import math
import textwrap
from abc import ABC, abstractmethod
import numpy as np
from scipy import spatial
from .._shared.utils import safe_as_int
from .._shared.compat import NP_COPY_IF_NEEDED
def _affine_matrix_from_vector(v):
"""Affine matrix from linearized (d, d + 1) matrix entries."""
nparam = v.size
# solve for d in: d * (d + 1) = nparam
d = (1 + np.sqrt(1 + 4 * nparam)) / 2 - 1
dimensionality = int(np.round(d)) # round to prevent approx errors
if d != dimensionality:
raise ValueError(
'Invalid number of elements for ' f'linearized matrix: {nparam}'
)
matrix = np.eye(dimensionality + 1)
matrix[:-1, :] = np.reshape(v, (dimensionality, dimensionality + 1))
return matrix
def _center_and_normalize_points(points):
"""Center and normalize image points.
The points are transformed in a two-step procedure that is expressed
as a transformation matrix. The matrix of the resulting points is usually
better conditioned than the matrix of the original points.
Center the image points, such that the new coordinate system has its
origin at the centroid of the image points.
Normalize the image points, such that the mean distance from the points
to the origin of the coordinate system is sqrt(D).
If the points are all identical, the returned values will contain nan.
Parameters
----------
points : (N, D) array
The coordinates of the image points.
Returns
-------
matrix : (D+1, D+1) array_like
The transformation matrix to obtain the new points.
new_points : (N, D) array
The transformed image points.
References
----------
.. [1] Hartley, Richard I. "In defense of the eight-point algorithm."
Pattern Analysis and Machine Intelligence, IEEE Transactions on 19.6
(1997): 580-593.
"""
n, d = points.shape
centroid = np.mean(points, axis=0)
centered = points - centroid
rms = np.sqrt(np.sum(centered**2) / n)
# if all the points are the same, the transformation matrix cannot be
# created. We return an equivalent matrix with np.nans as sentinel values.
# This obviates the need for try/except blocks in functions calling this
# one, and those are only needed when actual 0 is reached, rather than some
# small value; ie, we don't need to worry about numerical stability here,
# only actual 0.
if rms == 0:
return np.full((d + 1, d + 1), np.nan), np.full_like(points, np.nan)
norm_factor = np.sqrt(d) / rms
part_matrix = norm_factor * np.concatenate(
(np.eye(d), -centroid[:, np.newaxis]), axis=1
)
matrix = np.concatenate(
(
part_matrix,
[
[
0,
]
* d
+ [1]
],
),
axis=0,
)
points_h = np.vstack([points.T, np.ones(n)])
new_points_h = (matrix @ points_h).T
new_points = new_points_h[:, :d]
new_points /= new_points_h[:, d:]
return matrix, new_points
def _umeyama(src, dst, estimate_scale):
"""Estimate N-D similarity transformation with or without scaling.
Parameters
----------
src : (M, N) array_like
Source coordinates.
dst : (M, N) array_like
Destination coordinates.
estimate_scale : bool
Whether to estimate scaling factor.
Returns
-------
T : (N + 1, N + 1)
The homogeneous similarity transformation matrix. The matrix contains
NaN values only if the problem is not well-conditioned.
References
----------
.. [1] "Least-squares estimation of transformation parameters between two
point patterns", Shinji Umeyama, PAMI 1991, :DOI:`10.1109/34.88573`
"""
src = np.asarray(src)
dst = np.asarray(dst)
num = src.shape[0]
dim = src.shape[1]
# Compute mean of src and dst.
src_mean = src.mean(axis=0)
dst_mean = dst.mean(axis=0)
# Subtract mean from src and dst.
src_demean = src - src_mean
dst_demean = dst - dst_mean
# Eq. (38).
A = dst_demean.T @ src_demean / num
# Eq. (39).
d = np.ones((dim,), dtype=np.float64)
if np.linalg.det(A) < 0:
d[dim - 1] = -1
T = np.eye(dim + 1, dtype=np.float64)
U, S, V = np.linalg.svd(A)
# Eq. (40) and (43).
rank = np.linalg.matrix_rank(A)
if rank == 0:
return np.nan * T
elif rank == dim - 1:
if np.linalg.det(U) * np.linalg.det(V) > 0:
T[:dim, :dim] = U @ V
else:
s = d[dim - 1]
d[dim - 1] = -1
T[:dim, :dim] = U @ np.diag(d) @ V
d[dim - 1] = s
else:
T[:dim, :dim] = U @ np.diag(d) @ V
if estimate_scale:
# Eq. (41) and (42).
scale = 1.0 / src_demean.var(axis=0).sum() * (S @ d)
else:
scale = 1.0
T[:dim, dim] = dst_mean - scale * (T[:dim, :dim] @ src_mean.T)
T[:dim, :dim] *= scale
return T
class _GeometricTransform(ABC):
"""Abstract base class for geometric transformations."""
@abstractmethod
def __call__(self, coords):
"""Apply forward transformation.
Parameters
----------
coords : (N, 2) array_like
Source coordinates.
Returns
-------
coords : (N, 2) array
Destination coordinates.
"""
@property
@abstractmethod
def inverse(self):
"""Return a transform object representing the inverse."""
def residuals(self, src, dst):
"""Determine residuals of transformed destination coordinates.
For each transformed source coordinate the Euclidean distance to the
respective destination coordinate is determined.
Parameters
----------
src : (N, 2) array
Source coordinates.
dst : (N, 2) array
Destination coordinates.
Returns
-------
residuals : (N,) array
Residual for coordinate.
"""
return np.sqrt(np.sum((self(src) - dst) ** 2, axis=1))
class FundamentalMatrixTransform(_GeometricTransform):
"""Fundamental matrix transformation.
The fundamental matrix relates corresponding points between a pair of
uncalibrated images. The matrix transforms homogeneous image points in one
image to epipolar lines in the other image.
The fundamental matrix is only defined for a pair of moving images. In the
case of pure rotation or planar scenes, the homography describes the
geometric relation between two images (`ProjectiveTransform`). If the
intrinsic calibration of the images is known, the essential matrix describes
the metric relation between the two images (`EssentialMatrixTransform`).
References
----------
.. [1] Hartley, Richard, and Andrew Zisserman. Multiple view geometry in
computer vision. Cambridge university press, 2003.
