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util/flexicubes.py

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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