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# Copyright (c) 2023, NVIDIA CORPORATION & AFFILIATES. All rights reserved.
#
# NVIDIA CORPORATION & AFFILIATES and its licensors retain all intellectual property
# and proprietary rights in and to this software, related documentation
# and any modifications thereto. Any use, reproduction, disclosure or
# distribution of this software and related documentation without an express
# license agreement from NVIDIA CORPORATION & AFFILIATES is strictly prohibited.
import torch
from util.tables import *
__all__ = [
'FlexiCubes'
]
class FlexiCubes:
"""
This class implements the FlexiCubes method for extracting meshes from scalar fields.
It maintains a series of lookup tables and indices to support the mesh extraction process.
FlexiCubes, a differentiable variant of the Dual Marching Cubes (DMC) scheme, enhances
the geometric fidelity and mesh quality of reconstructed meshes by dynamically adjusting
the surface representation through gradient-based optimization.
During instantiation, the class loads DMC tables from a file and transforms them into
PyTorch tensors on the specified device.
Attributes:
device (str): Specifies the computational device (default is "cuda").
dmc_table (torch.Tensor): Dual Marching Cubes (DMC) table that encodes the edges
associated with each dual vertex in 256 Marching Cubes (MC) configurations.
num_vd_table (torch.Tensor): Table holding the number of dual vertices in each of
the 256 MC configurations.
check_table (torch.Tensor): Table resolving ambiguity in cases C16 and C19
of the DMC configurations.
tet_table (torch.Tensor): Lookup table used in tetrahedralizing the isosurface.
quad_split_1 (torch.Tensor): Indices for splitting a quad into two triangles
along one diagonal.
quad_split_2 (torch.Tensor): Alternative indices for splitting a quad into
two triangles along the other diagonal.
quad_split_train (torch.Tensor): Indices for splitting a quad into four triangles
during training by connecting all edges to their midpoints.
cube_corners (torch.Tensor): Defines the positions of a standard unit cube's
eight corners in 3D space, ordered starting from the origin (0,0,0),
moving along the x-axis, then y-axis, and finally z-axis.
Used as a blueprint for generating a voxel grid.
cube_corners_idx (torch.Tensor): Cube corners indexed as powers of 2, used
to retrieve the case id.
cube_edges (torch.Tensor): Edge connections in a cube, listed in pairs.
Used to retrieve edge vertices in DMC.
edge_dir_table (torch.Tensor): A mapping tensor that associates edge indices with
their corresponding axis. For instance, edge_dir_table[0] = 0 indicates that the
first edge is oriented along the x-axis.
dir_faces_table (torch.Tensor): A tensor that maps the corresponding axis of shared edges
across four adjacent cubes to the shared faces of these cubes. For instance,
dir_faces_table[0] = [5, 4] implies that for four cubes sharing an edge along
the x-axis, the first and second cubes share faces indexed as 5 and 4, respectively.
This tensor is only utilized during isosurface tetrahedralization.
adj_pairs (torch.Tensor):
A tensor containing index pairs that correspond to neighboring cubes that share the same edge.
qef_reg_scale (float):
The scaling factor applied to the regularization loss to prevent issues with singularity
when solving the QEF. This parameter is only used when a 'grad_func' is specified.
weight_scale (float):
The scale of weights in FlexiCubes. Should be between 0 and 1.
