py3d_tools.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.

import sys
import math
import warnings
from typing import List, Optional, Sequence, Tuple, Union, Any

import numpy as np
import torch
import torch.nn.functional as F

import copy
import inspect
import torch.nn as nn

Device = Union[str, torch.device]

# Default values for rotation and translation matrices.
_R = torch.eye(3)[None]  # (1, 3, 3)
_T = torch.zeros(1, 3)  # (1, 3)


# Provide get_origin and get_args even in Python 3.7.

if sys.version_info >= (3, 8, 0):
    from typing import get_args, get_origin
elif sys.version_info >= (3, 7, 0):

    def get_origin(cls):  # pragma: no cover
        return getattr(cls, "__origin__", None)

    def get_args(cls):  # pragma: no cover
        return getattr(cls, "__args__", None)


else:
    raise ImportError("This module requires Python 3.7+")

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class Transform3d:
    """
    A Transform3d object encapsulates a batch of N 3D transformations, and knows
    how to transform points and normal vectors. Suppose that t is a Transform3d;
    then we can do the following:

    .. code-block:: python

        N = len(t)
        points = torch.randn(N, P, 3)
        normals = torch.randn(N, P, 3)
        points_transformed = t.transform_points(points)    # => (N, P, 3)
        normals_transformed = t.transform_normals(normals)  # => (N, P, 3)


    BROADCASTING
    Transform3d objects supports broadcasting. Suppose that t1 and tN are
    Transform3d objects with len(t1) == 1 and len(tN) == N respectively. Then we
    can broadcast transforms like this:

    .. code-block:: python

        t1.transform_points(torch.randn(P, 3))     # => (P, 3)
        t1.transform_points(torch.randn(1, P, 3))  # => (1, P, 3)
        t1.transform_points(torch.randn(M, P, 3))  # => (M, P, 3)
        tN.transform_points(torch.randn(P, 3))     # => (N, P, 3)
        tN.transform_points(torch.randn(1, P, 3))  # => (N, P, 3)


    COMBINING TRANSFORMS
    Transform3d objects can be combined in two ways: composing and stacking.
    Composing is function composition. Given Transform3d objects t1, t2, t3,
    the following all compute the same thing:

    .. code-block:: python

        y1 = t3.transform_points(t2.transform_points(t1.transform_points(x)))
        y2 = t1.compose(t2).compose(t3).transform_points(x)
        y3 = t1.compose(t2, t3).transform_points(x)


    Composing transforms should broadcast.

    .. code-block:: python

        if len(t1) == 1 and len(t2) == N, then len(t1.compose(t2)) == N.

    We can also stack a sequence of Transform3d objects, which represents
    composition along the batch dimension; then the following should compute the
    same thing.

    .. code-block:: python

        N, M = len(tN), len(tM)
        xN = torch.randn(N, P, 3)
        xM = torch.randn(M, P, 3)
        y1 = torch.cat([tN.transform_points(xN), tM.transform_points(xM)], dim=0)
        y2 = tN.stack(tM).transform_points(torch.cat([xN, xM], dim=0))

    BUILDING TRANSFORMS
    We provide convenience methods for easily building Transform3d objects
    as compositions of basic transforms.

    .. code-block:: python

        # Scale by 0.5, then translate by (1, 2, 3)
        t1 = Transform3d().scale(0.5).translate(1, 2, 3)

        # Scale each axis by a different amount, then translate, then scale
        t2 = Transform3d().scale(1, 3, 3).translate(2, 3, 1).scale(2.0)

        t3 = t1.compose(t2)
        tN = t1.stack(t3, t3)


    BACKPROP THROUGH TRANSFORMS
    When building transforms, we can also parameterize them by Torch tensors;
    in this case we can backprop through the construction and application of
    Transform objects, so they could be learned via gradient descent or
    predicted by a neural network.

    .. code-block:: python

        s1_params = torch.randn(N, requires_grad=True)
        t_params = torch.randn(N, 3, requires_grad=True)
        s2_params = torch.randn(N, 3, requires_grad=True)

        t = Transform3d().scale(s1_params).translate(t_params).scale(s2_params)
        x = torch.randn(N, 3)
        y = t.transform_points(x)
        loss = compute_loss(y)
        loss.backward()

        with torch.no_grad():
            s1_params -= lr * s1_params.grad
            t_params -= lr * t_params.grad
            s2_params -= lr * s2_params.grad

    CONVENTIONS
    We adopt a right-hand coordinate system, meaning that rotation about an axis
    with a positive angle results in a counter clockwise rotation.

    This class assumes that transformations are applied on inputs which
    are row vectors. The internal representation of the Nx4x4 transformation
    matrix is of the form:

    .. code-block:: python

        M = [
                [Rxx, Ryx, Rzx, 0],
                [Rxy, Ryy, Rzy, 0],
                [Rxz, Ryz, Rzz, 0],
                [Tx,  Ty,  Tz,  1],
            ]

    To apply the transformation to points which are row vectors, the M matrix
    can be pre multiplied by the points:

    .. code-block:: python

        points = [[0, 1, 2]]  # (1 x 3) xyz coordinates of a point
        transformed_points = points * M

    """

    def __init__(
        self,
        dtype: torch.dtype = torch.float32,
        device: Device = "cpu",
        matrix: Optional[torch.Tensor] = None,
    ) -> None:
        """
        Args:
            dtype: The data type of the transformation matrix.
                to be used if `matrix = None`.
            device: The device for storing the implemented transformation.
                If `matrix != None`, uses the device of input `matrix`.
            matrix: A tensor of shape (4, 4) or of shape (minibatch, 4, 4)
                representing the 4x4 3D transformation matrix.
                If `None`, initializes with identity using
                the specified `device` and `dtype`.
        """

        if matrix is None:
            self._matrix = torch.eye(4, dtype=dtype, device=device).view(1, 4, 4)
        else:
            if matrix.ndim not in (2, 3):
                raise ValueError('"matrix" has to be a 2- or a 3-dimensional tensor.')
            if matrix.shape[-2] != 4 or matrix.shape[-1] != 4:
                raise ValueError(
                    '"matrix" has to be a tensor of shape (minibatch, 4, 4)'
                )
            # set dtype and device from matrix
            dtype = matrix.dtype
            device = matrix.device
            self._matrix = matrix.view(-1, 4, 4)

        self._transforms = []  # store transforms to compose
        self._lu = None
        self.device = make_device(device)
        self.dtype = dtype

    def __len__(self) -> int:
        return self.get_matrix().shape[0]

    def __getitem__(
        self, index: Union[int, List[int], slice, torch.Tensor]
    ) -> "Transform3d":
        """
        Args:
            index: Specifying the index of the transform to retrieve.
                Can be an int, slice, list of ints, boolean, long tensor.
                Supports negative indices.

        Returns:
            Transform3d object with selected transforms. The tensors are not cloned.
        """
        if isinstance(index, int):
            index = [index]
        return self.__class__(matrix=self.get_matrix()[index])

    def compose(self, *others: "Transform3d") -> "Transform3d":
        """
        Return a new Transform3d representing the composition of self with the
        given other transforms, which will be stored as an internal list.

