# Source code for pytorch3d.ops.cameras_alignment

```
# 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.
from typing import TYPE_CHECKING
import torch
from .. import ops
if TYPE_CHECKING:
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": # pragma: no cover
"""
.. 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,
): # pragma: no cover
"""
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,
): # pragma: no cover
"""
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
```