mirror of
https://github.com/wassname/scikit-image.git
synced 2026-06-29 04:44:43 +08:00
Johannes Schönberger
676 lines
19 KiB
Python
676 lines
19 KiB
Python
import math
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import numpy as np
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from scipy import ndimage
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from skimage.util import img_as_float
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def _stackcopy(a, b):
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"""Copy b into each color layer of a, such that::
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a[:,:,0] = a[:,:,1] = ... = b
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Parameters
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----------
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a : (M, N) or (M, N, P) ndarray
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Target array.
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b : (M, N)
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Source array.
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Notes
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-----
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Color images are stored as an ``(M, N, 3)`` or ``(M, N, 4)`` arrays.
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"""
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if a.ndim == 3:
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a[:] = b[:, :, np.newaxis]
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else:
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a[:] = b
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class GeometricTransform(object):
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"""Perform geometric transformations on a set of coordinates.
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"""
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def __call__(self, coords):
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"""Apply forward transformation.
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Parameters
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----------
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coords : (N, 2) array
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source coordinates
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Returns
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-------
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coords : (N, 2) array
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transformed coordinates
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"""
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raise NotImplementedError()
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def inverse(self, coords):
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"""Apply inverse transformation.
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Parameters
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----------
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coords : (N, 2) array
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source coordinates
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Returns
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-------
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coords : (N, 2) array
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transformed coordinates
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"""
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raise NotImplementedError()
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def __add__(self, other):
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"""Combine this transformation with another.
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"""
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raise NotImplementedError()
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class ProjectiveTransform(GeometricTransform):
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"""Matrix transformation.
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Apply a projective transformation (homography) on coordinates.
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For each homogeneous coordinate :math:`\mathbf{x} = [x, y, 1]^T`, its
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target position is calculated by multiplying with the given matrix,
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:math:`H`, to give :math:`H \mathbf{x}`. E.g., to rotate by theta degrees
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clockwise, the matrix should be::
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[[cos(theta) -sin(theta) 0]
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[sin(theta) cos(theta) 0]
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[0 0 1]]
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or, to translate x by 10 and y by 20::
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[[1 0 10]
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[0 1 20]
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[0 0 1 ]].
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Parameters
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----------
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matrix : (3, 3) array, optional
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Homogeneous transformation matrix.
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"""
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coeffs = range(8)
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def __init__(self, matrix=None):
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self._matrix = matrix
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@property
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def _inv_matrix(self):
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return np.linalg.inv(self._matrix)
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def _apply_mat(self, coords, matrix):
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coords = np.array(coords, copy=False, ndmin=2)
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x, y = np.transpose(coords)
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src = np.vstack((x, y, np.ones_like(x)))
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dst = np.dot(src.transpose(), matrix.transpose())
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# rescale to homogeneous coordinates
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dst[:, 0] /= dst[:, 2]
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dst[:, 1] /= dst[:, 2]
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return dst[:, :2]
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def __call__(self, coords):
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return self._apply_mat(coords, self._matrix)
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def inverse(self, coords):
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return self._apply_mat(coords, self._inv_matrix)
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def estimate(self, src, dst):
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"""Set the transformation matrix with the explicit transformation
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parameters.
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You can determine the over-, well- and under-determined parameters
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with the total least-squares method.
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Number of source and destination coordinates must match.
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Parameters
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----------
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src : (N, 2) array
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source coordinates
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dst : (N, 2) array
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destination coordinates
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"""
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xs = src[:, 0]
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ys = src[:, 1]
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xd = dst[:, 0]
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yd = dst[:, 1]
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rows = src.shape[0]
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# params: a0, a1, a2, b0, b1, b2, c0, c1
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A = np.zeros((rows * 2, 9))
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A[:rows, 0] = xs
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A[:rows, 1] = ys
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A[:rows, 2] = 1
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A[:rows, 6] = - xd * xs
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A[:rows, 7] = - xd * ys
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A[rows:, 3] = xs
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A[rows:, 4] = ys
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A[rows:, 5] = 1
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A[rows:, 6] = - yd * xs
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A[rows:, 7] = - yd * ys
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A[:rows, 8] = xd
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A[rows:, 8] = yd
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# Select relevant columns, depending on params
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A = A[:, self.coeffs + [8]]
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_, _, V = np.linalg.svd(A)
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H = np.zeros((3, 3))
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# solution is right singular vector that corresponds to smallest
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# singular value and normed by c3
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H.flat[self.coeffs + [8]] = - V[-1, :-1] / V[-1, -1]
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H[2, 2] = 1
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self._matrix = H
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def __add__(self, other):
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"""Combine this transformation with another.