        Args:
            *others: Any number of Transform3d objects

        Returns:
            A new Transform3d with the stored transforms
        """
        out = Transform3d(dtype=self.dtype, device=self.device)
        out._matrix = self._matrix.clone()
        for other in others:
            if not isinstance(other, Transform3d):
                msg = "Only possible to compose Transform3d objects; got %s"
                raise ValueError(msg % type(other))
        out._transforms = self._transforms + list(others)
        return out

    def get_matrix(self) -> torch.Tensor:
        """
        Return a matrix which is the result of composing this transform
        with others stored in self.transforms. Where necessary transforms
        are broadcast against each other.
        For example, if self.transforms contains transforms t1, t2, and t3, and
        given a set of points x, the following should be true:

        .. code-block:: python

            y1 = t1.compose(t2, t3).transform(x)
            y2 = t3.transform(t2.transform(t1.transform(x)))
            y1.get_matrix() == y2.get_matrix()

        Returns:
            A transformation matrix representing the composed inputs.
        """
        composed_matrix = self._matrix.clone()
        if len(self._transforms) > 0:
            for other in self._transforms:
                other_matrix = other.get_matrix()
                composed_matrix = _broadcast_bmm(composed_matrix, other_matrix)
        return composed_matrix

    def _get_matrix_inverse(self) -> torch.Tensor:
        """
        Return the inverse of self._matrix.
        """
        return torch.inverse(self._matrix)

    def inverse(self, invert_composed: bool = False) -> "Transform3d":
        """
        Returns a new Transform3d object that represents an inverse of the
        current transformation.

        Args:
            invert_composed:
                - True: First compose the list of stored transformations
                  and then apply inverse to the result. This is
                  potentially slower for classes of transformations
                  with inverses that can be computed efficiently
                  (e.g. rotations and translations).
                - False: Invert the individual stored transformations
                  independently without composing them.

        Returns:
            A new Transform3d object containing the inverse of the original
            transformation.
        """

        tinv = Transform3d(dtype=self.dtype, device=self.device)

        if invert_composed:
            # first compose then invert
            tinv._matrix = torch.inverse(self.get_matrix())
        else:
            # self._get_matrix_inverse() implements efficient inverse
            # of self._matrix
            i_matrix = self._get_matrix_inverse()

            # 2 cases:
            if len(self._transforms) > 0:
                # a) Either we have a non-empty list of transforms:
                # Here we take self._matrix and append its inverse at the
                # end of the reverted _transforms list. After composing
                # the transformations with get_matrix(), this correctly
                # right-multiplies by the inverse of self._matrix
                # at the end of the composition.
                tinv._transforms = [t.inverse() for t in reversed(self._transforms)]
                last = Transform3d(dtype=self.dtype, device=self.device)
                last._matrix = i_matrix
                tinv._transforms.append(last)
            else:
                # b) Or there are no stored transformations
                # we just set inverted matrix
                tinv._matrix = i_matrix

        return tinv

    def stack(self, *others: "Transform3d") -> "Transform3d":
        """
        Return a new batched Transform3d representing the batch elements from
        self and all the given other transforms all batched together.

        Args:
            *others: Any number of Transform3d objects

        Returns:
            A new Transform3d.
        """
        transforms = [self] + list(others)
        matrix = torch.cat([t.get_matrix() for t in transforms], dim=0)
        out = Transform3d(dtype=self.dtype, device=self.device)
        out._matrix = matrix
        return out

    def transform_points(self, points, eps: Optional[float] = None) -> torch.Tensor:
        """
        Use this transform to transform a set of 3D points. Assumes row major
        ordering of the input points.

        Args:
            points: Tensor of shape (P, 3) or (N, P, 3)
            eps: If eps!=None, the argument is used to clamp the
                last coordinate before performing the final division.
                The clamping corresponds to:
                last_coord := (last_coord.sign() + (last_coord==0)) *
                torch.clamp(last_coord.abs(), eps),
                i.e. the last coordinates that are exactly 0 will
                be clamped to +eps.

        Returns:
            points_out: points of shape (N, P, 3) or (P, 3) depending
            on the dimensions of the transform
        """
        points_batch = points.clone()
        if points_batch.dim() == 2:
            points_batch = points_batch[None]  # (P, 3) -> (1, P, 3)
        if points_batch.dim() != 3:
            msg = "Expected points to have dim = 2 or dim = 3: got shape %r"
            raise ValueError(msg % repr(points.shape))

        N, P, _3 = points_batch.shape
        ones = torch.ones(N, P, 1, dtype=points.dtype, device=points.device)
        points_batch = torch.cat([points_batch, ones], dim=2)

        composed_matrix = self.get_matrix()
        points_out = _broadcast_bmm(points_batch, composed_matrix)
        denom = points_out[..., 3:]  # denominator
        if eps is not None:
            denom_sign = denom.sign() + (denom == 0.0).type_as(denom)
            denom = denom_sign * torch.clamp(denom.abs(), eps)
        points_out = points_out[..., :3] / denom

        # When transform is (1, 4, 4) and points is (P, 3) return
        # points_out of shape (P, 3)
        if points_out.shape[0] == 1 and points.dim() == 2:
            points_out = points_out.reshape(points.shape)

        return points_out

    def transform_normals(self, normals) -> torch.Tensor:
        """
        Use this transform to transform a set of normal vectors.

        Args:
            normals: Tensor of shape (P, 3) or (N, P, 3)

        Returns:
            normals_out: Tensor of shape (P, 3) or (N, P, 3) depending
            on the dimensions of the transform
        """
        if normals.dim() not in [2, 3]:
            msg = "Expected normals to have dim = 2 or dim = 3: got shape %r"
            raise ValueError(msg % (normals.shape,))
        composed_matrix = self.get_matrix()

        # TODO: inverse is bad! Solve a linear system instead
        mat = composed_matrix[:, :3, :3]
        normals_out = _broadcast_bmm(normals, mat.transpose(1, 2).inverse())

        # This doesn't pass unit tests. TODO investigate further
        # if self._lu is None:
        #     self._lu = self._matrix[:, :3, :3].transpose(1, 2).lu()
        # normals_out = normals.lu_solve(*self._lu)

        # When transform is (1, 4, 4) and normals is (P, 3) return
        # normals_out of shape (P, 3)
        if normals_out.shape[0] == 1 and normals.dim() == 2:
            normals_out = normals_out.reshape(normals.shape)

        return normals_out

    def translate(self, *args, **kwargs) -> "Transform3d":
        return self.compose(
            Translate(device=self.device, dtype=self.dtype, *args, **kwargs)
        )

    def scale(self, *args, **kwargs) -> "Transform3d":
        return self.compose(
            Scale(device=self.device, dtype=self.dtype, *args, **kwargs)
        )

    def rotate(self, *args, **kwargs) -> "Transform3d":
        return self.compose(
            Rotate(device=self.device, dtype=self.dtype, *args, **kwargs)
        )

    def rotate_axis_angle(self, *args, **kwargs) -> "Transform3d":
        return self.compose(
            RotateAxisAngle(device=self.device, dtype=self.dtype, *args, **kwargs)
        )

    def clone(self) -> "Transform3d":
        """
        Deep copy of Transforms object. All internal tensors are cloned
        individually.

        Returns:
            new Transforms object.
        """
        other = Transform3d(dtype=self.dtype, device=self.device)
        if self._lu is not None:
            other._lu = [elem.clone() for elem in self._lu]
        other._matrix = self._matrix.clone()
        other._transforms = [t.clone() for t in self._transforms]
        return other

    def to(
        self,
        device: Device,
        copy: bool = False,
        dtype: Optional[torch.dtype] = None,
    ) -> "Transform3d":
        """
        Match functionality of torch.Tensor.to()
        If copy = True or the self Tensor is on a different device, the
        returned tensor is a copy of self with the desired torch.device.
        If copy = False and the self Tensor already has the correct torch.device,
        then self is returned.

        Args:
          device: Device (as str or torch.device) for the new tensor.
          copy: Boolean indicator whether or not to clone self. Default False.
          dtype: If not None, casts the internal tensor variables
              to a given torch.dtype.

