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Marching Cubes cuda extension
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Summary:
Torch CUDA extension for Marching Cubes
- MC involving 3 steps:
  - 1st forward pass to collect vertices and occupied state for each voxel
  - Compute compactVoxelArray to skip non-empty voxels
  - 2nd pass to genereate interpolated vertex positions and faces by marching through the grid
- In contrast to existing MC:
   - Bind each interpolated vertex with a global edge_id to address floating-point precision
   - Added deduplication process to remove redundant vertices and faces

Benchmarks (ms):

| N / V(^3)      | python          | C++             |   CUDA   | Speedup |
| 2 / 20          |    12176873  |       24338     |     4363   | 2790x/5x|
| 1 / 100          |     -             |    3070511     |   27126   |    113x    |
| 2 / 100          |     -             |    5968934     |   53129   |    112x    |
| 1 / 256          |     -             |  61278092     | 430900   |    142x    |
| 2 / 256          |     -             |125687930     | 856941   |    146x   |

Reviewed By: kjchalup

Differential Revision: D39644248

fbshipit-source-id: d679c0c79d67b98b235d12296f383d760a00042a
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Jiali Duan authored and facebook-github-bot committed Nov 16, 2022
1 parent 9a0b0c2 commit 8b82918
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559 changes: 559 additions & 0 deletions pytorch3d/csrc/marching_cubes/marching_cubes.cu
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31 changes: 27 additions & 4 deletions pytorch3d/csrc/marching_cubes/marching_cubes.h
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Expand Up @@ -23,17 +23,40 @@
// the points are within a volume.
//
// Returns:
// vertices: List of N FloatTensors of vertices
// faces: List of N LongTensors of faces
// vertices: (N_verts, 3) FloatTensor of vertices
// faces: (N_faces, 3) LongTensor of faces
// ids: (N_verts,) LongTensor used to identify each vertex and deduplication
// to avoid floating point precision issues.
// For Cuda, will be used to dedupe redundant vertices.
// For cpp implementation, this tensor is just a placeholder.

// CPU implementation
std::tuple<at::Tensor, at::Tensor> MarchingCubesCpu(
std::tuple<at::Tensor, at::Tensor, at::Tensor> MarchingCubesCpu(
const at::Tensor& vol,
const float isolevel);

// CUDA implementation
std::tuple<at::Tensor, at::Tensor, at::Tensor> MarchingCubesCuda(
const at::Tensor& vol,
const float isolevel);

// Implementation which is exposed
inline std::tuple<at::Tensor, at::Tensor> MarchingCubes(
inline std::tuple<at::Tensor, at::Tensor, at::Tensor> MarchingCubes(
const at::Tensor& vol,
const float isolevel) {
if (vol.is_cuda()) {
#ifdef WITH_CUDA
CHECK_CUDA(vol);
const int D = vol.size(0);
const int H = vol.size(1);
const int W = vol.size(2);
if (D > 1024 || H > 1024 || W > 1024) {
AT_ERROR("Maximum volume size allowed 1K x 1K x 1K");
}
return MarchingCubesCuda(vol.contiguous(), isolevel);
#else
AT_ERROR("Not compiled with GPU support.");
#endif
}
return MarchingCubesCpu(vol.contiguous(), isolevel);
}
15 changes: 9 additions & 6 deletions pytorch3d/csrc/marching_cubes/marching_cubes_cpu.cpp
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Expand Up @@ -13,6 +13,7 @@
#include <unordered_map>
#include <vector>
#include "marching_cubes/marching_cubes_utils.h"
#include "marching_cubes/tables.h"

// Cpu implementation for Marching Cubes
// Args:
Expand All @@ -21,10 +22,11 @@
// whether points are within a volume.
//
// Returns:
// vertices: a float tensor of shape (N, 3) for positions of the mesh
// faces: a long tensor of shape (N, 3) for indices of the face vertices
// vertices: a float tensor of shape (N_verts, 3) for positions of the mesh
// faces: a long tensor of shape (N_faces, 3) for indices of the face
// ids: a long tensor of shape (N_verts) as placeholder
//
std::tuple<at::Tensor, at::Tensor> MarchingCubesCpu(
std::tuple<at::Tensor, at::Tensor, at::Tensor> MarchingCubesCpu(
const at::Tensor& vol,
const float isolevel) {
// volume shapes
Expand All @@ -48,7 +50,7 @@ std::tuple<at::Tensor, at::Tensor> MarchingCubesCpu(
for (int x = 0; x < W - 1; x++) {
Cube cube(x, y, z, vol_a, isolevel);
// Cube is entirely in/out of the surface
if (_FACE_TABLE[cube.cubeindex][0] == -1) {
if (_FACE_TABLE[cube.cubeindex][0] == 255) {
continue;
}
// store all boundary vertices that intersect with the edges
Expand All @@ -58,7 +60,7 @@ std::tuple<at::Tensor, at::Tensor> MarchingCubesCpu(
std::vector<Vertex> ps;