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"""
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if isinstance(other, ProjectiveTransform):
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# combination of the same types result in a transformation of this
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# type again, otherwise use general projective transformation
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if type(self) == type(other):
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tform = self.__class__
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else:
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tform = ProjectiveTransform
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return tform(other._matrix.dot(self._matrix))
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else:
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raise TypeError("Cannot combine transformations of differing "
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"types.")
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class AffineTransform(ProjectiveTransform):
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"""2D affine transformation of the form::
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X = a0*x + a1*y + a2 =
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= sx*x*cos(rotation) - sy*y*sin(rotation + shear) + a2
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Y = b0*x + b1*y + b2 =
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= sx*x*sin(rotation) + sy*y*cos(rotation + shear) + b2
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where ``sx`` and ``sy`` are zoom factors in the x and y directions,
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and the homogeneous transformation matrix is::
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[[a0 a1 a2]
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[b0 b1 b2]
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[0 0 1]]
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Parameters
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----------
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matrix : (3, 3) array, optional
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Homogeneous transformation matrix.
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"""
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coeffs = range(6)
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def compose_implicit(self, scale=None, rotation=None, shear=None,
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translation=None):
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"""Set the transformation matrix with the implicit transformation
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parameters.
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Parameters
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----------
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scale : (sx, sy) as array, list or tuple
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scale factors
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rotation : float
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rotation angle in counter-clockwise direction
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shear : float
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shear angle in counter-clockwise direction
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translation : (tx, ty) as array, list or tuple
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translation parameters
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"""
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if scale is None:
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scale = (1, 1)
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if rotation is None:
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rotation = 0
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if shear is None:
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shear = 0
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if translation is None:
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translation = (0, 0)
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sx, sy = scale
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self._matrix = np.array([
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[sx * math.cos(rotation), - sy * math.sin(rotation + shear), 0],
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[sx * math.sin(rotation), sy * math.cos(rotation + shear), 0],
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[ 0, 0, 1]
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])
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self._matrix[0:2, 2] = translation
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@property
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def scale(self):
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sx = math.sqrt(self._matrix[0, 0] ** 2 + self._matrix[1, 0] ** 2)
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sy = math.sqrt(self._matrix[0, 1] ** 2 + self._matrix[1, 1] ** 2)
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return sx, sy
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@property
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def rotation(self):
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return math.atan2(self._matrix[1, 0], self._matrix[0, 0])
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@property
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def shear(self):
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beta = math.atan2(- self._matrix[0, 1], self._matrix[1, 1])
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return beta - self.rotation
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@property
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def translation(self):
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return self._matrix[0:2, 2]
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class SimilarityTransform(ProjectiveTransform):
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"""2D similarity transformation of the form::
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X = a0*x + b0*y + a1 =
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= m*x*cos(rotation) + m*y*sin(rotation) + a1
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Y = b0*x + a0*y + b1 =
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= m*x*sin(rotation) + m*y*cos(rotation) + b1
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where ``m`` is a zoom factor and the homogeneous transformation matrix is::
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[[a0 b0 a1]
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[b0 a0 b1]
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[0 0 1]]
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Parameters
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----------
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matrix : (3, 3) array, optional
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Homogeneous transformation matrix.
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"""
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def estimate(self, src, dst):
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"""Set the transformation matrix with the explicit parameters.
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You can determine the over-, well- and under-determined parameters
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with the total least-squares method.
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Number of source and destination coordinates must match.
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Parameters
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----------
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src : (N, 2) array
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source coordinates
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dst : (N, 2) array
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destination coordinates
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"""
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xs = src[:, 0]
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ys = src[:, 1]
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xd = dst[:, 0]
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yd = dst[:, 1]
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rows = src.shape[0]
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# params: a0, a1, b0, b1
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A = np.zeros((rows * 2, 5))
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A[:rows, 0] = xs
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A[:rows, 2] = - ys
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A[:rows, 1] = 1
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A[rows:, 2] = xs
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A[rows:, 0] = ys
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A[rows:, 3] = 1
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A[:rows, 4] = xd
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A[rows:, 4] = yd
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_, _, V = np.linalg.svd(A)
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# solution is right singular vector that corresponds to smallest
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# singular value and normed by c3
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a0, a1, b0, b1 = - V[-1, :-1] / V[-1, -1]
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self._matrix = np.array([[a0, -b0, a1],
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[b0, a0, b1],
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[ 0, 0, 1]])
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def compose_implicit(self, scale=None, rotation=None, translation=None):
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"""Set the transformation matrix with the implicit transformation
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parameters.