Parameters
----------
matrix : (3, 3) array_like, optional
Fundamental matrix.
Attributes
----------
params : (3, 3) array
Fundamental matrix.
Examples
--------
>>> import numpy as np
>>> import skimage as ski
>>> tform_matrix = ski.transform.FundamentalMatrixTransform()
Define source and destination points:
>>> src = np.array([1.839035, 1.924743,
... 0.543582, 0.375221,
... 0.473240, 0.142522,
... 0.964910, 0.598376,
... 0.102388, 0.140092,
... 15.994343, 9.622164,
... 0.285901, 0.430055,
... 0.091150, 0.254594]).reshape(-1, 2)
>>> dst = np.array([1.002114, 1.129644,
... 1.521742, 1.846002,
... 1.084332, 0.275134,
... 0.293328, 0.588992,
... 0.839509, 0.087290,
... 1.779735, 1.116857,
... 0.878616, 0.602447,
... 0.642616, 1.028681]).reshape(-1, 2)
Estimate the transformation matrix:
>>> tform_matrix.estimate(src, dst)
True
>>> tform_matrix.params
array([[-0.21785884, 0.41928191, -0.03430748],
[-0.07179414, 0.04516432, 0.02160726],
[ 0.24806211, -0.42947814, 0.02210191]])
Compute the Sampson distance:
>>> tform_matrix.residuals(src, dst)
array([0.0053886 , 0.00526101, 0.08689701, 0.01850534, 0.09418259,
0.00185967, 0.06160489, 0.02655136])
Apply inverse transformation:
>>> tform_matrix.inverse(dst)
array([[-0.0513591 , 0.04170974, 0.01213043],
[-0.21599496, 0.29193419, 0.00978184],
[-0.0079222 , 0.03758889, -0.00915389],
[ 0.14187184, -0.27988959, 0.02476507],
[ 0.05890075, -0.07354481, -0.00481342],
[-0.21985267, 0.36717464, -0.01482408],
[ 0.01339569, -0.03388123, 0.00497605],
[ 0.03420927, -0.1135812 , 0.02228236]])
"""
def __init__(self, matrix=None, *, dimensionality=2):
if matrix is None:
# default to an identity transform
matrix = np.eye(dimensionality + 1)
else:
matrix = np.asarray(matrix)
dimensionality = matrix.shape[0] - 1
if matrix.shape != (dimensionality + 1, dimensionality + 1):
raise ValueError("Invalid shape of transformation matrix")
self.params = matrix
if dimensionality != 2:
raise NotImplementedError(
f'{self.__class__} is only implemented for 2D coordinates '
'(i.e. 3D transformation matrices).'
)
def __call__(self, coords):
"""Apply forward transformation.
Parameters
----------
coords : (N, 2) array_like
Source coordinates.
Returns
-------
coords : (N, 3) array
Epipolar lines in the destination image.
"""
coords = np.asarray(coords)
coords_homogeneous = np.column_stack([coords, np.ones(coords.shape[0])])
return coords_homogeneous @ self.params.T
@property
def inverse(self):
"""Return a transform object representing the inverse.
See Hartley & Zisserman, Ch. 8: Epipolar Geometry and the Fundamental
Matrix, for an explanation of why F.T gives the inverse.
"""
return type(self)(matrix=self.params.T)
def _setup_constraint_matrix(self, src, dst):
"""Setup and solve the homogeneous epipolar constraint matrix::
dst' * F * src = 0.
Parameters
----------
src : (N, 2) array_like
Source coordinates.
dst : (N, 2) array_like
Destination coordinates.
Returns
-------
F_normalized : (3, 3) array
The normalized solution to the homogeneous system. If the system
is not well-conditioned, this matrix contains NaNs.
src_matrix : (3, 3) array
The transformation matrix to obtain the normalized source
coordinates.
dst_matrix : (3, 3) array
The transformation matrix to obtain the normalized destination
coordinates.
"""
src = np.asarray(src)
dst = np.asarray(dst)
if src.shape != dst.shape:
raise ValueError('src and dst shapes must be identical.')
if src.shape[0] < 8:
raise ValueError('src.shape[0] must be equal or larger than 8.')
# Center and normalize image points for better numerical stability.
try:
src_matrix, src = _center_and_normalize_points(src)
dst_matrix, dst = _center_and_normalize_points(dst)
except ZeroDivisionError:
self.params = np.full((3, 3), np.nan)
return 3 * [np.full((3, 3), np.nan)]
# Setup homogeneous linear equation as dst' * F * src = 0.
A = np.ones((src.shape[0], 9))
A[:, :2] = src
A[:, :3] *= dst[:, 0, np.newaxis]
A[:, 3:5] = src
A[:, 3:6] *= dst[:, 1, np.newaxis]
A[:, 6:8] = src
# Solve for the nullspace of the constraint matrix.
_, _, V = np.linalg.svd(A)
F_normalized = V[-1, :].reshape(3, 3)
return F_normalized, src_matrix, dst_matrix
def estimate(self, src, dst):
"""Estimate fundamental matrix using 8-point algorithm.
The 8-point algorithm requires at least 8 corresponding point pairs for
a well-conditioned solution, otherwise the over-determined solution is
estimated.
Parameters
----------
src : (N, 2) array_like
Source coordinates.
dst : (N, 2) array_like
Destination coordinates.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
F_normalized, src_matrix, dst_matrix = self._setup_constraint_matrix(src, dst)
# Enforcing the internal constraint that two singular values must be
# non-zero and one must be zero.
U, S, V = np.linalg.svd(F_normalized)
S[2] = 0
F = U @ np.diag(S) @ V
self.params = dst_matrix.T @ F @ src_matrix
return True
def residuals(self, src, dst):
"""Compute the Sampson distance.
The Sampson distance is the first approximation to the geometric error.
Parameters
----------
src : (N, 2) array
Source coordinates.
dst : (N, 2) array
Destination coordinates.
Returns
-------
residuals : (N,) array
Sampson distance.