// Interpolate the vertices where the surface intersects with the cube
for (int j = 0; _FACE_TABLE[cube.cubeindex][j] != -1; j++) {
for (int j = 0; _FACE_TABLE[cube.cubeindex][j] != 255; j++) {
const int e = _FACE_TABLE[cube.cubeindex][j];
interp_points[e] = cube.VertexInterp(isolevel, e, vol_a);

Expand Down Expand Up @@ -95,6 +97,7 @@ std::tuple<at::Tensor, at::Tensor> MarchingCubesCpu(
const int n_vertices = verts.size();
const int64_t n_faces = (int64_t)faces.size() / 3;
auto vert_tensor = torch::zeros({n_vertices, 3}, torch::kFloat);
auto id_tensor = torch::zeros({n_vertices}, torch::kInt64); // placeholder
auto face_tensor = torch::zeros({n_faces, 3}, torch::kInt64);

auto vert_a = vert_tensor.accessor<float, 2>();
Expand All @@ -111,5 +114,5 @@ std::tuple<at::Tensor, at::Tensor> MarchingCubesCpu(
face_a[i][2] = faces.at(i * 3 + 2);
}

return std::make_tuple(vert_tensor, face_tensor);
return std::make_tuple(vert_tensor, face_tensor, id_tensor);
}
282 changes: 1 addition & 281 deletions pytorch3d/csrc/marching_cubes/marching_cubes_utils.h
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294 changes: 294 additions & 0 deletions pytorch3d/csrc/marching_cubes/tables.h
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@@ -0,0 +1,294 @@
/*
* Copyright (c) Meta Platforms, Inc. and affiliates.
* All rights reserved.
*
* This source code is licensed under the BSD-style license found in the
* LICENSE file in the root directory of this source tree.
*/

#pragma once
using uint = unsigned int;

