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Umeyama
Summary: Umeyama estimates a rigid motion between two sets of corresponding points. Benchmark output for `bm_points_alignment` ``` Arguments key: [<allow_reflection>_<batch_size>_<dim>_<estimate_scale>_<n_points>_<use_pointclouds>] Benchmark Avg Time(μs) Peak Time(μs) Iterations -------------------------------------------------------------------------------- CorrespodingPointsAlignment_True_1_3_True_100_False 7382 9833 68 CorrespodingPointsAlignment_True_1_3_True_10000_False 8183 10500 62 CorrespodingPointsAlignment_True_1_3_False_100_False 7301 9263 69 CorrespodingPointsAlignment_True_1_3_False_10000_False 7945 9746 64 CorrespodingPointsAlignment_True_1_20_True_100_False 13706 41623 37 CorrespodingPointsAlignment_True_1_20_True_10000_False 11044 33766 46 CorrespodingPointsAlignment_True_1_20_False_100_False 9908 28791 51 CorrespodingPointsAlignment_True_1_20_False_10000_False 9523 18680 53 CorrespodingPointsAlignment_True_10_3_True_100_False 29585 32026 17 CorrespodingPointsAlignment_True_10_3_True_10000_False 29626 36324 18 CorrespodingPointsAlignment_True_10_3_False_100_False 26013 29253 20 CorrespodingPointsAlignment_True_10_3_False_10000_False 25000 33820 20 CorrespodingPointsAlignment_True_10_20_True_100_False 40955 41592 13 CorrespodingPointsAlignment_True_10_20_True_10000_False 42087 42393 12 CorrespodingPointsAlignment_True_10_20_False_100_False 39863 40381 13 CorrespodingPointsAlignment_True_10_20_False_10000_False 40813 41699 13 CorrespodingPointsAlignment_True_100_3_True_100_False 183146 194745 3 CorrespodingPointsAlignment_True_100_3_True_10000_False 213789 231466 3 CorrespodingPointsAlignment_True_100_3_False_100_False 177805 180796 3 CorrespodingPointsAlignment_True_100_3_False_10000_False 184963 185695 3 CorrespodingPointsAlignment_True_100_20_True_100_False 347181 347325 2 CorrespodingPointsAlignment_True_100_20_True_10000_False 363259 363613 2 CorrespodingPointsAlignment_True_100_20_False_100_False 351769 352496 2 CorrespodingPointsAlignment_True_100_20_False_10000_False 375629 379818 2 CorrespodingPointsAlignment_False_1_3_True_100_False 11155 13770 45 CorrespodingPointsAlignment_False_1_3_True_10000_False 10743 13938 47 CorrespodingPointsAlignment_False_1_3_False_100_False 9578 11511 53 CorrespodingPointsAlignment_False_1_3_False_10000_False 9549 11984 53 CorrespodingPointsAlignment_False_1_20_True_100_False 13809 14183 37 CorrespodingPointsAlignment_False_1_20_True_10000_False 14084 15082 36 CorrespodingPointsAlignment_False_1_20_False_100_False 12765 14177 40 CorrespodingPointsAlignment_False_1_20_False_10000_False 12811 13096 40 CorrespodingPointsAlignment_False_10_3_True_100_False 28823 39384 18 CorrespodingPointsAlignment_False_10_3_True_10000_False 27135 27525 19 CorrespodingPointsAlignment_False_10_3_False_100_False 26236 28980 20 CorrespodingPointsAlignment_False_10_3_False_10000_False 42324 45123 12 CorrespodingPointsAlignment_False_10_20_True_100_False 723902 723902 1 CorrespodingPointsAlignment_False_10_20_True_10000_False 220007 252886 3 CorrespodingPointsAlignment_False_10_20_False_100_False 55593 71636 9 CorrespodingPointsAlignment_False_10_20_False_10000_False 44419 71861 12 CorrespodingPointsAlignment_False_100_3_True_100_False 184768 185199 3 CorrespodingPointsAlignment_False_100_3_True_10000_False 198657 213868 3 CorrespodingPointsAlignment_False_100_3_False_100_False 224598 309645 3 CorrespodingPointsAlignment_False_100_3_False_10000_False 197863 202002 3 CorrespodingPointsAlignment_False_100_20_True_100_False 293484 309459 2 CorrespodingPointsAlignment_False_100_20_True_10000_False 327253 366644 2 CorrespodingPointsAlignment_False_100_20_False_100_False 420793 422194 2 CorrespodingPointsAlignment_False_100_20_False_10000_False 462634 485542 2 CorrespodingPointsAlignment_True_1_3_True_100_True 7664 9909 66 CorrespodingPointsAlignment_True_1_3_True_10000_True 7190 8366 70 CorrespodingPointsAlignment_True_1_3_False_100_True 6549 8316 77 CorrespodingPointsAlignment_True_1_3_False_10000_True 6534 7710 77 CorrespodingPointsAlignment_True_10_3_True_100_True 29052 32940 18 CorrespodingPointsAlignment_True_10_3_True_10000_True 30526 33453 17 CorrespodingPointsAlignment_True_10_3_False_100_True 28708 32993 18 CorrespodingPointsAlignment_True_10_3_False_10000_True 30630 35973 17 CorrespodingPointsAlignment_True_100_3_True_100_True 264909 320820 3 CorrespodingPointsAlignment_True_100_3_True_10000_True 310902 322604 2 CorrespodingPointsAlignment_True_100_3_False_100_True 246832 250634 3 CorrespodingPointsAlignment_True_100_3_False_10000_True 276006 289061 2 CorrespodingPointsAlignment_False_1_3_True_100_True 11421 13757 44 CorrespodingPointsAlignment_False_1_3_True_10000_True 11199 12532 45 CorrespodingPointsAlignment_False_1_3_False_100_True 11474 15841 44 CorrespodingPointsAlignment_False_1_3_False_10000_True 10384 13188 49 CorrespodingPointsAlignment_False_10_3_True_100_True 36599 47340 14 CorrespodingPointsAlignment_False_10_3_True_10000_True 40702 50754 13 CorrespodingPointsAlignment_False_10_3_False_100_True 41277 52149 13 CorrespodingPointsAlignment_False_10_3_False_10000_True 34286 37091 15 CorrespodingPointsAlignment_False_100_3_True_100_True 254991 258578 2 CorrespodingPointsAlignment_False_100_3_True_10000_True 257999 261285 2 CorrespodingPointsAlignment_False_100_3_False_100_True 247511 248693 3 CorrespodingPointsAlignment_False_100_3_False_10000_True 251807 263865 3 ``` Reviewed By: gkioxari Differential Revision: D19808389 fbshipit-source-id: 83305a58627d2fc5dcaf3c3015132d8148f28c29
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‎pytorch3d/ops/__init__.py