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Parameters
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----------
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scale : float, optional
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scale factor
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rotation : float, optional
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rotation angle in counter-clockwise direction
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translation : (tx, ty) as array, list or tuple, optional
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x, y translation parameters
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"""
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if scale is None:
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scale = 1
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if rotation is None:
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rotation = 0
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if translation is None:
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translation = (0, 0)
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self._matrix = np.array([
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[math.cos(rotation), - math.sin(rotation), 0],
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[math.sin(rotation), math.cos(rotation), 0],
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[ 0, 0, 1]
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])
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self._matrix *= scale
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self._matrix[0:2, 2] = translation
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@property
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def scale(self):
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if math.cos(self.rotation) == 0:
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# sin(self.rotation) == 1
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scale = self._matrix[0, 1]
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else:
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scale = self._matrix[0, 0] / math.cos(self.rotation)
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return scale
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@property
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def rotation(self):
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return math.atan2(self._matrix[1, 0], self._matrix[1, 1])
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@property
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def translation(self):
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return self._matrix[0:2, 2]
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class PolynomialTransform(GeometricTransform):
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"""2D transformation of the form::
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X = sum[j=0:order]( sum[i=0:j]( a_ji * x**(j - i) * y**i ))
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Y = sum[j=0:order]( sum[i=0:j]( b_ji * x**(j - i) * y**i ))
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Parameters
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----------
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params : (2, N) array, optional
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Polynomial coefficients where `N * 2 = (order + 1) * (order + 2)`. So,
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a_ji is defined in `params[0, :]` and b_ji in `params[1, :]`.
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"""
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def __init__(self, params=None):
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self._params = params
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def estimate(self, src, dst, order):
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"""Set the transformation matrix with the explicit transformation
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parameters.
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You can determine the over-, well- and under-determined parameters
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with the total least-squares method.
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Number of source and destination coordinates must match.
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Parameters
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----------
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src : (N, 2) array
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source coordinates
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dst : (N, 2) array
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destination coordinates
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order : int
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polynomial order (number of coefficients is order + 1)
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"""
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xs = src[:, 0]
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ys = src[:, 1]
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xd = dst[:, 0]
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yd = dst[:, 1]
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rows = src.shape[0]
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# number of unknown polynomial coefficients
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u = (order + 1) * (order + 2)
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A = np.zeros((rows * 2, u + 1))
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pidx = 0
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for j in range(order + 1):
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for i in range(j + 1):
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A[:rows, pidx] = xs ** (j - i) * ys ** i
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A[rows:, pidx + u / 2] = xs ** (j - i) * ys ** i
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pidx += 1
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A[:rows, -1] = xd
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A[rows:, -1] = yd
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_, _, V = np.linalg.svd(A)
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# solution is right singular vector that corresponds to smallest
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# singular value and normed by c3
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params = - V[-1, :-1] / V[-1, -1]
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self._params = params.reshape((2, u / 2))
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def __call__(self, coords):
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"""Apply forward transformation.
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Parameters
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----------
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coords : (N, 2) array
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source coordinates
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Returns
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-------
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coords : (N, 2) array
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transformed coordinates
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"""
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x = coords[:, 0]
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y = coords[:, 1]
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u = len(self._params.ravel())
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# number of coefficients -> u = (order + 1) * (order + 2)
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order = int((- 3 + math.sqrt(9 - 4 * (2 - u))) / 2)
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dst = np.zeros(coords.shape)
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pidx = 0
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for j in range(order + 1):
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for i in range(j + 1):
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dst[:, 0] += self._params[0, pidx] * x ** (j - i) * y ** i
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dst[:, 1] += self._params[1, pidx] * x ** (j - i) * y ** i
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pidx += 1
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return dst
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def inverse(self, coords):
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raise Exception(
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'There is no explicit way to do the inverse polynomial '
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'transformation. Instead, estimate the inverse transformation '
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'parameters by exchanging source and destination coordinates,'
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'then apply the forward transformation.')
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TRANSFORMATIONS = {
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'similarity': SimilarityTransform,
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'affine': AffineTransform,
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'projective': ProjectiveTransform,
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'polynomial': PolynomialTransform,
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}
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def estimate_transform(ttype, src, dst, **kwargs):
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"""Estimate 2D geometric transformation parameters.
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You can determine the over-, well- and under-determined parameters
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with the total least-squares method.
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Number of source and destination coordinates must match.
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Parameters
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----------
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ttype : {'similarity', 'affine', 'projective', 'polynomial'}
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Type of transform.
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kwargs : array or int
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Function parameters (src, dst, n, angle)::
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NAME / TTYPE FUNCTION PARAMETERS
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'similarity' `src, `dst`
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'affine' `src, `dst`
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'projective' `src, `dst`
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'polynomial' `src, `dst`, `order` (polynomial order)
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Also see examples below.
