_marching_cubes_classic.py

import numpy as np
import scipy.ndimage as ndi
from .._shared.utils import warn
from . import _marching_cubes_classic_cy


def marching_cubes_classic(volume, level=None, spacing=(1., 1., 1.),
                           gradient_direction='descent'):
    """
    Classic marching cubes algorithm to find surfaces in 3d volumetric data.

    Note that the ``marching_cubes()`` algorithm is recommended over
    this algorithm, because it's faster and produces better results.

    Parameters
    ----------
    volume : (M, N, P) array of doubles
        Input data volume to find isosurfaces. Will be cast to `np.float64`.
    level : float
        Contour value to search for isosurfaces in `volume`. If not
        given or None, the average of the min and max of vol is used.
    spacing : length-3 tuple of floats
        Voxel spacing in spatial dimensions corresponding to numpy array
        indexing dimensions (M, N, P) as in `volume`.
    gradient_direction : string
        Controls if the mesh was generated from an isosurface with gradient
        descent toward objects of interest (the default), or the opposite.
        The two options are:
        * descent : Object was greater than exterior
        * ascent : Exterior was greater than object

    Returns
    -------
    verts : (V, 3) array
        Spatial coordinates for V unique mesh vertices. Coordinate order
        matches input `volume` (M, N, P).
    faces : (F, 3) array
        Define triangular faces via referencing vertex indices from ``verts``.
        This algorithm specifically outputs triangles, so each face has
        exactly three indices.

    Notes
    -----
    The marching cubes algorithm is implemented as described in [1]_.
    A simple explanation is available here::

      http://www.essi.fr/~lingrand/MarchingCubes/algo.html

    There are several known ambiguous cases in the marching cubes algorithm.
    Using point labeling as in [1]_, Figure 4, as shown::

           v8 ------ v7
          / |       / |        y
         /  |      /  |        ^  z
       v4 ------ v3   |        | /
        |  v5 ----|- v6        |/          (note: NOT right handed!)
        |  /      |  /          ----> x
        | /       | /
       v1 ------ v2

    Most notably, if v4, v8, v2, and v6 are all >= `level` (or any
    generalization of this case) two parallel planes are generated by this
    algorithm, separating v4 and v8 from v2 and v6. An equally valid
    interpretation would be a single connected thin surface enclosing all
    four points. This is the best known ambiguity, though there are others.

    This algorithm does not attempt to resolve such ambiguities; it is a naive
    implementation of marching cubes as in [1]_, but may be a good beginning
    for work with more recent techniques (Dual Marching Cubes, Extended
    Marching Cubes, Cubic Marching Squares, etc.).

    Because of interactions between neighboring cubes, the isosurface(s)
    generated by this algorithm are NOT guaranteed to be closed, particularly
    for complicated contours. Furthermore, this algorithm does not guarantee
    a single contour will be returned. Indeed, ALL isosurfaces which cross
    `level` will be found, regardless of connectivity.

    The output is a triangular mesh consisting of a set of unique vertices and
    connecting triangles. The order of these vertices and triangles in the
    output list is determined by the position of the smallest ``x,y,z`` (in
    lexicographical order) coordinate in the contour.  This is a side-effect
    of how the input array is traversed, but can be relied upon.

    The generated mesh guarantees coherent orientation as of version 0.12.

    To quantify the area of an isosurface generated by this algorithm, pass
    outputs directly into `skimage.measure.mesh_surface_area`.

    References
    ----------
    .. [1] Lorensen, William and Harvey E. Cline. Marching Cubes: A High
           Resolution 3D Surface Construction Algorithm. Computer Graphics
           (SIGGRAPH 87 Proceedings) 21(4) July 1987, p. 163-170).
           DOI: 10.1145/37401.37422

    See Also
    --------
    skimage.measure.marching_cubes
    skimage.measure.mesh_surface_area
    """
    # Check inputs and ensure `volume` is C-contiguous for memoryviews
    if volume.ndim != 3:
        raise ValueError("Input volume must have 3 dimensions.")
    if level is None:
        level = 0.5 * (volume.min() + volume.max())
    else:
        level = float(level)
        if level < volume.min() or level > volume.max():
            raise ValueError("Surface level must be within volume data range.")
    if len(spacing) != 3:
        raise ValueError("`spacing` must consist of three floats.")

    volume = np.array(volume, dtype=np.float64, order="C")

