Source code for pytorch3d.ops.cameras_alignment

# Copyright (c) Facebook, Inc. and its affiliates. All rights reserved.

from typing import TYPE_CHECKING

import torch

from .. import ops

    from pytorch3d.renderer.cameras import CamerasBase

[docs]def corresponding_cameras_alignment( cameras_src: "CamerasBase", cameras_tgt: "CamerasBase", estimate_scale: bool = True, mode: str = "extrinsics", eps: float = 1e-9, ) -> "CamerasBase": """ .. warning:: The `corresponding_cameras_alignment` API is experimental and subject to change! Estimates a single similarity transformation between two sets of cameras `cameras_src` and `cameras_tgt` and returns an aligned version of `cameras_src`. Given source cameras [(R_1, T_1), (R_2, T_2), ..., (R_N, T_N)] and target cameras [(R_1', T_1'), (R_2', T_2'), ..., (R_N', T_N')], where (R_i, T_i) is a 2-tuple of the camera rotation and translation matrix respectively, the algorithm finds a global rotation, translation and scale (R_A, T_A, s_A) which aligns all source cameras with the target cameras such that the following holds: Under the change of coordinates using a similarity transform (R_A, T_A, s_A) a 3D point X' is mapped to X with: ``` X = (X' R_A + T_A) / s_A ``` Then, for all cameras `i`, we assume that the following holds: ``` X R_i + T_i = s' (X' R_i' + T_i'), ``` i.e. an adjusted point X' is mapped by a camera (R_i', T_i') to the same point as imaged from camera (R_i, T_i) after resolving the scale ambiguity with a global scalar factor s'. Substituting for X above gives rise to the following: ``` (X' R_A + T_A) / s_A R_i + T_i = s' (X' R_i' + T_i') // · s_A (X' R_A + T_A) R_i + T_i s_A = (s' s_A) (X' R_i' + T_i') s' := 1 / s_A # without loss of generality (X' R_A + T_A) R_i + T_i s_A = X' R_i' + T_i' X' R_A R_i + T_A R_i + T_i s_A = X' R_i' + T_i' ^^^^^^^ ^^^^^^^^^^^^^^^^^ ~= R_i' ~= T_i' ``` i.e. after estimating R_A, T_A, s_A, the aligned source cameras have extrinsics: `cameras_src_align = (R_A R_i, T_A R_i + T_i s_A) ~= (R_i', T_i')` We support two ways `R_A, T_A, s_A` can be estimated: 1) `mode=='centers'` Estimates the similarity alignment between camera centers using Umeyama's algorithm (see `pytorch3d.ops.corresponding_points_alignment` for details) and transforms camera extrinsics accordingly. 2) `mode=='extrinsics'` Defines the alignment problem as a system of the following equations: ``` for all i: [ R_A 0 ] x [ R_i 0 ] = [ R_i' 0 ] [ T_A^T 1 ] [ (s_A T_i^T) 1 ] [ T_i' 1 ] ``` `R_A, T_A` and `s_A` are then obtained by solving the system in the least squares sense. The estimated camera transformation is a true similarity transform, i.e. it cannot be a reflection. Args: cameras_src: `N` cameras to be aligned. cameras_tgt: `N` target cameras. estimate_scale: Controls whether the alignment transform is rigid (`estimate_scale=False`), or a similarity (`estimate_scale=True`). `s_A` is set to `1` if `estimate_scale==False`. mode: Controls the alignment algorithm. Can be one either `'centers'` or `'extrinsics'`. Please refer to the description above for details. eps: A scalar for clamping to avoid dividing by zero. Active when `estimate_scale==True`. Returns: cameras_src_aligned: `cameras_src` after applying the alignment transform. """ if cameras_src.R.shape[0] != cameras_tgt.R.shape[0]: raise ValueError( "cameras_src and cameras_tgt have to contain the same number of cameras!" ) if mode == "centers": align_fun = _align_camera_centers elif mode == "extrinsics": align_fun = _align_camera_extrinsics else: raise ValueError("mode has to be one of (centers, extrinsics)") align_t_R, align_t_T, align_t_s = align_fun( cameras_src, cameras_tgt, estimate_scale=estimate_scale, eps=eps ) # create a new cameras object and set the R and T accordingly cameras_src_aligned = cameras_src.clone() cameras_src_aligned.R = torch.bmm(align_t_R.expand_as(cameras_src.R), cameras_src.R) cameras_src_aligned.T = ( torch.bmm( align_t_T[:, None].repeat(cameras_src.R.shape[0], 1, 1), cameras_src.R )[:, 0] + cameras_src.T * align_t_s ) return cameras_src_aligned
def _align_camera_centers( cameras_src: "CamerasBase", cameras_tgt: "CamerasBase", estimate_scale: bool = True, eps: float = 1e-9, ): """ Use Umeyama's algorithm to align the camera centers. """ centers_src = cameras_src.get_camera_center() centers_tgt = cameras_tgt.get_camera_center() align_t = ops.corresponding_points_alignment( centers_src[None], centers_tgt[None], estimate_scale=estimate_scale, allow_reflection=False, eps=eps, ) # the camera transform is the inverse of the estimated transform between centers align_t_R = align_t.R.permute(0, 2, 1) align_t_T = -(torch.bmm(align_t.T[:, None], align_t_R))[:, 0] align_t_s = align_t.s[0] return align_t_R, align_t_T, align_t_s def _align_camera_extrinsics( cameras_src: "CamerasBase", cameras_tgt: "CamerasBase", estimate_scale: bool = True, eps: float = 1e-9, ): """ Get the global rotation R_A with svd of cov(RR^T): ``` R_A R_i = R_i' for all i R_A [R_1 R_2 ... R_N] = [R_1' R_2' ... R_N'] U, _, V = svd([R_1 R_2 ... R_N]^T [R_1' R_2' ... R_N']) R_A = (U V^T)^T ``` """ RRcov = torch.bmm(cameras_src.R, cameras_tgt.R.transpose(2, 1)).mean(0) U, _, V = torch.svd(RRcov) align_t_R = V @ U.t() """ The translation + scale `T_A` and `s_A` is computed by finding a translation and scaling that aligns two tensors `A, B` defined as follows: ``` T_A R_i + s_A T_i = T_i' ; for all i // · R_i^T s_A T_i R_i^T + T_A = T_i' R_i^T ; for all i ^^^^^^^^^ ^^^^^^^^^^ A_i B_i A_i := T_i R_i^T A = [A_1 A_2 ... A_N] B_i := T_i' R_i^T B = [B_1 B_2 ... B_N] ``` The scale s_A can be retrieved by matching the correlations of the points sets A and B: ``` s_A = (A-mean(A))*(B-mean(B)).sum() / ((A-mean(A))**2).sum() ``` The translation `T_A` is then defined as: ``` T_A = mean(B) - mean(A) * s_A ``` """ # pyre-fixme[29]: # `Union[BoundMethod[typing.Callable(torch.Tensor.__getitem__)[[Named(self, # torch.Tensor), Named(item, typing.Any)], typing.Any], torch.Tensor], # torch.Tensor, torch.nn.Module]` is not a function. A = torch.bmm(cameras_src.R, cameras_src.T[:, :, None])[:, :, 0] # pyre-fixme[29]: # `Union[BoundMethod[typing.Callable(torch.Tensor.__getitem__)[[Named(self, # torch.Tensor), Named(item, typing.Any)], typing.Any], torch.Tensor], # torch.Tensor, torch.nn.Module]` is not a function. B = torch.bmm(cameras_src.R, cameras_tgt.T[:, :, None])[:, :, 0] Amu = A.mean(0, keepdim=True) Bmu = B.mean(0, keepdim=True) if estimate_scale and A.shape[0] > 1: # get the scaling component by matching covariances # of centered A and centered B Ac = A - Amu Bc = B - Bmu align_t_s = (Ac * Bc).mean() / (Ac ** 2).mean().clamp(eps) else: # set the scale to identity align_t_s = 1.0 # get the translation as the difference between the means of A and B align_t_T = Bmu - align_t_s * Amu return align_t_R, align_t_T, align_t_s