refactor laplacian matrices

Summary:
Refactor of all functions to compute laplacian matrices in one file.
Support for:
* Standard Laplacian
* Cotangent Laplacian
* Norm Laplacian

Reviewed By: nikhilaravi

Differential Revision: D29297466

fbshipit-source-id: b96b88915ce8ef0c2f5693ec9b179fd27b70abf9

pytorch3d/ops/__init__.py

@@ -9,6 +9,7 @@ from .cubify import cubify
 from .graph_conv import GraphConv
 from .interp_face_attrs import interpolate_face_attributes
 from .knn import knn_gather, knn_points
+from .laplacian_matrices import laplacian, cot_laplacian, norm_laplacian
 from .mesh_face_areas_normals import mesh_face_areas_normals
 from .mesh_filtering import taubin_smoothing
 from .packed_to_padded import packed_to_padded, padded_to_packed

pytorch3d/ops/laplacian_matrices.py

@@ -0,0 +1,170 @@
+# Copyright (c) Facebook, Inc. and its 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.
+
+from typing import Tuple
+
+import torch
+
+# ------------------------ Laplacian Matrices ------------------------ #
+# This file contains implementations of differentiable laplacian matrices.
+# These include
+# 1) Standard Laplacian matrix
+# 2) Cotangent Laplacian matrix
+# 3) Norm Laplacian matrix
+# -------------------------------------------------------------------- #
+
+
+def laplacian(verts: torch.Tensor, edges: torch.Tensor) -> torch.Tensor:
+    """
+    Computes the laplacian matrix.
+    The definition of the laplacian is
+    L[i, j] =    -1       , if i == j
+    L[i, j] = 1 / deg(i)  , if (i, j) is an edge
+    L[i, j] =    0        , otherwise
+    where deg(i) is the degree of the i-th vertex in the graph.
+
+    Args:
+        verts: tensor of shape (V, 3) containing the vertices of the graph
+        edges: tensor of shape (E, 2) containing the vertex indices of each edge
+    Returns:
+        L: Sparse FloatTensor of shape (V, V)
+    """
+    V = verts.shape[0]
+
+    e0, e1 = edges.unbind(1)
+
+    idx01 = torch.stack([e0, e1], dim=1)  # (E, 2)
+    idx10 = torch.stack([e1, e0], dim=1)  # (E, 2)
+    idx = torch.cat([idx01, idx10], dim=0).t()  # (2, 2*E)
+
+    # First, we construct the adjacency matrix,
+    # i.e. A[i, j] = 1 if (i,j) is an edge, or
+    # A[e0, e1] = 1 &  A[e1, e0] = 1
+    ones = torch.ones(idx.shape[1], dtype=torch.float32, device=verts.device)
+    A = torch.sparse.FloatTensor(idx, ones, (V, V))
+
+    # the sum of i-th row of A gives the degree of the i-th vertex
+    deg = torch.sparse.sum(A, dim=1).to_dense()
+
+    # We construct the Laplacian matrix by adding the non diagonal values
+    # i.e. L[i, j] = 1 ./ deg(i) if (i, j) is an edge
+    deg0 = deg[e0]
+    deg0 = torch.where(deg0 > 0.0, 1.0 / deg0, deg0)
+    deg1 = deg[e1]
+    deg1 = torch.where(deg1 > 0.0, 1.0 / deg1, deg1)
+    val = torch.cat([deg0, deg1])
+    L = torch.sparse.FloatTensor(idx, val, (V, V))
+
+    # Then we add the diagonal values L[i, i] = -1.
+    idx = torch.arange(V, device=verts.device)
+    idx = torch.stack([idx, idx], dim=0)
+    ones = torch.ones(idx.shape[1], dtype=torch.float32, device=verts.device)
+    L -= torch.sparse.FloatTensor(idx, ones, (V, V))
+
+    return L
+
+
+def cot_laplacian(
+    verts: torch.Tensor, faces: torch.Tensor, eps: float = 1e-12
+) -> Tuple[torch.Tensor, torch.Tensor]:
+    """
+    Returns the Laplacian matrix with cotangent weights and the inverse of the
+    face areas.
+
+    Args:
+        verts: tensor of shape (V, 3) containing the vertices of the graph
+        faces: tensor of shape (F, 3) containing the vertex indices of each face
+    Returns:
+        2-element tuple containing
+        - **L**: Sparse FloatTensor of shape (V,V) for the Laplacian matrix.
+           Here, L[i, j] = cot a_ij + cot b_ij iff (i, j) is an edge in meshes.
+           See the description above for more clarity.
+        - **inv_areas**: FloatTensor of shape (V,) containing the inverse of sum of
+           face areas containing each vertex
+    """
+    V, F = verts.shape[0], faces.shape[0]
+
+    face_verts = verts[faces]
+    v0, v1, v2 = face_verts[:, 0], face_verts[:, 1], face_verts[:, 2]
+
+    # Side lengths of each triangle, of shape (sum(F_n),)
+    # A is the side opposite v1, B is opposite v2, and C is opposite v3
+    A = (v1 - v2).norm(dim=1)
+    B = (v0 - v2).norm(dim=1)
+    C = (v0 - v1).norm(dim=1)
+
+    # Area of each triangle (with Heron's formula); shape is (sum(F_n),)
+    s = 0.5 * (A + B + C)
+    # note that the area can be negative (close to 0) causing nans after sqrt()
+    # we clip it to a small positive value
+    area = (s * (s - A) * (s - B) * (s - C)).clamp_(min=eps).sqrt()
+
+    # Compute cotangents of angles, of shape (sum(F_n), 3)
+    A2, B2, C2 = A * A, B * B, C * C
+    cota = (B2 + C2 - A2) / area
+    cotb = (A2 + C2 - B2) / area
+    cotc = (A2 + B2 - C2) / area
+    cot = torch.stack([cota, cotb, cotc], dim=1)
+    cot /= 4.0
+
+    # Construct a sparse matrix by basically doing:
+    # L[v1, v2] = cota
+    # L[v2, v0] = cotb
+    # L[v0, v1] = cotc
+    ii = faces[:, [1, 2, 0]]
+    jj = faces[:, [2, 0, 1]]
+    idx = torch.stack([ii, jj], dim=0).view(2, F * 3)
+    L = torch.sparse.FloatTensor(idx, cot.view(-1), (V, V))
+
+    # Make it symmetric; this means we are also setting
+    # L[v2, v1] = cota
+    # L[v0, v2] = cotb
+    # L[v1, v0] = cotc
+    L += L.t()
+
+    # For each vertex, compute the sum of areas for triangles containing it.
+    idx = faces.view(-1)
+    inv_areas = torch.zeros(V, dtype=torch.float32, device=verts.device)
+    val = torch.stack([area] * 3, dim=1).view(-1)
+    inv_areas.scatter_add_(0, idx, val)
+    idx = inv_areas > 0
+    inv_areas[idx] = 1.0 / inv_areas[idx]
+    inv_areas = inv_areas.view(-1, 1)
+
+    return L, inv_areas
+
+
+def norm_laplacian(
+    verts: torch.Tensor, edges: torch.Tensor, eps: float = 1e-12
+) -> torch.Tensor:
+    """
+    Norm laplacian computes a variant of the laplacian matrix which weights each
+    affinity with the normalized distance of the neighboring nodes.
+    More concretely,
+    L[i, j] = 1. / wij where wij = ||vi - vj|| if (vi, vj) are neighboring nodes
+
+    Args:
+        verts: tensor of shape (V, 3) containing the vertices of the graph
+        edges: tensor of shape (E, 2) containing the vertex indices of each edge
+    Returns:
+        L: Sparse FloatTensor of shape (V, V)
+    """
+    edge_verts = verts[edges]  # (E, 2, 3)
+    v0, v1 = edge_verts[:, 0], edge_verts[:, 1]
+
+    # Side lengths of each edge, of shape (E,)
+    w01 = 1.0 / ((v0 - v1).norm(dim=1) + eps)
+
+    # Construct a sparse matrix by basically doing:
+    # L[v0, v1] = w01
+    # L[v1, v0] = w01
+    e01 = edges.t()  # (2, E)
+
+    V = verts.shape[0]
+    L = torch.sparse.FloatTensor(e01, w01, (V, V))
+    L = L + L.t()
+
+    return L