// A table mapping from cubeindex to a list of face configurations.
// Each list contains at most 5 faces, where each face is represented with
// 3 consecutive numbers
// Table adapted from http://paulbourke.net/geometry/polygonise/
//
#define X 255
const unsigned char _FACE_TABLE[256][16] = {
{X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X},
{0, 8, 3, X, X, X, X, X, X, X, X, X, X, X, X, X},
{0, 1, 9, X, X, X, X, X, X, X, X, X, X, X, X, X},
{1, 8, 3, 9, 8, 1, X, X, X, X, X, X, X, X, X, X},
{1, 2, 10, X, X, X, X, X, X, X, X, X, X, X, X, X},
{0, 8, 3, 1, 2, 10, X, X, X, X, X, X, X, X, X, X},
{9, 2, 10, 0, 2, 9, X, X, X, X, X, X, X, X, X, X},
{2, 8, 3, 2, 10, 8, 10, 9, 8, X, X, X, X, X, X, X},
{3, 11, 2, X, X, X, X, X, X, X, X, X, X, X, X, X},
{0, 11, 2, 8, 11, 0, X, X, X, X, X, X, X, X, X, X},
{1, 9, 0, 2, 3, 11, X, X, X, X, X, X, X, X, X, X},
{1, 11, 2, 1, 9, 11, 9, 8, 11, X, X, X, X, X, X, X},
{3, 10, 1, 11, 10, 3, X, X, X, X, X, X, X, X, X, X},
{0, 10, 1, 0, 8, 10, 8, 11, 10, X, X, X, X, X, X, X},
{3, 9, 0, 3, 11, 9, 11, 10, 9, X, X, X, X, X, X, X},
{9, 8, 10, 10, 8, 11, X, X, X, X, X, X, X, X, X, X},
{4, 7, 8, X, X, X, X, X, X, X, X, X, X, X, X, X},
{4, 3, 0, 7, 3, 4, X, X, X, X, X, X, X, X, X, X},
{0, 1, 9, 8, 4, 7, X, X, X, X, X, X, X, X, X, X},
{4, 1, 9, 4, 7, 1, 7, 3, 1, X, X, X, X, X, X, X},
{1, 2, 10, 8, 4, 7, X, X, X, X, X, X, X, X, X, X},
{3, 4, 7, 3, 0, 4, 1, 2, 10, X, X, X, X, X, X, X},
{9, 2, 10, 9, 0, 2, 8, 4, 7, X, X, X, X, X, X, X},
{2, 10, 9, 2, 9, 7, 2, 7, 3, 7, 9, 4, X, X, X, X},
{8, 4, 7, 3, 11, 2, X, X, X, X, X, X, X, X, X, X},
{11, 4, 7, 11, 2, 4, 2, 0, 4, X, X, X, X, X, X, X},
{9, 0, 1, 8, 4, 7, 2, 3, 11, X, X, X, X, X, X, X},
{4, 7, 11, 9, 4, 11, 9, 11, 2, 9, 2, 1, X, X, X, X},
{3, 10, 1, 3, 11, 10, 7, 8, 4, X, X, X, X, X, X, X},
{1, 11, 10, 1, 4, 11, 1, 0, 4, 7, 11, 4, X, X, X, X},
{4, 7, 8, 9, 0, 11, 9, 11, 10, 11, 0, 3, X, X, X, X},
{4, 7, 11, 4, 11, 9, 9, 11, 10, X, X, X, X, X, X, X},
{9, 5, 4, X, X, X, X, X, X, X, X, X, X, X, X, X},
{9, 5, 4, 0, 8, 3, X, X, X, X, X, X, X, X, X, X},
{0, 5, 4, 1, 5, 0, X, X, X, X, X, X, X, X, X, X},
{8, 5, 4, 8, 3, 5, 3, 1, 5, X, X, X, X, X, X, X},
{1, 2, 10, 9, 5, 4, X, X, X, X, X, X, X, X, X, X},
{3, 0, 8, 1, 2, 10, 4, 9, 5, X, X, X, X, X, X, X},
{5, 2, 10, 5, 4, 2, 4, 0, 2, X, X, X, X, X, X, X},
