_skeletonize.py

"""
Algorithms for computing the skeleton of a binary image
"""
import numpy as np
from scipy import ndimage as ndi

from .._shared.utils import check_nD, deprecate_kwarg
from ..util import crop, img_as_ubyte
from ._skeletonize_3d_cy import _compute_thin_image
from ._skeletonize_cy import _fast_skeletonize, _skeletonize_loop, _table_lookup_index


def skeletonize(image, *, method=None):
    """Compute the skeleton of a binary image.

    Thinning is used to reduce each connected component in a binary image
    to a single-pixel wide skeleton.

    Parameters
    ----------
    image : ndarray, 2D or 3D
        An image containing the objects to be skeletonized. Zeros
        represent background, nonzero values are foreground.
    method : {'zhang', 'lee'}, optional
        Which algorithm to use. Zhang's algorithm [Zha84]_ only works for
        2D images, and is the default for 2D. Lee's algorithm [Lee94]_
        works for 2D or 3D images and is the default for 3D.

    Returns
    -------
    skeleton : ndarray
        The thinned image.

    See Also
    --------
    medial_axis

    References
    ----------
    .. [Lee94] T.-C. Lee, R.L. Kashyap and C.-N. Chu, Building skeleton models
           via 3-D medial surface/axis thinning algorithms.
           Computer Vision, Graphics, and Image Processing, 56(6):462-478, 1994.

    .. [Zha84] A fast parallel algorithm for thinning digital patterns,
           T. Y. Zhang and C. Y. Suen, Communications of the ACM,
           March 1984, Volume 27, Number 3.

    Examples
    --------
    >>> X, Y = np.ogrid[0:9, 0:9]
    >>> ellipse = (1./3 * (X - 4)**2 + (Y - 4)**2 < 3**2).astype(np.uint8)
    >>> ellipse
    array([[0, 0, 0, 1, 1, 1, 0, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 0, 1, 1, 1, 0, 0, 0]], dtype=uint8)
    >>> skel = skeletonize(ellipse)
    >>> skel.astype(np.uint8)
    array([[0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=uint8)

    """
    if method not in {'zhang', 'lee', None}:
        raise ValueError(
            f'skeletonize method should be either "lee" or "zhang", ' f'got {method}.'
        )
    if image.ndim == 2 and (method is None or method == 'zhang'):
        skeleton = skeletonize_2d(image.astype(bool, copy=False))
    elif image.ndim == 3 and method == 'zhang':
        raise ValueError('skeletonize method "zhang" only works for 2D ' 'images.')
    elif image.ndim == 3 or (image.ndim == 2 and method == 'lee'):
        skeleton = skeletonize_3d(image)
    else:
        raise ValueError(
            f'skeletonize requires a 2D or 3D image as input, ' f'got {image.ndim}D.'
        )
    return skeleton


def skeletonize_2d(image):
    """Return the skeleton of a 2D binary image.

    Thinning is used to reduce each connected component in a binary image
    to a single-pixel wide skeleton.

    Parameters
    ----------
    image : numpy.ndarray
        A binary image containing the objects to be skeletonized. '1'
        represents foreground, and '0' represents background. It
        also accepts arrays of boolean values where True is foreground.

    Returns
    -------
    skeleton : ndarray
        A matrix containing the thinned image.

    See Also
    --------
    medial_axis

    Notes
    -----
    The algorithm [Zha84]_ works by making successive passes of the image,
    removing pixels on object borders. This continues until no
    more pixels can be removed.  The image is correlated with a
    mask that assigns each pixel a number in the range [0...255]
    corresponding to each possible pattern of its 8 neighboring
    pixels. A look up table is then used to assign the pixels a
    value of 0, 1, 2 or 3, which are selectively removed during
    the iterations.

    Note that this algorithm will give different results than a
    medial axis transform, which is also often referred to as
    "skeletonization".

    References
    ----------
    .. [Zha84] A fast parallel algorithm for thinning digital patterns,
           T. Y. Zhang and C. Y. Suen, Communications of the ACM,
           March 1984, Volume 27, Number 3.