"""
def __init__(self, device="cuda", qef_reg_scale=1e-3, weight_scale=0.99):
self.device = device
self.dmc_table = torch.tensor(dmc_table, dtype=torch.long, device=device, requires_grad=False)
self.num_vd_table = torch.tensor(num_vd_table,
dtype=torch.long, device=device, requires_grad=False)
self.check_table = torch.tensor(
check_table,
dtype=torch.long, device=device, requires_grad=False)
self.tet_table = torch.tensor(tet_table, dtype=torch.long, device=device, requires_grad=False)
self.quad_split_1 = torch.tensor([0, 1, 2, 0, 2, 3], dtype=torch.long, device=device, requires_grad=False)
self.quad_split_2 = torch.tensor([0, 1, 3, 3, 1, 2], dtype=torch.long, device=device, requires_grad=False)
self.quad_split_train = torch.tensor(
[0, 1, 1, 2, 2, 3, 3, 0], dtype=torch.long, device=device, requires_grad=False)
self.cube_corners = torch.tensor([[0, 0, 0], [1, 0, 0], [0, 1, 0], [1, 1, 0], [0, 0, 1], [
1, 0, 1], [0, 1, 1], [1, 1, 1]], dtype=torch.float, device=device)
self.cube_corners_idx = torch.pow(2, torch.arange(8, requires_grad=False))
self.cube_edges = torch.tensor([0, 1, 1, 5, 4, 5, 0, 4, 2, 3, 3, 7, 6, 7, 2, 6,
2, 0, 3, 1, 7, 5, 6, 4], dtype=torch.long, device=device, requires_grad=False)
self.edge_dir_table = torch.tensor([0, 2, 0, 2, 0, 2, 0, 2, 1, 1, 1, 1],
dtype=torch.long, device=device)
self.dir_faces_table = torch.tensor([
[[5, 4], [3, 2], [4, 5], [2, 3]],
[[5, 4], [1, 0], [4, 5], [0, 1]],
[[3, 2], [1, 0], [2, 3], [0, 1]]
], dtype=torch.long, device=device)
self.adj_pairs = torch.tensor([0, 1, 1, 3, 3, 2, 2, 0], dtype=torch.long, device=device)
self.qef_reg_scale = qef_reg_scale
self.weight_scale = weight_scale
def construct_voxel_grid(self, res):
"""
Generates a voxel grid based on the specified resolution.
Args:
res (int or list[int]): The resolution of the voxel grid. If an integer
is provided, it is used for all three dimensions. If a list or tuple
of 3 integers is provided, they define the resolution for the x,
y, and z dimensions respectively.
Returns:
(torch.Tensor, torch.Tensor): Returns the vertices and the indices of the
cube corners (index into vertices) of the constructed voxel grid.
The vertices are centered at the origin, with the length of each
dimension in the grid being one.
"""
base_cube_f = torch.arange(8).to(self.device)
if isinstance(res, int):
res = (res, res, res)
voxel_grid_template = torch.ones(res, device=self.device)
res = torch.tensor([res], dtype=torch.float, device=self.device)
coords = torch.nonzero(voxel_grid_template).float() / res # N, 3
verts = (self.cube_corners.unsqueeze(0) / res + coords.unsqueeze(1)).reshape(-1, 3)
cubes = (base_cube_f.unsqueeze(0) +
torch.arange(coords.shape[0], device=self.device).unsqueeze(1) * 8).reshape(-1)
verts_rounded = torch.round(verts * 10**5) / (10**5)
verts_unique, inverse_indices = torch.unique(verts_rounded, dim=0, return_inverse=True)
cubes = inverse_indices[cubes.reshape(-1)].reshape(-1, 8)
return verts_unique - 0.5, cubes
def __call__(self, x_nx3, s_n, cube_fx8, res, beta_fx12=None, alpha_fx8=None,
gamma_f=None, training=False, output_tetmesh=False, grad_func=None):
r"""
Main function for mesh extraction from scalar field using FlexiCubes. This function converts
discrete signed distance fields, encoded on voxel grids and additional per-cube parameters,
to triangle or tetrahedral meshes using a differentiable operation as described in
`Flexible Isosurface Extraction for Gradient-Based Mesh Optimization`_. FlexiCubes enhances
mesh quality and geometric fidelity by adjusting the surface representation based on gradient
optimization. The output surface is differentiable with respect to the input vertex positions,
scalar field values, and weight parameters.
If you intend to extract a surface mesh from a fixed Signed Distance Field without the
optimization of parameters, it is suggested to provide the "grad_func" which should
return the surface gradient at any given 3D position. When grad_func is provided, the process
to determine the dual vertex position adapts to solve a Quadratic Error Function (QEF), as
described in the `Manifold Dual Contouring`_ paper, and employs an smart splitting strategy.
Please note, this approach is non-differentiable.
For more details and example usage in optimization, refer to the
`Flexible Isosurface Extraction for Gradient-Based Mesh Optimization`_ SIGGRAPH 2023 paper.