"""
src_homogeneous = np.column_stack([src, np.ones(src.shape[0])])
dst_homogeneous = np.column_stack([dst, np.ones(dst.shape[0])])
F_src = self.params @ src_homogeneous.T
Ft_dst = self.params.T @ dst_homogeneous.T
dst_F_src = np.sum(dst_homogeneous * F_src.T, axis=1)
return np.abs(dst_F_src) / np.sqrt(
F_src[0] ** 2 + F_src[1] ** 2 + Ft_dst[0] ** 2 + Ft_dst[1] ** 2
)
class EssentialMatrixTransform(FundamentalMatrixTransform):
"""Essential matrix transformation.
The essential matrix relates corresponding points between a pair of
calibrated images. The matrix transforms normalized, homogeneous image
points in one image to epipolar lines in the other image.
The essential matrix is only defined for a pair of moving images capturing a
non-planar scene. In the case of pure rotation or planar scenes, the
homography describes the geometric relation between two images
(`ProjectiveTransform`). If the intrinsic calibration of the images is
unknown, the fundamental matrix describes the projective relation between
the two images (`FundamentalMatrixTransform`).
References
----------
.. [1] Hartley, Richard, and Andrew Zisserman. Multiple view geometry in
computer vision. Cambridge university press, 2003.
Parameters
----------
rotation : (3, 3) array_like, optional
Rotation matrix of the relative camera motion.
translation : (3, 1) array_like, optional
Translation vector of the relative camera motion. The vector must
have unit length.
matrix : (3, 3) array_like, optional
Essential matrix.
Attributes
----------
params : (3, 3) array
Essential matrix.
Examples
--------
>>> import numpy as np
>>> import skimage as ski
>>>
>>> tform_matrix = ski.transform.EssentialMatrixTransform(
... rotation=np.eye(3), translation=np.array([0, 0, 1])
... )
>>> tform_matrix.params
array([[ 0., -1., 0.],
[ 1., 0., 0.],
[ 0., 0., 0.]])
>>> src = np.array([[ 1.839035, 1.924743],
... [ 0.543582, 0.375221],
... [ 0.47324 , 0.142522],
... [ 0.96491 , 0.598376],
... [ 0.102388, 0.140092],
... [15.994343, 9.622164],
... [ 0.285901, 0.430055],
... [ 0.09115 , 0.254594]])
>>> dst = np.array([[1.002114, 1.129644],
... [1.521742, 1.846002],
... [1.084332, 0.275134],
... [0.293328, 0.588992],
... [0.839509, 0.08729 ],
... [1.779735, 1.116857],
... [0.878616, 0.602447],
... [0.642616, 1.028681]])
>>> tform_matrix.estimate(src, dst)
True
>>> tform_matrix.residuals(src, dst)
array([0.42455187, 0.01460448, 0.13847034, 0.12140951, 0.27759346,
0.32453118, 0.00210776, 0.26512283])
"""
def __init__(
self, rotation=None, translation=None, matrix=None, *, dimensionality=2
):
super().__init__(matrix=matrix, dimensionality=dimensionality)
if rotation is not None:
rotation = np.asarray(rotation)
if translation is None:
raise ValueError("Both rotation and translation required")
translation = np.asarray(translation)
if rotation.shape != (3, 3):
raise ValueError("Invalid shape of rotation matrix")
if abs(np.linalg.det(rotation) - 1) > 1e-6:
raise ValueError("Rotation matrix must have unit determinant")
if translation.size != 3:
raise ValueError("Invalid shape of translation vector")
if abs(np.linalg.norm(translation) - 1) > 1e-6:
raise ValueError("Translation vector must have unit length")
# Matrix representation of the cross product for t.
t_x = np.array(
[
0,
-translation[2],
translation[1],
translation[2],
0,
-translation[0],
-translation[1],
translation[0],
0,
]
).reshape(3, 3)
self.params = t_x @ rotation
elif matrix is not None:
matrix = np.asarray(matrix)
if matrix.shape != (3, 3):
raise ValueError("Invalid shape of transformation matrix")
self.params = matrix
else:
# default to an identity transform
self.params = np.eye(3)
def estimate(self, src, dst):
"""Estimate essential matrix using 8-point algorithm.
The 8-point algorithm requires at least 8 corresponding point pairs for
a well-conditioned solution, otherwise the over-determined solution is
estimated.
Parameters
----------
src : (N, 2) array_like
Source coordinates.
dst : (N, 2) array_like
Destination coordinates.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
E_normalized, src_matrix, dst_matrix = self._setup_constraint_matrix(src, dst)
# Enforcing the internal constraint that two singular values must be
# equal and one must be zero.
U, S, V = np.linalg.svd(E_normalized)
S[0] = (S[0] + S[1]) / 2.0
S[1] = S[0]
S[2] = 0
E = U @ np.diag(S) @ V
self.params = dst_matrix.T @ E @ src_matrix
return True
class ProjectiveTransform(_GeometricTransform):
r"""Projective transformation.
Apply a projective transformation (homography) on coordinates.
For each homogeneous coordinate :math:`\mathbf{x} = [x, y, 1]^T`, its
target position is calculated by multiplying with the given matrix,
:math:`H`, to give :math:`H \mathbf{x}`::
[[a0 a1 a2]
[b0 b1 b2]
[c0 c1 1 ]].
E.g., to rotate by theta degrees clockwise, the matrix should be::
[[cos(theta) -sin(theta) 0]
[sin(theta) cos(theta) 0]
[0 0 1]]
or, to translate x by 10 and y by 20::
[[1 0 10]
[0 1 20]
[0 0 1 ]].
Parameters
----------
matrix : (D+1, D+1) array_like, optional
Homogeneous transformation matrix.
dimensionality : int, optional
The number of dimensions of the transform. This is ignored if
``matrix`` is not None.
Attributes
----------
params : (D+1, D+1) array
Homogeneous transformation matrix.
"""
def __init__(self, matrix=None, *, dimensionality=2):
if matrix is None:
# default to an identity transform
matrix = np.eye(dimensionality + 1)
else:
matrix = np.asarray(matrix)
dimensionality = matrix.shape[0] - 1
if matrix.shape != (dimensionality + 1, dimensionality + 1):
raise ValueError("invalid shape of transformation matrix")
self.params = matrix
self._coeffs = range(matrix.size - 1)
@property
def _inv_matrix(self):
return np.linalg.inv(self.params)
def _apply_mat(self, coords, matrix):
ndim = matrix.shape[0] - 1
coords = np.array(coords, copy=NP_COPY_IF_NEEDED, ndmin=2)
src = np.concatenate([coords, np.ones((coords.shape[0], 1))], axis=1)
dst = src @ matrix.T
# below, we will divide by the last dimension of the homogeneous
# coordinate matrix. In order to avoid division by zero,
# we replace exact zeros in this column with a very small number.
dst[dst[:, ndim] == 0, ndim] = np.finfo(float).eps
# rescale to homogeneous coordinates
dst[:, :ndim] /= dst[:, ndim : ndim + 1]
return dst[:, :ndim]
def __array__(self, dtype=None):
if dtype is None:
return self.params
else:
return self.params.astype(dtype)
def __call__(self, coords):
"""Apply forward transformation.