    # Extract raw triangles using marching cubes in Cython
    #   Returns a list of length-3 lists, each sub-list containing three
    #   tuples. The tuples hold (x, y, z) coordinates for triangle vertices.
    # Note: this algorithm is fast, but returns degenerate "triangles" which
    #   have repeated vertices - and equivalent vertices are redundantly
    #   placed in every triangle they connect with.
    raw_faces = _marching_cubes_classic_cy.iterate_and_store_3d(volume,
                                                                float(level))

    # Find and collect unique vertices, storing triangle verts as indices.
    # Returns a true mesh with no degenerate faces.
    verts, faces = _marching_cubes_classic_cy.unpack_unique_verts(raw_faces)

    verts = np.asarray(verts)
    faces = np.asarray(faces)

    # Fancy indexing to define two vector arrays from triangle vertices
    faces = _correct_mesh_orientation(volume, verts[faces], faces, spacing,
                                      gradient_direction)

    # Adjust for non-isotropic spacing in `verts` at time of return
    return verts * np.r_[spacing], faces


def mesh_surface_area(verts, faces):
    """
    Compute surface area, given vertices & triangular faces

    Parameters
    ----------
    verts : (V, 3) array of floats
        Array containing (x, y, z) coordinates for V unique mesh vertices.
    faces : (F, 3) array of ints
        List of length-3 lists of integers, referencing vertex coordinates as
        provided in `verts`

    Returns
    -------
    area : float
        Surface area of mesh. Units now [coordinate units] ** 2.

    Notes
    -----
    The arguments expected by this function are the first two outputs from
    `skimage.measure.marching_cubes`. For unit correct output, ensure correct
    `spacing` was passed to `skimage.measure.marching_cubes`.

    This algorithm works properly only if the ``faces`` provided are all
    triangles.

    See Also
    --------
    skimage.measure.marching_cubes
    skimage.measure.marching_cubes_classic
    skimage.measure.correct_mesh_orientation

    """
    # Fancy indexing to define two vector arrays from triangle vertices
    actual_verts = verts[faces]
    a = actual_verts[:, 0, :] - actual_verts[:, 1, :]
    b = actual_verts[:, 0, :] - actual_verts[:, 2, :]
    del actual_verts

    # Area of triangle in 3D = 1/2 * Euclidean norm of cross product
    return ((np.cross(a, b) ** 2).sum(axis=1) ** 0.5).sum() / 2.


def correct_mesh_orientation(volume, verts, faces, spacing=(1., 1., 1.),
                             gradient_direction='descent'):
    """
    Correct orientations of mesh faces.

    Parameters
    ----------
    volume : (M, N, P) array of doubles
        Input data volume to find isosurfaces. Will be cast to `np.float64`.
    verts : (V, 3) array of floats
        Array containing (x, y, z) coordinates for V unique mesh vertices.
    faces : (F, 3) array of ints
        List of length-3 lists of integers, referencing vertex coordinates as
        provided in `verts`.
    spacing : length-3 tuple of floats
        Voxel spacing in spatial dimensions corresponding to numpy array
        indexing dimensions (M, N, P) as in `volume`.
    gradient_direction : string
        Controls if the mesh was generated from an isosurface with gradient
        descent toward objects of interest (the default), or the opposite.
        The two options are:
        * descent : Object was greater than exterior
        * ascent : Exterior was greater than object

    Returns
    -------
    faces_corrected (F, 3) array of ints
        Corrected list of faces referencing vertex coordinates in `verts`.

    Notes
    -----
    Certain applications and mesh processing algorithms require all faces
    to be oriented in a consistent way. Generally, this means a normal vector
    points "out" of the meshed shapes. This algorithm corrects the output from
    `skimage.measure.marching_cubes_classic` by flipping the orientation of
    mis-oriented faces.

    Because marching cubes could be used to find isosurfaces either on
    gradient descent (where the desired object has greater values than the
    exterior) or ascent (where the desired object has lower values than the
    exterior), the ``gradient_direction`` kwarg allows the user to inform this
    algorithm which is correct. If the resulting mesh appears to be oriented
    completely incorrectly, try changing this option.

    The arguments expected by this function are the exact outputs from
    `skimage.measure.marching_cubes_classic`. Only `faces` is corrected and
    returned, as the vertices do not change; only the order in which they are
    referenced.

    This algorithm assumes ``faces`` provided are all triangles.

