pytorch3d/pytorch3d/loss/mesh_laplacian_smoothing.py

# 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.

# pyre-unsafe


import torch
from pytorch3d.ops import cot_laplacian


def mesh_laplacian_smoothing(meshes, method: str = "uniform"):
    r"""
    Computes the laplacian smoothing objective for a batch of meshes.
    This function supports three variants of Laplacian smoothing,
    namely with uniform weights("uniform"), with cotangent weights ("cot"),
    and cotangent curvature ("cotcurv").For more details read [1, 2].

    Args:
        meshes: Meshes object with a batch of meshes.
        method: str specifying the method for the laplacian.
    Returns:
        loss: Average laplacian smoothing loss across the batch.
        Returns 0 if meshes contains no meshes or all empty meshes.

    Consider a mesh M = (V, F), with verts of shape Nx3 and faces of shape Mx3.
    The Laplacian matrix L is a NxN tensor such that LV gives a tensor of vectors:
    for a uniform Laplacian, LuV[i] points to the centroid of its neighboring
    vertices, a cotangent Laplacian LcV[i] is known to be an approximation of
    the surface normal, while the curvature variant LckV[i] scales the normals
    by the discrete mean curvature. For vertex i, assume S[i] is the set of
    neighboring vertices to i, a_ij and b_ij are the "outside" angles in the
    two triangles connecting vertex v_i and its neighboring vertex v_j
    for j in S[i], as seen in the diagram below.

    .. code-block:: python

               a_ij
                /\
               /  \
              /    \
             /      \
        v_i /________\ v_j
            \        /
             \      /
              \    /
               \  /
                \/
               b_ij

        The definition of the Laplacian is LV[i] = sum_j w_ij (v_j - v_i)
        For the uniform variant,    w_ij = 1 / |S[i]|
        For the cotangent variant,
            w_ij = (cot a_ij + cot b_ij) / (sum_k cot a_ik + cot b_ik)
        For the cotangent curvature, w_ij = (cot a_ij + cot b_ij) / (4 A[i])
        where A[i] is the sum of the areas of all triangles containing vertex v_i.

    There is a nice trigonometry identity to compute cotangents. Consider a triangle
    with side lengths A, B, C and angles a, b, c.

    .. code-block:: python

               c
              /|\
             / | \
            /  |  \
         B /  H|   \ A
          /    |    \
         /     |     \
        /a_____|_____b\
               C

        Then cot a = (B^2 + C^2 - A^2) / 4 * area
        We know that area = CH/2, and by the law of cosines we have

        A^2 = B^2 + C^2 - 2BC cos a => B^2 + C^2 - A^2 = 2BC cos a

        Putting these together, we get:

        B^2 + C^2 - A^2     2BC cos a
        _______________  =  _________ = (B/H) cos a = cos a / sin a = cot a
           4 * area            2CH


    [1] Desbrun et al, "Implicit fairing of irregular meshes using diffusion
    and curvature flow", SIGGRAPH 1999.

    [2] Nealan et al, "Laplacian Mesh Optimization", Graphite 2006.
    """

    if meshes.isempty():
        return torch.tensor(
            [0.0], dtype=torch.float32, device=meshes.device, requires_grad=True
        )

    N = len(meshes)
    verts_packed = meshes.verts_packed()  # (sum(V_n), 3)
    faces_packed = meshes.faces_packed()  # (sum(F_n), 3)
    num_verts_per_mesh = meshes.num_verts_per_mesh()  # (N,)
    verts_packed_idx = meshes.verts_packed_to_mesh_idx()  # (sum(V_n),)
    weights = num_verts_per_mesh.gather(0, verts_packed_idx)  # (sum(V_n),)
    weights = 1.0 / weights.float()

    # We don't want to backprop through the computation of the Laplacian;
    # just treat it as a magic constant matrix that is used to transform
    # verts into normals
    with torch.no_grad():
        if method == "uniform":
            L = meshes.laplacian_packed()
        elif method in ["cot", "cotcurv"]:
            L, inv_areas = cot_laplacian(verts_packed, faces_packed)
            if method == "cot":
                norm_w = torch.sparse.sum(L, dim=1).to_dense().view(-1, 1)
                idx = norm_w > 0
                norm_w[idx] = torch.reciprocal(norm_w[idx])
            else:
                L_sum = torch.sparse.sum(L, dim=1).to_dense().view(-1, 1)
                norm_w = 0.25 * inv_areas
        else:
            raise ValueError("Method should be one of {uniform, cot, cotcurv}")

    if method == "uniform":
        loss = L.mm(verts_packed)
    elif method == "cot":
        # pyre-fixme[61]: `norm_w` is undefined, or not always defined.
        loss = L.mm(verts_packed) * norm_w - verts_packed
    elif method == "cotcurv":
        # pyre-fixme[61]: `norm_w` may not be initialized here.
        loss = (L.mm(verts_packed) - L_sum * verts_packed) * norm_w
    loss = loss.norm(dim=1)

    loss = loss * weights
    return loss.sum() / N