utils3d/numpy/utils.py

import numpy as np
from typing import *
from numbers import Number
import warnings
import functools

from ._helpers import batched
from . import transforms
from . import mesh

__all__ = [
    'sliding_window_1d',
    'sliding_window_nd',
    'sliding_window_2d',
    'max_pool_1d',
    'max_pool_2d',
    'max_pool_nd',
    'depth_edge',
    'normals_edge',
    'depth_aliasing',
    'interpolate',
    'image_scrcoord',
    'image_uv',
    'image_pixel_center',
    'image_pixel',
    'image_mesh',
    'image_mesh_from_depth',
    'points_to_normals',
    'points_to_normals',
    'chessboard',
    'cube',
    'icosahedron',
    'square',
    'camera_frustum',
    'to4x4'
]


def no_runtime_warnings(fn):
    """
    Disable runtime warnings in numpy.
    """
    @functools.wraps(fn)
    def wrapper(*args, **kwargs):
        with warnings.catch_warnings():
            warnings.simplefilter("ignore")
            return fn(*args, **kwargs)
    return wrapper


def sliding_window_1d(x: np.ndarray, window_size: int, stride: int, axis: int = -1):
    """
    Return x view of the input array with x sliding window of the given kernel size and stride.
    The sliding window is performed over the given axis, and the window dimension is append to the end of the output array's shape.

    Args:
        x (np.ndarray): input array with shape (..., axis_size, ...)
        kernel_size (int): size of the sliding window
        stride (int): stride of the sliding window
        axis (int): axis to perform sliding window over
    
    Returns:
        a_sliding (np.ndarray): view of the input array with shape (..., n_windows, ..., kernel_size), where n_windows = (axis_size - kernel_size + 1) // stride
    """
    assert x.shape[axis] >= window_size, f"kernel_size ({window_size}) is larger than axis_size ({x.shape[axis]})"
    axis = axis % x.ndim
    shape = (*x.shape[:axis], (x.shape[axis] - window_size + 1) // stride, *x.shape[axis + 1:], window_size)
    strides = (*x.strides[:axis], stride * x.strides[axis], *x.strides[axis + 1:], x.strides[axis])
    x_sliding = np.lib.stride_tricks.as_strided(x, shape=shape, strides=strides)
    return x_sliding


def sliding_window_nd(x: np.ndarray, window_size: Tuple[int,...], stride: Tuple[int,...], axis: Tuple[int,...]) -> np.ndarray:
    axis = [axis[i] % x.ndim for i in range(len(axis))]
    for i in range(len(axis)):
        x = sliding_window_1d(x, window_size[i], stride[i], axis[i])
    return x


def sliding_window_2d(x: np.ndarray, window_size: Union[int, Tuple[int, int]], stride: Union[int, Tuple[int, int]], axis: Tuple[int, int] = (-2, -1)) -> np.ndarray:
    if isinstance(window_size, int):
        window_size = (window_size, window_size)
    if isinstance(stride, int):
        stride = (stride, stride)
    return sliding_window_nd(x, window_size, stride, axis)


def max_pool_1d(x: np.ndarray, kernel_size: int, stride: int, padding: int = 0, axis: int = -1):
    axis = axis % x.ndim
    if padding > 0:
        fill_value = np.nan if x.dtype.kind == 'f' else np.iinfo(x.dtype).min
        padding_arr = np.full((*x.shape[:axis], padding, *x.shape[axis + 1:]), fill_value=fill_value, dtype=x.dtype)
        x = np.concatenate([padding_arr, x, padding_arr], axis=axis)
    a_sliding = sliding_window_1d(x, kernel_size, stride, axis)
    max_pool = np.nanmax(a_sliding, axis=-1)
    return max_pool


def max_pool_nd(x: np.ndarray, kernel_size: Tuple[int,...], stride: Tuple[int,...], padding: Tuple[int,...], axis: Tuple[int,...]) -> np.ndarray:
    for i in range(len(axis)):
        x = max_pool_1d(x, kernel_size[i], stride[i], padding[i], axis[i])
    return x