        Returns:
          Transform3d object.
        """
        device_ = make_device(device)
        dtype_ = self.dtype if dtype is None else dtype
        skip_to = self.device == device_ and self.dtype == dtype_

        if not copy and skip_to:
            return self

        other = self.clone()

        if skip_to:
            return other

        other.device = device_
        other.dtype = dtype_
        other._matrix = other._matrix.to(device=device_, dtype=dtype_)
        other._transforms = [
            t.to(device_, copy=copy, dtype=dtype_) for t in other._transforms
        ]
        return other

    def cpu(self) -> "Transform3d":
        return self.to("cpu")

    def cuda(self) -> "Transform3d":
        return self.to("cuda")

class Translate(Transform3d):
    def __init__(
        self,
        x,
        y=None,
        z=None,
        dtype: torch.dtype = torch.float32,
        device: Optional[Device] = None,
    ) -> None:
        """
        Create a new Transform3d representing 3D translations.

        Option I: Translate(xyz, dtype=torch.float32, device='cpu')
            xyz should be a tensor of shape (N, 3)

        Option II: Translate(x, y, z, dtype=torch.float32, device='cpu')
            Here x, y, and z will be broadcast against each other and
            concatenated to form the translation. Each can be:
                - A python scalar
                - A torch scalar
                - A 1D torch tensor
        """
        xyz = _handle_input(x, y, z, dtype, device, "Translate")
        super().__init__(device=xyz.device, dtype=dtype)
        N = xyz.shape[0]

        mat = torch.eye(4, dtype=dtype, device=self.device)
        mat = mat.view(1, 4, 4).repeat(N, 1, 1)
        mat[:, 3, :3] = xyz
        self._matrix = mat

    def _get_matrix_inverse(self) -> torch.Tensor:
        """
        Return the inverse of self._matrix.
        """
        inv_mask = self._matrix.new_ones([1, 4, 4])
        inv_mask[0, 3, :3] = -1.0
        i_matrix = self._matrix * inv_mask
        return i_matrix

class Rotate(Transform3d):
    def __init__(
        self,
        R: torch.Tensor,
        dtype: torch.dtype = torch.float32,
        device: Optional[Device] = None,
        orthogonal_tol: float = 1e-5,
    ) -> None:
        """
        Create a new Transform3d representing 3D rotation using a rotation
        matrix as the input.

        Args:
            R: a tensor of shape (3, 3) or (N, 3, 3)
            orthogonal_tol: tolerance for the test of the orthogonality of R

        """
        device_ = get_device(R, device)
        super().__init__(device=device_, dtype=dtype)
        if R.dim() == 2:
            R = R[None]
        if R.shape[-2:] != (3, 3):
            msg = "R must have shape (3, 3) or (N, 3, 3); got %s"
            raise ValueError(msg % repr(R.shape))
        R = R.to(device=device_, dtype=dtype)
        _check_valid_rotation_matrix(R, tol=orthogonal_tol)
        N = R.shape[0]
        mat = torch.eye(4, dtype=dtype, device=device_)
        mat = mat.view(1, 4, 4).repeat(N, 1, 1)
        mat[:, :3, :3] = R
        self._matrix = mat

    def _get_matrix_inverse(self) -> torch.Tensor:
        """
        Return the inverse of self._matrix.
        """
        return self._matrix.permute(0, 2, 1).contiguous()

class TensorAccessor(nn.Module):
    """
    A helper class to be used with the __getitem__ method. This can be used for
    getting/setting the values for an attribute of a class at one particular
    index.  This is useful when the attributes of a class are batched tensors
    and one element in the batch needs to be modified.
    """

    def __init__(self, class_object, index: Union[int, slice]) -> None:
        """
        Args:
            class_object: this should be an instance of a class which has
                attributes which are tensors representing a batch of
                values.
            index: int/slice, an index indicating the position in the batch.
                In __setattr__ and __getattr__ only the value of class
                attributes at this index will be accessed.
        """
        self.__dict__["class_object"] = class_object
        self.__dict__["index"] = index

    def __setattr__(self, name: str, value: Any):
        """
        Update the attribute given by `name` to the value given by `value`
        at the index specified by `self.index`.
        Args:
            name: str, name of the attribute.
            value: value to set the attribute to.
        """
        v = getattr(self.class_object, name)
        if not torch.is_tensor(v):
            msg = "Can only set values on attributes which are tensors; got %r"
            raise AttributeError(msg % type(v))

        # Convert the attribute to a tensor if it is not a tensor.
        if not torch.is_tensor(value):
            value = torch.tensor(
                value, device=v.device, dtype=v.dtype, requires_grad=v.requires_grad
            )

        # Check the shapes match the existing shape and the shape of the index.
        if v.dim() > 1 and value.dim() > 1 and value.shape[1:] != v.shape[1:]:
            msg = "Expected value to have shape %r; got %r"
            raise ValueError(msg % (v.shape, value.shape))
        if (
            v.dim() == 0
            and isinstance(self.index, slice)
            and len(value) != len(self.index)
        ):
            msg = "Expected value to have len %r; got %r"
            raise ValueError(msg % (len(self.index), len(value)))
        self.class_object.__dict__[name][self.index] = value

    def __getattr__(self, name: str):
        """
        Return the value of the attribute given by "name" on self.class_object
        at the index specified in self.index.
        Args:
            name: string of the attribute name
        """
        if hasattr(self.class_object, name):
            return self.class_object.__dict__[name][self.index]
        else:
            msg = "Attribute %s not found on %r"
            return AttributeError(msg % (name, self.class_object.__name__))

BROADCAST_TYPES = (float, int, list, tuple, torch.Tensor, np.ndarray)

class TensorProperties(nn.Module):
    """
    A mix-in class for storing tensors as properties with helper methods.
    """

    def __init__(
        self,
        dtype: torch.dtype = torch.float32,
        device: Device = "cpu",
        **kwargs,
    ) -> None:
        """
        Args:
            dtype: data type to set for the inputs
            device: Device (as str or torch.device)
            kwargs: any number of keyword arguments. Any arguments which are
                of type (float/int/list/tuple/tensor/array) are broadcasted and
                other keyword arguments are set as attributes.
        """
        super().__init__()
        self.device = make_device(device)
        self._N = 0
        if kwargs is not None:

            # broadcast all inputs which are float/int/list/tuple/tensor/array
            # set as attributes anything else e.g. strings, bools
            args_to_broadcast = {}
            for k, v in kwargs.items():
                if v is None or isinstance(v, (str, bool)):
                    setattr(self, k, v)
                elif isinstance(v, BROADCAST_TYPES):
                    args_to_broadcast[k] = v
                else:
                    msg = "Arg %s with type %r is not broadcastable"
                    warnings.warn(msg % (k, type(v)))

            names = args_to_broadcast.keys()
            # convert from type dict.values to tuple
            values = tuple(v for v in args_to_broadcast.values())

            if len(values) > 0:
                broadcasted_values = convert_to_tensors_and_broadcast(
                    *values, device=device
                )

                # Set broadcasted values as attributes on self.
                for i, n in enumerate(names):
                    setattr(self, n, broadcasted_values[i])
                    if self._N == 0:
                        self._N = broadcasted_values[i].shape[0]

    def __len__(self) -> int:
        return self._N

    def isempty(self) -> bool:
        return self._N == 0

    def __getitem__(self, index: Union[int, slice]) -> TensorAccessor:
        """
        Args:
            index: an int or slice used to index all the fields.
        Returns:
            if `index` is an index int/slice return a TensorAccessor class
            with getattribute/setattribute methods which return/update the value
            at the index in the original class.
        """
        if isinstance(index, (int, slice)):
            return TensorAccessor(class_object=self, index=index)

        msg = "Expected index of type int or slice; got %r"
        raise ValueError(msg % type(index))

    # pyre-fixme[14]: `to` overrides method defined in `Module` inconsistently.
    def to(self, device: Device = "cpu") -> "TensorProperties":
        """
        In place operation to move class properties which are tensors to a
        specified device. If self has a property "device", update this as well.
        """
        device_ = make_device(device)
        for k in dir(self):
            v = getattr(self, k)
            if k == "device":
                setattr(self, k, device_)
            if torch.is_tensor(v) and v.device != device_:
                setattr(self, k, v.to(device_))
        return self

    def cpu(self) -> "TensorProperties":
        return self.to("cpu")

