3d_jit(frustum[np.newaxis, ...])
    indices = points_in_convex_polygon_3d_jit(points[:, :3], frustum_surfaces)
    points = points[indices.reshape([-1])]
    return points


def get_frustum(bbox_image, C, near_clip=0.001, far_clip=100):
    """Get frustum corners in camera coordinates.

    Args:
        bbox_image (list[int]): box in image coordinates.
        C (np.ndarray): Intrinsics.
        near_clip (float, optional): Nearest distance of frustum.
            Defaults to 0.001.
        far_clip (float, optional): Farthest distance of frustum.
            Defaults to 100.

    Returns:
        np.ndarray, shape=[8, 3]: coordinates of frustum corners.
    """
    fku = C[0, 0]
    fkv = -C[1, 1]
    u0v0 = C[0:2, 2]
    z_points = np.array(
        [near_clip] * 4 + [far_clip] * 4, dtype=C.dtype)[:, np.newaxis]
    b = bbox_image
    box_corners = np.array(
        [[b[0], b[1]], [b[0], b[3]], [b[2], b[3]], [b[2], b[1]]],
        dtype=C.dtype)
    near_box_corners = (box_corners - u0v0) / np.array(
        [fku / near_clip, -fkv / near_clip], dtype=C.dtype)
    far_box_corners = (box_corners - u0v0) / np.array(
        [fku / far_clip, -fkv / far_clip], dtype=C.dtype)
    ret_xy = np.concatenate([near_box_corners, far_box_corners],
                            axis=0)  # [8, 2]
    ret_xyz = np.concatenate([ret_xy, z_points], axis=1)
    return ret_xyz


def surface_equ_3d(polygon_surfaces):
    """

    Args:
        polygon_surfaces (np.ndarray): Polygon surfaces with shape of
            [num_polygon, max_num_surfaces, max_num_points_of_surface, 3].
            All surfaces' normal vector must direct to internal.
            Max_num_points_of_surface must at least 3.

    Returns:
        tuple: normal vector and its direction.
    """
    # return [a, b, c], d in ax+by+cz+d=0
    # polygon_surfaces: [num_polygon, num_surfaces, num_points_of_polygon, 3]
    surface_vec = polygon_surfaces[:, :, :2, :] - \
        polygon_surfaces[:, :, 1:3, :]
    # normal_vec: [..., 3]
    normal_vec = np.cross(surface_vec[:, :, 0, :], surface_vec[:, :, 1, :])
    # print(normal_vec.shape, points[..., 0, :].shape)
    # d = -np.inner(normal_vec, points[..., 0, :])
    d = np.einsum('aij, aij->ai', normal_vec, polygon_surfaces[:, :, 0, :])
    return normal_vec, -d


@numba.njit
def _points_in_convex_polygon_3d_jit(points, polygon_surfaces, normal_vec, d,
                                     num_surfaces):
    """
    Args:
        points (np.ndarray): Input points with shape of (num_points, 3).
        polygon_surfaces (np.ndarray): Polygon surfaces with shape of
            (num_polygon, max_num_surfaces, max_num_points_of_surface, 3).
            All surfaces' normal vector must direct to internal.
            Max_num_points_of_surface must at least 3.
        normal_vec (np.ndarray): Normal vector of polygon_surfaces.
        d (int): Directions of normal vector.
        num_surfaces (np.ndarray): Number of surfaces a polygon contains
            shape of (num_polygon).

    Returns:
        np.ndarray: Result matrix with the shape of [num_points, num_polygon].
    """
    max_num_surfaces, max_num_points_of_surface = polygon_surfaces.shape[1:3]
    num_points = points.shape[0]
    num_polygons = polygon_surfaces.shape[0]
    ret = np.ones((num_points, num_polygons), dtype=np.bool_)
    sign = 0.0
    for i in range(num_points):
        for j in range(num_polygons):
            for k in range(max_num_surfaces):
                if k > num_surfaces[j]:
                    break
                sign = (
                    points[i, 0] * normal_vec[j, k, 0] +
                    points[i, 1] * normal_vec[j, k, 1] +
                    points[i, 2] * normal_vec[j, k, 2] + d[j, k])
                if sign >= 0:
                    ret[i, j] = False
                    break
    return ret


def points_in_convex_polygon_3d_jit(points,
                                    polygon_surfaces,
                                    num_surfaces=None):
    """Check points is in 3d convex polygons.

    Args:
        points (np.ndarray): Input points with shape of (num_points, 3).
        polygon_surfaces (np.ndarray): Polygon surfaces with shape of
            (num_polygon, max_num_surfaces, max_num_points_of_surface, 3).
            All surfaces' normal vector must direct to internal.
            Max_num_points_of_surface must at least 3.
        num_surfaces (np.ndarray, optional): Number of surfaces a polygon
            contains shape of (num_polygon). Defaults to None.