{2, 10, 5, 3, 2, 5, 3, 5, 4, 3, 4, 8, X, X, X, X},
{9, 5, 4, 2, 3, 11, X, X, X, X, X, X, X, X, X, X},
{0, 11, 2, 0, 8, 11, 4, 9, 5, X, X, X, X, X, X, X},
{0, 5, 4, 0, 1, 5, 2, 3, 11, X, X, X, X, X, X, X},
{2, 1, 5, 2, 5, 8, 2, 8, 11, 4, 8, 5, X, X, X, X},
{10, 3, 11, 10, 1, 3, 9, 5, 4, X, X, X, X, X, X, X},
{4, 9, 5, 0, 8, 1, 8, 10, 1, 8, 11, 10, X, X, X, X},
{5, 4, 0, 5, 0, 11, 5, 11, 10, 11, 0, 3, X, X, X, X},
{5, 4, 8, 5, 8, 10, 10, 8, 11, X, X, X, X, X, X, X},
{9, 7, 8, 5, 7, 9, X, X, X, X, X, X, X, X, X, X},
{9, 3, 0, 9, 5, 3, 5, 7, 3, X, X, X, X, X, X, X},
{0, 7, 8, 0, 1, 7, 1, 5, 7, X, X, X, X, X, X, X},
{1, 5, 3, 3, 5, 7, X, X, X, X, X, X, X, X, X, X},
{9, 7, 8, 9, 5, 7, 10, 1, 2, X, X, X, X, X, X, X},
{10, 1, 2, 9, 5, 0, 5, 3, 0, 5, 7, 3, X, X, X, X},
{8, 0, 2, 8, 2, 5, 8, 5, 7, 10, 5, 2, X, X, X, X},
{2, 10, 5, 2, 5, 3, 3, 5, 7, X, X, X, X, X, X, X},
{7, 9, 5, 7, 8, 9, 3, 11, 2, X, X, X, X, X, X, X},
{9, 5, 7, 9, 7, 2, 9, 2, 0, 2, 7, 11, X, X, X, X},
{2, 3, 11, 0, 1, 8, 1, 7, 8, 1, 5, 7, X, X, X, X},
{11, 2, 1, 11, 1, 7, 7, 1, 5, X, X, X, X, X, X, X},
{9, 5, 8, 8, 5, 7, 10, 1, 3, 10, 3, 11, X, X, X, X},
{5, 7, 0, 5, 0, 9, 7, 11, 0, 1, 0, 10, 11, 10, 0, X},
{11, 10, 0, 11, 0, 3, 10, 5, 0, 8, 0, 7, 5, 7, 0, X},
{11, 10, 5, 7, 11, 5, X, X, X, X, X, X, X, X, X, X},
{10, 6, 5, X, X, X, X, X, X, X, X, X, X, X, X, X},
{0, 8, 3, 5, 10, 6, X, X, X, X, X, X, X, X, X, X},
{9, 0, 1, 5, 10, 6, X, X, X, X, X, X, X, X, X, X},
{1, 8, 3, 1, 9, 8, 5, 10, 6, X, X, X, X, X, X, X},
{1, 6, 5, 2, 6, 1, X, X, X, X, X, X, X, X, X, X},
{1, 6, 5, 1, 2, 6, 3, 0, 8, X, X, X, X, X, X, X},
{9, 6, 5, 9, 0, 6, 0, 2, 6, X, X, X, X, X, X, X},
{5, 9, 8, 5, 8, 2, 5, 2, 6, 3, 2, 8, X, X, X, X},
{2, 3, 11, 10, 6, 5, X, X, X, X, X, X, X, X, X, X},
{11, 0, 8, 11, 2, 0, 10, 6, 5, X, X, X, X, X, X, X},
{0, 1, 9, 2, 3, 11, 5, 10, 6, X, X, X, X, X, X, X},
{5, 10, 6, 1, 9, 2, 9, 11, 2, 9, 8, 11, X, X, X, X},
{6, 3, 11, 6, 5, 3, 5, 1, 3, X, X, X, X, X, X, X},
{0, 8, 11, 0, 11, 5, 0, 5, 1, 5, 11, 6, X, X, X, X},
{3, 11, 6, 0, 3, 6, 0, 6, 5, 0, 5, 9, X, X, X, X},
{6, 5, 9, 6, 9, 11, 11, 9, 8, X, X, X, X, X, X, X},
{5, 10, 6, 4, 7, 8, X, X, X, X, X, X, X, X, X, X},
{4, 3, 0, 4, 7, 3, 6, 5, 10, X, X, X, X, X, X, X},
{1, 9, 0, 5, 10, 6, 8, 4, 7, X, X, X, X, X, X, X},
{10, 6, 5, 1, 9, 7, 1, 7, 3, 7, 9, 4, X, X, X, X},