    # pyre-fixme[14]: `cuda` overrides method defined in `Module` inconsistently.
    def cuda(self, device: Optional[int] = None) -> "TensorProperties":
        return self.to(f"cuda:{device}" if device is not None else "cuda")

    def clone(self, other) -> "TensorProperties":
        """
        Update the tensor properties of other with the cloned properties of self.
        """
        for k in dir(self):
            v = getattr(self, k)
            if inspect.ismethod(v) or k.startswith("__"):
                continue
            if torch.is_tensor(v):
                v_clone = v.clone()
            else:
                v_clone = copy.deepcopy(v)
            setattr(other, k, v_clone)
        return other

    def gather_props(self, batch_idx) -> "TensorProperties":
        """
        This is an in place operation to reformat all tensor class attributes
        based on a set of given indices using torch.gather. This is useful when
        attributes which are batched tensors e.g. shape (N, 3) need to be
        multiplied with another tensor which has a different first dimension
        e.g. packed vertices of shape (V, 3).
        Example
        .. code-block:: python
            self.specular_color = (N, 3) tensor of specular colors for each mesh
        A lighting calculation may use
        .. code-block:: python
            verts_packed = meshes.verts_packed()  # (V, 3)
        To multiply these two tensors the batch dimension needs to be the same.
        To achieve this we can do
        .. code-block:: python
            batch_idx = meshes.verts_packed_to_mesh_idx()  # (V)
        This gives index of the mesh for each vertex in verts_packed.
        .. code-block:: python
            self.gather_props(batch_idx)
            self.specular_color = (V, 3) tensor with the specular color for
                                     each packed vertex.
        torch.gather requires the index tensor to have the same shape as the
        input tensor so this method takes care of the reshaping of the index
        tensor to use with class attributes with arbitrary dimensions.
        Args:
            batch_idx: shape (B, ...) where `...` represents an arbitrary
                number of dimensions
        Returns:
            self with all properties reshaped. e.g. a property with shape (N, 3)
            is transformed to shape (B, 3).
        """
        # Iterate through the attributes of the class which are tensors.
        for k in dir(self):
            v = getattr(self, k)
            if torch.is_tensor(v):
                if v.shape[0] > 1:
                    # There are different values for each batch element
                    # so gather these using the batch_idx.
                    # First clone the input batch_idx tensor before
                    # modifying it.
                    _batch_idx = batch_idx.clone()
                    idx_dims = _batch_idx.shape
                    tensor_dims = v.shape
                    if len(idx_dims) > len(tensor_dims):
                        msg = "batch_idx cannot have more dimensions than %s. "
                        msg += "got shape %r and %s has shape %r"
                        raise ValueError(msg % (k, idx_dims, k, tensor_dims))
                    if idx_dims != tensor_dims:
                        # To use torch.gather the index tensor (_batch_idx) has
                        # to have the same shape as the input tensor.
                        new_dims = len(tensor_dims) - len(idx_dims)
                        new_shape = idx_dims + (1,) * new_dims
                        expand_dims = (-1,) + tensor_dims[1:]
                        _batch_idx = _batch_idx.view(*new_shape)
                        _batch_idx = _batch_idx.expand(*expand_dims)

                    v = v.gather(0, _batch_idx)
                    setattr(self, k, v)
        return self

class CamerasBase(TensorProperties):
    """
    `CamerasBase` implements a base class for all cameras.
    For cameras, there are four different coordinate systems (or spaces)
    - World coordinate system: This is the system the object lives - the world.
    - Camera view coordinate system: This is the system that has its origin on the camera
        and the and the Z-axis perpendicular to the image plane.
        In PyTorch3D, we assume that +X points left, and +Y points up and
        +Z points out from the image plane.
        The transformation from world --> view happens after applying a rotation (R)
        and translation (T)
    - NDC coordinate system: This is the normalized coordinate system that confines
        in a volume the rendered part of the object or scene. Also known as view volume.
        For square images, given the PyTorch3D convention, (+1, +1, znear)
        is the top left near corner, and (-1, -1, zfar) is the bottom right far
        corner of the volume.
        The transformation from view --> NDC happens after applying the camera
        projection matrix (P) if defined in NDC space.
        For non square images, we scale the points such that smallest side
        has range [-1, 1] and the largest side has range [-u, u], with u > 1.
    - Screen coordinate system: This is another representation of the view volume with
        the XY coordinates defined in image space instead of a normalized space.
    A better illustration of the coordinate systems can be found in
    pytorch3d/docs/notes/cameras.md.
    It defines methods that are common to all camera models:
        - `get_camera_center` that returns the optical center of the camera in
            world coordinates
        - `get_world_to_view_transform` which returns a 3D transform from
            world coordinates to the camera view coordinates (R, T)
        - `get_full_projection_transform` which composes the projection
            transform (P) with the world-to-view transform (R, T)
        - `transform_points` which takes a set of input points in world coordinates and
            projects to the space the camera is defined in (NDC or screen)
        - `get_ndc_camera_transform` which defines the transform from screen/NDC to
            PyTorch3D's NDC space
        - `transform_points_ndc` which takes a set of points in world coordinates and
            projects them to PyTorch3D's NDC space
        - `transform_points_screen` which takes a set of points in world coordinates and
            projects them to screen space
    For each new camera, one should implement the `get_projection_transform`
    routine that returns the mapping from camera view coordinates to camera
    coordinates (NDC or screen).
    Another useful function that is specific to each camera model is
    `unproject_points` which sends points from camera coordinates (NDC or screen)
    back to camera view or world coordinates depending on the `world_coordinates`
    boolean argument of the function.
    """

    # Used in __getitem__ to index the relevant fields
    # When creating a new camera, this should be set in the __init__
    _FIELDS: Tuple[str, ...] = ()

    # Names of fields which are a constant property of the whole batch, rather
    # than themselves a batch of data.
    # When joining objects into a batch, they will have to agree.
    _SHARED_FIELDS: Tuple[str, ...] = ()

    def get_projection_transform(self):
        """
        Calculate the projective transformation matrix.
        Args:
            **kwargs: parameters for the projection can be passed in as keyword
                arguments to override the default values set in `__init__`.
        Return:
            a `Transform3d` object which represents a batch of projection
            matrices of shape (N, 3, 3)
        """
        raise NotImplementedError()

    def unproject_points(self, xy_depth: torch.Tensor, **kwargs):
        """
        Transform input points from camera coodinates (NDC or screen)
        to the world / camera coordinates.
        Each of the input points `xy_depth` of shape (..., 3) is
        a concatenation of the x, y location and its depth.
        For instance, for an input 2D tensor of shape `(num_points, 3)`
        `xy_depth` takes the following form:
            `xy_depth[i] = [x[i], y[i], depth[i]]`,
        for a each point at an index `i`.
        The following example demonstrates the relationship between
        `transform_points` and `unproject_points`:
        .. code-block:: python
            cameras = # camera object derived from CamerasBase
            xyz = # 3D points of shape (batch_size, num_points, 3)
            # transform xyz to the camera view coordinates
            xyz_cam = cameras.get_world_to_view_transform().transform_points(xyz)
            # extract the depth of each point as the 3rd coord of xyz_cam
            depth = xyz_cam[:, :, 2:]
            # project the points xyz to the camera
            xy = cameras.transform_points(xyz)[:, :, :2]
            # append depth to xy
            xy_depth = torch.cat((xy, depth), dim=2)
            # unproject to the world coordinates
            xyz_unproj_world = cameras.unproject_points(xy_depth, world_coordinates=True)
            print(torch.allclose(xyz, xyz_unproj_world)) # True
            # unproject to the camera coordinates
            xyz_unproj = cameras.unproject_points(xy_depth, world_coordinates=False)
            print(torch.allclose(xyz_cam, xyz_unproj)) # True
        Args:
            xy_depth: torch tensor of shape (..., 3).
            world_coordinates: If `True`, unprojects the points back to world
                coordinates using the camera extrinsics `R` and `T`.
                `False` ignores `R` and `T` and unprojects to
                the camera view coordinates.
            from_ndc: If `False` (default), assumes xy part of input is in
                NDC space if self.in_ndc(), otherwise in screen space. If
                `True`, assumes xy is in NDC space even if the camera
                is defined in screen space.
        Returns
            new_points: unprojected points with the same shape as `xy_depth`.
        """
        raise NotImplementedError()