def max_pool_2d(x: np.ndarray, kernel_size: Union[int, Tuple[int, int]], stride: Union[int, Tuple[int, int]], padding: Union[int, Tuple[int, int]], axis: Tuple[int, int] = (-2, -1)):
    if isinstance(kernel_size, Number):
        kernel_size = (kernel_size, kernel_size)
    if isinstance(stride, Number):
        stride = (stride, stride)
    if isinstance(padding, Number):
        padding = (padding, padding)
    axis = tuple(axis)
    return max_pool_nd(x, kernel_size, stride, padding, axis)


@no_runtime_warnings
def depth_edge(depth: np.ndarray, atol: float = None, rtol: float = None, kernel_size: int = 3, mask: np.ndarray = None) -> np.ndarray:
    """
    Compute the edge mask from depth map. The edge is defined as the pixels whose neighbors have large difference in depth.
    
    Args:
        depth (np.ndarray): shape (..., height, width), linear depth map
        atol (float): absolute tolerance
        rtol (float): relative tolerance

    Returns:
        edge (np.ndarray): shape (..., height, width) of dtype torch.bool
    """
    if mask is None:
        diff = (max_pool_2d(depth, kernel_size, stride=1, padding=kernel_size // 2) + max_pool_2d(-depth, kernel_size, stride=1, padding=kernel_size // 2))
    else:
        diff = (max_pool_2d(np.where(mask, depth, -np.inf), kernel_size, stride=1, padding=kernel_size // 2) + max_pool_2d(np.where(mask, -depth, -np.inf), kernel_size, stride=1, padding=kernel_size // 2))

    edge = np.zeros_like(depth, dtype=bool)
    if atol is not None:
        edge |= diff > atol
    
    with warnings.catch_warnings():
        warnings.simplefilter("ignore", category=RuntimeWarning)
        if rtol is not None:
            edge |= diff / depth > rtol
    return edge


@no_runtime_warnings
def depth_aliasing(depth: np.ndarray, atol: float = None, rtol: float = None, kernel_size: int = 3, mask: np.ndarray = None) -> np.ndarray:
    """
    Compute the map that indicates the aliasing of x depth map. The aliasing is defined as the pixels which neither close to the maximum nor the minimum of its neighbors.
    Args:
        depth (np.ndarray): shape (..., height, width), linear depth map
        atol (float): absolute tolerance
        rtol (float): relative tolerance

    Returns:
        edge (np.ndarray): shape (..., height, width) of dtype torch.bool
    """
    if mask is None:
        diff_max = max_pool_2d(depth, kernel_size, stride=1, padding=kernel_size // 2) - depth
        diff_min = max_pool_2d(-depth, kernel_size, stride=1, padding=kernel_size // 2) + depth
    else:
        diff_max = max_pool_2d(np.where(mask, depth, -np.inf), kernel_size, stride=1, padding=kernel_size // 2) - depth
        diff_min = max_pool_2d(np.where(mask, -depth, -np.inf), kernel_size, stride=1, padding=kernel_size // 2) + depth
    diff = np.minimum(diff_max, diff_min)

    edge = np.zeros_like(depth, dtype=bool)
    if atol is not None:
        edge |= diff > atol
    if rtol is not None:
        edge |= diff / depth > rtol
    return edge


@no_runtime_warnings
def normals_edge(normals: np.ndarray, tol: float, kernel_size: int = 3, mask: np.ndarray = None) -> np.ndarray:
    """
    Compute the edge mask from normal map.