{6, 1, 2, 6, 5, 1, 4, 7, 8, X, X, X, X, X, X, X},
{1, 2, 5, 5, 2, 6, 3, 0, 4, 3, 4, 7, X, X, X, X},
{8, 4, 7, 9, 0, 5, 0, 6, 5, 0, 2, 6, X, X, X, X},
{7, 3, 9, 7, 9, 4, 3, 2, 9, 5, 9, 6, 2, 6, 9, X},
{3, 11, 2, 7, 8, 4, 10, 6, 5, X, X, X, X, X, X, X},
{5, 10, 6, 4, 7, 2, 4, 2, 0, 2, 7, 11, X, X, X, X},
{0, 1, 9, 4, 7, 8, 2, 3, 11, 5, 10, 6, X, X, X, X},
{9, 2, 1, 9, 11, 2, 9, 4, 11, 7, 11, 4, 5, 10, 6, X},
{8, 4, 7, 3, 11, 5, 3, 5, 1, 5, 11, 6, X, X, X, X},
{5, 1, 11, 5, 11, 6, 1, 0, 11, 7, 11, 4, 0, 4, 11, X},
{0, 5, 9, 0, 6, 5, 0, 3, 6, 11, 6, 3, 8, 4, 7, X},
{6, 5, 9, 6, 9, 11, 4, 7, 9, 7, 11, 9, X, X, X, X},
{10, 4, 9, 6, 4, 10, X, X, X, X, X, X, X, X, X, X},
{4, 10, 6, 4, 9, 10, 0, 8, 3, X, X, X, X, X, X, X},
{10, 0, 1, 10, 6, 0, 6, 4, 0, X, X, X, X, X, X, X},
{8, 3, 1, 8, 1, 6, 8, 6, 4, 6, 1, 10, X, X, X, X},
{1, 4, 9, 1, 2, 4, 2, 6, 4, X, X, X, X, X, X, X},
{3, 0, 8, 1, 2, 9, 2, 4, 9, 2, 6, 4, X, X, X, X},
{0, 2, 4, 4, 2, 6, X, X, X, X, X, X, X, X, X, X},
{8, 3, 2, 8, 2, 4, 4, 2, 6, X, X, X, X, X, X, X},
{10, 4, 9, 10, 6, 4, 11, 2, 3, X, X, X, X, X, X, X},
{0, 8, 2, 2, 8, 11, 4, 9, 10, 4, 10, 6, X, X, X, X},
{3, 11, 2, 0, 1, 6, 0, 6, 4, 6, 1, 10, X, X, X, X},
{6, 4, 1, 6, 1, 10, 4, 8, 1, 2, 1, 11, 8, 11, 1, X},
{9, 6, 4, 9, 3, 6, 9, 1, 3, 11, 6, 3, X, X, X, X},
{8, 11, 1, 8, 1, 0, 11, 6, 1, 9, 1, 4, 6, 4, 1, X},
{3, 11, 6, 3, 6, 0, 0, 6, 4, X, X, X, X, X, X, X},
{6, 4, 8, 11, 6, 8, X, X, X, X, X, X, X, X, X, X},
{7, 10, 6, 7, 8, 10, 8, 9, 10, X, X, X, X, X, X, X},
{0, 7, 3, 0, 10, 7, 0, 9, 10, 6, 7, 10, X, X, X, X},
{10, 6, 7, 1, 10, 7, 1, 7, 8, 1, 8, 0, X, X, X, X},
{10, 6, 7, 10, 7, 1, 1, 7, 3, X, X, X, X, X, X, X},
{1, 2, 6, 1, 6, 8, 1, 8, 9, 8, 6, 7, X, X, X, X},
{2, 6, 9, 2, 9, 1, 6, 7, 9, 0, 9, 3, 7, 3, 9, X},
{7, 8, 0, 7, 0, 6, 6, 0, 2, X, X, X, X, X, X, X},
{7, 3, 2, 6, 7, 2, X, X, X, X, X, X, X, X, X, X},
{2, 3, 11, 10, 6, 8, 10, 8, 9, 8, 6, 7, X, X, X, X},
{2, 0, 7, 2, 7, 11, 0, 9, 7, 6, 7, 10, 9, 10, 7, X},
{1, 8, 0, 1, 7, 8, 1, 10, 7, 6, 7, 10, 2, 3, 11, X},
{11, 2, 1, 11, 1, 7, 10, 6, 1, 6, 7, 1, X, X, X, X},
{8, 9, 6, 8, 6, 7, 9, 1, 6, 11, 6, 3, 1, 3, 6, X},
{0, 9, 1, 11, 6, 7, X, X, X, X, X, X, X, X, X, X},
{7, 8, 0, 7, 0, 6, 3, 11, 0, 11, 6, 0, X, X, X, X},
{7, 11, 6, X, X, X, X, X, X, X, X, X, X, X, X, X},
{7, 6, 11, X, X, X, X, X, X, X, X, X, X, X, X, X},