    Examples
    --------
    >>> X, Y = np.ogrid[0:9, 0:9]
    >>> ellipse = (1./3 * (X - 4)**2 + (Y - 4)**2 < 3**2).astype(np.uint8)
    >>> ellipse
    array([[0, 0, 0, 1, 1, 1, 0, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 1, 1, 0, 0],
           [0, 0, 0, 1, 1, 1, 0, 0, 0]], dtype=uint8)
    >>> skel = skeletonize(ellipse)
    >>> skel.astype(np.uint8)
    array([[0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype=uint8)

    """

    if image.ndim != 2:
        raise ValueError("Zhang's skeletonize method requires a 2D array")

    return _fast_skeletonize(image)


# --------- Skeletonization and thinning based on Guo and Hall 1989 ---------


def _generate_thin_luts():
    """generate LUTs for thinning algorithm (for reference)"""

    def nabe(n):
        return np.array([n >> i & 1 for i in range(0, 9)]).astype(bool)

    def G1(n):
        s = 0
        bits = nabe(n)
        for i in (0, 2, 4, 6):
            if not (bits[i]) and (bits[i + 1] or bits[(i + 2) % 8]):
                s += 1
        return s == 1

    g1_lut = np.array([G1(n) for n in range(256)])

    def G2(n):
        n1, n2 = 0, 0
        bits = nabe(n)
        for k in (1, 3, 5, 7):
            if bits[k] or bits[k - 1]:
                n1 += 1
            if bits[k] or bits[(k + 1) % 8]:
                n2 += 1
        return min(n1, n2) in [2, 3]

    g2_lut = np.array([G2(n) for n in range(256)])

    g12_lut = g1_lut & g2_lut

    def G3(n):
        bits = nabe(n)
        return not ((bits[1] or bits[2] or not (bits[7])) and bits[0])

    def G3p(n):
        bits = nabe(n)
        return not ((bits[5] or bits[6] or not (bits[3])) and bits[4])

    g3_lut = np.array([G3(n) for n in range(256)])
    g3p_lut = np.array([G3p(n) for n in range(256)])

    g123_lut = g12_lut & g3_lut
    g123p_lut = g12_lut & g3p_lut

    return g123_lut, g123p_lut


G123_LUT = np.array(
    [
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
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        0,
        1,
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        1,
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        1,
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        1,
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        1,
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        1,
        0,
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        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        1,
        0,
        0,
        1,
        0,
        0,
        0,
        1,
        1,
        0,
        0,
        1,
        0,
        0,
        0,
        1,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        1,
        0,
        0,
        1,
        0,
        0,
        0,
        1,
        1,
        0,
        0,
        1,
        0,
        0,
        0,
    ],
    dtype=bool,
)

G123P_LUT = np.array(
    [
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        1,
        1,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        1,
        1,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        1,
        0,
        1,
        0,
        1,
        0,
        1,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
        0,
    ],
    dtype=bool,
)


def thin(image, max_num_iter=None):
    """
    Perform morphological thinning of a binary image.

    Parameters
    ----------
    image : binary (M, N) ndarray
        The image to be thinned.
    max_num_iter : int, number of iterations, optional
        Regardless of the value of this parameter, the thinned image
        is returned immediately if an iteration produces no change.
        If this parameter is specified it thus sets an upper bound on
        the number of iterations performed.

    Returns
    -------
    out : ndarray of bool
        Thinned image.

    See Also
    --------
    skeletonize, medial_axis

    Notes
    -----
    This algorithm [1]_ works by making multiple passes over the image,
    removing pixels matching a set of criteria designed to thin
    connected regions while preserving eight-connected components and
    2 x 2 squares [2]_. In each of the two sub-iterations the algorithm
    correlates the intermediate skeleton image with a neighborhood mask,
    then looks up each neighborhood in a lookup table indicating whether
    the central pixel should be deleted in that sub-iteration.