Parameters
----------
coords : (N, D) array_like
Source coordinates.
Returns
-------
coords_out : (N, D) array
Destination coordinates.
"""
return self._apply_mat(coords, self.params)
@property
def inverse(self):
"""Return a transform object representing the inverse."""
return type(self)(matrix=self._inv_matrix)
def estimate(self, src, dst, weights=None):
"""Estimate the transformation from a set of corresponding points.
You can determine the over-, well- and under-determined parameters
with the total least-squares method.
Number of source and destination coordinates must match.
The transformation is defined as::
X = (a0*x + a1*y + a2) / (c0*x + c1*y + 1)
Y = (b0*x + b1*y + b2) / (c0*x + c1*y + 1)
These equations can be transformed to the following form::
0 = a0*x + a1*y + a2 - c0*x*X - c1*y*X - X
0 = b0*x + b1*y + b2 - c0*x*Y - c1*y*Y - Y
which exist for each set of corresponding points, so we have a set of
N * 2 equations. The coefficients appear linearly so we can write
A x = 0, where::
A = [[x y 1 0 0 0 -x*X -y*X -X]
[0 0 0 x y 1 -x*Y -y*Y -Y]
...
...
]
x.T = [a0 a1 a2 b0 b1 b2 c0 c1 c3]
In case of total least-squares the solution of this homogeneous system
of equations is the right singular vector of A which corresponds to the
smallest singular value normed by the coefficient c3.
Weights can be applied to each pair of corresponding points to
indicate, particularly in an overdetermined system, if point pairs have
higher or lower confidence or uncertainties associated with them. From
the matrix treatment of least squares problems, these weight values are
normalised, square-rooted, then built into a diagonal matrix, by which
A is multiplied.
In case of the affine transformation the coefficients c0 and c1 are 0.
Thus the system of equations is::
A = [[x y 1 0 0 0 -X]
[0 0 0 x y 1 -Y]
...
...
]
x.T = [a0 a1 a2 b0 b1 b2 c3]
Parameters
----------
src : (N, 2) array_like
Source coordinates.
dst : (N, 2) array_like
Destination coordinates.
weights : (N,) array_like, optional
Relative weight values for each pair of points.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
src = np.asarray(src)
dst = np.asarray(dst)
n, d = src.shape
src_matrix, src = _center_and_normalize_points(src)
dst_matrix, dst = _center_and_normalize_points(dst)
if not np.all(np.isfinite(src_matrix + dst_matrix)):
self.params = np.full((d + 1, d + 1), np.nan)
return False
# params: a0, a1, a2, b0, b1, b2, c0, c1
A = np.zeros((n * d, (d + 1) ** 2))
# fill the A matrix with the appropriate block matrices; see docstring
# for 2D example — this can be generalised to more blocks in the 3D and
# higher-dimensional cases.
for ddim in range(d):
A[ddim * n : (ddim + 1) * n, ddim * (d + 1) : ddim * (d + 1) + d] = src
A[ddim * n : (ddim + 1) * n, ddim * (d + 1) + d] = 1
A[ddim * n : (ddim + 1) * n, -d - 1 : -1] = src
A[ddim * n : (ddim + 1) * n, -1] = -1
A[ddim * n : (ddim + 1) * n, -d - 1 :] *= -dst[:, ddim : (ddim + 1)]
# Select relevant columns, depending on params
A = A[:, list(self._coeffs) + [-1]]
# Get the vectors that correspond to singular values, also applying
# the weighting if provided
if weights is None:
_, _, V = np.linalg.svd(A)
else:
weights = np.asarray(weights)
W = np.diag(np.tile(np.sqrt(weights / np.max(weights)), d))
_, _, V = np.linalg.svd(W @ A)
# if the last element of the vector corresponding to the smallest
# singular value is close to zero, this implies a degenerate case
# because it is a rank-defective transform, which would map points
# to a line rather than a plane.
if np.isclose(V[-1, -1], 0):
self.params = np.full((d + 1, d + 1), np.nan)
return False
H = np.zeros((d + 1, d + 1))
# solution is right singular vector that corresponds to smallest
# singular value
H.flat[list(self._coeffs) + [-1]] = -V[-1, :-1] / V[-1, -1]
H[d, d] = 1
# De-center and de-normalize
H = np.linalg.inv(dst_matrix) @ H @ src_matrix
# Small errors can creep in if points are not exact, causing the last
# element of H to deviate from unity. Correct for that here.
H /= H[-1, -1]
self.params = H
return True
def __add__(self, other):
"""Combine this transformation with another."""
if isinstance(other, ProjectiveTransform):
# combination of the same types result in a transformation of this
# type again, otherwise use general projective transformation
if type(self) == type(other):
tform = self.__class__
else:
tform = ProjectiveTransform
return tform(other.params @ self.params)
else:
raise TypeError("Cannot combine transformations of differing " "types.")
def __nice__(self):
"""common 'paramstr' used by __str__ and __repr__"""
npstring = np.array2string(self.params, separator=', ')
paramstr = 'matrix=\n' + textwrap.indent(npstring, ' ')
return paramstr
def __repr__(self):
"""Add standard repr formatting around a __nice__ string"""
paramstr = self.__nice__()
classname = self.__class__.__name__
classstr = classname
return f'<{classstr}({paramstr}) at {hex(id(self))}>'
def __str__(self):
"""Add standard str formatting around a __nice__ string"""
paramstr = self.__nice__()
classname = self.__class__.__name__
classstr = classname
return f'<{classstr}({paramstr})>'
@property
def dimensionality(self):
"""The dimensionality of the transformation."""
return self.params.shape[0] - 1
class AffineTransform(ProjectiveTransform):
"""Affine transformation.