    Returns:
        np.ndarray: Result matrix with the shape of [num_points, num_polygon].
    """
    max_num_surfaces, max_num_points_of_surface = polygon_surfaces.shape[1:3]
    # num_points = points.shape[0]
    num_polygons = polygon_surfaces.shape[0]
    if num_surfaces is None:
        num_surfaces = np.full((num_polygons, ), 9999999, dtype=np.int64)
    normal_vec, d = surface_equ_3d(polygon_surfaces[:, :, :3, :])
    # normal_vec: [num_polygon, max_num_surfaces, 3]
    # d: [num_polygon, max_num_surfaces]
    return _points_in_convex_polygon_3d_jit(points, polygon_surfaces,
                                            normal_vec, d, num_surfaces)


@numba.njit
def points_in_convex_polygon_jit(points, polygon, clockwise=False):
    """Check points is in 2d convex polygons. True when point in polygon.

    Args:
        points (np.ndarray): Input points with the shape of [num_points, 2].
        polygon (np.ndarray): Input polygon with the shape of
            [num_polygon, num_points_of_polygon, 2].
        clockwise (bool, optional): Indicate polygon is clockwise. Defaults
            to True.

    Returns:
        np.ndarray: Result matrix with the shape of [num_points, num_polygon].
    """
    # first convert polygon to directed lines
    num_points_of_polygon = polygon.shape[1]
    num_points = points.shape[0]
    num_polygons = polygon.shape[0]
    # vec for all the polygons
    if clockwise:
        vec1 = polygon - polygon[:,
                                 np.array([num_points_of_polygon - 1] + list(
                                     range(num_points_of_polygon - 1))), :]
    else:
        vec1 = polygon[:,
                       np.array([num_points_of_polygon - 1] +
                                list(range(num_points_of_polygon -
                                           1))), :] - polygon
    ret = np.zeros((num_points, num_polygons), dtype=np.bool_)
    success = True
    cross = 0.0
    for i in range(num_points):
        for j in range(num_polygons):
            success = True
            for k in range(num_points_of_polygon):
                vec = vec1[j, k]
                cross = vec[1] * (polygon[j, k, 0] - points[i, 0])
                cross -= vec[0] * (polygon[j, k, 1] - points[i, 1])
                if cross >= 0:
                    success = False
                    break
            ret[i, j] = success
    return ret


def boxes3d_to_corners3d_lidar(boxes3d, bottom_center=True):
    """Convert kitti center boxes to corners.

        7 -------- 4
       /|         /|
      6 -------- 5 .
      | |        | |
      . 3 -------- 0
      |/         |/
      2 -------- 1

    Note:
        This function is for LiDAR boxes only.

    Args:
        boxes3d (np.ndarray): Boxes with shape of (N, 7)
            [x, y, z, x_size, y_size, z_size, ry] in LiDAR coords,
            see the definition of ry in KITTI dataset.
        bottom_center (bool, optional): Whether z is on the bottom center
            of object. Defaults to True.

    Returns:
        np.ndarray: Box corners with the shape of [N, 8, 3].
    """
    boxes_num = boxes3d.shape[0]
    x_size, y_size, z_size = boxes3d[:, 3], boxes3d[:, 4], boxes3d[:, 5]
    x_corners = np.array([
        x_size / 2., -x_size / 2., -x_size / 2., x_size / 2., x_size / 2.,
        -x_size / 2., -x_size / 2., x_size / 2.
    ],
                         dtype=np.float32).T
    y_corners = np.array([
        -y_size / 2., -y_size / 2., y_size / 2., y_size / 2., -y_size / 2.,
        -y_size / 2., y_size / 2., y_size / 2.
    ],
                         dtype=np.float32).T
    if bottom_center:
        z_corners = np.zeros((boxes_num, 8), dtype=np.float32)
        z_corners[:, 4:8] = z_size.reshape(boxes_num, 1).repeat(
            4, axis=1)  # (N, 8)
    else:
        z_corners = np.array([
            -z_size / 2., -z_size / 2., -z_size / 2., -z_size / 2.,
            z_size / 2., z_size / 2., z_size / 2., z_size / 2.
        ],
                             dtype=np.float32).T

    ry = boxes3d[:, 6]
    zeros, ones = np.zeros(
        ry.size, dtype=np.float32), np.ones(
            ry.size, dtype=np.float32)
    rot_list = np.array([[np.cos(ry), np.sin(ry), zeros],
                         [-np.sin(ry), np.cos(ry), zeros],
                         [zeros, zeros, ones]])  # (3, 3, N)
    R_list = np.transpose(rot_list, (2, 0, 1))  # (N, 3, 3)

    temp_corners = np.concatenate((x_corners.reshape(
        -1, 8, 1), y_corners.reshape(-1, 8, 1), z_corners.reshape(-1, 8, 1)),
                                  axis=2)  # (N, 8, 3)
    rotated_corners = np.matmul(temp_corners, R_list)  # (N, 8, 3)
    x_corners = rotated_corners[:, :, 0]
    y_corners = rotated_corners[:, :, 1]
    z_corners = rotated_corners[:, :, 2]

    x_loc, y_loc, z_loc = boxes3d[:, 0], boxes3d[:, 1], boxes3d[:, 2]

    x = x_loc.reshape(-1, 1) + x_corners.reshape(-1, 8)
    y = y_loc.reshape(-1, 1) + y_corners.reshape(-1, 8)
    z = z_loc.reshape(-1, 1) + z_corners.reshape(-1, 8)

    corners = np.concatenate(
        (x.reshape(-1, 8, 1), y.reshape(-1, 8, 1), z.reshape(-1, 8, 1)),
        axis=2)

    return corners.astype(np.float32)