{3, 0, 8, 11, 7, 6, X, X, X, X, X, X, X, X, X, X},
{0, 1, 9, 11, 7, 6, X, X, X, X, X, X, X, X, X, X},
{8, 1, 9, 8, 3, 1, 11, 7, 6, X, X, X, X, X, X, X},
{10, 1, 2, 6, 11, 7, X, X, X, X, X, X, X, X, X, X},
{1, 2, 10, 3, 0, 8, 6, 11, 7, X, X, X, X, X, X, X},
{2, 9, 0, 2, 10, 9, 6, 11, 7, X, X, X, X, X, X, X},
{6, 11, 7, 2, 10, 3, 10, 8, 3, 10, 9, 8, X, X, X, X},
{7, 2, 3, 6, 2, 7, X, X, X, X, X, X, X, X, X, X},
{7, 0, 8, 7, 6, 0, 6, 2, 0, X, X, X, X, X, X, X},
{2, 7, 6, 2, 3, 7, 0, 1, 9, X, X, X, X, X, X, X},
{1, 6, 2, 1, 8, 6, 1, 9, 8, 8, 7, 6, X, X, X, X},
{10, 7, 6, 10, 1, 7, 1, 3, 7, X, X, X, X, X, X, X},
{10, 7, 6, 1, 7, 10, 1, 8, 7, 1, 0, 8, X, X, X, X},
{0, 3, 7, 0, 7, 10, 0, 10, 9, 6, 10, 7, X, X, X, X},
{7, 6, 10, 7, 10, 8, 8, 10, 9, X, X, X, X, X, X, X},
{6, 8, 4, 11, 8, 6, X, X, X, X, X, X, X, X, X, X},
{3, 6, 11, 3, 0, 6, 0, 4, 6, X, X, X, X, X, X, X},
{8, 6, 11, 8, 4, 6, 9, 0, 1, X, X, X, X, X, X, X},
{9, 4, 6, 9, 6, 3, 9, 3, 1, 11, 3, 6, X, X, X, X},
{6, 8, 4, 6, 11, 8, 2, 10, 1, X, X, X, X, X, X, X},
{1, 2, 10, 3, 0, 11, 0, 6, 11, 0, 4, 6, X, X, X, X},
{4, 11, 8, 4, 6, 11, 0, 2, 9, 2, 10, 9, X, X, X, X},
{10, 9, 3, 10, 3, 2, 9, 4, 3, 11, 3, 6, 4, 6, 3, X},
{8, 2, 3, 8, 4, 2, 4, 6, 2, X, X, X, X, X, X, X},
{0, 4, 2, 4, 6, 2, X, X, X, X, X, X, X, X, X, X},
{1, 9, 0, 2, 3, 4, 2, 4, 6, 4, 3, 8, X, X, X, X},
{1, 9, 4, 1, 4, 2, 2, 4, 6, X, X, X, X, X, X, X},
{8, 1, 3, 8, 6, 1, 8, 4, 6, 6, 10, 1, X, X, X, X},
{10, 1, 0, 10, 0, 6, 6, 0, 4, X, X, X, X, X, X, X},
{4, 6, 3, 4, 3, 8, 6, 10, 3, 0, 3, 9, 10, 9, 3, X},
{10, 9, 4, 6, 10, 4, X, X, X, X, X, X, X, X, X, X},
{4, 9, 5, 7, 6, 11, X, X, X, X, X, X, X, X, X, X},
{0, 8, 3, 4, 9, 5, 11, 7, 6, X, X, X, X, X, X, X},
{5, 0, 1, 5, 4, 0, 7, 6, 11, X, X, X, X, X, X, X},
{11, 7, 6, 8, 3, 4, 3, 5, 4, 3, 1, 5, X, X, X, X},
{9, 5, 4, 10, 1, 2, 7, 6, 11, X, X, X, X, X, X, X},
{6, 11, 7, 1, 2, 10, 0, 8, 3, 4, 9, 5, X, X, X, X},
{7, 6, 11, 5, 4, 10, 4, 2, 10, 4, 0, 2, X, X, X, X},
{3, 4, 8, 3, 5, 4, 3, 2, 5, 10, 5, 2, 11, 7, 6, X},
{7, 2, 3, 7, 6, 2, 5, 4, 9, X, X, X, X, X, X, X},
{9, 5, 4, 0, 8, 6, 0, 6, 2, 6, 8, 7, X, X, X, X},
{3, 6, 2, 3, 7, 6, 1, 5, 0, 5, 4, 0, X, X, X, X},
{6, 2, 8, 6, 8, 7, 2, 1, 8, 4, 8, 5, 1, 5, 8, X},
{9, 5, 4, 10, 1, 6, 1, 7, 6, 1, 3, 7, X, X, X, X},
{1, 6, 10, 1, 7, 6, 1, 0, 7, 8, 7, 0, 9, 5, 4, X},