    def get_camera_center(self, **kwargs) -> torch.Tensor:
        """
        Return the 3D location of the camera optical center
        in the world coordinates.
        Args:
            **kwargs: parameters for the camera extrinsics can be passed in
                as keyword arguments to override the default values
                set in __init__.
        Setting T here will update the values set in init as this
        value may be needed later on in the rendering pipeline e.g. for
        lighting calculations.
        Returns:
            C: a batch of 3D locations of shape (N, 3) denoting
            the locations of the center of each camera in the batch.
        """
        w2v_trans = self.get_world_to_view_transform(**kwargs)
        P = w2v_trans.inverse().get_matrix()
        # the camera center is the translation component (the first 3 elements
        # of the last row) of the inverted world-to-view
        # transform (4x4 RT matrix)
        C = P[:, 3, :3]
        return C

    def get_world_to_view_transform(self, **kwargs) -> Transform3d:
        """
        Return the world-to-view transform.
        Args:
            **kwargs: parameters for the camera extrinsics can be passed in
                as keyword arguments to override the default values
                set in __init__.
        Setting R and T here will update the values set in init as these
        values may be needed later on in the rendering pipeline e.g. for
        lighting calculations.
        Returns:
            A Transform3d object which represents a batch of transforms
            of shape (N, 3, 3)
        """
        R: torch.Tensor = kwargs.get("R", self.R)
        T: torch.Tensor = kwargs.get("T", self.T)
        self.R = R  # pyre-ignore[16]
        self.T = T  # pyre-ignore[16]
        world_to_view_transform = get_world_to_view_transform(R=R, T=T)
        return world_to_view_transform

    def get_full_projection_transform(self, **kwargs) -> Transform3d:
        """
        Return the full world-to-camera transform composing the
        world-to-view and view-to-camera transforms.
        If camera is defined in NDC space, the projected points are in NDC space.
        If camera is defined in screen space, the projected points are in screen space.
        Args:
            **kwargs: parameters for the projection transforms can be passed in
                as keyword arguments to override the default values
                set in __init__.
        Setting R and T here will update the values set in init as these
        values may be needed later on in the rendering pipeline e.g. for
        lighting calculations.
        Returns:
            a Transform3d object which represents a batch of transforms
            of shape (N, 3, 3)
        """
        self.R: torch.Tensor = kwargs.get("R", self.R)  # pyre-ignore[16]
        self.T: torch.Tensor = kwargs.get("T", self.T)  # pyre-ignore[16]
        world_to_view_transform = self.get_world_to_view_transform(R=self.R, T=self.T)
        view_to_proj_transform = self.get_projection_transform(**kwargs)
        return world_to_view_transform.compose(view_to_proj_transform)

    def transform_points(
        self, points, eps: Optional[float] = None, **kwargs
    ) -> torch.Tensor:
        """
        Transform input points from world to camera space with the
        projection matrix defined by the camera.
        For `CamerasBase.transform_points`, setting `eps > 0`
        stabilizes gradients since it leads to avoiding division
        by excessively low numbers for points close to the camera plane.
        Args:
            points: torch tensor of shape (..., 3).
            eps: If eps!=None, the argument is used to clamp the
                divisor in the homogeneous normalization of the points
                transformed to the ndc space. Please see
                `transforms.Transform3d.transform_points` for details.
                For `CamerasBase.transform_points`, setting `eps > 0`
                stabilizes gradients since it leads to avoiding division
                by excessively low numbers for points close to the
                camera plane.
        Returns
            new_points: transformed points with the same shape as the input.
        """
        world_to_proj_transform = self.get_full_projection_transform(**kwargs)
        return world_to_proj_transform.transform_points(points, eps=eps)

    def get_ndc_camera_transform(self, **kwargs) -> Transform3d:
        """
        Returns the transform from camera projection space (screen or NDC) to NDC space.
        For cameras that can be specified in screen space, this transform
        allows points to be converted from screen to NDC space.
        The default transform scales the points from [0, W]x[0, H]
        to [-1, 1]x[-u, u] or [-u, u]x[-1, 1] where u > 1 is the aspect ratio of the image.
        This function should be modified per camera definitions if need be,
        e.g. for Perspective/Orthographic cameras we provide a custom implementation.
        This transform assumes PyTorch3D coordinate system conventions for
        both the NDC space and the input points.
        This transform interfaces with the PyTorch3D renderer which assumes
        input points to the renderer to be in NDC space.
        """
        if self.in_ndc():
            return Transform3d(device=self.device, dtype=torch.float32)
        else:
            # For custom cameras which can be defined in screen space,
            # users might might have to implement the screen to NDC transform based
            # on the definition of the camera parameters.
            # See PerspectiveCameras/OrthographicCameras for an example.
            # We don't flip xy because we assume that world points are in
            # PyTorch3D coordinates, and thus conversion from screen to ndc
            # is a mere scaling from image to [-1, 1] scale.
            image_size = kwargs.get("image_size", self.get_image_size())
            return get_screen_to_ndc_transform(
                self, with_xyflip=False, image_size=image_size
            )

    def transform_points_ndc(
        self, points, eps: Optional[float] = None, **kwargs
    ) -> torch.Tensor:
        """
        Transforms points from PyTorch3D world/camera space to NDC space.
        Input points follow the PyTorch3D coordinate system conventions: +X left, +Y up.
        Output points are in NDC space: +X left, +Y up, origin at image center.
        Args:
            points: torch tensor of shape (..., 3).
            eps: If eps!=None, the argument is used to clamp the
                divisor in the homogeneous normalization of the points
                transformed to the ndc space. Please see
                `transforms.Transform3d.transform_points` for details.
                For `CamerasBase.transform_points`, setting `eps > 0`
                stabilizes gradients since it leads to avoiding division
                by excessively low numbers for points close to the
                camera plane.
        Returns
            new_points: transformed points with the same shape as the input.
        """
        world_to_ndc_transform = self.get_full_projection_transform(**kwargs)
        if not self.in_ndc():
            to_ndc_transform = self.get_ndc_camera_transform(**kwargs)
            world_to_ndc_transform = world_to_ndc_transform.compose(to_ndc_transform)

        return world_to_ndc_transform.transform_points(points, eps=eps)

    def transform_points_screen(
        self, points, eps: Optional[float] = None, **kwargs
    ) -> torch.Tensor:
        """
        Transforms points from PyTorch3D world/camera space to screen space.
        Input points follow the PyTorch3D coordinate system conventions: +X left, +Y up.
        Output points are in screen space: +X right, +Y down, origin at top left corner.
        Args:
            points: torch tensor of shape (..., 3).
            eps: If eps!=None, the argument is used to clamp the
                divisor in the homogeneous normalization of the points
                transformed to the ndc space. Please see
                `transforms.Transform3d.transform_points` for details.
                For `CamerasBase.transform_points`, setting `eps > 0`
                stabilizes gradients since it leads to avoiding division
                by excessively low numbers for points close to the
                camera plane.
        Returns
            new_points: transformed points with the same shape as the input.
        """
        points_ndc = self.transform_points_ndc(points, eps=eps, **kwargs)
        image_size = kwargs.get("image_size", self.get_image_size())
        return get_ndc_to_screen_transform(
            self, with_xyflip=True, image_size=image_size
        ).transform_points(points_ndc, eps=eps)