Args:
x_nx3 (torch.Tensor): Coordinates of the voxel grid vertices, can be deformed.
s_n (torch.Tensor): Scalar field values at each vertex of the voxel grid. Negative values
denote that the corresponding vertex resides inside the isosurface. This affects
the directions of the extracted triangle faces and volume to be tetrahedralized.
cube_fx8 (torch.Tensor): Indices of 8 vertices for each cube in the voxel grid.
res (int or list[int]): The resolution of the voxel grid. If an integer is provided, it
is used for all three dimensions. If a list or tuple of 3 integers is provided, they
specify the resolution for the x, y, and z dimensions respectively.
beta_fx12 (torch.Tensor, optional): Weight parameters for the cube edges to adjust dual
vertices positioning. Defaults to uniform value for all edges.
alpha_fx8 (torch.Tensor, optional): Weight parameters for the cube corners to adjust dual
vertices positioning. Defaults to uniform value for all vertices.
gamma_f (torch.Tensor, optional): Weight parameters to control the splitting of
quadrilaterals into triangles. Defaults to uniform value for all cubes.
training (bool, optional): If set to True, applies differentiable quad splitting for
training. Defaults to False.
output_tetmesh (bool, optional): If set to True, outputs a tetrahedral mesh, otherwise,
outputs a triangular mesh. Defaults to False.
grad_func (callable, optional): A function to compute the surface gradient at specified
3D positions (input: Nx3 positions). The function should return gradients as an Nx3
tensor. If None, the original FlexiCubes algorithm is utilized. Defaults to None.
Returns:
(torch.Tensor, torch.LongTensor, torch.Tensor): Tuple containing:
- Vertices for the extracted triangular/tetrahedral mesh.
- Faces for the extracted triangular/tetrahedral mesh.
- Regularizer L_dev, computed per dual vertex.
.. _Flexible Isosurface Extraction for Gradient-Based Mesh Optimization:
https://research.nvidia.com/labs/toronto-ai/flexicubes/
.. _Manifold Dual Contouring:
https://people.engr.tamu.edu/schaefer/research/dualsimp_tvcg.pdf
"""
surf_cubes, occ_fx8 = self._identify_surf_cubes(s_n, cube_fx8)
if surf_cubes.sum() == 0:
return torch.zeros(
(0, 3),
device=self.device), torch.zeros(
(0, 4),
dtype=torch.long, device=self.device) if output_tetmesh else torch.zeros(
(0, 3),
dtype=torch.long, device=self.device), torch.zeros(
(0),
device=self.device)
beta_fx12, alpha_fx8, gamma_f = self._normalize_weights(beta_fx12, alpha_fx8, gamma_f, surf_cubes)
case_ids = self._get_case_id(occ_fx8, surf_cubes, res)
surf_edges, idx_map, edge_counts, surf_edges_mask = self._identify_surf_edges(s_n, cube_fx8, surf_cubes)
vd, L_dev, vd_gamma, vd_idx_map = self._compute_vd(
x_nx3, cube_fx8[surf_cubes], surf_edges, s_n, case_ids, beta_fx12, alpha_fx8, gamma_f, idx_map, grad_func)
vertices, faces, s_edges, edge_indices = self._triangulate(
s_n, surf_edges, vd, vd_gamma, edge_counts, idx_map, vd_idx_map, surf_edges_mask, training, grad_func)
if not output_tetmesh:
return vertices, faces, L_dev
else:
vertices, tets = self._tetrahedralize(
x_nx3, s_n, cube_fx8, vertices, faces, surf_edges, s_edges, vd_idx_map, case_ids, edge_indices,
surf_cubes, training)
return vertices, tets, L_dev
def _compute_reg_loss(self, vd, ue, edge_group_to_vd, vd_num_edges):
"""
Regularizer L_dev as in Equation 8
"""
dist = torch.norm(ue - torch.index_select(input=vd, index=edge_group_to_vd, dim=0), dim=-1)
mean_l2 = torch.zeros_like(vd[:, 0])
mean_l2 = (mean_l2).index_add_(0, edge_group_to_vd, dist) / vd_num_edges.squeeze(1).float()
mad = (dist - torch.index_select(input=mean_l2, index=edge_group_to_vd, dim=0)).abs()
return mad
def _normalize_weights(self, beta_fx12, alpha_fx8, gamma_f, surf_cubes):
"""
Normalizes the given weights to be non-negative. If input weights are None, it creates and returns a set of weights of ones.
