9i82SSKrSSKrSSKJr SSKJr SSSSSSSS .S jjrS rS r \R"SS/SS/SS/SS/SS/SS/SS/SS/SS/SS/SS/SS// S 5r \R"SS/SS/SS/SS/SS/SS/SS/SS/SS/SS/SS/SS// S 5r \R"SS/SS/SS/SS/SS/SS/SS/SS/SS/SS/SS/SS// S 5r SrSrg)N)_marching_cubes_lewiner_luts)_marching_cubes_lewiner_cy)?rrdescentTlewiner)spacinggradient_direction step_sizeallow_degeneratemethodmaskc ZSnUS:XaSnOUS:wa [S5e[UUUUUUUUS9$)ahMarching 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. 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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 Nrrr)axisrg@)r crosssum)vertsr6 actual_vertsabs rmesh_surface_arear;s>rs 4( K    K\OHd XX1qeQqE1Q%!A!uaUAa5STUVRWYZ[\X]_`ab^cefghdikmstXX1qeQqE1Q%!A!uaUAa5STUVRWYZ[\X]_`ab^cefghdikmstXX1qeQqE1Q%!A!uaUAa5STUVRWYZ[\X]_`ab^cefghdikmst9x%Br