    Args:
        normal (np.ndarray): shape (..., height, width, 3), normal map
        tol (float): tolerance in degrees
   
    Returns:
        edge (np.ndarray): shape (..., height, width) of dtype torch.bool
    """
    assert normals.ndim >= 3 and normals.shape[-1] == 3, "normal should be of shape (..., height, width, 3)"
    normals = normals / (np.linalg.norm(normals, axis=-1, keepdims=True) + 1e-12)
    
    padding = kernel_size // 2
    normals_window = sliding_window_2d(
        np.pad(normals, (*([(0, 0)] * (normals.ndim - 3)), (padding, padding), (padding, padding), (0, 0)), mode='edge'), 
        window_size=kernel_size, 
        stride=1, 
        axis=(-3, -2)
    )
    if mask is None:
        angle_diff = np.acos((normals[..., None, None] * normals_window).sum(axis=-3)).max(axis=(-2, -1))
    else:
        mask_window = sliding_window_2d(
            np.pad(mask, (*([(0, 0)] * (mask.ndim - 3)), (padding, padding), (padding, padding)), mode='edge'), 
            window_size=kernel_size, 
            stride=1, 
            axis=(-3, -2)
        )
        angle_diff = np.where(mask_window, np.acos((normals[..., None, None] * normals_window).sum(axis=-3)), 0).max(axis=(-2, -1))

    angle_diff = max_pool_2d(angle_diff, kernel_size, stride=1, padding=kernel_size // 2)
    edge = angle_diff > np.deg2rad(tol)
    return edge

@no_runtime_warnings
def points_to_normals(point: np.ndarray, mask: np.ndarray = None) -> np.ndarray:
    """
    Calculate normal map from point map. Value range is [-1, 1]. Normal direction in OpenGL identity camera's coordinate system.

    Args:
        point (np.ndarray): shape (height, width, 3), point map
    Returns:
        normal (np.ndarray): shape (height, width, 3), normal map. 
    """
    height, width = point.shape[-3:-1]
    has_mask = mask is not None

    if mask is None:
        mask = np.ones_like(point[..., 0], dtype=bool)
    mask_pad = np.zeros((height + 2, width + 2), dtype=bool)
    mask_pad[1:-1, 1:-1] = mask
    mask = mask_pad

    pts = np.zeros((height + 2, width + 2, 3), dtype=point.dtype)
    pts[1:-1, 1:-1, :] = point
    up = pts[:-2, 1:-1, :] - pts[1:-1, 1:-1, :]
    left = pts[1:-1, :-2, :] - pts[1:-1, 1:-1, :]
    down = pts[2:, 1:-1, :] - pts[1:-1, 1:-1, :]
    right = pts[1:-1, 2:, :] - pts[1:-1, 1:-1, :]
    normal = np.stack([
        np.cross(up, left, axis=-1),
        np.cross(left, down, axis=-1),
        np.cross(down, right, axis=-1),
        np.cross(right, up, axis=-1),
    ])
    normal = normal / (np.linalg.norm(normal, axis=-1, keepdims=True) + 1e-12)
    valid = np.stack([
        mask[:-2, 1:-1] & mask[1:-1, :-2],
        mask[1:-1, :-2] & mask[2:, 1:-1],
        mask[2:, 1:-1] & mask[1:-1, 2:],
        mask[1:-1, 2:] & mask[:-2, 1:-1],
    ]) & mask[None, 1:-1, 1:-1]
    normal = (normal * valid[..., None]).sum(axis=0)
    normal = normal / (np.linalg.norm(normal, axis=-1, keepdims=True) + 1e-12)
    
    if has_mask:
        normal_mask =  valid.any(axis=0)
        normal = np.where(normal_mask[..., None], normal, 0)
        return normal, normal_mask
    else:
        return normal


def depth_to_normals(depth: np.ndarray, intrinsics: np.ndarray, mask: np.ndarray = None) -> np.ndarray:
    """
    Calculate normal map from depth map. Value range is [-1, 1]. Normal direction in OpenGL identity camera's coordinate system.