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from .mesh_face_areas_normals import mesh_face_areas_normals
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from .nearest_neighbor_points import nn_points_idx
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from .packed_to_padded import packed_to_padded, padded_to_packed
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from .points_alignment import corresponding_points_alignment
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from .sample_points_from_meshes import sample_points_from_meshes
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from .subdivide_meshes import SubdivideMeshes
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from .vert_align import vert_align

‎pytorch3d/ops/points_alignment.py

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#!/usr/bin/env python3
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# Copyright (c) Facebook, Inc. and its affiliates. All rights reserved.
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import warnings
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from typing import Tuple, Union
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import torch
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from pytorch3d.structures.pointclouds import Pointclouds
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def corresponding_points_alignment(
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X: Union[torch.Tensor, Pointclouds],
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Y: Union[torch.Tensor, Pointclouds],
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estimate_scale: bool = False,
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allow_reflection: bool = False,
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eps: float = 1e-8,
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) -> Tuple[torch.Tensor, torch.Tensor, torch.Tensor]:
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"""
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Finds a similarity transformation (rotation `R`, translation `T`
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and optionally scale `s`) between two given sets of corresponding
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`d`-dimensional points `X` and `Y` such that:
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`s[i] X[i] R[i] + T[i] = Y[i]`,
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for all batch indexes `i` in the least squares sense.
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The algorithm is also known as Umeyama [1].
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Args:
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X: Batch of `d`-dimensional points of shape `(minibatch, num_point, d)`
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or a `Pointclouds` object.
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Y: Batch of `d`-dimensional points of shape `(minibatch, num_point, d)`
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or a `Pointclouds` object.
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estimate_scale: If `True`, also estimates a scaling component `s`
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of the transformation. Otherwise assumes an identity
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scale and returns a tensor of ones.
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allow_reflection: If `True`, allows the algorithm to return `R`
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which is orthonormal but has determinant==-1.
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eps: A scalar for clamping to avoid dividing by zero. Active for the
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code that estimates the output scale `s`.
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Returns:
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3-element tuple containing
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- **R**: Batch of orthonormal matrices of shape `(minibatch, d, d)`.
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- **T**: Batch of translations of shape `(minibatch, d)`.
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- **s**: batch of scaling factors of shape `(minibatch, )`.
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References:
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[1] Shinji Umeyama: Least-Suqares Estimation of
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Transformation Parameters Between Two Point Patterns
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"""
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# make sure we convert input Pointclouds structures to tensors
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Xt, num_points = _convert_point_cloud_to_tensor(X)
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Yt, num_points_Y = _convert_point_cloud_to_tensor(Y)
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if (Xt.shape != Yt.shape) or (num_points != num_points_Y).any():
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raise ValueError(
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"Point sets X and Y have to have the same \
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number of batches, points and dimensions."
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)
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b, n, dim = Xt.shape
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# compute the centroids of the point sets
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Xmu = Xt.sum(1) / torch.clamp(num_points[:, None], 1)
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Ymu = Yt.sum(1) / torch.clamp(num_points[:, None], 1)
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# mean-center the point sets
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Xc = Xt - Xmu[:, None]
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Yc = Yt - Ymu[:, None]
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if (num_points < Xt.shape[1]).any() or (num_points < Yt.shape[1]).any():
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# in case we got Pointclouds as input, mask the unused entries in Xc, Yc
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mask = (
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torch.arange(n, dtype=torch.int64, device=Xc.device)[None]
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< num_points[:, None]
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).type_as(Xc)
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Xc *= mask[:, :, None]
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Yc *= mask[:, :, None]
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if (num_points < (dim + 1)).any():
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warnings.warn(
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"The size of one of the point clouds is <= dim+1. "
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+ "corresponding_points_alignment can't return a unique solution."
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)
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# compute the covariance XYcov between the point sets Xc, Yc