"""
n_cubes = surf_cubes.shape[0]
if beta_fx12 is not None:
beta_fx12 = (torch.tanh(beta_fx12) * self.weight_scale + 1)
else:
beta_fx12 = torch.ones((n_cubes, 12), dtype=torch.float, device=self.device)
if alpha_fx8 is not None:
alpha_fx8 = (torch.tanh(alpha_fx8) * self.weight_scale + 1)
else:
alpha_fx8 = torch.ones((n_cubes, 8), dtype=torch.float, device=self.device)
if gamma_f is not None:
gamma_f = torch.sigmoid(gamma_f) * self.weight_scale + (1 - self.weight_scale)/2
else:
gamma_f = torch.ones((n_cubes), dtype=torch.float, device=self.device)
return beta_fx12[surf_cubes], alpha_fx8[surf_cubes], gamma_f[surf_cubes]
@torch.no_grad()
def _get_case_id(self, occ_fx8, surf_cubes, res):
"""
Obtains the ID of topology cases based on cell corner occupancy. This function resolves the
ambiguity in the Dual Marching Cubes (DMC) configurations as described in Section 1.3 of the
supplementary material. It should be noted that this function assumes a regular grid.
"""
case_ids = (occ_fx8[surf_cubes] * self.cube_corners_idx.to(self.device).unsqueeze(0)).sum(-1)
problem_config = self.check_table.to(self.device)[case_ids]
to_check = problem_config[..., 0] == 1
problem_config = problem_config[to_check]
if not isinstance(res, (list, tuple)):
res = [res, res, res]
# The 'problematic_configs' only contain configurations for surface cubes. Next, we construct a 3D array,
# 'problem_config_full', to store configurations for all cubes (with default config for non-surface cubes).
# This allows efficient checking on adjacent cubes.
problem_config_full = torch.zeros(list(res) + [5], device=self.device, dtype=torch.long)
vol_idx = torch.nonzero(problem_config_full[..., 0] == 0) # N, 3
vol_idx_problem = vol_idx[surf_cubes][to_check]
problem_config_full[vol_idx_problem[..., 0], vol_idx_problem[..., 1], vol_idx_problem[..., 2]] = problem_config
vol_idx_problem_adj = vol_idx_problem + problem_config[..., 1:4]
within_range = (
vol_idx_problem_adj[..., 0] >= 0) & (
vol_idx_problem_adj[..., 0] < res[0]) & (
vol_idx_problem_adj[..., 1] >= 0) & (
vol_idx_problem_adj[..., 1] < res[1]) & (
vol_idx_problem_adj[..., 2] >= 0) & (
vol_idx_problem_adj[..., 2] < res[2])
vol_idx_problem = vol_idx_problem[within_range]
vol_idx_problem_adj = vol_idx_problem_adj[within_range]
problem_config = problem_config[within_range]
problem_config_adj = problem_config_full[vol_idx_problem_adj[..., 0],
vol_idx_problem_adj[..., 1], vol_idx_problem_adj[..., 2]]
# If two cubes with cases C16 and C19 share an ambiguous face, both cases are inverted.
to_invert = (problem_config_adj[..., 0] == 1)
idx = torch.arange(case_ids.shape[0], device=self.device)[to_check][within_range][to_invert]
case_ids.index_put_((idx,), problem_config[to_invert][..., -1])
return case_ids
@torch.no_grad()
def _identify_surf_edges(self, s_n, cube_fx8, surf_cubes):
"""
Identifies grid edges that intersect with the underlying surface by checking for opposite signs. As each edge
can be shared by multiple cubes, this function also assigns a unique index to each surface-intersecting edge
and marks the cube edges with this index.
"""
occ_n = s_n < 0
all_edges = cube_fx8[surf_cubes][:, self.cube_edges].reshape(-1, 2)
unique_edges, _idx_map, counts = torch.unique(all_edges, dim=0, return_inverse=True, return_counts=True)
unique_edges = unique_edges.long()
mask_edges = occ_n[unique_edges.reshape(-1)].reshape(-1, 2).sum(-1) == 1
surf_edges_mask = mask_edges[_idx_map]
counts = counts[_idx_map]
mapping = torch.ones((unique_edges.shape[0]), dtype=torch.long, device=cube_fx8.device) * -1
mapping[mask_edges] = torch.arange(mask_edges.sum(), device=cube_fx8.device)
# Shaped as [number of cubes x 12 edges per cube]. This is later used to map a cube edge to the unique index
# for a surface-intersecting edge. Non-surface-intersecting edges are marked with -1.
idx_map = mapping[_idx_map]
surf_edges = unique_edges[mask_edges]
return surf_edges, idx_map, counts, surf_edges_mask
@torch.no_grad()
def _identify_surf_cubes(self, s_n, cube_fx8):
"""
Identifies grid cubes that intersect with the underlying surface by checking if the signs at
all corners are not identical.