    Args:
        depth (np.ndarray): shape (height, width), linear depth map
        intrinsics (np.ndarray): shape (3, 3), intrinsics matrix
    Returns:
        normal (np.ndarray): shape (height, width, 3), normal map. 
    """
    has_mask = mask is not None

    height, width = depth.shape[-2:]
    if mask is None:
        mask = np.ones_like(depth, dtype=bool)

    uv = image_uv(width=width, height=height, dtype=np.float32)
    pts = transforms.unproject_cv(uv, depth, intrinsics=intrinsics, extrinsics=None)
    
    return points_to_normals(pts, mask)

def interpolate(bary: np.ndarray, tri_id: np.ndarray, attr: np.ndarray, faces: np.ndarray) -> np.ndarray:
    """Interpolate with given barycentric coordinates and triangle indices

    Args:
        bary (np.ndarray): shape (..., 3), barycentric coordinates
        tri_id (np.ndarray): int array of shape (...), triangle indices
        attr (np.ndarray): shape (N, M), vertices attributes
        faces (np.ndarray): int array of shape (T, 3), face vertex indices

    Returns:
        np.ndarray: shape (..., M) interpolated result
    """
    faces_ = np.concatenate([np.zeros((1, 3), dtype=faces.dtype), faces + 1], axis=0)
    attr_ = np.concatenate([np.zeros((1, attr.shape[1]), dtype=attr.dtype), attr], axis=0)
    return np.sum(bary[..., None] * attr_[faces_[tri_id + 1]], axis=-2)


def image_scrcoord(
    width: int,
    height: int,
) -> np.ndarray:
    """
    Get OpenGL's screen space coordinates, ranging in [0, 1].
    [0, 0] is the bottom-left corner of the image.

    Args:
        width (int): image width
        height (int): image height

    Returns:
        (np.ndarray): shape (height, width, 2)
    """
    x, y = np.meshgrid(
        np.linspace(0.5 / width, 1 - 0.5 / width, width, dtype=np.float32),
        np.linspace(1 - 0.5 / height, 0.5 / height, height, dtype=np.float32),
        indexing='xy'
    )
    return np.stack([x, y], axis=2)


def image_uv(
    height: int,
    width: int,
    left: int = None,
    top: int = None,
    right: int = None,
    bottom: int = None,
    dtype: np.dtype = np.float32
) -> np.ndarray:
    """
    Get image space UV grid, ranging in [0, 1]. 

    >>> image_uv(10, 10):
    [[[0.05, 0.05], [0.15, 0.05], ..., [0.95, 0.05]],
     [[0.05, 0.15], [0.15, 0.15], ..., [0.95, 0.15]],
      ...             ...                  ...
     [[0.05, 0.95], [0.15, 0.95], ..., [0.95, 0.95]]]

    Args:
        width (int): image width
        height (int): image height

    Returns:
        np.ndarray: shape (height, width, 2)
    """
    if left is None: left = 0
    if top is None: top = 0
    if right is None: right = width
    if bottom is None: bottom = height
    u = np.linspace((left + 0.5) / width, (right - 0.5) / width, right - left, dtype=dtype)
    v = np.linspace((top + 0.5) / height, (bottom - 0.5) / height, bottom - top, dtype=dtype)
    u, v = np.meshgrid(u, v, indexing='xy')
    return np.stack([u, v], axis=2)


def image_pixel_center(
    height: int,
    width: int,
    left: int = None,
    top: int = None,
    right: int = None,
    bottom: int = None,
    dtype: np.dtype = np.float32
) -> np.ndarray:
    """
    Get image pixel center coordinates, ranging in [0, width] and [0, height].
    `image[i, j]` has pixel center coordinates `(j + 0.5, i + 0.5)`.

