content: import torch
  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.int)
  cube_neighbor = torch.tensor([[1, 0, 0], [-1, 0, 0], [0, 1, 0], [0, -1, 0], [0, 0, 1], [0, 0, -1]])
  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, requires_grad=False)
       
  def construct_dense_grid(res, device='cuda'):
      '''construct a dense grid based on resolution'''
      res_v = res + 1
      vertsid = torch.arange(res_v ** 3, device=device)
      coordsid = vertsid.reshape(res_v, res_v, res_v)[:res, :res, :res].flatten()
      cube_corners_bias = (cube_corners[:, 0] * res_v + cube_corners[:, 1]) * res_v + cube_corners[:, 2]
      cube_fx8 = (coordsid.unsqueeze(1) + cube_corners_bias.unsqueeze(0).to(device))
      verts = torch.stack([vertsid // (res_v ** 2), (vertsid // res_v) % res_v, vertsid % res_v], dim=1)
      return verts, cube_fx8
  
  
  def construct_voxel_grid(coords):
      verts = (cube_corners.unsqueeze(0).to(coords) + coords.unsqueeze(1)).reshape(-1, 3)
      verts_unique, inverse_indices = torch.unique(verts, dim=0, return_inverse=True)
      cubes = inverse_indices.reshape(-1, 8)
      return verts_unique, cubes
  
  
  def cubes_to_verts(num_verts, cubes, value, reduce='mean'):
      """
      Args:
          cubes [Vx8] verts index for each cube
          value [Vx8xM] value to be scattered
      Operation:
          reduced[cubes[i][j]][k] += value[i][k]
      """
      M = value.shape[2] # number of channels
      reduced = torch.zeros(num_verts, M, device=cubes.device)
      return torch.scatter_reduce(reduced, 0, 
          cubes.unsqueeze(-1).expand(-1, -1, M).flatten(0, 1), 
          value.flatten(0, 1), reduce=reduce, include_self=False)
      
  def sparse_cube2verts(coords, feats, training=True):
      new_coords, cubes = construct_voxel_grid(coords)
      new_feats = cubes_to_verts(new_coords.shape[0], cubes, feats)
      if training:
          con_loss = torch.mean((feats - new_feats[cubes]) ** 2)
      else:
          con_loss = 0.0
      return new_coords, new_feats, con_loss
      
  
  def get_dense_attrs(coords : torch.Tensor, feats : torch.Tensor, res : int, sdf_init=True):
      F = feats.shape[-1]
      dense_attrs = torch.zeros([res] * 3 + [F], device=feats.device)
      if sdf_init:
          dense_attrs[..., 0] = 1 # initial outside sdf value
      dense_attrs[coords[:, 0], coords[:, 1], coords[:, 2], :] = feats
      return dense_attrs.reshape(-1, F)
  
  
  def get_defomed_verts(v_pos : torch.Tensor, deform : torch.Tensor, res):
      return v_pos / res - 0.5 + (1 - 1e-8) / (res * 2) * torch.tanh(deform)
          