import base64 import numpy as np from . import _marching_cubes_lewiner_luts as mcluts from . import _marching_cubes_lewiner_cy def marching_cubes( volume, level=None, *, spacing=(1.0, 1.0, 1.0), gradient_direction='descent', step_size=1, allow_degenerate=True, method='lewiner', mask=None, ): """Marching cubes algorithm to find surfaces in 3d volumetric data. In contrast with Lorensen et al. approach [2]_, Lewiner et al. algorithm is faster, resolves ambiguities, and guarantees topologically correct results. Therefore, this algorithm generally a better choice. Parameters ---------- volume : (M, N, P) ndarray Input data volume to find isosurfaces. Will internally be converted to float32 if necessary. level : float, optional 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, optional Voxel spacing in spatial dimensions corresponding to numpy array indexing dimensions (M, N, P) as in `volume`. gradient_direction : string, optional Controls if the mesh was generated from an isosurface with gradient descent toward objects of interest (the default), or the opposite, considering the *left-hand* rule. The two options are: * descent : Object was greater than exterior * ascent : Exterior was greater than object step_size : int, optional Step size in voxels. Default 1. Larger steps yield faster but coarser results. The result will always be topologically correct though. allow_degenerate : bool, optional Whether to allow degenerate (i.e. zero-area) triangles in the end-result. Default True. If False, degenerate triangles are removed, at the cost of making the algorithm slower. method: {'lewiner', 'lorensen'}, optional Whether the method of Lewiner et al. or Lorensen et al. will be used. mask : (M, N, P) array, optional Boolean array. The marching cube algorithm will be computed only on True elements. This will save computational time when interfaces are located within certain region of the volume M, N, P-e.g. the top half of the cube-and also allow to compute finite surfaces-i.e. open surfaces that do not end at the border of the cube. Returns ------- verts : (V, 3) array Spatial coordinates for V unique mesh vertices. Coordinate order matches input `volume` (M, N, P). If ``allow_degenerate`` is set to True, then the presence of degenerate triangles in the mesh can make this array have duplicate vertices. 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. normals : (V, 3) array The normal direction at each vertex, as calculated from the data. values : (V,) array Gives a measure for the maximum value of the data in the local region near each vertex. This can be used by visualization tools to apply a colormap to the mesh. See Also -------- skimage.measure.mesh_surface_area skimage.measure.find_contours Notes ----- The algorithm [1]_ is an improved version of Chernyaev's Marching Cubes 33 algorithm. It is an efficient algorithm that relies on heavy use of lookup tables to handle the many different cases, keeping the algorithm relatively easy. This implementation is written in Cython, ported from Lewiner's C++ implementation. To quantify the area of an isosurface generated by this algorithm, pass verts and faces to `skimage.measure.mesh_surface_area`. Regarding visualization of algorithm output, to contour a volume named `myvolume` about the level 0.0, using the ``mayavi`` package:: >>> >> from mayavi import mlab >> verts, faces, _, _ = marching_cubes(myvolume, 0.0) >> mlab.triangular_mesh([vert[0] for vert in verts], [vert[1] for vert in verts], [vert[2] for vert in verts], faces) >> mlab.show() Similarly using the ``visvis`` package:: >>> >> import visvis as vv >> verts, faces, normals, values = marching_cubes(myvolume, 0.0) >> vv.mesh(np.fliplr(verts), faces, normals, values) >> vv.use().Run() To reduce the number of triangles in the mesh for better performance, see this `example `_ using the ``mayavi`` package. References ---------- .. [1] Thomas Lewiner, Helio Lopes, Antonio Wilson Vieira and Geovan Tavares. Efficient implementation of Marching Cubes' cases with topological guarantees. Journal of Graphics Tools 8(2) pp. 1-15 (december 2003). :DOI:`10.1080/10867651.2003.10487582` .. [2] 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` """ use_classic = False if method == 'lorensen': use_classic = True elif method != 'lewiner': raise ValueError("method should be either 'lewiner' or 'lorensen'") return _marching_cubes_lewiner( volume, level, spacing, gradient_direction, step_size, allow_degenerate, use_classic=use_classic, mask=mask, ) def _marching_cubes_lewiner( volume, level, spacing, gradient_direction, step_size, allow_degenerate, use_classic, mask, ): """Lewiner et al. algorithm