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XYcov = torch.bmm(Xc.transpose(2, 1), Yc)
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XYcov = XYcov / torch.clamp(num_points[:, None, None], 1)
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# decompose the covariance matrix XYcov
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U, S, V = torch.svd(XYcov)
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# identity matrix used for fixing reflections
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E = torch.eye(dim, dtype=XYcov.dtype, device=XYcov.device)[None].repeat(
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b, 1, 1
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)
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if not allow_reflection:
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# reflection test:
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# checks whether the estimated rotation has det==1,
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# if not, finds the nearest rotation s.t. det==1 by
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# flipping the sign of the last singular vector U
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R_test = torch.bmm(U, V.transpose(2, 1))
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E[:, -1, -1] = torch.det(R_test)
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# find the rotation matrix by composing U and V again
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R = torch.bmm(torch.bmm(U, E), V.transpose(2, 1))
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if estimate_scale:
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# estimate the scaling component of the transformation
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trace_ES = (torch.diagonal(E, dim1=1, dim2=2) * S).sum(1)
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Xcov = (Xc * Xc).sum((1, 2)) / torch.clamp(num_points, 1)
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# the scaling component
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s = trace_ES / torch.clamp(Xcov, eps)
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# translation component
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T = Ymu - s[:, None] * torch.bmm(Xmu[:, None], R)[:, 0, :]
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else:
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# translation component
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T = Ymu - torch.bmm(Xmu[:, None], R)[:, 0]
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# unit scaling since we do not estimate scale
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s = T.new_ones(b)
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return R, T, s
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def _convert_point_cloud_to_tensor(pcl: Union[torch.Tensor, Pointclouds]):
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"""
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If `type(pcl)==Pointclouds`, converts a `pcl` object to a
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padded representation and returns it together with the number of points
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per batch. Otherwise, returns the input itself with the number of points
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set to the size of the second dimension of `pcl`.
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"""
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if isinstance(pcl, Pointclouds):
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X = pcl.points_padded()
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num_points = pcl.num_points_per_cloud()
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elif torch.is_tensor(pcl):
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X = pcl
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num_points = X.shape[1] * torch.ones(
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X.shape[0], device=X.device, dtype=torch.int64
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)
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else:
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raise ValueError(
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"The inputs X, Y should be either Pointclouds objects or tensors."
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)
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return X, num_points

‎tests/bm_points_alignment.py

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#!/usr/bin/env python3
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# Copyright (c) Facebook, Inc. and its affiliates. All rights reserved.
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from copy import deepcopy
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from itertools import product
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from fvcore.common.benchmark import benchmark
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from test_points_alignment import TestCorrespondingPointsAlignment
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def bm_corresponding_points_alignment() -> None:
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case_grid = {
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"allow_reflection": [True, False],
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"batch_size": [1, 10, 100],
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"dim": [3, 20],
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"estimate_scale": [True, False],
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"n_points": [100, 10000],
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"use_pointclouds": [False],
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}
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test_args = sorted(case_grid.keys())
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test_cases = product(*[case_grid[k] for k in test_args])
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kwargs_list = [dict(zip(test_args, case)) for case in test_cases]
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# add the use_pointclouds=True test cases whenever we have dim==3
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kwargs_to_add = []
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for entry in kwargs_list:
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if entry["dim"] == 3:
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entry_add = deepcopy(entry)
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entry_add["use_pointclouds"] = True
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kwargs_to_add.append(entry_add)
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kwargs_list.extend(kwargs_to_add)
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benchmark(
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TestCorrespondingPointsAlignment.corresponding_points_alignment,
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"CorrespodingPointsAlignment",
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kwargs_list,
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warmup_iters=1,
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)

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