"""
occ_n = s_n < 0
occ_fx8 = occ_n[cube_fx8.reshape(-1)].reshape(-1, 8)
_occ_sum = torch.sum(occ_fx8, -1)
surf_cubes = (_occ_sum > 0) & (_occ_sum < 8)
return surf_cubes, occ_fx8
def _linear_interp(self, edges_weight, edges_x):
"""
Computes the location of zero-crossings on 'edges_x' using linear interpolation with 'edges_weight'.
"""
edge_dim = edges_weight.dim() - 2
assert edges_weight.shape[edge_dim] == 2
edges_weight = torch.cat([torch.index_select(input=edges_weight, index=torch.tensor(1, device=self.device), dim=edge_dim), -
torch.index_select(input=edges_weight, index=torch.tensor(0, device=self.device), dim=edge_dim)], edge_dim)
denominator = edges_weight.sum(edge_dim)
ue = (edges_x * edges_weight).sum(edge_dim) / denominator
return ue
def _solve_vd_QEF(self, p_bxnx3, norm_bxnx3, c_bx3=None):
p_bxnx3 = p_bxnx3.reshape(-1, 7, 3)
norm_bxnx3 = norm_bxnx3.reshape(-1, 7, 3)
c_bx3 = c_bx3.reshape(-1, 3)
A = norm_bxnx3
B = ((p_bxnx3) * norm_bxnx3).sum(-1, keepdims=True)
A_reg = (torch.eye(3, device=p_bxnx3.device) * self.qef_reg_scale).unsqueeze(0).repeat(p_bxnx3.shape[0], 1, 1)
B_reg = (self.qef_reg_scale * c_bx3).unsqueeze(-1)
A = torch.cat([A, A_reg], 1)
B = torch.cat([B, B_reg], 1)
dual_verts = torch.linalg.lstsq(A, B).solution.squeeze(-1)
return dual_verts
def _compute_vd(self, x_nx3, surf_cubes_fx8, surf_edges, s_n, case_ids, beta_fx12, alpha_fx8, gamma_f, idx_map, grad_func):
"""
Computes the location of dual vertices as described in Section 4.2
"""
alpha_nx12x2 = torch.index_select(input=alpha_fx8, index=self.cube_edges, dim=1).reshape(-1, 12, 2)
surf_edges_x = torch.index_select(input=x_nx3, index=surf_edges.reshape(-1), dim=0).reshape(-1, 2, 3)
surf_edges_s = torch.index_select(input=s_n, index=surf_edges.reshape(-1), dim=0).reshape(-1, 2, 1)
zero_crossing = self._linear_interp(surf_edges_s, surf_edges_x)
idx_map = idx_map.reshape(-1, 12)
num_vd = torch.index_select(input=self.num_vd_table, index=case_ids, dim=0)
edge_group, edge_group_to_vd, edge_group_to_cube, vd_num_edges, vd_gamma = [], [], [], [], []
total_num_vd = 0
vd_idx_map = torch.zeros((case_ids.shape[0], 12), dtype=torch.long, device=self.device, requires_grad=False)
if grad_func is not None:
normals = torch.nn.functional.normalize(grad_func(zero_crossing), dim=-1)
vd = []
for num in torch.unique(num_vd):
cur_cubes = (num_vd == num) # consider cubes with the same numbers of vd emitted (for batching)
curr_num_vd = cur_cubes.sum() * num
curr_edge_group = self.dmc_table[case_ids[cur_cubes], :num].reshape(-1, num * 7)
curr_edge_group_to_vd = torch.arange(
curr_num_vd, device=self.device).unsqueeze(-1).repeat(1, 7) + total_num_vd
total_num_vd += curr_num_vd
curr_edge_group_to_cube = torch.arange(idx_map.shape[0], device=self.device)[
cur_cubes].unsqueeze(-1).repeat(1, num * 7).reshape_as(curr_edge_group)
curr_mask = (curr_edge_group != -1)
edge_group.append(torch.masked_select(curr_edge_group, curr_mask))
edge_group_to_vd.append(torch.masked_select(curr_edge_group_to_vd.reshape_as(curr_edge_group), curr_mask))