    >>> image_pixel_center(10, 10):
    [[[0.5, 0.5], [1.5, 0.5], ..., [9.5, 0.5]],
     [[0.5, 1.5], [1.5, 1.5], ..., [9.5, 1.5]],
      ...             ...                  ...
    [[0.5, 9.5], [1.5, 9.5], ..., [9.5, 9.5]]]

    Args:
        width (int): image width
        height (int): image height

    Returns:
        np.ndarray: shape (height, width, 2)
    """
    if left is None: left = 0
    if top is None: top = 0
    if right is None: right = width
    if bottom is None: bottom = height
    u = np.linspace(left + 0.5, right - 0.5, right - left, dtype=dtype)
    v = np.linspace(top + 0.5, bottom - 0.5, bottom - top, dtype=dtype)
    u, v = np.meshgrid(u, v, indexing='xy')
    return np.stack([u, v], axis=2)

def image_pixel(
    height: int,
    width: int,
    left: int = None,
    top: int = None,
    right: int = None,
    bottom: int = None,
    dtype: np.dtype = np.int32
) -> np.ndarray:
    """
    Get image pixel coordinates grid, ranging in [0, width - 1] and [0, height - 1].
    `image[i, j]` has pixel center coordinates `(j, i)`.

    >>> image_pixel_center(10, 10):
    [[[0, 0], [1, 0], ..., [9, 0]],
     [[0, 1.5], [1, 1], ..., [9, 1]],
      ...             ...                  ...
    [[0, 9.5], [1, 9], ..., [9, 9 ]]]

    Args:
        width (int): image width
        height (int): image height

    Returns:
        np.ndarray: shape (height, width, 2)
    """
    if left is None: left = 0
    if top is None: top = 0
    if right is None: right = width
    if bottom is None: bottom = height
    u = np.arange(left, right, dtype=dtype)
    v = np.arange(top, bottom, dtype=dtype)
    u, v = np.meshgrid(u, v, indexing='xy')
    return np.stack([u, v], axis=2)


def image_mesh(
    *image_attrs: np.ndarray,
    mask: np.ndarray = None,
    tri: bool = False,
    return_indices: bool = False
) -> Tuple[np.ndarray, ...]:
    """
    Get a mesh regarding image pixel uv coordinates as vertices and image grid as faces.

    Args:
        *image_attrs (np.ndarray): image attributes in shape (height, width, [channels])
        mask (np.ndarray, optional): binary mask of shape (height, width), dtype=bool. Defaults to None.

    Returns:
        faces (np.ndarray): faces connecting neighboring pixels. shape (T, 4) if tri is False, else (T, 3)
        *vertex_attrs (np.ndarray): vertex attributes in corresponding order with input image_attrs
        indices (np.ndarray, optional): indices of vertices in the original mesh
    """
    assert (len(image_attrs) > 0) or (mask is not None), "At least one of image_attrs or mask should be provided"
    height, width = next(image_attrs).shape[:2] if mask is None else mask.shape
    assert all(img.shape[:2] == (height, width) for img in image_attrs), "All image_attrs should have the same shape"
    
    row_faces = np.stack([np.arange(0, width - 1, dtype=np.int32), np.arange(width, 2 * width - 1, dtype=np.int32), np.arange(1 + width, 2 * width, dtype=np.int32), np.arange(1, width, dtype=np.int32)], axis=1)
    faces = (np.arange(0, (height - 1) * width, width, dtype=np.int32)[:, None, None] + row_faces[None, :, :]).reshape((-1, 4))
    if mask is None:
        if tri:
            faces = mesh.triangulate(faces)
        ret = [faces, *(img.reshape(-1, *img.shape[2:]) for img in image_attrs)]
        if return_indices:
            ret.append(np.arange(height * width, dtype=np.int32))
        return tuple(ret)
    else:
        quad_mask = (mask[:-1, :-1] & mask[1:, :-1] & mask[1:, 1:] & mask[:-1, 1:]).ravel()
        faces = faces[quad_mask]
        if tri:
            faces = mesh.triangulate(faces)
        return mesh.remove_unreferenced_vertices(
            faces, 
            *(x.reshape(-1, *x.shape[2:]) for x in image_attrs), 
            return_indices=return_indices
        )

def image_mesh_from_depth(
    depth: np.ndarray,
    extrinsics: np.ndarray = None,
    intrinsics: np.ndarray = None,
    *vertice_attrs: np.ndarray,
    atol: float = None,
    rtol: float = None,
    remove_by_depth: bool = False,
    return_uv: bool = False,
    return_indices: bool = False
) -> Tuple[np.ndarray, ...]:
    """
    Get x triangle mesh by lifting depth map to 3D.