    def clone(self):
        """
        Returns a copy of `self`.
        """
        cam_type = type(self)
        other = cam_type(device=self.device)
        return super().clone(other)

    def is_perspective(self):
        raise NotImplementedError()

    def in_ndc(self):
        """
        Specifies whether the camera is defined in NDC space
        or in screen (image) space
        """
        raise NotImplementedError()

    def get_znear(self):
        return self.znear if hasattr(self, "znear") else None

    def get_image_size(self):
        """
        Returns the image size, if provided, expected in the form of (height, width)
        The image size is used for conversion of projected points to screen coordinates.
        """
        return self.image_size if hasattr(self, "image_size") else None

    def __getitem__(
        self, index: Union[int, List[int], torch.LongTensor]
    ) -> "CamerasBase":
        """
        Override for the __getitem__ method in TensorProperties which needs to be
        refactored.
        Args:
            index: an int/list/long tensor used to index all the fields in the cameras given by
                self._FIELDS.
        Returns:
            if `index` is an index int/list/long tensor return an instance of the current
            cameras class with only the values at the selected index.
        """

        kwargs = {}

        if not isinstance(index, (int, list, torch.LongTensor, torch.cuda.LongTensor)):
            msg = "Invalid index type, expected int, List[int] or torch.LongTensor; got %r"
            raise ValueError(msg % type(index))

        if isinstance(index, int):
            index = [index]

        if max(index) >= len(self):
            raise ValueError(f"Index {max(index)} is out of bounds for select cameras")

        for field in self._FIELDS:
            val = getattr(self, field, None)
            if val is None:
                continue

            # e.g. "in_ndc" is set as attribute "_in_ndc" on the class
            # but provided as "in_ndc" on initialization
            if field.startswith("_"):
                field = field[1:]

            if isinstance(val, (str, bool)):
                kwargs[field] = val
            elif isinstance(val, torch.Tensor):
                # In the init, all inputs will be converted to
                # tensors before setting as attributes
                kwargs[field] = val[index]
            else:
                raise ValueError(f"Field {field} type is not supported for indexing")

        kwargs["device"] = self.device
        return self.__class__(**kwargs)

class FoVPerspectiveCameras(CamerasBase):
    """
    A class which stores a batch of parameters to generate a batch of
    projection matrices by specifying the field of view.
    The definition of the parameters follow the OpenGL perspective camera.

    The extrinsics of the camera (R and T matrices) can also be set in the
    initializer or passed in to `get_full_projection_transform` to get
    the full transformation from world -> ndc.

    The `transform_points` method calculates the full world -> ndc transform
    and then applies it to the input points.

    The transforms can also be returned separately as Transform3d objects.

    * Setting the Aspect Ratio for Non Square Images *

    If the desired output image size is non square (i.e. a tuple of (H, W) where H != W)
    the aspect ratio needs special consideration: There are two aspect ratios
    to be aware of:
        - the aspect ratio of each pixel
        - the aspect ratio of the output image
    The `aspect_ratio` setting in the FoVPerspectiveCameras sets the
    pixel aspect ratio. When using this camera with the differentiable rasterizer
    be aware that in the rasterizer we assume square pixels, but allow
    variable image aspect ratio (i.e rectangle images).

    In most cases you will want to set the camera `aspect_ratio=1.0`
    (i.e. square pixels) and only vary the output image dimensions in pixels
    for rasterization.
    """

    # For __getitem__
    _FIELDS = (
        "K",
        "znear",
        "zfar",
        "aspect_ratio",
        "fov",
        "R",
        "T",
        "degrees",
    )

    _SHARED_FIELDS = ("degrees",)

    def __init__(
        self,
        znear=1.0,
        zfar=100.0,
        aspect_ratio=1.0,
        fov=60.0,
        degrees: bool = True,
        R: torch.Tensor = _R,
        T: torch.Tensor = _T,
        K: Optional[torch.Tensor] = None,
        device: Device = "cpu",
    ) -> None:
        """

        Args:
            znear: near clipping plane of the view frustrum.
            zfar: far clipping plane of the view frustrum.
            aspect_ratio: aspect ratio of the image pixels.
                1.0 indicates square pixels.
            fov: field of view angle of the camera.
            degrees: bool, set to True if fov is specified in degrees.
            R: Rotation matrix of shape (N, 3, 3)
            T: Translation matrix of shape (N, 3)
            K: (optional) A calibration matrix of shape (N, 4, 4)
                If provided, don't need znear, zfar, fov, aspect_ratio, degrees
            device: Device (as str or torch.device)
        """
        # The initializer formats all inputs to torch tensors and broadcasts
        # all the inputs to have the same batch dimension where necessary.
        super().__init__(
            device=device,
            znear=znear,
            zfar=zfar,
            aspect_ratio=aspect_ratio,
            fov=fov,
            R=R,
            T=T,
            K=K,
        )

        # No need to convert to tensor or broadcast.
        self.degrees = degrees

    def compute_projection_matrix(
        self, znear, zfar, fov, aspect_ratio, degrees: bool
    ) -> torch.Tensor:
        """
        Compute the calibration matrix K of shape (N, 4, 4)

        Args:
            znear: near clipping plane of the view frustrum.
            zfar: far clipping plane of the view frustrum.
            fov: field of view angle of the camera.
            aspect_ratio: aspect ratio of the image pixels.
                1.0 indicates square pixels.
            degrees: bool, set to True if fov is specified in degrees.

        Returns:
            torch.FloatTensor of the calibration matrix with shape (N, 4, 4)
        """
        K = torch.zeros((self._N, 4, 4), device=self.device, dtype=torch.float32)
        ones = torch.ones((self._N), dtype=torch.float32, device=self.device)
        if degrees:
            fov = (np.pi / 180) * fov

        if not torch.is_tensor(fov):
            fov = torch.tensor(fov, device=self.device)
        tanHalfFov = torch.tan((fov / 2))
        max_y = tanHalfFov * znear
        min_y = -max_y
        max_x = max_y * aspect_ratio
        min_x = -max_x

        # NOTE: In OpenGL the projection matrix changes the handedness of the
        # coordinate frame. i.e the NDC space positive z direction is the
        # camera space negative z direction. This is because the sign of the z
        # in the projection matrix is set to -1.0.
        # In pytorch3d we maintain a right handed coordinate system throughout
        # so the so the z sign is 1.0.
        z_sign = 1.0

        K[:, 0, 0] = 2.0 * znear / (max_x - min_x)
        K[:, 1, 1] = 2.0 * znear / (max_y - min_y)
        K[:, 0, 2] = (max_x + min_x) / (max_x - min_x)
        K[:, 1, 2] = (max_y + min_y) / (max_y - min_y)
        K[:, 3, 2] = z_sign * ones

        # NOTE: This maps the z coordinate from [0, 1] where z = 0 if the point
        # is at the near clipping plane and z = 1 when the point is at the far
        # clipping plane.
        K[:, 2, 2] = z_sign * zfar / (zfar - znear)
        K[:, 2, 3] = -(zfar * znear) / (zfar - znear)

        return K

    def get_projection_transform(self, **kwargs) -> Transform3d:
        """
        Calculate the perspective projection matrix with a symmetric
        viewing frustrum. Use column major order.
        The viewing frustrum will be projected into ndc, s.t.
        (max_x, max_y) -> (+1, +1)
        (min_x, min_y) -> (-1, -1)

        Args:
            **kwargs: parameters for the projection can be passed in as keyword
                arguments to override the default values set in `__init__`.