edge_group_to_cube.append(torch.masked_select(curr_edge_group_to_cube, curr_mask))
vd_num_edges.append(curr_mask.reshape(-1, 7).sum(-1, keepdims=True))
vd_gamma.append(torch.masked_select(gamma_f, cur_cubes).unsqueeze(-1).repeat(1, num).reshape(-1))
if grad_func is not None:
with torch.no_grad():
cube_e_verts_idx = idx_map[cur_cubes]
curr_edge_group[~curr_mask] = 0
verts_group_idx = torch.gather(input=cube_e_verts_idx, dim=1, index=curr_edge_group)
verts_group_idx[verts_group_idx == -1] = 0
verts_group_pos = torch.index_select(
input=zero_crossing, index=verts_group_idx.reshape(-1), dim=0).reshape(-1, num.item(), 7, 3)
v0 = x_nx3[surf_cubes_fx8[cur_cubes][:, 0]].reshape(-1, 1, 1, 3).repeat(1, num.item(), 1, 1)
curr_mask = curr_mask.reshape(-1, num.item(), 7, 1)
verts_centroid = (verts_group_pos * curr_mask).sum(2) / (curr_mask.sum(2))
normals_bx7x3 = torch.index_select(input=normals, index=verts_group_idx.reshape(-1), dim=0).reshape(
-1, num.item(), 7,
3)
curr_mask = curr_mask.squeeze(2)
vd.append(self._solve_vd_QEF((verts_group_pos - v0) * curr_mask, normals_bx7x3 * curr_mask,
verts_centroid - v0.squeeze(2)) + v0.reshape(-1, 3))
edge_group = torch.cat(edge_group)
edge_group_to_vd = torch.cat(edge_group_to_vd)
edge_group_to_cube = torch.cat(edge_group_to_cube)
vd_num_edges = torch.cat(vd_num_edges)
vd_gamma = torch.cat(vd_gamma)
if grad_func is not None:
vd = torch.cat(vd)
L_dev = torch.zeros([1], device=self.device)
else:
vd = torch.zeros((total_num_vd, 3), device=self.device)
beta_sum = torch.zeros((total_num_vd, 1), device=self.device)
idx_group = torch.gather(input=idx_map.reshape(-1), dim=0, index=edge_group_to_cube * 12 + edge_group)
x_group = torch.index_select(input=surf_edges_x, index=idx_group.reshape(-1), dim=0).reshape(-1, 2, 3)
s_group = torch.index_select(input=surf_edges_s, index=idx_group.reshape(-1), dim=0).reshape(-1, 2, 1)
zero_crossing_group = torch.index_select(
input=zero_crossing, index=idx_group.reshape(-1), dim=0).reshape(-1, 3)
alpha_group = torch.index_select(input=alpha_nx12x2.reshape(-1, 2), dim=0,
index=edge_group_to_cube * 12 + edge_group).reshape(-1, 2, 1)
ue_group = self._linear_interp(s_group * alpha_group, x_group)
beta_group = torch.gather(input=beta_fx12.reshape(-1), dim=0,
index=edge_group_to_cube * 12 + edge_group).reshape(-1, 1)
beta_sum = beta_sum.index_add_(0, index=edge_group_to_vd, source=beta_group)
vd = vd.index_add_(0, index=edge_group_to_vd, source=ue_group * beta_group) / beta_sum
L_dev = self._compute_reg_loss(vd, zero_crossing_group, edge_group_to_vd, vd_num_edges)
v_idx = torch.arange(vd.shape[0], device=self.device) # + total_num_vd
vd_idx_map = (vd_idx_map.reshape(-1)).scatter(dim=0, index=edge_group_to_cube *
12 + edge_group, src=v_idx[edge_group_to_vd])
return vd, L_dev, vd_gamma, vd_idx_map
def _triangulate(self, s_n, surf_edges, vd, vd_gamma, edge_counts, idx_map, vd_idx_map, surf_edges_mask, training, grad_func):
"""
Connects four neighboring dual vertices to form a quadrilateral. The quadrilaterals are then split into
triangles based on the gamma parameter, as described in Section 4.3.
