import abc
import copy
import numpy as np
from menpo.base import DX, DP
from .base import Homogeneous, HomogFamilyAlignment
[docs]class Affine(Homogeneous, DP, DX):
r"""Base class for all n-dimensional affine transformations. Provides
methods to break the transform down into it's 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)
@classmethod
def identity(cls, n_dims):
return Affine(np.eye(n_dims + 1))
@property
def h_matrix(self):
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
[docs] 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
[docs] 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::
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 : string
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
[docs] def n_parameters(self):
r"""
`n_dims * (n_dims + 1)` parameters - every element of the matrix bar
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')
[docs] 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):
return Affine
def _build_pseudoinverse(self):
# Skip the checks as we know inverse of a homogeneous is a homogeneous
return Affine(np.linalg.inv(self.h_matrix), copy=False,
skip_checks=True)
[docs] def d_dp(self, points):
r"""The first order derivative of this Affine transform wrt parameter
changes evaluated at points.
The Jacobian generated (for 2D) is of the form::
x 0 y 0 1 0
0 x 0 y 0 1
This maintains a parameter order of::
W(x;p) = [1 + p1 p3 p5] [x]
[p2 1 + p4 p6] [y]
[1]
Parameters
----------
points : (n_points, n_dims) ndarray
The set of points to calculate the jacobian for.
Returns
-------
(n_points, n_params, n_dims) ndarray
The jacobian wrt parametrization
"""
n_points, points_n_dim = points.shape
if points_n_dim != self.n_dims:
raise ValueError(
"Trying to sample jacobian in incorrect dimensions "
"(transform is {0}D, sampling at {1}D)".format(
self.n_dims, points_n_dim))
# prealloc the jacobian
jac = np.zeros((n_points, self.n_parameters, self.n_dims))
# a mask that we can apply at each iteration
dim_mask = np.eye(self.n_dims, dtype=np.bool)
for i, s in enumerate(
range(0, self.n_dims * self.n_dims, self.n_dims)):
# i is current axis
# s is slicing offset
# make a mask for a single points jacobian
full_mask = np.zeros((self.n_parameters, self.n_dims), dtype=bool)
# fill the mask in for the ith axis
full_mask[slice(s, s + self.n_dims)] = dim_mask
# assign the ith axis points to this mask, broadcasting over all
# points
jac[:, full_mask] = points[:, i][..., None]
# finally, just repeat the same but for the ones at the end
full_mask = np.zeros((self.n_parameters, self.n_dims), dtype=bool)
full_mask[slice(s + self.n_dims, s + 2 * self.n_dims)] = dim_mask
jac[:, full_mask] = 1
return jac
[docs] def d_dx(self, points):
r"""
The first order derivative of this Affine transform wrt spatial changes
evaluated at points.
The Jacobian for a given point (for 2D) is of the form::
Jx = [(1 + a), -b ]
Jy = [ b, (1 + a)]
J = [Jx, Jy] = [[(1 + a), -b], [b, (1 + a)]]
where a and b come from:
W(x;p) = [1 + a -b tx] [x]
[b 1 + a ty] [y]
[1]
Hence it is simply the linear component of the transform.
Parameters
----------
points: ndarray shape (n_points, n_dims)
The spatial points at which the derivative should be evaluated.
Returns
-------
d_dx: (1, n_dims, n_dims) ndarray
The jacobian wrt spatial changes.
d_dx[0, j, k] is the scalar differential change that the
j'th dimension of the i'th point experiences due to a first order
change in the k'th dimension.
Note that because the jacobian is constant across space the first
axis is length 1 to allow for broadcasting.
"""
return self.linear_component[None, ...]
[docs]class AlignmentAffine(HomogFamilyAlignment, Affine):
r"""
Constructs an 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
M a = b
where `a` and `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.
"""
a = source.h_points
b = target.h_points
return np.linalg.solve(np.dot(a, a.T), np.dot(a, b.T)).T
[docs] 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``.
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 verification for performance.
"""
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 affine without it's 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 able to invertable.
Make sure you inherit from :class:`DiscreteAffine` first,
for optimal `decompose()` behavior.
"""
__metaclass__ = abc.ABCMeta
[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()]