for marching cubes. See marching_cubes_lewiner for documentation. """ # Check volume and ensure its in the format that the alg needs if not isinstance(volume, np.ndarray) or (volume.ndim != 3): raise ValueError('Input volume should be a 3D numpy array.') if volume.shape[0] < 2 or volume.shape[1] < 2 or volume.shape[2] < 2: raise ValueError("Input array must be at least 2x2x2.") volume = np.ascontiguousarray(volume, np.float32) # no copy if not necessary # Check/convert other inputs: # level 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.") # spacing if len(spacing) != 3: raise ValueError("`spacing` must consist of three floats.") # step_size step_size = int(step_size) if step_size < 1: raise ValueError('step_size must be at least one.') # use_classic use_classic = bool(use_classic) # Get LutProvider class (reuse if possible) L = _get_mc_luts() # Check if a mask array is passed if mask is not None: if not mask.shape == volume.shape: raise ValueError('volume and mask must have the same shape.') # Apply algorithm func = _marching_cubes_lewiner_cy.marching_cubes vertices, faces, normals, values = func( volume, level, L, step_size, use_classic, mask ) if not len(vertices): raise RuntimeError('No surface found at the given iso value.') # Output in z-y-x order, as is common in skimage vertices = np.fliplr(vertices) normals = np.fliplr(normals) # Finishing touches to output faces.shape = -1, 3 if gradient_direction == 'descent': # MC implementation is right-handed, but gradient_direction is # left-handed faces = np.fliplr(faces) elif not gradient_direction == 'ascent': raise ValueError( f"Incorrect input {gradient_direction} in `gradient_direction`, " "see docstring." ) if not np.array_equal(spacing, (1, 1, 1)): vertices = vertices * np.r_[spacing] if allow_degenerate: return vertices, faces, normals, values else: fun = _marching_cubes_lewiner_cy.remove_degenerate_faces return fun(vertices.astype(np.float32), faces, normals, values) def _to_array(args): shape, text = args byts = base64.decodebytes(text.encode('utf-8')) ar = np.frombuffer(byts, dtype='int8') ar.shape = shape return ar # Map an edge-index to two relative pixel positions. The edge index # represents a point that lies somewhere in between these pixels. # Linear interpolation should be used to determine where it is exactly. # 0 # 3 1 -> 0x # 2 xx # fmt: off EDGETORELATIVEPOSX = np.array([ [0,1],[1,1],[1,0],[0,0], [0,1],[1,1],[1,0],[0,0], [0,0],[1,1],[1,1],[0,0] ], 'int8') EDGETORELATIVEPOSY = np.array([ [0,0],[0,1],[1,1],[1,0], [0,0],[0,1],[1,1],[1,0], [0,0],[0,0],[1,1],[1,1] ], 'int8') EDGETORELATIVEPOSZ = np.array([ [0,0],[0,0],[0,0],[0,0], [1,1],[1,1],[1,1],[1,1], [0,1],[0,1],[0,1],[0,1] ], 'int8') # fmt: on def _get_mc_luts(): """Kind of lazy obtaining of the luts.""" if not hasattr(mcluts, 'THE_LUTS'): mcluts.THE_LUTS = _marching_cubes_lewiner_cy.LutProvider( EDGETORELATIVEPOSX, EDGETORELATIVEPOSY, EDGETORELATIVEPOSZ, _to_array(mcluts.CASESCLASSIC), _to_array(mcluts.CASES), _to_array(mcluts.TILING1), _to_array(mcluts.TILING2), _to_array(mcluts.TILING3_1), _to_array(mcluts.TILING3_2), _to_array(mcluts.TILING4_1), _to_array(mcluts.TILING4_2), _to_array(mcluts.TILING5), _to_array(mcluts.TILING6_1_1), _to_array(mcluts.TILING6_1_2), _to_array(mcluts.TILING6_2), _to_array(mcluts.TILING7_1), _to_array(mcluts.TILING7_2), _to_array(mcluts.TILING7_3), _to_array(mcluts.TILING7_4_1), _to_array(mcluts.TILING7_4_2), _to_array(mcluts.TILING8), _to_array(mcluts.TILING9), _to_array(mcluts.TILING10_1_1), _to_array(mcluts.TILING10_1_1_), _to_array(mcluts.TILING10_1_2), _to_array(mcluts.TILING10_2), _to_array(mcluts.TILING10_2_), _to_array(mcluts.TILING11), _to_array(mcluts.TILING12_1_1), _to_array(mcluts.TILING12_1_1_), _to_array(mcluts.TILING12_1_2), _to_array(mcluts.TILING12_2), _to_array(mcluts.TILING12_2_), _to_array(mcluts.TILING13_1), _to_array(mcluts.TILING13_1_), _to_array(mcluts.TILING13_2), _to_array(mcluts.TILING13_2_), _to_array(mcluts.TILING13_3), _to_array(mcluts.TILING13_3_), _to_array(mcluts.TILING13_4), _to_array(mcluts.TILING13_5_1), _to_array(mcluts.TILING13_5_2), _to_array(mcluts.TILING14), _to_array(mcluts.TEST3), _to_array(mcluts.TEST4), _to_array(mcluts.TEST6), _to_array(mcluts.TEST7), _to_array(mcluts.TEST10), _to_array(mcluts.TEST12), _to_array(mcluts.TEST13), _to_array(mcluts.SUBCONFIG13), ) return mcluts.THE_LUTS def mesh_surface_area(verts, faces): """Compute surface area, given vertices and triangular faces. Parameters ---------- verts : (V, 3) array of floats Array containing 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 """ # 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() / 2.0