        Return:
            a Transform3d object which represents a batch of projection
            matrices of shape (N, 4, 4)

        .. code-block:: python

            h1 = (max_y + min_y)/(max_y - min_y)
            w1 = (max_x + min_x)/(max_x - min_x)
            tanhalffov = tan((fov/2))
            s1 = 1/tanhalffov
            s2 = 1/(tanhalffov * (aspect_ratio))

            # To map z to the range [0, 1] use:
            f1 =  far / (far - near)
            f2 = -(far * near) / (far - near)

            # Projection matrix
            K = [
                    [s1,   0,   w1,   0],
                    [0,   s2,   h1,   0],
                    [0,    0,   f1,  f2],
                    [0,    0,    1,   0],
            ]
        """
        K = kwargs.get("K", self.K)
        if K is not None:
            if K.shape != (self._N, 4, 4):
                msg = "Expected K to have shape of (%r, 4, 4)"
                raise ValueError(msg % (self._N))
        else:
            K = self.compute_projection_matrix(
                kwargs.get("znear", self.znear),
                kwargs.get("zfar", self.zfar),
                kwargs.get("fov", self.fov),
                kwargs.get("aspect_ratio", self.aspect_ratio),
                kwargs.get("degrees", self.degrees),
            )

        # Transpose the projection matrix as PyTorch3D transforms use row vectors.
        transform = Transform3d(
            matrix=K.transpose(1, 2).contiguous(), device=self.device
        )
        return transform

    def unproject_points(
        self,
        xy_depth: torch.Tensor,
        world_coordinates: bool = True,
        scaled_depth_input: bool = False,
        **kwargs,
    ) -> torch.Tensor:
        """>!
        FoV cameras further allow for passing depth in world units
        (`scaled_depth_input=False`) or in the [0, 1]-normalized units
        (`scaled_depth_input=True`)

        Args:
            scaled_depth_input: If `True`, assumes the input depth is in
                the [0, 1]-normalized units. If `False` the input depth is in
                the world units.
        """

        # obtain the relevant transformation to ndc
        if world_coordinates:
            to_ndc_transform = self.get_full_projection_transform()
        else:
            to_ndc_transform = self.get_projection_transform()

        if scaled_depth_input:
            # the input is scaled depth, so we don't have to do anything
            xy_sdepth = xy_depth
        else:
            # parse out important values from the projection matrix
            K_matrix = self.get_projection_transform(**kwargs.copy()).get_matrix()
            # parse out f1, f2 from K_matrix
            unsqueeze_shape = [1] * xy_depth.dim()
            unsqueeze_shape[0] = K_matrix.shape[0]
            f1 = K_matrix[:, 2, 2].reshape(unsqueeze_shape)
            f2 = K_matrix[:, 3, 2].reshape(unsqueeze_shape)
            # get the scaled depth
            sdepth = (f1 * xy_depth[..., 2:3] + f2) / xy_depth[..., 2:3]
            # concatenate xy + scaled depth
            xy_sdepth = torch.cat((xy_depth[..., 0:2], sdepth), dim=-1)

        # unproject with inverse of the projection
        unprojection_transform = to_ndc_transform.inverse()
        return unprojection_transform.transform_points(xy_sdepth)

    def is_perspective(self):
        return True

    def in_ndc(self):
        return True

#######################################################################################
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def make_device(device: Device) -> torch.device:
    """
    Makes an actual torch.device object from the device specified as
    either a string or torch.device object. If the device is `cuda` without
    a specific index, the index of the current device is assigned.
    Args:
        device: Device (as str or torch.device)
    Returns:
        A matching torch.device object
    """
    device = torch.device(device) if isinstance(device, str) else device
    if device.type == "cuda" and device.index is None:  # pyre-ignore[16]
        # If cuda but with no index, then the current cuda device is indicated.
        # In that case, we fix to that device
        device = torch.device(f"cuda:{torch.cuda.current_device()}")
    return device

def get_device(x, device: Optional[Device] = None) -> torch.device:
    """
    Gets the device of the specified variable x if it is a tensor, or
    falls back to a default CPU device otherwise. Allows overriding by
    providing an explicit device.
    Args:
        x: a torch.Tensor to get the device from or another type
        device: Device (as str or torch.device) to fall back to
    Returns:
        A matching torch.device object
    """

    # User overrides device
    if device is not None:
        return make_device(device)

    # Set device based on input tensor
    if torch.is_tensor(x):
        return x.device

    # Default device is cpu
    return torch.device("cpu")

def _axis_angle_rotation(axis: str, angle: torch.Tensor) -> torch.Tensor:
    """
    Return the rotation matrices for one of the rotations about an axis
    of which Euler angles describe, for each value of the angle given.

    Args:
        axis: Axis label "X" or "Y or "Z".
        angle: any shape tensor of Euler angles in radians

    Returns:
        Rotation matrices as tensor of shape (..., 3, 3).
    """

    cos = torch.cos(angle)
    sin = torch.sin(angle)
    one = torch.ones_like(angle)
    zero = torch.zeros_like(angle)

    if axis == "X":
        R_flat = (one, zero, zero, zero, cos, -sin, zero, sin, cos)
    elif axis == "Y":
        R_flat = (cos, zero, sin, zero, one, zero, -sin, zero, cos)
    elif axis == "Z":
        R_flat = (cos, -sin, zero, sin, cos, zero, zero, zero, one)
    else:
        raise ValueError("letter must be either X, Y or Z.")

    return torch.stack(R_flat, -1).reshape(angle.shape + (3, 3))

def euler_angles_to_matrix(euler_angles: torch.Tensor, convention: str) -> torch.Tensor:
    """
    Convert rotations given as Euler angles in radians to rotation matrices.

    Args:
        euler_angles: Euler angles in radians as tensor of shape (..., 3).
        convention: Convention string of three uppercase letters from
            {"X", "Y", and "Z"}.

    Returns:
        Rotation matrices as tensor of shape (..., 3, 3).
    """
    if euler_angles.dim() == 0 or euler_angles.shape[-1] != 3:
        raise ValueError("Invalid input euler angles.")
    if len(convention) != 3:
        raise ValueError("Convention must have 3 letters.")
    if convention[1] in (convention[0], convention[2]):
        raise ValueError(f"Invalid convention {convention}.")
    for letter in convention:
        if letter not in ("X", "Y", "Z"):
            raise ValueError(f"Invalid letter {letter} in convention string.")
    matrices = [
        _axis_angle_rotation(c, e)
        for c, e in zip(convention, torch.unbind(euler_angles, -1))
    ]
    # return functools.reduce(torch.matmul, matrices)
    return torch.matmul(torch.matmul(matrices[0], matrices[1]), matrices[2])

def _broadcast_bmm(a, b) -> torch.Tensor:
    """
    Batch multiply two matrices and broadcast if necessary.

    Args:
        a: torch tensor of shape (P, K) or (M, P, K)
        b: torch tensor of shape (N, K, K)

    Returns:
        a and b broadcast multiplied. The output batch dimension is max(N, M).