{4, 0, 10, 4, 10, 5, 0, 3, 10, 6, 10, 7, 3, 7, 10, X},
{7, 6, 10, 7, 10, 8, 5, 4, 10, 4, 8, 10, X, X, X, X},
{6, 9, 5, 6, 11, 9, 11, 8, 9, X, X, X, X, X, X, X},
{3, 6, 11, 0, 6, 3, 0, 5, 6, 0, 9, 5, X, X, X, X},
{0, 11, 8, 0, 5, 11, 0, 1, 5, 5, 6, 11, X, X, X, X},
{6, 11, 3, 6, 3, 5, 5, 3, 1, X, X, X, X, X, X, X},
{1, 2, 10, 9, 5, 11, 9, 11, 8, 11, 5, 6, X, X, X, X},
{0, 11, 3, 0, 6, 11, 0, 9, 6, 5, 6, 9, 1, 2, 10, X},
{11, 8, 5, 11, 5, 6, 8, 0, 5, 10, 5, 2, 0, 2, 5, X},
{6, 11, 3, 6, 3, 5, 2, 10, 3, 10, 5, 3, X, X, X, X},
{5, 8, 9, 5, 2, 8, 5, 6, 2, 3, 8, 2, X, X, X, X},
{9, 5, 6, 9, 6, 0, 0, 6, 2, X, X, X, X, X, X, X},
{1, 5, 8, 1, 8, 0, 5, 6, 8, 3, 8, 2, 6, 2, 8, X},
{1, 5, 6, 2, 1, 6, X, X, X, X, X, X, X, X, X, X},
{1, 3, 6, 1, 6, 10, 3, 8, 6, 5, 6, 9, 8, 9, 6, X},
{10, 1, 0, 10, 0, 6, 9, 5, 0, 5, 6, 0, X, X, X, X},
{0, 3, 8, 5, 6, 10, X, X, X, X, X, X, X, X, X, X},
{10, 5, 6, X, X, X, X, X, X, X, X, X, X, X, X, X},
{11, 5, 10, 7, 5, 11, X, X, X, X, X, X, X, X, X, X},
{11, 5, 10, 11, 7, 5, 8, 3, 0, X, X, X, X, X, X, X},
{5, 11, 7, 5, 10, 11, 1, 9, 0, X, X, X, X, X, X, X},
{10, 7, 5, 10, 11, 7, 9, 8, 1, 8, 3, 1, X, X, X, X},
{11, 1, 2, 11, 7, 1, 7, 5, 1, X, X, X, X, X, X, X},
{0, 8, 3, 1, 2, 7, 1, 7, 5, 7, 2, 11, X, X, X, X},
{9, 7, 5, 9, 2, 7, 9, 0, 2, 2, 11, 7, X, X, X, X},
{7, 5, 2, 7, 2, 11, 5, 9, 2, 3, 2, 8, 9, 8, 2, X},
{2, 5, 10, 2, 3, 5, 3, 7, 5, X, X, X, X, X, X, X},
{8, 2, 0, 8, 5, 2, 8, 7, 5, 10, 2, 5, X, X, X, X},
{9, 0, 1, 5, 10, 3, 5, 3, 7, 3, 10, 2, X, X, X, X},
{9, 8, 2, 9, 2, 1, 8, 7, 2, 10, 2, 5, 7, 5, 2, X},
{1, 3, 5, 3, 7, 5, X, X, X, X, X, X, X, X, X, X},
{0, 8, 7, 0, 7, 1, 1, 7, 5, X, X, X, X, X, X, X},
{9, 0, 3, 9, 3, 5, 5, 3, 7, X, X, X, X, X, X, X},
{9, 8, 7, 5, 9, 7, X, X, X, X, X, X, X, X, X, X},
{5, 8, 4, 5, 10, 8, 10, 11, 8, X, X, X, X, X, X, X},
{5, 0, 4, 5, 11, 0, 5, 10, 11, 11, 3, 0, X, X, X, X},
{0, 1, 9, 8, 4, 10, 8, 10, 11, 10, 4, 5, X, X, X, X},
{10, 11, 4, 10, 4, 5, 11, 3, 4, 9, 4, 1, 3, 1, 4, X},
{2, 5, 1, 2, 8, 5, 2, 11, 8, 4, 5, 8, X, X, X, X},
{0, 4, 11, 0, 11, 3, 4, 5, 11, 2, 11, 1, 5, 1, 11, X},
{0, 2, 5, 0, 5, 9, 2, 11, 5, 4, 5, 8, 11, 8, 5, X},
{9, 4, 5, 2, 11, 3, X, X, X, X, X, X, X, X, X, X},
{2, 5, 10, 3, 5, 2, 3, 4, 5, 3, 8, 4, X, X, X, X},
{5, 10, 2, 5, 2, 4, 4, 2, 0, X, X, X, X, X, X, X},
{3, 10, 2, 3, 5, 10, 3, 8, 5, 4, 5, 8, 0, 1, 9, X},