"""
with torch.no_grad():
group_mask = (edge_counts == 4) & surf_edges_mask # surface edges shared by 4 cubes.
group = idx_map.reshape(-1)[group_mask]
vd_idx = vd_idx_map[group_mask]
edge_indices, indices = torch.sort(group, stable=True)
quad_vd_idx = vd_idx[indices].reshape(-1, 4)
# Ensure all face directions point towards the positive SDF to maintain consistent winding.
s_edges = s_n[surf_edges[edge_indices.reshape(-1, 4)[:, 0]].reshape(-1)].reshape(-1, 2)
flip_mask = s_edges[:, 0] > 0
quad_vd_idx = torch.cat((quad_vd_idx[flip_mask][:, [0, 1, 3, 2]],
quad_vd_idx[~flip_mask][:, [2, 3, 1, 0]]))
if grad_func is not None:
# when grad_func is given, split quadrilaterals along the diagonals with more consistent gradients.
with torch.no_grad():
vd_gamma = torch.nn.functional.normalize(grad_func(vd), dim=-1)
quad_gamma = torch.index_select(input=vd_gamma, index=quad_vd_idx.reshape(-1), dim=0).reshape(-1, 4, 3)
gamma_02 = (quad_gamma[:, 0] * quad_gamma[:, 2]).sum(-1, keepdims=True)
gamma_13 = (quad_gamma[:, 1] * quad_gamma[:, 3]).sum(-1, keepdims=True)
else:
quad_gamma = torch.index_select(input=vd_gamma, index=quad_vd_idx.reshape(-1), dim=0).reshape(-1, 4)
gamma_02 = torch.index_select(input=quad_gamma, index=torch.tensor(
0, device=self.device), dim=1) * torch.index_select(input=quad_gamma, index=torch.tensor(2, device=self.device), dim=1)
gamma_13 = torch.index_select(input=quad_gamma, index=torch.tensor(
1, device=self.device), dim=1) * torch.index_select(input=quad_gamma, index=torch.tensor(3, device=self.device), dim=1)
if not training:
mask = (gamma_02 > gamma_13).squeeze(1)
faces = torch.zeros((quad_gamma.shape[0], 6), dtype=torch.long, device=quad_vd_idx.device)
faces[mask] = quad_vd_idx[mask][:, self.quad_split_1]
faces[~mask] = quad_vd_idx[~mask][:, self.quad_split_2]
faces = faces.reshape(-1, 3)
else:
vd_quad = torch.index_select(input=vd, index=quad_vd_idx.reshape(-1), dim=0).reshape(-1, 4, 3)
vd_02 = (torch.index_select(input=vd_quad, index=torch.tensor(0, device=self.device), dim=1) +
torch.index_select(input=vd_quad, index=torch.tensor(2, device=self.device), dim=1)) / 2
vd_13 = (torch.index_select(input=vd_quad, index=torch.tensor(1, device=self.device), dim=1) +
torch.index_select(input=vd_quad, index=torch.tensor(3, device=self.device), dim=1)) / 2
weight_sum = (gamma_02 + gamma_13) + 1e-8
vd_center = ((vd_02 * gamma_02.unsqueeze(-1) + vd_13 * gamma_13.unsqueeze(-1)) /
weight_sum.unsqueeze(-1)).squeeze(1)
vd_center_idx = torch.arange(vd_center.shape[0], device=self.device) + vd.shape[0]
vd = torch.cat([vd, vd_center])
faces = quad_vd_idx[:, self.quad_split_train].reshape(-1, 4, 2)
faces = torch.cat([faces, vd_center_idx.reshape(-1, 1, 1).repeat(1, 4, 1)], -1).reshape(-1, 3)
return vd, faces, s_edges, edge_indices
def _tetrahedralize(
self, x_nx3, s_n, cube_fx8, vertices, faces, surf_edges, s_edges, vd_idx_map, case_ids, edge_indices,
surf_cubes, training):
"""
Tetrahedralizes the interior volume to produce a tetrahedral mesh, as described in Section 4.5.