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Returns
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-------
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tform : :class:`GeometricTransform`
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Transform object containing the transformation parameters and providing
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access to forward and inverse transformation functions.
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Examples
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--------
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>>> import numpy as np
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>>> from skimage import transform as tf
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>>> # estimate transformation parameters
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>>> src = np.array([0, 0, 10, 10]).reshape((2, 2))
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>>> dst = np.array([12, 14, 1, -20]).reshape((2, 2))
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>>> tform = tf.estimate_transform('similarity', src, dst)
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>>> tform.inverse(tform(src)) # == src
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>>> # warp image using the estimated transformation
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>>> from skimage import data
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>>> image = data.camera()
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>>> warp(image, inverse_map=tform.inverse)
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>>> # create transformation with explicit parameters
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>>> tform2 = tf.SimilarityTransform()
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>>> tform2.compose_implicit(scale=1.1, rotation=1, translation=(10, 20))
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>>> # unite transformations, applied in order from left to right
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>>> tform3 = tform + tform2
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>>> tform3(src) # == tform2(tform(src))
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"""
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ttype = ttype.lower()
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if ttype not in TRANSFORMATIONS:
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raise ValueError('the transformation type \'%s\' is not'
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'implemented' % ttype)
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tform = TRANSFORMATIONS[ttype]()
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tform.estimate(src, dst, **kwargs)
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return tform
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def matrix_transform(coords, matrix):
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"""Apply 2D matrix transform.
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Parameters
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----------
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coords : (N, 2) array
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x, y coordinates to transform
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matrix : (3, 3) array
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Homogeneous transformation matrix.
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Returns
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-------
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coords : (N, 2) array
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transformed coordinates
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"""
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return ProjectiveTransform(matrix)(coords)
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def warp(image, inverse_map=None, map_args={}, output_shape=None, order=1,
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mode='constant', cval=0., reverse_map=None):
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"""Warp an image according to a given coordinate transformation.
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Parameters
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----------
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image : 2-D array
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Input image.
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inverse_map : transformation object, callable xy = f(xy, **kwargs)
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Inverse coordinate map. A function that transforms a (N, 2) array of
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``(x, y)`` coordinates in the *output image* into their corresponding
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coordinates in the *source image*. In case of a transformation object
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its `inverse` method will be used as transformation function. Also see
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examples below.
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map_args : dict, optional
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Keyword arguments passed to `inverse_map`.
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output_shape : tuple (rows, cols)
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Shape of the output image generated.
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order : int
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Order of splines used in interpolation. See
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`scipy.ndimage.map_coordinates` for detail.
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mode : string
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How to handle values outside the image borders. See
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`scipy.ndimage.map_coordinates` for detail.
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cval : string
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Used in conjunction with mode 'constant', the value outside
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the image boundaries.
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Examples
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--------
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Shift an image to the right:
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>>> from skimage import data
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>>> image = data.camera()
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>>>
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>>> def shift_right(xy):
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... xy[:, 0] -= 10
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... return xy
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>>>
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>>> warp(image, shift_right)
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"""
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# Backward API compatibility
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if reverse_map is not None:
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inverse_map = reverse_map
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if image.ndim < 2:
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raise ValueError("Input must have more than 1 dimension.")
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image = np.atleast_3d(img_as_float(image))
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ishape = np.array(image.shape)
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bands = ishape[2]
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if output_shape is None:
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output_shape = ishape
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coords = np.empty(np.r_[3, output_shape], dtype=float)
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## Construct transformed coordinates
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rows, cols = output_shape[:2]
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# Reshape grid coordinates into a (P, 2) array of (x, y) pairs
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tf_coords = np.indices((cols, rows), dtype=float).reshape(2, -1).T
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# Map each (x, y) pair to the source image according to
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# the user-provided mapping
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if callable(getattr(inverse_map, 'inverse', None)):
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inverse_map = inverse_map.inverse
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tf_coords = inverse_map(tf_coords, **map_args)
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# Reshape back to a (2, M, N) coordinate grid
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tf_coords = tf_coords.T.reshape((-1, cols, rows)).swapaxes(1, 2)
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# Place the y-coordinate mapping
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_stackcopy(coords[1, ...], tf_coords[0, ...])
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# Place the x-coordinate mapping
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_stackcopy(coords[0, ...], tf_coords[1, ...])
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# colour-coordinate mapping
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coords[2, ...] = range(bands)
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# Prefilter not necessary for order 1 interpolation
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prefilter = order > 1
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mapped = ndimage.map_coordinates(image, coords, prefilter=prefilter,
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mode=mode, order=order, cval=cval)
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# The spline filters sometimes return results outside [0, 1],
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# so clip to ensure valid data
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return np.clip(mapped.squeeze(), 0, 1)
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