import numpy as np
from .base import Homogeneous, HomogFamilyAlignment
from functools import reduce
[docs]class Affine(Homogeneous):
r"""
Base class for all ``n``-dimensional affine transformations. Provides
methods to break the transform down into its constituent
scale/rotation/translation, to view the homogeneous matrix equivalent,
and to chain this transform with other affine transformations.
Parameters
----------
h_matrix : ``(n_dims + 1, n_dims + 1)`` `ndarray`
The homogeneous matrix of the affine transformation.
copy : `bool`, optional
If ``False`` avoid copying ``h_matrix`` for performance.
skip_checks : `bool`, optional
If ``True`` avoid sanity checks on ``h_matrix`` for performance.
"""
def __init__(self, h_matrix, copy=True, skip_checks=False):
Homogeneous.__init__(self, h_matrix, copy=copy, skip_checks=skip_checks)
[docs] @classmethod
def init_identity(cls, n_dims):
r"""
Creates an identity matrix Affine transform.
Parameters
----------
n_dims : `int`
The number of dimensions.
Returns
-------
identity : :class:`Affine`
The identity matrix transform.
"""
return cls(np.eye(n_dims + 1), copy=False, skip_checks=True)
[docs] @classmethod
def init_from_2d_shear(cls, phi, psi, degrees=True):
r"""
Convenience constructor for 2D shear transformations about the origin.
Parameters
----------
phi : `float`
The angle of shearing in the X direction.
psi : `float`
The angle of shearing in the Y direction.
degrees : `bool`, optional
If ``True`` phi and psi are interpreted as degrees.
If ``False``, phi and psi are interpreted as radians.
Returns
-------
shear_transform : :map:`Affine`
A 2D shear transform.
"""
if degrees:
phi = np.deg2rad(phi)
psi = np.deg2rad(psi)
# Create shear matrix
h_matrix = np.eye(3)
h_matrix[0, 1] = np.tan(phi)
h_matrix[1, 0] = np.tan(psi)
return cls(h_matrix, skip_checks=True)
@property
def h_matrix(self):
r"""
The homogeneous matrix defining this transform.
:type: ``(n_dims + 1, n_dims + 1)`` `ndarray`
"""
return self._h_matrix
def _set_h_matrix(self, value, copy=True, skip_checks=False):
r"""
Updates the `h_matrix`, performing sanity checks.
Parameters
----------
value : `ndarray`
The new homogeneous matrix to set
copy : `bool`, optional
If ``False`` do not copy the h_matrix. Useful for performance.
skip_checks : `bool`, optional
If ``True`` skip sanity checks on the matrix. Useful for performance.
"""
if not skip_checks:
shape = value.shape
if len(shape) != 2 or shape[0] != shape[1]:
raise ValueError("You need to provide a square homogeneous " "matrix")
if self.h_matrix is not None:
# already have a matrix set! The update better be the same size
if self.n_dims != shape[0] - 1:
raise ValueError(
"Trying to update the homogeneous "
"matrix to a different dimension"
)
if shape[0] - 1 not in [2, 3]:
raise ValueError("Affine Transforms can only be 2D or 3D")
if not (np.allclose(value[-1, :-1], 0) and np.allclose(value[-1, -1], 1)):
raise ValueError("Bottom row must be [0 0 0 1] or [0, 0, 1]")
if copy:
value = value.copy()
self._h_matrix = value
@property
def linear_component(self):
r"""
The linear component of this affine transform.
:type: ``(n_dims, n_dims)`` `ndarray`
"""
return self.h_matrix[:-1, :-1]
@property
def translation_component(self):
r"""
The translation component of this affine transform.
:type: ``(n_dims,)`` `ndarray`
"""
return self.h_matrix[:-1, -1]
[docs] def decompose(self):
r"""
Decompose this transform into discrete Affine Transforms.
Useful for understanding the effect of a complex composite transform.
Returns
-------
transforms : `list` of :map:`DiscreteAffine`
Equivalent to this affine transform, such that
.. code-block:: python
reduce(lambda x, y: x.chain(y), self.decompose()) == self
"""
from .rotation import Rotation
from .translation import Translation
from .scale import Scale
U, S, V = np.linalg.svd(self.linear_component)
rotation_2 = Rotation(U)
rotation_1 = Rotation(V)
scale = Scale(S)
translation = Translation(self.translation_component)
return [rotation_1, scale, rotation_2, translation]
def _transform_str(self):
r"""
A string representation explaining what this affine transform does.
Has to be implemented by base classes.
Returns
-------
str : `str`
String representation of transform.
"""
header = "Affine decomposing into:"
list_str = [t._transform_str() for t in self.decompose()]
return header + reduce(lambda x, y: x + "\n" + " " + y, list_str, " ")
def _apply(self, x, **kwargs):
r"""
Applies this transform to a new set of vectors.
Parameters
----------
x : ``(N, D)`` `ndarray`
Array to apply this transform to.
Returns
-------
transformed_x : ``(N, D)`` `ndarray`
The transformed array.
"""
return np.dot(x, self.linear_component.T) + self.translation_component
@property
def n_parameters(self):
r"""
``n_dims * (n_dims + 1)`` parameters - every element of the matrix but
the homogeneous part.