Has the following form::
X = a0 * x + a1 * y + a2
= sx * x * [cos(rotation) + tan(shear_y) * sin(rotation)]
- sy * y * [tan(shear_x) * cos(rotation) + sin(rotation)]
+ translation_x
Y = b0 * x + b1 * y + b2
= sx * x * [sin(rotation) - tan(shear_y) * cos(rotation)]
- sy * y * [tan(shear_x) * sin(rotation) - cos(rotation)]
+ translation_y
where ``sx`` and ``sy`` are scale factors in the x and y directions.
This is equivalent to applying the operations in the following order:
1. Scale
2. Shear
3. Rotate
4. Translate
The homogeneous transformation matrix is::
[[a0 a1 a2]
[b0 b1 b2]
[0 0 1]]
In 2D, the transformation parameters can be given as the homogeneous
transformation matrix, above, or as the implicit parameters, scale,
rotation, shear, and translation in x (a2) and y (b2). For 3D and higher,
only the matrix form is allowed.
In narrower transforms, such as the Euclidean (only rotation and
translation) or Similarity (rotation, translation, and a global scale
factor) transforms, it is possible to specify 3D transforms using implicit
parameters also.
Parameters
----------
matrix : (D+1, D+1) array_like, optional
Homogeneous transformation matrix. If this matrix is provided, it is an
error to provide any of scale, rotation, shear, or translation.
scale : {s as float or (sx, sy) as array, list or tuple}, optional
Scale factor(s). If a single value, it will be assigned to both
sx and sy. Only available for 2D.
.. versionadded:: 0.17
Added support for supplying a single scalar value.
rotation : float, optional
Rotation angle, clockwise, as radians. Only available for 2D.
shear : float or 2-tuple of float, optional
The x and y shear angles, clockwise, by which these axes are
rotated around the origin [2].
If a single value is given, take that to be the x shear angle, with
the y angle remaining 0. Only available in 2D.
translation : (tx, ty) as array, list or tuple, optional
Translation parameters. Only available for 2D.
dimensionality : int, optional
The dimensionality of the transform. This is not used if any other
parameters are provided.
Attributes
----------
params : (D+1, D+1) array
Homogeneous transformation matrix.
Raises
------
ValueError
If both ``matrix`` and any of the other parameters are provided.
Examples
--------
>>> import numpy as np
>>> import skimage as ski
>>> img = ski.data.astronaut()
Define source and destination points:
>>> src = np.array([[150, 150],
... [250, 100],
... [150, 200]])
>>> dst = np.array([[200, 200],
... [300, 150],
... [150, 400]])
Estimate the transformation matrix:
>>> tform = ski.transform.AffineTransform()
>>> tform.estimate(src, dst)
True
Apply the transformation:
>>> warped = ski.transform.warp(img, inverse_map=tform.inverse)
References
----------
.. [1] Wikipedia, "Affine transformation",
https://en.wikipedia.org/wiki/Affine_transformation#Image_transformation
.. [2] Wikipedia, "Shear mapping",
https://en.wikipedia.org/wiki/Shear_mapping
"""
def __init__(
self,
matrix=None,
scale=None,
rotation=None,
shear=None,
translation=None,
*,
dimensionality=2,
):
params = any(
param is not None for param in (scale, rotation, shear, translation)
)
# these parameters get overwritten if a higher-D matrix is given
self._coeffs = range(dimensionality * (dimensionality + 1))
if params and matrix is not None:
raise ValueError(
"You cannot specify the transformation matrix and"
" the implicit parameters at the same time."
)
if params and dimensionality > 2:
raise ValueError('Parameter input is only supported in 2D.')
elif matrix is not None:
matrix = np.asarray(matrix)
if matrix.ndim != 2 or matrix.shape[0] != matrix.shape[1]:
raise ValueError("Invalid shape of transformation matrix.")
else:
dimensionality = matrix.shape[0] - 1
nparam = dimensionality * (dimensionality + 1)
self._coeffs = range(nparam)
self.params = matrix
elif params: # note: 2D only
if scale is None:
scale = (1, 1)
if rotation is None:
rotation = 0
if shear is None:
shear = 0
if translation is None:
translation = (0, 0)
if np.isscalar(scale):
sx = sy = scale
else:
sx, sy = scale
if np.isscalar(shear):
shear_x, shear_y = (shear, 0)
else:
shear_x, shear_y = shear
a0 = sx * (math.cos(rotation) + math.tan(shear_y) * math.sin(rotation))
a1 = -sy * (math.tan(shear_x) * math.cos(rotation) + math.sin(rotation))
a2 = translation[0]
b0 = sx * (math.sin(rotation) - math.tan(shear_y) * math.cos(rotation))
b1 = -sy * (math.tan(shear_x) * math.sin(rotation) - math.cos(rotation))
b2 = translation[1]
self.params = np.array([[a0, a1, a2], [b0, b1, b2], [0, 0, 1]])
else:
# default to an identity transform
self.params = np.eye(dimensionality + 1)
@property
def scale(self):
if self.dimensionality != 2:
return np.sqrt(np.sum(self.params**2, axis=0))[: self.dimensionality]
else:
ss = np.sum(self.params**2, axis=0)
ss[1] = ss[1] / (math.tan(self.shear) ** 2 + 1)
return np.sqrt(ss)[: self.dimensionality]
@property
def rotation(self):
if self.dimensionality != 2:
raise NotImplementedError(
'The rotation property is only implemented for 2D transforms.'
)
return math.atan2(self.params[1, 0], self.params[0, 0])
@property
def shear(self):
if self.dimensionality != 2:
raise NotImplementedError(
'The shear property is only implemented for 2D transforms.'
)
beta = math.atan2(-self.params[0, 1], self.params[1, 1])
return beta - self.rotation
@property
def translation(self):
return self.params[0 : self.dimensionality, self.dimensionality]
class PiecewiseAffineTransform(_GeometricTransform):
"""Piecewise affine transformation.
Control points are used to define the mapping. The transform is based on
a Delaunay triangulation of the points to form a mesh. Each triangle is
used to find a local affine transform.
Attributes
----------
affines : list of AffineTransform objects
Affine transformations for each triangle in the mesh.
inverse_affines : list of AffineTransform objects
Inverse affine transformations for each triangle in the mesh.