    See Also
    --------
    skimage.measure.marching_cubes_classic
    skimage.measure.mesh_surface_area

    """
    warn(DeprecationWarning("`correct_mesh_orientation` is deprecated for "
                            "removal as `marching_cubes_classic` now "
                            "guarantees correct mesh orientation."))

    verts = verts.copy()
    verts[:, 0] /= spacing[0]
    verts[:, 1] /= spacing[1]
    verts[:, 2] /= spacing[2]

    # Fancy indexing to define two vector arrays from triangle vertices
    actual_verts = verts[faces]

    return _correct_mesh_orientation(volume, actual_verts, faces, spacing,
                                     gradient_direction)


def _correct_mesh_orientation(volume, actual_verts, faces,
                              spacing=(1., 1., 1.),
                              gradient_direction='descent'):
    """
    Correct orientations of mesh faces.

    Parameters
    ----------
    volume : (M, N, P) array of doubles
        Input data volume to find isosurfaces. Will be cast to `np.float64`.
    actual_verts : (F, 3, 3) array of floats
        Array with (face, vertex, coords) index coordinates.
    faces : (F, 3) array of ints
        List of length-3 lists of integers, referencing vertex coordinates as
        provided in `verts`.
    spacing : length-3 tuple of floats
        Voxel spacing in spatial dimensions corresponding to numpy array
        indexing dimensions (M, N, P) as in `volume`.
    gradient_direction : string
        Controls if the mesh was generated from an isosurface with gradient
        descent toward objects of interest (the default), or the opposite.
        The two options are:
        * descent : Object was greater than exterior
        * ascent : Exterior was greater than object

    Returns
    -------
    faces_corrected (F, 3) array of ints
        Corrected list of faces referencing vertex coordinates in `verts`.

    Notes
    -----
    Certain applications and mesh processing algorithms require all faces
    to be oriented in a consistent way. Generally, this means a normal vector
    points "out" of the meshed shapes. This algorithm corrects the output from
    `skimage.measure.marching_cubes_classic` by flipping the orientation of
    mis-oriented faces.

    Because marching cubes could be used to find isosurfaces either on
    gradient descent (where the desired object has greater values than the
    exterior) or ascent (where the desired object has lower values than the
    exterior), the ``gradient_direction`` kwarg allows the user to inform this
    algorithm which is correct. If the resulting mesh appears to be oriented
    completely incorrectly, try changing this option.

    The arguments expected by this function are the exact outputs from
    `skimage.measure.marching_cubes_classic` except `actual_verts`, which is an
    uncorrected version of the fancy indexing operation `verts[faces]`.
    Only `faces` is corrected and returned as the vertices do not change,
    only the order in which they are referenced.

    This algorithm assumes ``faces`` provided are exclusively triangles.

    See Also
    --------
    skimage.measure.marching_cubes_classic
    skimage.measure.mesh_surface_area

    """
    # Calculate gradient of `volume`, then interpolate to vertices in `verts`
    grad_x, grad_y, grad_z = np.gradient(volume)

    a = actual_verts[:, 0, :] - actual_verts[:, 1, :]
    b = actual_verts[:, 0, :] - actual_verts[:, 2, :]

    # Find triangle centroids
    centroids = (actual_verts.sum(axis=1) / 3.).T

    del actual_verts

    # Interpolate face centroids into each gradient axis
    grad_centroids_x = ndi.map_coordinates(grad_x, centroids)
    grad_centroids_y = ndi.map_coordinates(grad_y, centroids)
    grad_centroids_z = ndi.map_coordinates(grad_z, centroids)

    # Combine and normalize interpolated gradients
    grad_centroids = np.c_[grad_centroids_x, grad_centroids_y,
                           grad_centroids_z]
    grad_centroids = (grad_centroids /
                      (np.sum(grad_centroids ** 2,
                              axis=1) ** 0.5)[:, np.newaxis])

    # Find normal vectors for each face via cross product
    crosses = np.cross(a, b)
    crosses = crosses / (np.sum(crosses ** 2, axis=1) ** (0.5))[:, np.newaxis]

    # Take dot product
    dotproducts = (grad_centroids * crosses).sum(axis=1)

    # Find mis-oriented faces
    if 'descent' in gradient_direction:
        # Faces with incorrect orientations have dot product < 0
        indices = (dotproducts < 0).nonzero()[0]
    elif 'ascent' in gradient_direction:
        # Faces with incorrection orientation have dot product > 0
        indices = (dotproducts > 0).nonzero()[0]
    else:
        raise ValueError("Incorrect input %s in `gradient_direction`, see "
                         "docstring." % (gradient_direction))

    # Correct orientation and return, without modifying original data
    faces_corrected = faces.copy()
    faces_corrected[indices] = faces_corrected[indices, ::-1]

    return faces_corrected