    References
    ----------
    .. [1] Z. Guo and R. W. Hall, "Parallel thinning with
           two-subiteration algorithms," Comm. ACM, vol. 32, no. 3,
           pp. 359-373, 1989. :DOI:`10.1145/62065.62074`
    .. [2] Lam, L., Seong-Whan Lee, and Ching Y. Suen, "Thinning
           Methodologies-A Comprehensive Survey," IEEE Transactions on
           Pattern Analysis and Machine Intelligence, Vol 14, No. 9,
           p. 879, 1992. :DOI:`10.1109/34.161346`

    Examples
    --------
    >>> square = np.zeros((7, 7), dtype=np.uint8)
    >>> square[1:-1, 2:-2] = 1
    >>> square[0, 1] =  1
    >>> square
    array([[0, 1, 0, 0, 0, 0, 0],
           [0, 0, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 0, 0],
           [0, 0, 0, 0, 0, 0, 0]], dtype=uint8)
    >>> skel = thin(square)
    >>> skel.astype(np.uint8)
    array([[0, 1, 0, 0, 0, 0, 0],
           [0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 1, 0, 0, 0],
           [0, 0, 0, 1, 0, 0, 0],
           [0, 0, 0, 1, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0]], dtype=uint8)
    """
    # check that image is 2d
    check_nD(image, 2)

    # convert image to uint8 with values in {0, 1}
    skel = np.asanyarray(image, dtype=bool).astype(np.uint8)

    # neighborhood mask
    mask = np.array([[8, 4, 2], [16, 0, 1], [32, 64, 128]], dtype=np.uint8)

    # iterate until convergence, up to the iteration limit
    max_num_iter = max_num_iter or np.inf
    num_iter = 0
    n_pts_old, n_pts_new = np.inf, np.sum(skel)
    while n_pts_old != n_pts_new and num_iter < max_num_iter:
        n_pts_old = n_pts_new

        # perform the two "subiterations" described in the paper
        for lut in [G123_LUT, G123P_LUT]:
            # correlate image with neighborhood mask
            N = ndi.correlate(skel, mask, mode='constant')
            # take deletion decision from this subiteration's LUT
            D = np.take(lut, N)
            # perform deletion
            skel[D] = 0

        n_pts_new = np.sum(skel)  # count points after thinning
        num_iter += 1

    return skel.astype(bool)


# --------- Skeletonization by medial axis transform --------

_eight_connect = ndi.generate_binary_structure(2, 2)


@deprecate_kwarg(
    {'random_state': 'rng'}, deprecated_version='0.21', removed_version='0.23'
)
def medial_axis(image, mask=None, return_distance=False, *, rng=None):
    """Compute the medial axis transform of a binary image.

    Parameters
    ----------
    image : binary ndarray, shape (M, N)
        The image of the shape to be skeletonized.
    mask : binary ndarray, shape (M, N), optional
        If a mask is given, only those elements in `image` with a true
        value in `mask` are used for computing the medial axis.
    return_distance : bool, optional
        If true, the distance transform is returned as well as the skeleton.
    rng : {`numpy.random.Generator`, int}, optional
        Pseudo-random number generator.
        By default, a PCG64 generator is used (see :func:`numpy.random.default_rng`).
        If `rng` is an int, it is used to seed the generator.

        The PRNG determines the order in which pixels are processed for
        tiebreaking.

        .. versionadded:: 0.19

    Returns
    -------
    out : ndarray of bools
        Medial axis transform of the image
    dist : ndarray of ints, optional
        Distance transform of the image (only returned if `return_distance`
        is True)

    See Also
    --------
    skeletonize

    Notes
    -----
    This algorithm computes the medial axis transform of an image
    as the ridges of its distance transform.

    The different steps of the algorithm are as follows
     * A lookup table is used, that assigns 0 or 1 to each configuration of
       the 3x3 binary square, whether the central pixel should be removed
       or kept. We want a point to be removed if it has more than one neighbor
       and if removing it does not change the number of connected components.

     * The distance transform to the background is computed, as well as
       the cornerness of the pixel.

     * The foreground (value of 1) points are ordered by
       the distance transform, then the cornerness.