"""
def __init__(self):
self._tesselation = None
self._inverse_tesselation = None
self.affines = None
self.inverse_affines = None
def estimate(self, src, dst):
"""Estimate the transformation from a set of corresponding points.
Number of source and destination coordinates must match.
Parameters
----------
src : (N, D) array_like
Source coordinates.
dst : (N, D) array_like
Destination coordinates.
Returns
-------
success : bool
True, if all pieces of the model are successfully estimated.
"""
src = np.asarray(src)
dst = np.asarray(dst)
ndim = src.shape[1]
# forward piecewise affine
# triangulate input positions into mesh
self._tesselation = spatial.Delaunay(src)
success = True
# find affine mapping from source positions to destination
self.affines = []
for tri in self._tesselation.simplices:
affine = AffineTransform(dimensionality=ndim)
success &= affine.estimate(src[tri, :], dst[tri, :])
self.affines.append(affine)
# inverse piecewise affine
# triangulate input positions into mesh
self._inverse_tesselation = spatial.Delaunay(dst)
# find affine mapping from source positions to destination
self.inverse_affines = []
for tri in self._inverse_tesselation.simplices:
affine = AffineTransform(dimensionality=ndim)
success &= affine.estimate(dst[tri, :], src[tri, :])
self.inverse_affines.append(affine)
return success
def __call__(self, coords):
"""Apply forward transformation.
Coordinates outside of the mesh will be set to `- 1`.
Parameters
----------
coords : (N, D) array_like
Source coordinates.
Returns
-------
coords : (N, 2) array
Transformed coordinates.
"""
coords = np.asarray(coords)
out = np.empty_like(coords, np.float64)
# determine triangle index for each coordinate
simplex = self._tesselation.find_simplex(coords)
# coordinates outside of mesh
out[simplex == -1, :] = -1
for index in range(len(self._tesselation.simplices)):
# affine transform for triangle
affine = self.affines[index]
# all coordinates within triangle
index_mask = simplex == index
out[index_mask, :] = affine(coords[index_mask, :])
return out
@property
def inverse(self):
"""Return a transform object representing the inverse."""
tform = type(self)()
tform._tesselation = self._inverse_tesselation
tform._inverse_tesselation = self._tesselation
tform.affines = self.inverse_affines
tform.inverse_affines = self.affines
return tform
def _euler_rotation(axis, angle):
"""Produce a single-axis Euler rotation matrix.
Parameters
----------
axis : int in {0, 1, 2}
The axis of rotation.
angle : float
The angle of rotation in radians.
Returns
-------
Ri : array of float, shape (3, 3)
The rotation matrix along axis `axis`.
"""
i = axis
s = (-1) ** i * np.sin(angle)
c = np.cos(angle)
R2 = np.array([[c, -s], [s, c]])
Ri = np.eye(3)
# We need the axes other than the rotation axis, in the right order:
# 0 -> (1, 2); 1 -> (0, 2); 2 -> (0, 1).
axes = sorted({0, 1, 2} - {axis})
# We then embed the 2-axis rotation matrix into the full matrix.
# (1, 2) -> R[1:3:1, 1:3:1] = R2, (0, 2) -> R[0:3:2, 0:3:2] = R2, etc.
sl = slice(axes[0], axes[1] + 1, axes[1] - axes[0])
Ri[sl, sl] = R2
return Ri
def _euler_rotation_matrix(angles, axes=None):
"""Produce an Euler rotation matrix from the given angles.
The matrix will have dimension equal to the number of angles given.
Parameters
----------
angles : array of float, shape (3,)
The transformation angles in radians.
axes : list of int
The axes about which to produce the rotation. Defaults to 0, 1, 2.
Returns
-------
R : array of float, shape (3, 3)
The Euler rotation matrix.
"""
if axes is None:
axes = range(3)
dim = len(angles)
R = np.eye(dim)
for i, angle in zip(axes, angles):
R = R @ _euler_rotation(i, angle)
return R
class EuclideanTransform(ProjectiveTransform):
"""Euclidean transformation, also known as a rigid transform.
Has the following form::
X = a0 * x - b0 * y + a1 =
= x * cos(rotation) - y * sin(rotation) + a1
Y = b0 * x + a0 * y + b1 =
= x * sin(rotation) + y * cos(rotation) + b1
where the homogeneous transformation matrix is::
[[a0 -b0 a1]
[b0 a0 b1]
[0 0 1 ]]
The Euclidean transformation is a rigid transformation with rotation and
translation parameters. The similarity transformation extends the Euclidean
transformation with a single scaling factor.
In 2D and 3D, the transformation parameters may be provided either via
`matrix`, the homogeneous transformation matrix, above, or via the
implicit parameters `rotation` and/or `translation` (where `a1` is the
translation along `x`, `b1` along `y`, etc.). Beyond 3D, if the
transformation is only a translation, you may use the implicit parameter
`translation`; otherwise, you must use `matrix`.
Parameters
----------
matrix : (D+1, D+1) array_like, optional
Homogeneous transformation matrix.
rotation : float or sequence of float, optional
Rotation angle, clockwise, as radians. If given as
a vector, it is interpreted as Euler rotation angles [1]_. Only 2D
(single rotation) and 3D (Euler rotations) values are supported. For
higher dimensions, you must provide or estimate the transformation
matrix.
translation : (x, y[, z, ...]) sequence of float, length D, optional
Translation parameters for each axis.
dimensionality : int, optional
The dimensionality of the transform.
Attributes
----------
params : (D+1, D+1) array
Homogeneous transformation matrix.
References
----------
.. [1] https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions
"""
def __init__(
self, matrix=None, rotation=None, translation=None, *, dimensionality=2
):
params_given = rotation is not None or translation is not None
if params_given and matrix is not None:
raise ValueError(
"You cannot specify the transformation matrix and"
" the implicit parameters at the same time."