{5, 10, 2, 5, 2, 4, 1, 9, 2, 9, 4, 2, X, X, X, X},
{8, 4, 5, 8, 5, 3, 3, 5, 1, X, X, X, X, X, X, X},
{0, 4, 5, 1, 0, 5, X, X, X, X, X, X, X, X, X, X},
{8, 4, 5, 8, 5, 3, 9, 0, 5, 0, 3, 5, X, X, X, X},
{9, 4, 5, X, X, X, X, X, X, X, X, X, X, X, X, X},
{4, 11, 7, 4, 9, 11, 9, 10, 11, X, X, X, X, X, X, X},
{0, 8, 3, 4, 9, 7, 9, 11, 7, 9, 10, 11, X, X, X, X},
{1, 10, 11, 1, 11, 4, 1, 4, 0, 7, 4, 11, X, X, X, X},
{3, 1, 4, 3, 4, 8, 1, 10, 4, 7, 4, 11, 10, 11, 4, X},
{4, 11, 7, 9, 11, 4, 9, 2, 11, 9, 1, 2, X, X, X, X},
{9, 7, 4, 9, 11, 7, 9, 1, 11, 2, 11, 1, 0, 8, 3, X},
{11, 7, 4, 11, 4, 2, 2, 4, 0, X, X, X, X, X, X, X},
{11, 7, 4, 11, 4, 2, 8, 3, 4, 3, 2, 4, X, X, X, X},
{2, 9, 10, 2, 7, 9, 2, 3, 7, 7, 4, 9, X, X, X, X},
{9, 10, 7, 9, 7, 4, 10, 2, 7, 8, 7, 0, 2, 0, 7, X},
{3, 7, 10, 3, 10, 2, 7, 4, 10, 1, 10, 0, 4, 0, 10, X},
{1, 10, 2, 8, 7, 4, X, X, X, X, X, X, X, X, X, X},
{4, 9, 1, 4, 1, 7, 7, 1, 3, X, X, X, X, X, X, X},
{4, 9, 1, 4, 1, 7, 0, 8, 1, 8, 7, 1, X, X, X, X},
{4, 0, 3, 7, 4, 3, X, X, X, X, X, X, X, X, X, X},
{4, 8, 7, X, X, X, X, X, X, X, X, X, X, X, X, X},
{9, 10, 8, 10, 11, 8, X, X, X, X, X, X, X, X, X, X},
{3, 0, 9, 3, 9, 11, 11, 9, 10, X, X, X, X, X, X, X},
{0, 1, 10, 0, 10, 8, 8, 10, 11, X, X, X, X, X, X, X},
{3, 1, 10, 11, 3, 10, X, X, X, X, X, X, X, X, X, X},
{1, 2, 11, 1, 11, 9, 9, 11, 8, X, X, X, X, X, X, X},
{3, 0, 9, 3, 9, 11, 1, 2, 9, 2, 11, 9, X, X, X, X},
{0, 2, 11, 8, 0, 11, X, X, X, X, X, X, X, X, X, X},
{3, 2, 11, X, X, X, X, X, X, X, X, X, X, X, X, X},
{2, 3, 8, 2, 8, 10, 10, 8, 9, X, X, X, X, X, X, X},
{9, 10, 2, 0, 9, 2, X, X, X, X, X, X, X, X, X, X},
{2, 3, 8, 2, 8, 10, 0, 1, 8, 1, 10, 8, X, X, X, X},
{1, 10, 2, X, X, X, X, X, X, X, X, X, X, X, X, X},
{1, 3, 8, 9, 1, 8, X, X, X, X, X, X, X, X, X, X},
{0, 9, 1, X, X, X, X, X, X, X, X, X, X, X, X, X},
{0, 3, 8, X, X, X, X, X, X, X, X, X, X, X, X, X},
{X, X, X, X, X, X, X, X, X, X, X, X, X, X, X, X}};
#undef X

// Table mapping each edge to the corresponding cube vertices offsets
const uint _EDGE_TO_VERTICES[12][2] = {
{0, 1},
{1, 5},
{4, 5},
{0, 4},
{2, 3},
{3, 7},
{6, 7},
{2, 6},
{0, 2},
{1, 3},
{5, 7},
{4, 6},
};

// Table mapping from 0-7 to v0-v7 in cube.vertices
const int _INDEX_TABLE[8] = {0, 1, 5, 4, 2, 3, 7, 6};
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