     * A cython function is called to reduce the image to its skeleton. It
       processes pixels in the order determined at the previous step, and
       removes or maintains a pixel according to the lookup table. Because
       of the ordering, it is possible to process all pixels in only one
       pass.

    Examples
    --------
    >>> square = np.zeros((7, 7), dtype=np.uint8)
    >>> square[1:-1, 2:-2] = 1
    >>> square
    array([[0, 0, 0, 0, 0, 0, 0],
           [0, 0, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 0, 0],
           [0, 0, 1, 1, 1, 0, 0],
           [0, 0, 0, 0, 0, 0, 0]], dtype=uint8)
    >>> medial_axis(square).astype(np.uint8)
    array([[0, 0, 0, 0, 0, 0, 0],
           [0, 0, 1, 0, 1, 0, 0],
           [0, 0, 0, 1, 0, 0, 0],
           [0, 0, 0, 1, 0, 0, 0],
           [0, 0, 0, 1, 0, 0, 0],
           [0, 0, 1, 0, 1, 0, 0],
           [0, 0, 0, 0, 0, 0, 0]], dtype=uint8)

    """
    global _eight_connect
    if mask is None:
        masked_image = image.astype(bool)
    else:
        masked_image = image.astype(bool).copy()
        masked_image[~mask] = False
    #
    # Build lookup table - three conditions
    # 1. Keep only positive pixels (center_is_foreground array).
    # AND
    # 2. Keep if removing the pixel results in a different connectivity
    # (if the number of connected components is different with and
    # without the central pixel)
    # OR
    # 3. Keep if # pixels in neighborhood is 2 or less
    # Note that table is independent of image
    center_is_foreground = (np.arange(512) & 2**4).astype(bool)
    table = (
        center_is_foreground  # condition 1.
        & (
            np.array(
                [
                    ndi.label(_pattern_of(index), _eight_connect)[1]
                    != ndi.label(_pattern_of(index & ~(2**4)), _eight_connect)[1]
                    for index in range(512)
                ]
            )  # condition 2
            | np.array([np.sum(_pattern_of(index)) < 3 for index in range(512)])
        )
        # condition 3
    )

    # Build distance transform
    distance = ndi.distance_transform_edt(masked_image)
    if return_distance:
        store_distance = distance.copy()

    # Corners
    # The processing order along the edge is critical to the shape of the
    # resulting skeleton: if you process a corner first, that corner will
    # be eroded and the skeleton will miss the arm from that corner. Pixels
    # with fewer neighbors are more "cornery" and should be processed last.
    # We use a cornerness_table lookup table where the score of a
    # configuration is the number of background (0-value) pixels in the
    # 3x3 neighborhood
    cornerness_table = np.array(
        [9 - np.sum(_pattern_of(index)) for index in range(512)]
    )
    corner_score = _table_lookup(masked_image, cornerness_table)

    # Define arrays for inner loop
    i, j = np.mgrid[0 : image.shape[0], 0 : image.shape[1]]
    result = masked_image.copy()
    distance = distance[result]
    i = np.ascontiguousarray(i[result], dtype=np.intp)
    j = np.ascontiguousarray(j[result], dtype=np.intp)
    result = np.ascontiguousarray(result, np.uint8)

    # Determine the order in which pixels are processed.
    # We use a random # for tiebreaking. Assign each pixel in the image a
    # predictable, random # so that masking doesn't affect arbitrary choices
    # of skeletons
    #
    generator = np.random.default_rng(rng)
    tiebreaker = generator.permutation(np.arange(masked_image.sum()))
    order = np.lexsort((tiebreaker, corner_score[masked_image], distance))
    order = np.ascontiguousarray(order, dtype=np.int32)

    table = np.ascontiguousarray(table, dtype=np.uint8)
    # Remove pixels not belonging to the medial axis
    _skeletonize_loop(result, i, j, order, table)

    result = result.astype(bool)
    if mask is not None:
        result[~mask] = image[~mask]
    if return_distance:
        return result, store_distance
    else:
        return result


def _pattern_of(index):
    """
    Return the pattern represented by an index value
    Byte decomposition of index
    """
    return np.array(
        [
            [index & 2**0, index & 2**1, index & 2**2],
            [index & 2**3, index & 2**4, index & 2**5],
            [index & 2**6, index & 2**7, index & 2**8],
        ],
        bool,
    )


def _table_lookup(image, table):
    """
    Perform a morphological transform on an image, directed by its
    neighbors

    Parameters
    ----------
    image : ndarray
        A binary image
    table : ndarray
        A 512-element table giving the transform of each pixel given
        the values of that pixel and its 8-connected neighbors.