)
elif matrix is not None:
matrix = np.asarray(matrix)
if matrix.shape[0] != matrix.shape[1]:
raise ValueError("Invalid shape of transformation matrix.")
self.params = matrix
elif params_given:
if rotation is None:
dimensionality = len(translation)
if dimensionality == 2:
rotation = 0
elif dimensionality == 3:
rotation = np.zeros(3)
else:
raise ValueError(
'Parameters cannot be specified for dimension '
f'{dimensionality} transforms'
)
else:
if not np.isscalar(rotation) and len(rotation) != 3:
raise ValueError(
'Parameters cannot be specified for dimension '
f'{dimensionality} transforms'
)
if translation is None:
translation = (0,) * dimensionality
if dimensionality == 2:
self.params = np.array(
[
[math.cos(rotation), -math.sin(rotation), 0],
[math.sin(rotation), math.cos(rotation), 0],
[0, 0, 1],
]
)
elif dimensionality == 3:
self.params = np.eye(dimensionality + 1)
self.params[:dimensionality, :dimensionality] = _euler_rotation_matrix(
rotation
)
self.params[0:dimensionality, dimensionality] = translation
else:
# default to an identity transform
self.params = np.eye(dimensionality + 1)
def estimate(self, src, dst):
"""Estimate the transformation from a set of corresponding points.
You can determine the over-, well- and under-determined parameters
with the total least-squares method.
Number of source and destination coordinates must match.
Parameters
----------
src : (N, 2) array_like
Source coordinates.
dst : (N, 2) array_like
Destination coordinates.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
self.params = _umeyama(src, dst, False)
# _umeyama will return nan if the problem is not well-conditioned.
return not np.any(np.isnan(self.params))
@property
def rotation(self):
if self.dimensionality == 2:
return math.atan2(self.params[1, 0], self.params[1, 1])
elif self.dimensionality == 3:
# Returning 3D Euler rotation matrix
return self.params[:3, :3]
else:
raise NotImplementedError(
'Rotation only implemented for 2D and 3D transforms.'
)
@property
def translation(self):
return self.params[0 : self.dimensionality, self.dimensionality]
class SimilarityTransform(EuclideanTransform):
"""Similarity transformation.
Has the following form in 2D::
X = a0 * x - b0 * y + a1 =
= s * x * cos(rotation) - s * y * sin(rotation) + a1
Y = b0 * x + a0 * y + b1 =
= s * x * sin(rotation) + s * y * cos(rotation) + b1
where ``s`` is a scale factor and the homogeneous transformation matrix is::
[[a0 -b0 a1]
[b0 a0 b1]
[0 0 1 ]]
The similarity transformation extends the Euclidean transformation with a
single scaling factor in addition to the rotation and translation
parameters.
Parameters
----------
matrix : (dim+1, dim+1) array_like, optional
Homogeneous transformation matrix.
scale : float, optional
Scale factor. Implemented only for 2D and 3D.
rotation : float, optional
Rotation angle, clockwise, as radians.
Implemented only for 2D and 3D. For 3D, this is given in ZYX Euler
angles.
translation : (dim,) array_like, optional
x, y[, z] translation parameters. Implemented only for 2D and 3D.
Attributes
----------
params : (dim+1, dim+1) array
Homogeneous transformation matrix.
"""
def __init__(
self,
matrix=None,
scale=None,
rotation=None,
translation=None,
*,
dimensionality=2,
):
self.params = None
params = any(param is not None for param in (scale, rotation, translation))
if params and matrix is not None:
raise ValueError(
"You cannot specify the transformation matrix and"
" the implicit parameters at the same time."
)
elif matrix is not None:
matrix = np.asarray(matrix)
if matrix.ndim != 2 or matrix.shape[0] != matrix.shape[1]:
raise ValueError("Invalid shape of transformation matrix.")
else:
self.params = matrix
dimensionality = matrix.shape[0] - 1
if params:
if dimensionality not in (2, 3):
raise ValueError('Parameters only supported for 2D and 3D.')
matrix = np.eye(dimensionality + 1, dtype=float)
if scale is None:
scale = 1
if rotation is None:
rotation = 0 if dimensionality == 2 else (0, 0, 0)
if translation is None:
translation = (0,) * dimensionality
if dimensionality == 2:
ax = (0, 1)
c, s = np.cos(rotation), np.sin(rotation)
matrix[ax, ax] = c
matrix[ax, ax[::-1]] = -s, s
else: # 3D rotation
matrix[:3, :3] = _euler_rotation_matrix(rotation)
matrix[:dimensionality, :dimensionality] *= scale
matrix[:dimensionality, dimensionality] = translation
self.params = matrix
elif self.params is None:
# default to an identity transform
self.params = np.eye(dimensionality + 1)
def estimate(self, src, dst):
"""Estimate the transformation from a set of corresponding points.
You can determine the over-, well- and under-determined parameters
with the total least-squares method.
Number of source and destination coordinates must match.
Parameters
----------
src : (N, 2) array_like
Source coordinates.
dst : (N, 2) array_like
Destination coordinates.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
self.params = _umeyama(src, dst, estimate_scale=True)
# _umeyama will return nan if the problem is not well-conditioned.
return not np.any(np.isnan(self.params))
@property
def scale(self):
# det = scale**(# of dimensions), therefore scale = det**(1/ndim)
if self.dimensionality == 2:
return np.sqrt(np.linalg.det(self.params))
elif self.dimensionality == 3:
return np.cbrt(np.linalg.det(self.params))
else:
raise NotImplementedError('Scale is only implemented for 2D and 3D.')
class PolynomialTransform(_GeometricTransform):
"""2D polynomial transformation.
Has the following form::
X = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i ))
Y = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i ))
Parameters
----------
params : (2, N) array_like, optional
Polynomial coefficients where `N * 2 = (order + 1) * (order + 2)`. So,
a_ji is defined in `params[0, :]` and b_ji in `params[1, :]`.
Attributes
----------
params : (2, N) array
Polynomial coefficients where `N * 2 = (order + 1) * (order + 2)`. So,
a_ji is defined in `params[0, :]` and b_ji in `params[1, :]`.
"""
def __init__(self, params=None, *, dimensionality=2):
if dimensionality != 2:
raise NotImplementedError(
'Polynomial transforms are only implemented for 2D.'
)
if params is None:
# default to transformation which preserves original coordinates
params = np.array([[0, 1, 0], [0, 0, 1]])
else:
params = np.asarray(params)
if params.shape[0] != 2:
raise ValueError("invalid shape of transformation parameters")
self.params = params
def estimate(self, src, dst, order=2, weights=None):
"""Estimate the transformation from a set of corresponding points.
You can determine the over-, well- and under-determined parameters
with the total least-squares method.