:type: int
Examples
--------
2D Affine: 6 parameters::
[p1, p3, p5]
[p2, p4, p6]
3D Affine: 12 parameters::
[p1, p4, p7, p10]
[p2, p5, p8, p11]
[p3, p6, p9, p12]
"""
return self.n_dims * (self.n_dims + 1)
def _as_vector(self):
r"""
Return the parameters of the transform as a 1D array. These parameters
are parametrised as deltas from the identity warp. This does not
include the homogeneous part of the warp. Note that it flattens using
Fortran ordering, to stay consistent with Matlab.
**2D**
========= ===========================================
parameter definition
========= ===========================================
p1 Affine parameter
p2 Affine parameter
p3 Affine parameter
p4 Affine parameter
p5 Translation in `x`
p6 Translation in `y`
========= ===========================================
3D and higher transformations follow a similar format to the 2D case.
Returns
-------
params : ``(n_parameters,)`` `ndarray`
The values that parametrise the transform.
"""
params = self.h_matrix - np.eye(self.n_dims + 1)
return params[: self.n_dims, :].ravel(order="F")
def _from_vector_inplace(self, p):
r"""
Updates this Affine in-place from the new parameters. See
from_vector for details of the parameter format
"""
h_matrix = None
if p.shape[0] == 6: # 2D affine
h_matrix = np.eye(3)
h_matrix[:2, :] += p.reshape((2, 3), order="F")
elif p.shape[0] == 12: # 3D affine
h_matrix = np.eye(4)
h_matrix[:3, :] += p.reshape((3, 4), order="F")
else:
ValueError(
"Only 2D (6 parameters) or 3D (12 parameters) "
"homogeneous matrices are supported."
)
self._set_h_matrix(h_matrix, copy=False, skip_checks=True)
@property
def composes_inplace_with(self):
r"""
:class:`Affine` can swallow composition with any other :class:`Affine`.
"""
return Affine
[docs]class AlignmentAffine(HomogFamilyAlignment, Affine):
r"""
Constructs an :class:`Affine` by finding the optimal affine transform to
align `source` to `target`.
Parameters
----------
source : :map:`PointCloud`
The source pointcloud instance used in the alignment
target : :map:`PointCloud`
The target pointcloud instance used in the alignment
Notes
-----
We want to find the optimal transform M which satisfies :math:`M a = b`
where :math:`a` and :math:`b` are the `source` and `target` homogeneous
vectors respectively. ::
(M a)' = b'
a' M' = b'
a a' M' = a b'
`a a'` is of shape `(n_dim + 1, n_dim + 1)` and so can be inverted
to solve for `M`.
This approach is the analytical linear least squares solution to the
problem at hand. It will have a solution as long as `(a a')` is
non-singular, which generally means at least 2 corresponding points are
required.
"""
def __init__(self, source, target):
# first, initialize the alignment
HomogFamilyAlignment.__init__(self, source, target)
# now, the Affine
optimal_h = self._build_alignment_h_matrix(source, target)
Affine.__init__(self, optimal_h, copy=False, skip_checks=True)
@staticmethod
def _build_alignment_h_matrix(source, target):
r"""
Returns the optimal alignment of `source` to `target`.
Parameters
----------
source : :map:`PointCloud`
The source pointcloud instance used in the alignment
target : :map:`PointCloud`
The target pointcloud instance used in the alignment
"""
a = source.h_points()
b = target.h_points()
return np.linalg.solve(np.dot(a, a.T), np.dot(a, b.T)).T
def _set_h_matrix(self, value, copy=True, skip_checks=False):
r"""
Updates ``h_matrix``, optionally performing sanity checks.
.. note::
Updating the ``h_matrix`` on an :map:`AlignmentAffine`
triggers a sync of the target.
Note that it won't always be possible to manually specify the
``h_matrix`` through this method, specifically if changing the
``h_matrix`` could change the nature of the transform. See
:attr:`h_matrix_is_mutable` for how you can discover if the
``h_matrix`` is allowed to be set for a given class.
Parameters
----------
value : `ndarray`
The new homogeneous matrix to set
copy : `bool`, optional
If ``False`` do not copy the h_matrix. Useful for performance.
skip_checks : `bool`, optional
If ``True`` skip checking. Useful for performance.
Raises
------
NotImplementedError
If :attr:`h_matrix_is_mutable` returns ``False``.
"""
Affine._set_h_matrix(self, value, copy=copy, skip_checks=skip_checks)
# now update the state
self._sync_target_from_state()
def _sync_state_from_target(self):
optimal_h = self._build_alignment_h_matrix(self.source, self.target)
# Use the pure Affine setter (so we don't get syncing)
# We know the resulting affine is correct so skip the checks
Affine._set_h_matrix(self, optimal_h, copy=False, skip_checks=True)
[docs] def as_non_alignment(self):
r"""
Returns a copy of this :map:`Affine` without its alignment nature.
Returns
-------
transform : :map:`Affine`
A version of this affine with the same transform behavior but
without the alignment logic.
"""
return Affine(self.h_matrix, skip_checks=True)
[docs]class DiscreteAffine(object):
r"""
A discrete Affine transform operation (such as a :meth:`Scale`,
:class:`Translation` or :meth:`Rotation`). Has to be invertable. Make sure
you inherit from :class:`DiscreteAffine` first, for optimal
`decompose()` behavior.
"""
[docs] def decompose(self):
r"""
A :class:`DiscreteAffine` is already maximally decomposed -
return a copy of self in a `list`.
Returns
-------
transform : :class:`DiscreteAffine`
Deep copy of `self`.
"""
return [self.copy()]