    Returns
    -------
    result : ndarray of same shape as `image`
        Transformed image

    Notes
    -----
    The pixels are numbered like this::

      0 1 2
      3 4 5
      6 7 8

    The index at a pixel is the sum of 2**<pixel-number> for pixels
    that evaluate to true.
    """
    #
    # We accumulate into the indexer to get the index into the table
    # at each point in the image
    #
    if image.shape[0] < 3 or image.shape[1] < 3:
        image = image.astype(bool)
        indexer = np.zeros(image.shape, int)
        indexer[1:, 1:] += image[:-1, :-1] * 2**0
        indexer[1:, :] += image[:-1, :] * 2**1
        indexer[1:, :-1] += image[:-1, 1:] * 2**2

        indexer[:, 1:] += image[:, :-1] * 2**3
        indexer[:, :] += image[:, :] * 2**4
        indexer[:, :-1] += image[:, 1:] * 2**5

        indexer[:-1, 1:] += image[1:, :-1] * 2**6
        indexer[:-1, :] += image[1:, :] * 2**7
        indexer[:-1, :-1] += image[1:, 1:] * 2**8
    else:
        indexer = _table_lookup_index(np.ascontiguousarray(image, np.uint8))
    image = table[indexer]
    return image


def skeletonize_3d(image):
    """Compute the skeleton of a binary image.

    Thinning is used to reduce each connected component in a binary image
    to a single-pixel wide skeleton.

    Parameters
    ----------
    image : ndarray, 2D or 3D
        A binary image containing the objects to be skeletonized. Zeros
        represent background, nonzero values are foreground.

    Returns
    -------
    skeleton : ndarray
        The thinned image.

    See Also
    --------
    skeletonize, medial_axis

    Notes
    -----
    The method of [Lee94]_ uses an octree data structure to examine a 3x3x3
    neighborhood of a pixel. The algorithm proceeds by iteratively sweeping
    over the image, and removing pixels at each iteration until the image
    stops changing. Each iteration consists of two steps: first, a list of
    candidates for removal is assembled; then pixels from this list are
    rechecked sequentially, to better preserve connectivity of the image.

    The algorithm this function implements is different from the algorithms
    used by either `skeletonize` or `medial_axis`, thus for 2D images the
    results produced by this function are generally different.

    References
    ----------
    .. [Lee94] T.-C. Lee, R.L. Kashyap and C.-N. Chu, Building skeleton models
           via 3-D medial surface/axis thinning algorithms.
           Computer Vision, Graphics, and Image Processing, 56(6):462-478, 1994.

    """
    # make sure the image is 3D or 2D
    if image.ndim < 2 or image.ndim > 3:
        raise ValueError(
            "skeletonize_3d can only handle 2D or 3D images; "
            f"got image.ndim = {image.ndim} instead."
        )
    image = np.ascontiguousarray(image)
    image = img_as_ubyte(image, force_copy=False)

    # make an in image 3D and pad it w/ zeros to simplify dealing w/ boundaries
    # NB: careful here to not clobber the original *and* minimize copying
    image_o = image
    if image.ndim == 2:
        image_o = image[np.newaxis, ...]
    image_o = np.pad(image_o, pad_width=1, mode='constant')

    # normalize to binary
    maxval = image_o.max()
    image_o[image_o != 0] = 1

    # do the computation
    image_o = np.asarray(_compute_thin_image(image_o))

    # crop it back and restore the original intensity range
    image_o = crop(image_o, crop_width=1)
    if image.ndim == 2:
        image_o = image_o[0]
    image_o *= maxval

    return image_o