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scikit-image/skimage/transform/_geometric.py
T
Johannes Schönberger eb6c3ede38 Fix RANSAC for invalid model estimation and confidence corner case 
Previously, estimators did not return whether the model estimation
was successful. RANSAC now tests whether the estimation was
successful and skips invalid models.

When the confidence/stop_probability of RANSAC was set to 1,
the iteration was falsely terminated early instead of running for
the maximum number of iterations.
2015-03-06 23:47:08 -05:00

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import six
import math
import warnings
import numpy as np
from scipy import ndimage, spatial
from .._shared.utils import get_bound_method_class, safe_as_int
from ..util import img_as_float
from ..exposure import rescale_intensity
from ._warps_cy import _warp_fast
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(2).
Parameters
----------
points : (N, 2) array
The coordinates of the image points.
Returns
-------
matrix : (3, 3) array
The transformation matrix to obtain the new points.
new_points : (N, 2) array
The transformed image points.
"""
centroid = np.mean(points, axis=0)
rms = math.sqrt(np.sum((points - centroid) ** 2) / points.shape[0])
norm_factor = math.sqrt(2) / rms
matrix = np.array([[norm_factor, 0, -norm_factor * centroid[0]],
[0, norm_factor, -norm_factor * centroid[1]],
[0, 0, 1]])
pointsh = np.row_stack([points.T, np.ones((points.shape[0]),)])
new_pointsh = np.dot(matrix, pointsh).T
new_points = new_pointsh[:, :2]
new_points[:, 0] /= new_pointsh[:, 2]
new_points[:, 1] /= new_pointsh[:, 2]
return matrix, new_points
class GeometricTransform(object):
"""Perform geometric transformations on a set of coordinates.
"""
def __call__(self, coords):
"""Apply forward transformation.
Parameters
----------
coords : (N, 2) array
Source coordinates.
Returns
-------
coords : (N, 2) array
Transformed coordinates.
"""
raise NotImplementedError()
def inverse(self, coords):
"""Apply inverse transformation.
Parameters
----------
coords : (N, 2) array
Source coordinates.
Returns
-------
coords : (N, 2) array
Transformed coordinates.
"""
raise NotImplementedError()
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))
def __add__(self, other):
"""Combine this transformation with another.
"""
raise NotImplementedError()
class ProjectiveTransform(GeometricTransform):
"""Matrix 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 : (3, 3) array, optional
Homogeneous transformation matrix.
Attributes
----------
params : (3, 3) array
Homogeneous transformation matrix.
"""
_coeffs = range(8)
def __init__(self, matrix=None):
if matrix is None:
# default to an identity transform
matrix = np.eye(3)
if matrix.shape != (3, 3):
raise ValueError("invalid shape of transformation matrix")
self.params = matrix
@property
def _matrix(self):
warnings.warn('`_matrix` attribute is deprecated, '
'use `params` instead.')
return self.params
@property
def _inv_matrix(self):
return np.linalg.inv(self.params)
def _apply_mat(self, coords, matrix):
coords = np.array(coords, copy=False, ndmin=2)
x, y = np.transpose(coords)
src = np.vstack((x, y, np.ones_like(x)))
dst = np.dot(src.transpose(), matrix.transpose())
# rescale to homogeneous coordinates
dst[:, 0] /= dst[:, 2]
dst[:, 1] /= dst[:, 2]
return dst[:, :2]
def __call__(self, coords):
return self._apply_mat(coords, self.params)
def inverse(self, coords):
"""Apply inverse transformation.
Parameters
----------
coords : (N, 2) array
Source coordinates.
Returns
-------
coords : (N, 2) array
Transformed coordinates.
"""
return self._apply_mat(coords, self._inv_matrix)
def estimate(self, src, dst):
"""Set the transformation matrix with the explicit 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.
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.
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
Source coordinates.
dst : (N, 2) array
Destination coordinates.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
try:
src_matrix, src = _center_and_normalize_points(src)
dst_matrix, dst = _center_and_normalize_points(dst)
except ZeroDivisionError:
self.params = np.nan * np.empty((3, 3))
return False
xs = src[:, 0]
ys = src[:, 1]
xd = dst[:, 0]
yd = dst[:, 1]
rows = src.shape[0]
# params: a0, a1, a2, b0, b1, b2, c0, c1
A = np.zeros((rows * 2, 9))
A[:rows, 0] = xs
A[:rows, 1] = ys
A[:rows, 2] = 1
A[:rows, 6] = - xd * xs
A[:rows, 7] = - xd * ys
A[rows:, 3] = xs
A[rows:, 4] = ys
A[rows:, 5] = 1
A[rows:, 6] = - yd * xs
A[rows:, 7] = - yd * ys
A[:rows, 8] = xd
A[rows:, 8] = yd
# Select relevant columns, depending on params
A = A[:, list(self._coeffs) + [8]]
_, _, V = np.linalg.svd(A)
H = np.zeros((3, 3))
# solution is right singular vector that corresponds to smallest
# singular value
H.flat[list(self._coeffs) + [8]] = - V[-1, :-1] / V[-1, -1]
H[2, 2] = 1
# De-center and de-normalize
H = np.dot(np.linalg.inv(dst_matrix), np.dot(H, src_matrix))
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.dot(self.params))
elif (hasattr(other, '__name__')
and other.__name__ == 'inverse'
and hasattr(get_bound_method_class(other), '_inv_matrix')):
return ProjectiveTransform(self._inv_matrix.dot(self.params))
else:
raise TypeError("Cannot combine transformations of differing "
"types.")
class AffineTransform(ProjectiveTransform):
"""2D affine transformation of the form:
..:math:
X = a0*x + a1*y + a2 =
= sx*x*cos(rotation) - sy*y*sin(rotation + shear) + a2
Y = b0*x + b1*y + b2 =
= sx*x*sin(rotation) + sy*y*cos(rotation + shear) + b2
where ``sx`` and ``sy`` are zoom factors in the x and y directions,
and the homogeneous transformation matrix is::
[[a0 a1 a2]
[b0 b1 b2]
[0 0 1]]
Parameters
----------
matrix : (3, 3) array, optional
Homogeneous transformation matrix.
scale : (sx, sy) as array, list or tuple, optional
Scale factors.
rotation : float, optional
Rotation angle in counter-clockwise direction as radians.
shear : float, optional
Shear angle in counter-clockwise direction as radians.
translation : (tx, ty) as array, list or tuple, optional
Translation parameters.
Attributes
----------
params : (3, 3) array
Homogeneous transformation matrix.
"""
_coeffs = range(6)
def __init__(self, matrix=None, scale=None, rotation=None, shear=None,
translation=None):
params = any(param is not None
for param in (scale, rotation, shear, 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:
if matrix.shape != (3, 3):
raise ValueError("Invalid shape of transformation matrix.")
self.params = matrix
elif params:
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)
sx, sy = scale
self.params = np.array([
[sx * math.cos(rotation), -sy * math.sin(rotation + shear), 0],
[sx * math.sin(rotation), sy * math.cos(rotation + shear), 0],
[ 0, 0, 1]
])
self.params[0:2, 2] = translation
else:
# default to an identity transform
self.params = np.eye(3)
@property
def scale(self):
sx = math.sqrt(self.params[0, 0] ** 2 + self.params[1, 0] ** 2)
sy = math.sqrt(self.params[0, 1] ** 2 + self.params[1, 1] ** 2)
return sx, sy
@property
def rotation(self):
return math.atan2(self.params[1, 0], self.params[0, 0])
@property
def shear(self):
beta = math.atan2(- self.params[0, 1], self.params[1, 1])
return beta - self.rotation
@property
def translation(self):
return self.params[0:2, 2]
class PiecewiseAffineTransform(GeometricTransform):
"""2D 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):
"""Set the control points with which to perform the piecewise mapping.
Number of source and destination coordinates must match.
Parameters
----------
src : (N, 2) array
Source coordinates.
dst : (N, 2) array
Destination coordinates.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
# forward piecewise affine
# triangulate input positions into mesh
self._tesselation = spatial.Delaunay(src)
# find affine mapping from source positions to destination
self.affines = []
for tri in self._tesselation.vertices:
affine = AffineTransform()
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.vertices:
affine = AffineTransform()
affine.estimate(dst[tri, :], src[tri, :])
self.inverse_affines.append(affine)
return True
def __call__(self, coords):
"""Apply forward transformation.
Coordinates outside of the mesh will be set to `- 1`.
Parameters
----------
coords : (N, 2) array
Source coordinates.
Returns
-------
coords : (N, 2) array
Transformed coordinates.
"""
out = np.empty_like(coords, np.double)
# 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.vertices)):
# 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
def inverse(self, coords):
"""Apply inverse transformation.
Coordinates outside of the mesh will be set to `- 1`.
Parameters
----------
coords : (N, 2) array
Source coordinates.
Returns
-------
coords : (N, 2) array
Transformed coordinates.
"""
out = np.empty_like(coords, np.double)
# determine triangle index for each coordinate
simplex = self._inverse_tesselation.find_simplex(coords)
# coordinates outside of mesh
out[simplex == -1, :] = -1
for index in range(len(self._inverse_tesselation.vertices)):
# affine transform for triangle
affine = self.inverse_affines[index]
# all coordinates within triangle
index_mask = simplex == index
out[index_mask, :] = affine(coords[index_mask, :])
return out
class SimilarityTransform(ProjectiveTransform):
"""2D similarity transformation of the form:
..:math:
X = a0 * x - b0 * y + a1 =
= m * x * cos(rotation) - m * y * sin(rotation) + a1
Y = b0 * x + a0 * y + b1 =
= m * x * sin(rotation) + m * y * cos(rotation) + b1
where ``m`` is a zoom factor and the homogeneous transformation matrix is::
[[a0 b0 a1]
[b0 a0 b1]
[0 0 1]]
Parameters
----------
matrix : (3, 3) array, optional
Homogeneous transformation matrix.
scale : float, optional
Scale factor.
rotation : float, optional
Rotation angle in counter-clockwise direction as radians.
translation : (tx, ty) as array, list or tuple, optional
x, y translation parameters.
Attributes
----------
params : (3, 3) array
Homogeneous transformation matrix.
"""
def __init__(self, matrix=None, scale=None, rotation=None,
translation=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:
if matrix.shape != (3, 3):
raise ValueError("Invalid shape of transformation matrix.")
self.params = matrix
elif params:
if scale is None:
scale = 1
if rotation is None:
rotation = 0
if translation is None:
translation = (0, 0)
self.params = np.array([
[math.cos(rotation), - math.sin(rotation), 0],
[math.sin(rotation), math.cos(rotation), 0],
[ 0, 0, 1]
])
self.params[0:2, 0:2] *= scale
self.params[0:2, 2] = translation
else:
# default to an identity transform
self.params = np.eye(3)
def estimate(self, src, dst):
"""Set the transformation matrix with the explicit 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.
The transformation is defined as::
X = a0 * x - b0 * y + a1
Y = b0 * x + a0 * y + b1
These equations can be transformed to the following form::
0 = a0 * x - b0 * y + a1 - X
0 = b0 * x + a0 * y + b1 - 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 1 -y 0 -X]
[y 0 x 1 -Y]
...
...
]
x.T = [a0 a1 b0 b1 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.
Parameters
----------
src : (N, 2) array
Source coordinates.
dst : (N, 2) array
Destination coordinates.
Returns
-------
success : bool
True, if model estimation succeeds.
"""
try:
src_matrix, src = _center_and_normalize_points(src)
dst_matrix, dst = _center_and_normalize_points(dst)
except ZeroDivisionError:
self.params = np.nan * np.empty((3, 3))
return False
xs = src[:, 0]
ys = src[:, 1]
xd = dst[:, 0]
yd = dst[:, 1]
rows = src.shape[0]
# params: a0, a1, b0, b1
A = np.zeros((rows * 2, 5))
A[:rows, 0] = xs
A[:rows, 2] = - ys
A[:rows, 1] = 1
A[rows:, 2] = xs
A[rows:, 0] = ys
A[rows:, 3] = 1
A[:rows, 4] = xd
A[rows:, 4] = yd
_, _, V = np.linalg.svd(A)
# solution is right singular vector that corresponds to smallest
# singular value
a0, a1, b0, b1 = - V[-1, :-1] / V[-1, -1]
S = np.array([[a0, -b0, a1],
[b0, a0, b1],
[ 0, 0, 1]])
# De-center and de-normalize
S = np.dot(np.linalg.inv(dst_matrix), np.dot(S, src_matrix))
self.params = S
return True
@property
def scale(self):
if abs(math.cos(self.rotation)) < np.spacing(1):
# sin(self.rotation) == 1
scale = self.params[1, 0]
else:
scale = self.params[0, 0] / math.cos(self.rotation)
return scale
@property
def rotation(self):
return math.atan2(self.params[1, 0], self.params[1, 1])
@property
def translation(self):
return self.params[0:2, 2]
class PolynomialTransform(GeometricTransform):
"""2D transformation of the form:
..:math:
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, 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):
if params is None:
# default to transformation which preserves original coordinates
params = np.array([[0, 1, 0], [0, 0, 1]])
if params.shape[0] != 2:
raise ValueError("invalid shape of transformation parameters")
self.params = params
@property
def _params(self):
warnings.warn('`_params` attribute is deprecated, '
'use `params` instead.')
return self.params
def estimate(self, src, dst, order=2):
"""Set the transformation matrix with the explicit 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.
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.
Parameters
----------
src : (N, 2) array
Source coordinates.
dst : (N, 2) array
Destination coordinates.
order : int, optional
Polynomial order (number of coefficients is order + 1).
Returns
-------
success : bool
True, if model estimation succeeds.
"""
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
_, _, V = np.linalg.svd(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
source coordinates
Returns
-------
coords : (N, 2) array
Transformed coordinates.
"""
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
def inverse(self, coords):
raise Exception(
'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 = {
'similarity': SimilarityTransform,
'affine': AffineTransform,
'piecewise-affine': PiecewiseAffineTransform,
'projective': ProjectiveTransform,
'polynomial': PolynomialTransform,
}
HOMOGRAPHY_TRANSFORMS = (
SimilarityTransform,
AffineTransform,
ProjectiveTransform
)
def estimate_transform(ttype, src, dst, **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 : {'similarity', 'affine', 'piecewise-affine', 'projective', \
'polynomial'}
Type of transform.
kwargs : array or int
Function parameters (src, dst, n, angle)::
NAME / TTYPE FUNCTION PARAMETERS
'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
>>> from skimage import transform as tf
>>> # estimate transformation parameters
>>> src = np.array([0, 0, 10, 10]).reshape((2, 2))
>>> dst = np.array([12, 14, 1, -20]).reshape((2, 2))
>>> tform = tf.estimate_transform('similarity', src, dst)
>>> np.allclose(tform.inverse(tform(src)), src)
True
>>> # warp image using the estimated transformation
>>> from skimage import data
>>> image = data.camera()
>>> warp(image, inverse_map=tform.inverse) # doctest: +SKIP
>>> # create transformation with explicit parameters
>>> tform2 = tf.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('the transformation type \'%s\' is not'
'implemented' % ttype)
tform = TRANSFORMS[ttype]()
tform.estimate(src, dst, **kwargs)
return tform
def matrix_transform(coords, matrix):
"""Apply 2D matrix transform.
Parameters
----------
coords : (N, 2) array
x, y coordinates to transform
matrix : (3, 3) array
Homogeneous transformation matrix.
Returns
-------
coords : (N, 2) array
Transformed coordinates.
"""
return ProjectiveTransform(matrix)(coords)
def _stackcopy(a, b):
"""Copy b into each color layer of a, such that::
a[:,:,0] = a[:,:,1] = ... = b
Parameters
----------
a : (M, N) or (M, N, P) ndarray
Target array.
b : (M, N)
Source array.
Notes
-----
Color images are stored as an ``(M, N, 3)`` or ``(M, N, 4)`` arrays.
"""
if a.ndim == 3:
a[:] = b[:, :, np.newaxis]
else:
a[:] = b
def warp_coords(coord_map, shape, dtype=np.float64):
"""Build the source coordinates for the output of a 2-D image warp.
Parameters
----------
coord_map : callable like GeometricTransform.inverse
Return input coordinates for given output coordinates.
Coordinates are in the shape (P, 2), where P is the number
of coordinates and each element is a ``(row, col)`` pair.
shape : tuple
Shape of output image ``(rows, cols[, bands])``.
dtype : np.dtype or string
dtype for return value (sane choices: float32 or float64).
Returns
-------
coords : (ndim, rows, cols[, bands]) array of dtype `dtype`
Coordinates for `scipy.ndimage.map_coordinates`, that will yield
an image of shape (orows, ocols, bands) by drawing from source
points according to the `coord_transform_fn`.
Notes
-----
This is a lower-level routine that produces the source coordinates for 2-D
images used by `warp()`.
It is provided separately from `warp` to give additional flexibility to
users who would like, for example, to re-use a particular coordinate
mapping, to use specific dtypes at various points along the the
image-warping process, or to implement different post-processing logic
than `warp` performs after the call to `ndimage.map_coordinates`.
Examples
--------
Produce a coordinate map that shifts an image up and to the right:
>>> from skimage import data
>>> from scipy.ndimage import map_coordinates
>>>
>>> def shift_up10_left20(xy):
... return xy - np.array([-20, 10])[None, :]
>>>
>>> image = data.astronaut().astype(np.float32)
>>> coords = warp_coords(shift_up10_left20, image.shape)
>>> warped_image = map_coordinates(image, coords)
"""
shape = safe_as_int(shape)
rows, cols = shape[0], shape[1]
coords_shape = [len(shape), rows, cols]
if len(shape) == 3:
coords_shape.append(shape[2])
coords = np.empty(coords_shape, dtype=dtype)
# Reshape grid coordinates into a (P, 2) array of (row, col) pairs
tf_coords = np.indices((cols, rows), dtype=dtype).reshape(2, -1).T
# Map each (row, col) pair to the source image according to
# the user-provided mapping
tf_coords = coord_map(tf_coords)
# Reshape back to a (2, M, N) coordinate grid
tf_coords = tf_coords.T.reshape((-1, cols, rows)).swapaxes(1, 2)
# Place the y-coordinate mapping
_stackcopy(coords[1, ...], tf_coords[0, ...])
# Place the x-coordinate mapping
_stackcopy(coords[0, ...], tf_coords[1, ...])
if len(shape) == 3:
coords[2, ...] = range(shape[2])
return coords
def _convert_warp_input(image, preserve_range):
"""Convert input image to double image with the appropriate range."""
if preserve_range:
image = image.astype(np.double)
else:
image = img_as_float(image)
return image
def _clip_warp_output(input_image, output_image, order, mode, cval, clip):
"""Clip output image to range of values of input image.
Note that this function modifies the values of `output_image` in-place
and it is only modified if ``clip=True``.
Parameters
----------
input_image : ndarray
Input image.
output_image : ndarray
Output image, which is modified in-place.
Other parameters
----------------
order : int, optional
The order of the spline interpolation, default is 1. The order has to
be in the range 0-5. See `skimage.transform.warp` for detail.
mode : string, optional
Points outside the boundaries of the input are filled according
to the given mode ('constant', 'nearest', 'reflect' or 'wrap').
cval : float, optional
Used in conjunction with mode 'constant', the value outside
the image boundaries.
clip : bool, optional
Whether to clip the output to the range of values of the input image.
This is enabled by default, since higher order interpolation may
produce values outside the given input range.
"""
if clip and order != 0:
min_val = input_image.min()
max_val = input_image.max()
preserve_cval = mode == 'constant' and not \
(min_val <= cval <= max_val)
if preserve_cval:
cval_mask = output_image == cval
np.clip(output_image, min_val, max_val, out=output_image)
if preserve_cval:
output_image[cval_mask] = cval
def warp(image, inverse_map=None, map_args={}, output_shape=None, order=1,
mode='constant', cval=0., clip=True, preserve_range=False):
"""Warp an image according to a given coordinate transformation.
Parameters
----------
image : ndarray
Input image.
inverse_map : transformation object, callable ``cr = f(cr, **kwargs)``, or ndarray
Inverse coordinate map, which transforms coordinates in the output
images into their corresponding coordinates in the input image.
There are a number of different options to define this map, depending
on the dimensionality of the input image. A 2-D image can have 2
dimensions for gray-scale images, or 3 dimensions with color
information.
- For 2-D images, you can directly pass a transformation object,
e.g. `skimage.transform.SimilarityTransform`, or its inverse.
- For 2-D images, you can pass a ``(3, 3)`` homogeneous
transformation matrix, e.g.
`skimage.transform.SimilarityTransform.params`.
- For 2-D images, a function that transforms a ``(M, 2)`` array of
``(col, row)`` coordinates in the output image to their
corresponding coordinates in the input image. Extra parameters to
the function can be specified through `map_args`.
- For N-D images, you can directly pass an array of coordinates.
The first dimension specifies the coordinates in the input image,
while the subsequent dimensions determine the position in the
output image. E.g. in case of 2-D images, you need to pass an array
of shape ``(2, rows, cols)``, where `rows` and `cols` determine the
shape of the output image, and the first dimension contains the
``(row, col)`` coordinate in the input image.
See `scipy.ndimage.map_coordinates` for further documentation.
Note, that a ``(3, 3)`` matrix is interpreted as a homogeneous
transformation matrix, so you cannot interpolate values from a 3-D
input, if the output is of shape ``(3,)``.
See example section for usage.
map_args : dict, optional
Keyword arguments passed to `inverse_map`.
output_shape : tuple (rows, cols), optional
Shape of the output image generated. By default the shape of the input
image is preserved. Note that, even for multi-band images, only rows
and columns need to be specified.
order : int, optional
The order of interpolation. The order has to be in the range 0-5:
- 0: Nearest-neighbor
- 1: Bi-linear (default)
- 2: Bi-quadratic
- 3: Bi-cubic
- 4: Bi-quartic
- 5: Bi-quintic
mode : string, optional
Points outside the boundaries of the input are filled according
to the given mode ('constant', 'nearest', 'reflect' or 'wrap').
cval : float, optional
Used in conjunction with mode 'constant', the value outside
the image boundaries.
clip : bool, optional
Whether to clip the output to the range of values of the input image.
This is enabled by default, since higher order interpolation may
produce values outside the given input range.
preserve_range : bool, optional
Whether to keep the original range of values. Otherwise, the input
image is converted according to the conventions of `img_as_float`.
Returns
-------
warped : double ndarray
The warped input image.
Notes
-----
- The input image is converted to a `double` image.
- In case of a `SimilarityTransform`, `AffineTransform` and
`ProjectiveTransform` and `order` in [0, 3] this function uses the
underlying transformation matrix to warp the image with a much faster
routine.
Examples
--------
>>> from skimage.transform import warp
>>> from skimage import data
>>> image = data.camera()
The following image warps are all equal but differ substantially in
execution time. The image is shifted to the bottom.
Use a geometric transform to warp an image (fast):
>>> from skimage.transform import SimilarityTransform
>>> tform = SimilarityTransform(translation=(0, -10))
>>> warped = warp(image, tform)
Use a callable (slow):
>>> def shift_down(xy):
... xy[:, 1] -= 10
... return xy
>>> warped = warp(image, shift_down)
Use a transformation matrix to warp an image (fast):
>>> matrix = np.array([[1, 0, 0], [0, 1, -10], [0, 0, 1]])
>>> warped = warp(image, matrix)
>>> from skimage.transform import ProjectiveTransform
>>> warped = warp(image, ProjectiveTransform(matrix=matrix))
You can also use the inverse of a geometric transformation (fast):
>>> warped = warp(image, tform.inverse)
For N-D images you can pass a coordinate array, that specifies the
coordinates in the input image for every element in the output image. E.g.
if you want to rescale a 3-D cube, you can do:
>>> cube_shape = np.array([30, 30, 30])
>>> cube = np.random.rand(*cube_shape)
Setup the coordinate array, that defines the scaling:
>>> scale = 0.1
>>> output_shape = (scale * cube_shape).astype(int)
>>> coords0, coords1, coords2 = np.mgrid[:output_shape[0],
... :output_shape[1], :output_shape[2]]
>>> coords = np.array([coords0, coords1, coords2])
Assume that the cube contains spatial data, where the first array element
center is at coordinate (0.5, 0.5, 0.5) in real space, i.e. we have to
account for this extra offset when scaling the image:
>>> coords = (coords + 0.5) / scale - 0.5
>>> warped = warp(cube, coords)
"""
image = _convert_warp_input(image, preserve_range)
input_shape = np.array(image.shape)
if output_shape is None:
output_shape = input_shape
else:
output_shape = safe_as_int(output_shape)
warped = None
if order == 2:
# When fixing this issue, make sure to fix the branches further
# below in this function
warnings.warn("Bi-quadratic interpolation behavior has changed due "
"to a bug in the implementation of scikit-image. "
"The new version now serves as a wrapper "
"around SciPy's interpolation functions, which itself "
"is not verified to be a correct implementation. Until "
"skimage's implementation is fixed, we recommend "
"to use bi-linear or bi-cubic interpolation instead.")
if order in (0, 1, 3) and not map_args:
# use fast Cython version for specific interpolation orders and input
matrix = None
if isinstance(inverse_map, np.ndarray) and inverse_map.shape == (3, 3):
# inverse_map is a transformation matrix as numpy array
matrix = inverse_map
elif isinstance(inverse_map, HOMOGRAPHY_TRANSFORMS):
# inverse_map is a homography
matrix = inverse_map.params
elif (hasattr(inverse_map, '__name__')
and inverse_map.__name__ == 'inverse'
and get_bound_method_class(inverse_map) \
in HOMOGRAPHY_TRANSFORMS):
# inverse_map is the inverse of a homography
matrix = np.linalg.inv(six.get_method_self(inverse_map).params)
if matrix is not None:
matrix = matrix.astype(np.double)
if image.ndim == 2:
warped = _warp_fast(image, matrix,
output_shape=output_shape,
order=order, mode=mode, cval=cval)
elif image.ndim == 3:
dims = []
for dim in range(image.shape[2]):
dims.append(_warp_fast(image[..., dim], matrix,
output_shape=output_shape,
order=order, mode=mode, cval=cval))
warped = np.dstack(dims)
if warped is None:
# use ndimage.map_coordinates
if (isinstance(inverse_map, np.ndarray)
and inverse_map.shape == (3, 3)):
# inverse_map is a transformation matrix as numpy array,
# this is only used for order >= 4.
inverse_map = ProjectiveTransform(matrix=inverse_map)
if isinstance(inverse_map, np.ndarray):
# inverse_map is directly given as coordinates
coords = inverse_map
else:
# inverse_map is given as function, that transforms (N, 2)
# destination coordinates to their corresponding source
# coordinates. This is only supported for 2(+1)-D images.
if image.ndim < 2 or image.ndim > 3:
raise ValueError("Only 2-D images (grayscale or color) are "
"supported, when providing a callable "
"`inverse_map`.")
def coord_map(*args):
return inverse_map(*args, **map_args)
if len(input_shape) == 3 and len(output_shape) == 2:
# Input image is 2D and has color channel, but output_shape is
# given for 2-D images. Automatically add the color channel
# dimensionality.
output_shape = (output_shape[0], output_shape[1],
input_shape[2])
coords = warp_coords(coord_map, output_shape)
# Pre-filtering not necessary for order 0, 1 interpolation
prefilter = order > 1
warped = ndimage.map_coordinates(image, coords, prefilter=prefilter,
mode=mode, order=order, cval=cval)
_clip_warp_output(image, warped, order, mode, cval, clip)
return warped
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