Source code for pytorch3d.loss.mesh_laplacian_smoothing

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import torch
from pytorch3d.ops import cot_laplacian


[docs]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] = 1.0 / 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": 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