    Args:
        depth (np.ndarray): [H, W] depth map
        extrinsics (np.ndarray, optional): [4, 4] extrinsics matrix. Defaults to None.
        intrinsics (np.ndarray, optional): [3, 3] intrinsics matrix. Defaults to None.
        *vertice_attrs (np.ndarray): [H, W, C] vertex attributes. Defaults to None.
        atol (float, optional): absolute tolerance. Defaults to None.
        rtol (float, optional): relative tolerance. Defaults to None.
            triangles with vertices having depth difference larger than atol + rtol * depth will be marked.
        remove_by_depth (bool, optional): whether to remove triangles with large depth difference. Defaults to True.
        return_uv (bool, optional): whether to return uv coordinates. Defaults to False.
        return_indices (bool, optional): whether to return indices of vertices in the original mesh. Defaults to False.

    Returns:
        vertices (np.ndarray): [N, 3] vertices
        faces (np.ndarray): [T, 3] faces
        *vertice_attrs (np.ndarray): [N, C] vertex attributes
        image_uv (np.ndarray, optional): [N, 2] uv coordinates
        ref_indices (np.ndarray, optional): [N] indices of vertices in the original mesh
    """
    height, width = depth.shape
    image_uv, image_face = image_mesh(height, width)
    depth = depth.reshape(-1)
    pts = transforms.unproject_cv(image_uv, depth, extrinsics, intrinsics)
    image_face = mesh.triangulate(image_face, vertices=pts)
    ref_indices = None
    ret = []
    if atol is not None or rtol is not None:
        atol = 0 if atol is None else atol
        rtol = 0 if rtol is None else rtol
        mean = depth[image_face].mean(axis=1)
        diff = np.max(np.abs(depth[image_face] - depth[image_face[:, [1, 2, 0]]]), axis=1)
        mask = (diff <= atol + rtol * mean)
        image_face_ = image_face[mask]
        image_face_, ref_indices = mesh.remove_unreferenced_vertices(image_face_, return_indices=True)

    remove = remove_by_depth and ref_indices is not None
    if remove:
        pts = pts[ref_indices]
        image_face = image_face_
    ret += [pts, image_face]
    for attr in vertice_attrs:
        ret.append(attr.reshape(-1, attr.shape[-1]) if not remove else attr.reshape(-1, attr.shape[-1])[ref_indices])
    if return_uv:
        ret.append(image_uv if not remove else image_uv[ref_indices])
    if return_indices and ref_indices is not None:
        ret.append(ref_indices)
    return tuple(ret)


def chessboard(width: int, height: int, grid_size: int, color_a: np.ndarray, color_b: np.ndarray) -> np.ndarray:
    """get x chessboard image

    Args:
        width (int): image width
        height (int): image height
        grid_size (int): size of chessboard grid
        color_a (np.ndarray): color of the grid at the top-left corner
        color_b (np.ndarray): color in complementary grid cells

    Returns:
        image (np.ndarray): shape (height, width, channels), chessboard image
    """
    x = np.arange(width) // grid_size
    y = np.arange(height) // grid_size
    mask = (x[None, :] + y[:, None]) % 2
    image = (1 - mask[..., None]) * color_a + mask[..., None] * color_b
    return image


def square(tri: bool = False) -> Tuple[np.ndarray, np.ndarray]: 
    """
    Get a square mesh of area 1 centered at origin in the xy-plane.