pytorch3d/ops/mesh_filtering.py

@@ -5,6 +5,7 @@
 # LICENSE file in the root directory of this source tree.
 
 import torch
+from pytorch3d.ops import norm_laplacian
 from pytorch3d.structures import Meshes, utils as struct_utils
 
 
@@ -19,35 +20,6 @@ from pytorch3d.structures import Meshes, utils as struct_utils
 # ----------------------- Taubin Smoothing ----------------------- #
 
 
-def norm_laplacian(verts: torch.Tensor, edges: torch.Tensor, eps: float = 1e-12):
-    """
-    Norm laplacian computes a variant of the laplacian matrix which weights each
-    affinity with the normalized distance of the neighboring nodes.
-    More concretely,
-    L[i, j] = 1. / wij where wij = ||vi - vj|| if (vi, vj) are neighboring nodes
-
-    Args:
-        verts: tensor of shape (V, 3) containing the vertices of the graph
-        edges: tensor of shape (E, 2) containing the vertex indices of each edge
-    """
-    edge_verts = verts[edges]  # (E, 2, 3)
-    v0, v1 = edge_verts[:, 0], edge_verts[:, 1]
-
-    # Side lengths of each edge, of shape (E,)
-    w01 = 1.0 / ((v0 - v1).norm(dim=1) + eps)
-
-    # Construct a sparse matrix by basically doing:
-    # L[v0, v1] = w01
-    # L[v1, v0] = w01
-    e01 = edges.t()  # (2, E)
-
-    V = verts.shape[0]
-    L = torch.sparse.FloatTensor(e01, w01, (V, V))
-    L = L + L.t()
-
-    return L
-
-
 def taubin_smoothing(
     meshes: Meshes, lambd: float = 0.53, mu: float = -0.53, num_iter: int = 10
 ) -> Meshes: