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Refactor geometric transforms.

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Stefan van der Walt
2012-07-20 17:57:15 -04:00
parent 8e8e2b99a0
commit a130b8d2d9
6 changed files with 830 additions and 914 deletions
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skimage/transform/_geometric.py
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import math
import numpy as np
from scipy import ndimage
from skimage.util import img_as_float
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 ``MxNx3`` or ``MxNx4`` arrays.
"""
if a.ndim == 3:
a[:] = b[:, :, np.newaxis]
else:
a[:] = b
class GeometricTransform(object):
"""Perform geometric transformations on a set of coordinates.
Parameters
----------
matrix : 3x3 array, optional
Homogeneous transformation matrix.
"""
def __call__(self, coords):
"""Apply forward transformation.
Parameters
----------
coords : Nx2 array
source coordinates
Returns
-------
coords : Nx2 array
transformed coordinates
"""
raise NotImplementedError()
def inverse(self, coords):
"""Apply inverse transformation.
Parameters
----------
coords : Nx2 array
source coordinates
Returns
-------
coords : Nx2 array
transformed coordinates
"""
raise NotImplementedError()
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}`. 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 ]].
"""
_coefs = range(8)
def __init__(self, matrix=None):
self._matrix = matrix
@property
def _inv_matrix(self):
return np.linalg.inv(self._matrix)
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._matrix)
def inverse(self, coords):
return self._apply_mat(coords, self._inv_matrix)
def estimate(self, src, dst):
"""Set the transformation matrix with the explicit transformation
parameters.
Parameters
----------
src : Nx2 array
source coordinates
dst : Nx2 array
destination coordinates
"""
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, 8))
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
# Select relevant columns, depending on coeffs
A = A[:, self._coefs]
b = np.hstack([xd, yd])
H = np.zeros((3, 3))
H.flat[self._coefs] = np.linalg.lstsq(A, b)[0]
H[2, 2] = 1
self._matrix = H
def __add__(self, other):
"""Combine this transformation with another.
"""
if isinstance(other, ProjectiveTransform):
return ProjectiveTransform(np.dot(other._matrix, self._matrix))
else:
raise TypeError("Cannot combine transformations of differing "
"types.")
class AffineTransform(ProjectiveTransform):
"""2D affine transformation of the form::
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
----------
scale : (sx, sy), floats
Scale factors.
rotation : float
Rotation angle in radians, counter-clockwise direction.
shear : float
Shear angle in radians, counter-clockwise direction.
translation : (tx, ty), floats
Translation in x and y.
"""
_coefs = range(6)
def __init__(self, scale=None, rotation=None, shear=None, translation=None):
ProjectiveTransform.__init__(self)
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)
a = rotation
sx, sy = scale
tx, ty = translation
self._matrix = np.array([
[sx * math.cos(a), - sy * math.sin(a + shear), tx],
[sx * math.sin(a), sy * math.cos(a + shear), ty],
[0, 0, 1]
])
class SimilarityTransform(AffineTransform):
"""2D similarity transformation of the form::
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
----------
scale : float, optional
Scale / zoom factor.
rotation : float, optional
Rotation angle, counter-clockwise, in radians.
translation : (tx, ty) of float
x, y translation parameters
"""
def __init__(self, scale=None, rotation=None, translation=None):
if scale is not None:
scale = (scale, scale)
AffineTransform.__init__(self, scale=scale,
rotation=rotation,
shear=0,
translation=translation)
class PolynomialTransform(GeometricTransform):
"""2D transformation of the form::
X = sum[j=0:n]( sum[i=0:j]( a_ji * x**(j - i) * y**i ))
Y = sum[j=0:n]( sum[i=0:j]( b_ji * x**(j - i) * y**i ))
TODO: Describe structure of coefficients.
Shall we store it as a (2, M) ndarray?
"""
def __init__(self, coeffs=None):
"""Create polynomial transformation.
Parameters
----------
coeffs : array, optional
polynomial coefficients
"""
self.coeffs = coeffs
def estimate(self, src, dst, order):
"""Set the transformation matrix with the explicit transformation
parameters.
Parameters
----------
src : Nx2 array
source coordinates
dst : Nx2 array
destination coordinates
order : int
polynomial order (number of coefficients is order + 1)
"""
xs = src[:, 0]
ys = src[:, 1]
xd = dst[:, 0]
yd = dst[:, 1]
rows = src.shape[0]
# number of unknown polynomial coefficients
u = (order + 1) * (order + 2)
A = np.zeros((rows * 2, u))
pidx = 0
for j in xrange(order + 1):
for i in xrange(j + 1):
A[:rows, pidx] = xs ** (j - i) * ys ** i
A[rows:, pidx + u / 2] = xs ** (j - i) * ys ** i
pidx += 1
b = np.hstack([xd, yd])
self.coeffs = np.linalg.lstsq(A, b)[0]
def __call__(self, coords):
"""Apply forward transformation.
Parameters
----------
coords : Nx2 array
source coordinates
Returns
-------
coords : Nx2 array
transformed coordinates
"""
x = coords[:, 0]
y = coords[:, 1]
u = len(self.coeffs.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 xrange(order + 1):
for i in xrange(j + 1):
dst[:, 0] += self.coeffs[pidx] * x ** (j - i) * y ** i
dst[:, 1] += self.coeffs[pidx + u / 2] * 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.')
TRANSFORMATIONS = {
'similarity': SimilarityTransform,
'affine': AffineTransform,
'projective': ProjectiveTransform,
'polynomial': PolynomialTransform,
}
def estimate_transform(ttype, src, dst, **kwargs):
"""Estimate 2D geometric transformation parameters.
You can determine the over-, well- and under-determined parameters
with the least-squares method.
Number of source must match number of destination coordinates.
Parameters
----------
ttype : {'similarity', '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`
'projective' `src, `dst`
'polynomial' `src, `dst`, `order` (polynomial order)
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)
>>> tform.inverse(tform.forward(src)) # == src
>>> # warp image using the estimated transformation
>>> from skimage import data
>>> image = data.camera()
>>> warp(image, inverse_map=tform.inverse)
>>> # create transformation with explicit parameters
>>> scale = 1.1
>>> rotation = 1
>>> translation = (10, 20)
>>>
>>> tform2 = tf.SimilarityTransform(scale, rotation, translation)
>>> # unite transformations, applied in order from left to right
>>> tform3 = tform + tform2
>>> tform3.forward(src) # == tform2.forward(tform.forward(src))
"""
ttype = ttype.lower()
if ttype not in TRANSFORMATIONS:
raise ValueError('the transformation type \'%s\' is not'
'implemented' % ttype)
tform = TRANSFORMATIONS[ttype]()
tform.estimate(src, dst, **kwargs)
return tform
def matrix_transform(coords, matrix):
"""Apply 2D matrix transform.
Parameters
----------
coords : Nx2 array
x, y coordinates to transform
matrix : 3x3 array
Homogeneous transformation matrix.
Returns
-------
coords : Nx2 array
transformed coordinates
"""
return ProjectiveTransform(matrix)(coords)
def warp(image, inverse_map=None, map_args={}, output_shape=None, order=1,
mode='constant', cval=0., reverse_map=None):
"""Warp an image according to a given coordinate transformation.
Parameters
----------
image : 2-D array
Input image.
inverse_map : transformation object, callable xy = f(xy, **kwargs)
Inverse coordinate map. A function that transforms a Px2 array of
``(x, y)`` coordinates in the *output image* into their corresponding
coordinates in the *source image*. In case of a transformation object
its `inverse` method will be used as transformation function. Also see
examples below.
map_args : dict, optional
Keyword arguments passed to `inverse_map`.
output_shape : tuple (rows, cols)
Shape of the output image generated.
order : int
Order of splines used in interpolation. See
`scipy.ndimage.map_coordinates` for detail.
mode : string
How to handle values outside the image borders. See
`scipy.ndimage.map_coordinates` for detail.
cval : string
Used in conjunction with mode 'constant', the value outside
the image boundaries.
Examples
--------
Shift an image to the right:
>>> from skimage import data
>>> image = data.camera()
>>>
>>> def shift_right(xy):
... xy[:, 0] -= 10
... return xy
>>>
>>> warp(image, shift_right)
"""
# Backward API compatibility
if reverse_map is not None:
inverse_map = reverse_map
if image.ndim < 2:
raise ValueError("Input must have more than 1 dimension.")
image = np.atleast_3d(img_as_float(image))
ishape = np.array(image.shape)
bands = ishape[2]
if output_shape is None:
output_shape = ishape
coords = np.empty(np.r_[3, output_shape], dtype=float)
## Construct transformed coordinates
rows, cols = output_shape[:2]
# Reshape grid coordinates into a (P, 2) array of (x, y) pairs
tf_coords = np.indices((cols, rows), dtype=float).reshape(2, -1).T
# Map each (x, y) pair to the source image according to
# the user-provided mapping
if callable(getattr(inverse_map, 'inverse', None)):
inverse_map = inverse_map.inverse
tf_coords = inverse_map(tf_coords, **map_args)
# 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, ...])
# colour-coordinate mapping
coords[2, ...] = range(bands)
# Prefilter not necessary for order 1 interpolation
prefilter = order > 1
mapped = ndimage.map_coordinates(image, coords, prefilter=prefilter,
mode=mode, order=order, cval=cval)
# The spline filters sometimes return results outside [0, 1],
# so clip to ensure valid data
return np.clip(mapped.squeeze(), 0, 1)