    To broadcast transforms across a batch dimension if M != N then
    expect that either M = 1 or N = 1. The tensor with batch dimension 1 is
    expanded to have shape N or M.
    """
    if a.dim() == 2:
        a = a[None]
    if len(a) != len(b):
        if not ((len(a) == 1) or (len(b) == 1)):
            msg = "Expected batch dim for bmm to be equal or 1; got %r, %r"
            raise ValueError(msg % (a.shape, b.shape))
        if len(a) == 1:
            a = a.expand(len(b), -1, -1)
        if len(b) == 1:
            b = b.expand(len(a), -1, -1)
    return a.bmm(b)

def _safe_det_3x3(t: torch.Tensor):
    """
    Fast determinant calculation for a batch of 3x3 matrices.
    Note, result of this function might not be the same as `torch.det()`.
    The differences might be in the last significant digit.
    Args:
        t: Tensor of shape (N, 3, 3).
    Returns:
        Tensor of shape (N) with determinants.
    """

    det = (
        t[..., 0, 0] * (t[..., 1, 1] * t[..., 2, 2] - t[..., 1, 2] * t[..., 2, 1])
        - t[..., 0, 1] * (t[..., 1, 0] * t[..., 2, 2] - t[..., 2, 0] * t[..., 1, 2])
        + t[..., 0, 2] * (t[..., 1, 0] * t[..., 2, 1] - t[..., 2, 0] * t[..., 1, 1])
    )

    return det

def get_world_to_view_transform(
    R: torch.Tensor = _R, T: torch.Tensor = _T
) -> Transform3d:
    """
    This function returns a Transform3d representing the transformation
    matrix to go from world space to view space by applying a rotation and
    a translation.
    PyTorch3D uses the same convention as Hartley & Zisserman.
    I.e., for camera extrinsic parameters R (rotation) and T (translation),
    we map a 3D point `X_world` in world coordinates to
    a point `X_cam` in camera coordinates with:
    `X_cam = X_world R + T`
    Args:
        R: (N, 3, 3) matrix representing the rotation.
        T: (N, 3) matrix representing the translation.
    Returns:
        a Transform3d object which represents the composed RT transformation.
    """
    # TODO: also support the case where RT is specified as one matrix
    # of shape (N, 4, 4).

    if T.shape[0] != R.shape[0]:
        msg = "Expected R, T to have the same batch dimension; got %r, %r"
        raise ValueError(msg % (R.shape[0], T.shape[0]))
    if T.dim() != 2 or T.shape[1:] != (3,):
        msg = "Expected T to have shape (N, 3); got %r"
        raise ValueError(msg % repr(T.shape))
    if R.dim() != 3 or R.shape[1:] != (3, 3):
        msg = "Expected R to have shape (N, 3, 3); got %r"
        raise ValueError(msg % repr(R.shape))

    # Create a Transform3d object
    T_ = Translate(T, device=T.device)
    R_ = Rotate(R, device=R.device)
    return R_.compose(T_)

def _check_valid_rotation_matrix(R, tol: float = 1e-7) -> None:
    """
    Determine if R is a valid rotation matrix by checking it satisfies the
    following conditions:

    ``RR^T = I and det(R) = 1``

    Args:
        R: an (N, 3, 3) matrix

    Returns:
        None

    Emits a warning if R is an invalid rotation matrix.
    """
    N = R.shape[0]
    eye = torch.eye(3, dtype=R.dtype, device=R.device)
    eye = eye.view(1, 3, 3).expand(N, -1, -1)
    orthogonal = torch.allclose(R.bmm(R.transpose(1, 2)), eye, atol=tol)
    det_R = _safe_det_3x3(R)
    no_distortion = torch.allclose(det_R, torch.ones_like(det_R))
    if not (orthogonal and no_distortion):
        msg = "R is not a valid rotation matrix"
        warnings.warn(msg)
    return

def format_tensor(
    input,
    dtype: torch.dtype = torch.float32,
    device: Device = "cpu",
) -> torch.Tensor:
    """
    Helper function for converting a scalar value to a tensor.
    Args:
        input: Python scalar, Python list/tuple, torch scalar, 1D torch tensor
        dtype: data type for the input
        device: Device (as str or torch.device) on which the tensor should be placed.
    Returns:
        input_vec: torch tensor with optional added batch dimension.
    """
    device_ = make_device(device)
    if not torch.is_tensor(input):
        input = torch.tensor(input, dtype=dtype, device=device_)

    if input.dim() == 0:
        input = input.view(1)

    if input.device == device_:
        return input

    input = input.to(device=device)
    return input

def convert_to_tensors_and_broadcast(
    *args,
    dtype: torch.dtype = torch.float32,
    device: Device = "cpu",
):
    """
    Helper function to handle parsing an arbitrary number of inputs (*args)
    which all need to have the same batch dimension.
    The output is a list of tensors.
    Args:
        *args: an arbitrary number of inputs
            Each of the values in `args` can be one of the following
                - Python scalar
                - Torch scalar
                - Torch tensor of shape (N, K_i) or (1, K_i) where K_i are
                  an arbitrary number of dimensions which can vary for each
                  value in args. In this case each input is broadcast to a
                  tensor of shape (N, K_i)
        dtype: data type to use when creating new tensors.
        device: torch device on which the tensors should be placed.
    Output:
        args: A list of tensors of shape (N, K_i)
    """
    # Convert all inputs to tensors with a batch dimension
    args_1d = [format_tensor(c, dtype, device) for c in args]

    # Find broadcast size
    sizes = [c.shape[0] for c in args_1d]
    N = max(sizes)

    args_Nd = []
    for c in args_1d:
        if c.shape[0] != 1 and c.shape[0] != N:
            msg = "Got non-broadcastable sizes %r" % sizes
            raise ValueError(msg)

        # Expand broadcast dim and keep non broadcast dims the same size
        expand_sizes = (N,) + (-1,) * len(c.shape[1:])
        args_Nd.append(c.expand(*expand_sizes))

    return args_Nd

def _handle_coord(c, dtype: torch.dtype, device: torch.device) -> torch.Tensor:
    """
    Helper function for _handle_input.

    Args:
        c: Python scalar, torch scalar, or 1D torch tensor

    Returns:
        c_vec: 1D torch tensor
    """
    if not torch.is_tensor(c):
        c = torch.tensor(c, dtype=dtype, device=device)
    if c.dim() == 0:
        c = c.view(1)
    if c.device != device or c.dtype != dtype:
        c = c.to(device=device, dtype=dtype)
    return c

def _handle_input(
    x,
    y,
    z,
    dtype: torch.dtype,
    device: Optional[Device],
    name: str,
    allow_singleton: bool = False,
) -> torch.Tensor:
    """
    Helper function to handle parsing logic for building transforms. The output
    is always a tensor of shape (N, 3), but there are several types of allowed
    input.

    Case I: Single Matrix
        In this case x is a tensor of shape (N, 3), and y and z are None. Here just
        return x.

    Case II: Vectors and Scalars
        In this case each of x, y, and z can be one of the following
            - Python scalar
            - Torch scalar
            - Torch tensor of shape (N, 1) or (1, 1)
        In this case x, y and z are broadcast to tensors of shape (N, 1)
        and concatenated to a tensor of shape (N, 3)

    Case III: Singleton (only if allow_singleton=True)
        In this case y and z are None, and x can be one of the following:
            - Python scalar
            - Torch scalar
            - Torch tensor of shape (N, 1) or (1, 1)
        Here x will be duplicated 3 times, and we return a tensor of shape (N, 3)

    Returns:
        xyz: Tensor of shape (N, 3)
    """
    device_ = get_device(x, device)
    # If x is actually a tensor of shape (N, 3) then just return it
    if torch.is_tensor(x) and x.dim() == 2:
        if x.shape[1] != 3:
            msg = "Expected tensor of shape (N, 3); got %r (in %s)"
            raise ValueError(msg % (x.shape, name))
        if y is not None or z is not None:
            msg = "Expected y and z to be None (in %s)" % name
            raise ValueError(msg)
        return x.to(device=device_, dtype=dtype)

    if allow_singleton and y is None and z is None:
        y = x
        z = x

    # Convert all to 1D tensors
    xyz = [_handle_coord(c, dtype, device_) for c in [x, y, z]]

    # Broadcast and concatenate
    sizes = [c.shape[0] for c in xyz]
    N = max(sizes)
    for c in xyz:
        if c.shape[0] != 1 and c.shape[0] != N:
            msg = "Got non-broadcastable sizes %r (in %s)" % (sizes, name)
            raise ValueError(msg)
    xyz = [c.expand(N) for c in xyz]
    xyz = torch.stack(xyz, dim=1)
    return xyz