Number of source and destination coordinates must match.
The transformation is defined as::
X = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i ))
Y = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i ))
These equations can be transformed to the following form::
0 = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i )) - X
0 = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i )) - Y
which exist for each set of corresponding points, so we have a set of
N * 2 equations. The coefficients appear linearly so we can write
A x = 0, where::
A = [[1 x y x**2 x*y y**2 ... 0 ... 0 -X]
[0 ... 0 1 x y x**2 x*y y**2 -Y]
...
...
]
x.T = [a00 a10 a11 a20 a21 a22 ... ann
b00 b10 b11 b20 b21 b22 ... bnn c3]
In case of total least-squares the solution of this homogeneous system
of equations is the right singular vector of A which corresponds to the
smallest singular value normed by the coefficient c3.
Weights can be applied to each pair of corresponding points to
indicate, particularly in an overdetermined system, if point pairs have
higher or lower confidence or uncertainties associated with them. From
the matrix treatment of least squares problems, these weight values are
normalised, square-rooted, then built into a diagonal matrix, by which
A is multiplied.
Parameters
----------
src : (N, 2) array_like
Source coordinates.
dst : (N, 2) array_like
Destination coordinates.
order : int, optional
Polynomial order (number of coefficients is order + 1).
weights : (N,) array_like, optional
Relative weight values for each pair of points.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
src = np.asarray(src)
dst = np.asarray(dst)
xs = src[:, 0]
ys = src[:, 1]
xd = dst[:, 0]
yd = dst[:, 1]
rows = src.shape[0]
# number of unknown polynomial coefficients
order = safe_as_int(order)
u = (order + 1) * (order + 2)
A = np.zeros((rows * 2, u + 1))
pidx = 0
for j in range(order + 1):
for i in range(j + 1):
A[:rows, pidx] = xs ** (j - i) * ys**i
A[rows:, pidx + u // 2] = xs ** (j - i) * ys**i
pidx += 1
A[:rows, -1] = xd
A[rows:, -1] = yd
# Get the vectors that correspond to singular values, also applying
# the weighting if provided
if weights is None:
_, _, V = np.linalg.svd(A)
else:
weights = np.asarray(weights)
W = np.diag(np.tile(np.sqrt(weights / np.max(weights)), 2))
_, _, V = np.linalg.svd(W @ A)
# solution is right singular vector that corresponds to smallest
# singular value
params = -V[-1, :-1] / V[-1, -1]
self.params = params.reshape((2, u // 2))
return True
def __call__(self, coords):
"""Apply forward transformation.
Parameters
----------
coords : (N, 2) array_like
source coordinates
Returns
-------
coords : (N, 2) array
Transformed coordinates.
"""
coords = np.asarray(coords)
x = coords[:, 0]
y = coords[:, 1]
u = len(self.params.ravel())
# number of coefficients -> u = (order + 1) * (order + 2)
order = int((-3 + math.sqrt(9 - 4 * (2 - u))) / 2)
dst = np.zeros(coords.shape)
pidx = 0
for j in range(order + 1):
for i in range(j + 1):
dst[:, 0] += self.params[0, pidx] * x ** (j - i) * y**i
dst[:, 1] += self.params[1, pidx] * x ** (j - i) * y**i
pidx += 1
return dst
@property
def inverse(self):
raise NotImplementedError(
'There is no explicit way to do the inverse polynomial '
'transformation. Instead, estimate the inverse transformation '
'parameters by exchanging source and destination coordinates,'
'then apply the forward transformation.'
)
TRANSFORMS = {
'euclidean': EuclideanTransform,
'similarity': SimilarityTransform,
'affine': AffineTransform,
'piecewise-affine': PiecewiseAffineTransform,
'projective': ProjectiveTransform,
'fundamental': FundamentalMatrixTransform,
'essential': EssentialMatrixTransform,
'polynomial': PolynomialTransform,
}
def estimate_transform(ttype, src, dst, *args, **kwargs):
"""Estimate 2D geometric transformation parameters.
You can determine the over-, well- and under-determined parameters
with the total least-squares method.
Number of source and destination coordinates must match.
Parameters
----------
ttype : {'euclidean', similarity', 'affine', 'piecewise-affine', \
'projective', 'polynomial'}
Type of transform.
kwargs : array_like or int
Function parameters (src, dst, n, angle)::
NAME / TTYPE FUNCTION PARAMETERS
'euclidean' `src, `dst`
'similarity' `src, `dst`
'affine' `src, `dst`
'piecewise-affine' `src, `dst`
'projective' `src, `dst`
'polynomial' `src, `dst`, `order` (polynomial order,
default order is 2)
Also see examples below.
Returns
-------
tform : :class:`_GeometricTransform`
Transform object containing the transformation parameters and providing
access to forward and inverse transformation functions.
Examples
--------
>>> import numpy as np
>>> import skimage as ski
>>> # estimate transformation parameters
>>> src = np.array([0, 0, 10, 10]).reshape((2, 2))
>>> dst = np.array([12, 14, 1, -20]).reshape((2, 2))
>>> tform = ski.transform.estimate_transform('similarity', src, dst)
>>> np.allclose(tform.inverse(tform(src)), src)
True
>>> # warp image using the estimated transformation
>>> image = ski.data.camera()
>>> ski.transform.warp(image, inverse_map=tform.inverse) # doctest: +SKIP
>>> # create transformation with explicit parameters
>>> tform2 = ski.transform.SimilarityTransform(scale=1.1, rotation=1,
... translation=(10, 20))
>>> # unite transformations, applied in order from left to right
>>> tform3 = tform + tform2
>>> np.allclose(tform3(src), tform2(tform(src)))
True
"""
ttype = ttype.lower()
if ttype not in TRANSFORMS:
raise ValueError(f'the transformation type \'{ttype}\' is not implemented')
tform = TRANSFORMS[ttype](dimensionality=src.shape[1])
tform.estimate(src, dst, *args, **kwargs)
return tform
def matrix_transform(coords, matrix):
"""Apply 2D matrix transform.
Parameters
----------
coords : (N, 2) array_like
x, y coordinates to transform
matrix : (3, 3) array_like
Homogeneous transformation matrix.
Returns
-------
coords : (N, 2) array
Transformed coordinates.
"""
return ProjectiveTransform(matrix)(coords)