"""
occ_n = s_n < 0
occ_fx8 = occ_n[cube_fx8.reshape(-1)].reshape(-1, 8)
occ_sum = torch.sum(occ_fx8, -1)
inside_verts = x_nx3[occ_n]
mapping_inside_verts = torch.ones((occ_n.shape[0]), dtype=torch.long, device=self.device) * -1
mapping_inside_verts[occ_n] = torch.arange(occ_n.sum(), device=self.device) + vertices.shape[0]
"""
For each grid edge connecting two grid vertices with different
signs, we first form a four-sided pyramid by connecting one
of the grid vertices with four mesh vertices that correspond
to the grid edge and then subdivide the pyramid into two tetrahedra
"""
inside_verts_idx = mapping_inside_verts[surf_edges[edge_indices.reshape(-1, 4)[:, 0]].reshape(-1, 2)[
s_edges < 0]]
if not training:
inside_verts_idx = inside_verts_idx.unsqueeze(1).expand(-1, 2).reshape(-1)
else:
inside_verts_idx = inside_verts_idx.unsqueeze(1).expand(-1, 4).reshape(-1)
tets_surface = torch.cat([faces, inside_verts_idx.unsqueeze(-1)], -1)
"""
For each grid edge connecting two grid vertices with the
same sign, the tetrahedron is formed by the two grid vertices
and two vertices in consecutive adjacent cells
"""
inside_cubes = (occ_sum == 8)
inside_cubes_center = x_nx3[cube_fx8[inside_cubes].reshape(-1)].reshape(-1, 8, 3).mean(1)
inside_cubes_center_idx = torch.arange(
inside_cubes_center.shape[0], device=inside_cubes.device) + vertices.shape[0] + inside_verts.shape[0]
surface_n_inside_cubes = surf_cubes | inside_cubes
edge_center_vertex_idx = torch.ones(((surface_n_inside_cubes).sum(), 13),
dtype=torch.long, device=x_nx3.device) * -1
surf_cubes = surf_cubes[surface_n_inside_cubes]
inside_cubes = inside_cubes[surface_n_inside_cubes]
edge_center_vertex_idx[surf_cubes, :12] = vd_idx_map.reshape(-1, 12)
edge_center_vertex_idx[inside_cubes, 12] = inside_cubes_center_idx
all_edges = cube_fx8[surface_n_inside_cubes][:, self.cube_edges].reshape(-1, 2)
unique_edges, _idx_map, counts = torch.unique(all_edges, dim=0, return_inverse=True, return_counts=True)
unique_edges = unique_edges.long()
mask_edges = occ_n[unique_edges.reshape(-1)].reshape(-1, 2).sum(-1) == 2
mask = mask_edges[_idx_map]
counts = counts[_idx_map]
mapping = torch.ones((unique_edges.shape[0]), dtype=torch.long, device=self.device) * -1
mapping[mask_edges] = torch.arange(mask_edges.sum(), device=self.device)
idx_map = mapping[_idx_map]
group_mask = (counts == 4) & mask
group = idx_map.reshape(-1)[group_mask]
edge_indices, indices = torch.sort(group)
cube_idx = torch.arange((_idx_map.shape[0] // 12), dtype=torch.long,
device=self.device).unsqueeze(1).expand(-1, 12).reshape(-1)[group_mask]
edge_idx = torch.arange((12), dtype=torch.long, device=self.device).unsqueeze(
0).expand(_idx_map.shape[0] // 12, -1).reshape(-1)[group_mask]
# Identify the face shared by the adjacent cells.
cube_idx_4 = cube_idx[indices].reshape(-1, 4)
edge_dir = self.edge_dir_table[edge_idx[indices]].reshape(-1, 4)[..., 0]
shared_faces_4x2 = self.dir_faces_table[edge_dir].reshape(-1)
cube_idx_4x2 = cube_idx_4[:, self.adj_pairs].reshape(-1)
# Identify an edge of the face with different signs and
# select the mesh vertex corresponding to the identified edge.
case_ids_expand = torch.ones((surface_n_inside_cubes).sum(), dtype=torch.long, device=x_nx3.device) * 255
case_ids_expand[surf_cubes] = case_ids
cases = case_ids_expand[cube_idx_4x2]
quad_edge = edge_center_vertex_idx[cube_idx_4x2, self.tet_table[cases, shared_faces_4x2]].reshape(-1, 2)
mask = (quad_edge == -1).sum(-1) == 0
inside_edge = mapping_inside_verts[unique_edges[mask_edges][edge_indices].reshape(-1)].reshape(-1, 2)
tets_inside = torch.cat([quad_edge, inside_edge], -1)[mask]
tets = torch.cat([tets_surface, tets_inside])
vertices = torch.cat([vertices, inside_verts, inside_cubes_center])
return vertices, tets
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