    ### Returns
        vertices (np.ndarray): shape (4, 3)
        faces (np.ndarray): shape (1, 4)
    """
    vertices = np.array([
        [-0.5, 0.5, 0],   [0.5, 0.5, 0],   [0.5, -0.5, 0],   [-0.5, -0.5, 0] # v0-v1-v2-v3
    ], dtype=np.float32)
    if tri:
        faces = np.array([[0, 1, 2], [0, 2, 3]], dtype=np.int32)
    else:
        faces = np.array([[0, 1, 2, 3]], dtype=np.int32)
    return vertices, faces  


def cube(tri: bool = False) -> Tuple[np.ndarray, np.ndarray]:
    """
    Get x cube mesh of size 1 centered at origin.

    ### Parameters
        tri (bool, optional): return triangulated mesh. Defaults to False, which returns quad mesh.

    ### Returns
        vertices (np.ndarray): shape (8, 3) 
        faces (np.ndarray): shape (12, 3)
    """
    vertices = np.array([
        [-0.5, 0.5, 0.5],   [0.5, 0.5, 0.5],   [0.5, -0.5, 0.5],   [-0.5, -0.5, 0.5], # v0-v1-v2-v3
        [-0.5, 0.5, -0.5],  [0.5, 0.5, -0.5],  [0.5, -0.5, -0.5],  [-0.5, -0.5, -0.5] # v4-v5-v6-v7
    ], dtype=np.float32).reshape((-1, 3))

    faces = np.array([
        [0, 1, 2, 3], # v0-v1-v2-v3 (front)
        [4, 5, 1, 0], # v4-v5-v1-v0 (top)
        [3, 2, 6, 7], # v3-v2-v6-v7 (bottom)
        [5, 4, 7, 6], # v5-v4-v7-v6 (back)
        [1, 5, 6, 2], # v1-v5-v6-v2 (right)
        [4, 0, 3, 7]  # v4-v0-v3-v7 (left)
    ], dtype=np.int32)

    if tri:
        faces = mesh.triangulate(faces, vertices=vertices)

    return vertices, faces


def camera_frustum(extrinsics: np.ndarray, intrinsics: np.ndarray, depth: float = 1.0) -> Tuple[np.ndarray, np.ndarray, np.ndarray]:
    """
    Get x triangle mesh of camera frustum.
    """
    assert extrinsics.shape == (4, 4) and intrinsics.shape == (3, 3)
    vertices = transforms.unproject_cv(
        np.array([[0, 0], [0, 0], [0, 1], [1, 1], [1, 0]], dtype=np.float32), 
        np.array([0] + [depth] * 4, dtype=np.float32), 
        extrinsics, 
        intrinsics
    ).astype(np.float32)
    edges = np.array([
        [0, 1], [0, 2], [0, 3], [0, 4], 
        [1, 2], [2, 3], [3, 4], [4, 1]
    ], dtype=np.int32)
    faces = np.array([
        [0, 1, 2],
        [0, 2, 3],
        [0, 3, 4],
        [0, 4, 1],
        [1, 2, 3],
        [1, 3, 4]
    ], dtype=np.int32)
    return vertices, edges, faces


def icosahedron():
    A = (1 + 5 ** 0.5) / 2
    vertices = np.array([
        [0, 1, A], [0, -1, A], [0, 1, -A], [0, -1, -A],
        [1, A, 0], [-1, A, 0], [1, -A, 0], [-1, -A, 0],
        [A, 0, 1], [A, 0, -1], [-A, 0, 1], [-A, 0, -1]
    ], dtype=np.float32)
    faces = np.array([
        [0, 1, 8], [0, 8, 4], [0, 4, 5], [0, 5, 10], [0, 10, 1],
        [3, 2, 9], [3, 9, 6], [3, 6, 7], [3, 7, 11], [3, 11, 2],
        [1, 6, 8], [8, 9, 4], [4, 2, 5], [5, 11, 10], [10, 7, 1],
        [2, 4, 9], [9, 8, 6], [6, 1, 7], [7, 10, 11], [11, 5, 2]
    ], dtype=np.int32)
    return vertices, faces