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from itertools import combinations_with_replacement
import numpy as np
from scipy import ndimage as ndi
from scipy import stats
from ..util import img_as_float
from ..feature import peak_local_max
from ..feature.util import _prepare_grayscale_input_2D
from ..feature.corner_cy import _corner_fast
from ._hessian_det_appx import _hessian_matrix_det
from ..transform import integral_image
from .._shared.utils import safe_as_int
from .corner_cy import _corner_moravec, _corner_orientations
from warnings import warn
def _compute_derivatives(image, mode='constant', cval=0):
"""Compute derivatives in x and y direction using the Sobel operator.
Parameters
----------
image : ndarray
Input image.
mode : {'constant', 'reflect', 'wrap', 'nearest', 'mirror'}, optional
How to handle values outside the image borders.
cval : float, optional
Used in conjunction with mode 'constant', the value outside
the image boundaries.
Returns
-------
imx : ndarray
Derivative in x-direction.
imy : ndarray
Derivative in y-direction.
"""
imy = ndi.sobel(image, axis=0, mode=mode, cval=cval)
imx = ndi.sobel(image, axis=1, mode=mode, cval=cval)
return imx, imy
def structure_tensor(image, sigma=1, mode='constant', cval=0):
"""Compute structure tensor using sum of squared differences.
The structure tensor A is defined as::
A = [Axx Axy]
[Axy Ayy]
which is approximated by the weighted sum of squared differences in a local
window around each pixel in the image.
Parameters
----------
image : ndarray
Input image.
sigma : float, optional
Standard deviation used for the Gaussian kernel, which is used as a
weighting function for the local summation of squared differences.
mode : {'constant', 'reflect', 'wrap', 'nearest', 'mirror'}, optional
How to handle values outside the image borders.
cval : float, optional
Used in conjunction with mode 'constant', the value outside
the image boundaries.
Returns
-------
Axx : ndarray
Element of the structure tensor for each pixel in the input image.
Axy : ndarray
Element of the structure tensor for each pixel in the input image.
Ayy : ndarray
Element of the structure tensor for each pixel in the input image.
Examples
--------
>>> from skimage.feature import structure_tensor
>>> square = np.zeros((5, 5))
>>> square[2, 2] = 1
>>> Axx, Axy, Ayy = structure_tensor(square, sigma=0.1)
>>> Axx
array([[ 0., 0., 0., 0., 0.],
[ 0., 1., 0., 1., 0.],
[ 0., 4., 0., 4., 0.],
[ 0., 1., 0., 1., 0.],
[ 0., 0., 0., 0., 0.]])
"""
image = _prepare_grayscale_input_2D(image)
imx, imy = _compute_derivatives(image, mode=mode, cval=cval)
# structure tensore
Axx = ndi.gaussian_filter(imx * imx, sigma, mode=mode, cval=cval)
Axy = ndi.gaussian_filter(imx * imy, sigma, mode=mode, cval=cval)
Ayy = ndi.gaussian_filter(imy * imy, sigma, mode=mode, cval=cval)
return Axx, Axy, Ayy
def hessian_matrix(image, sigma=1, mode='constant', cval=0, order=None):
"""Compute Hessian matrix.
The Hessian matrix is defined as::
H = [Hrr Hrc]
[Hrc Hcc]
which is computed by convolving the image with the second derivatives
of the Gaussian kernel in the respective x- and y-directions.
Parameters
----------
image : ndarray
Input image.
sigma : float
Standard deviation used for the Gaussian kernel, which is used as
weighting function for the auto-correlation matrix.
mode : {'constant', 'reflect', 'wrap', 'nearest', 'mirror'}, optional
How to handle values outside the image borders.
cval : float, optional
Used in conjunction with mode 'constant', the value outside
the image boundaries.
order : {'xy', 'rc'}, optional
This parameter allows for the use of reverse or forward order of
the image axes in gradient computation. 'xy' indicates the usage
of the last axis initially (Hxx, Hxy, Hyy), whilst 'rc' indicates
the use of the first axis initially (Hrr, Hrc, Hcc).
Returns
-------
Hrr : ndarray
Element of the Hessian matrix for each pixel in the input image.
Hrc : ndarray
Element of the Hessian matrix for each pixel in the input image.
Hcc : ndarray
Element of the Hessian matrix for each pixel in the input image.
Examples
--------
>>> from skimage.feature import hessian_matrix
>>> square = np.zeros((5, 5))
>>> square[2, 2] = 4
>>> Hrr, Hrc, Hcc = hessian_matrix(square, sigma=0.1, order = 'rc')
>>> Hrc
array([[ 0., 0., 0., 0., 0.],
[ 0., 1., 0., -1., 0.],
[ 0., 0., 0., 0., 0.],
[ 0., -1., 0., 1., 0.],
[ 0., 0., 0., 0., 0.]])
"""
image = img_as_float(image)
gaussian_filtered = ndi.gaussian_filter(image, sigma=sigma,
mode=mode, cval=cval)
if order is None:
if image.ndim == 2:
# The legacy 2D code followed (x, y) convention, so we swap the axis
# order to maintain compatibility with old code
warn('deprecation warning: the default order of the hessian matrix values '
'will be "row-column" instead of "xy" starting in skimage version 0.15. '
'Use order="rc" or order="xy" to set this explicitly')
order = 'xy'
else:
order = 'rc'
gradients = np.gradient(gaussian_filtered)
axes = range(image.ndim)
if order == 'rc':
axes = reversed(axes)
H_elems = [np.gradient(gradients[ax0], axis=ax1)
for ax0, ax1 in combinations_with_replacement(axes, 2)]
return H_elems
def _hessian_matrix_image(H_elems):
"""Convert the upper-diagonal elements of the Hessian matrix to a matrix.
Parameters
----------
H_elems : list of array
The upper-diagonal elements of the Hessian matrix, as returned by
`hessian_matrix`.
Returns
-------
hessian_image : array
An array of shape ``(M, N[, ...], image.ndim, image.ndim)``,
containing the Hessian matrix corresponding to each coordinate.
"""
image = H_elems[0]
hessian_image = np.zeros(image.shape + (image.ndim, image.ndim))
for idx, (row, col) in \
enumerate(combinations_with_replacement(range(image.ndim), 2)):
hessian_image[..., row, col] = H_elems[idx]
hessian_image[..., col, row] = H_elems[idx]
return hessian_image
def hessian_matrix_det(image, sigma=1, approximate=True):
"""Compute the approximate Hessian Determinant over an image.
The 2D approximate method uses box filters over integral images to
compute the approximate Hessian Determinant, as described in [1]_.
Parameters
----------
image : array
The image over which to compute Hessian Determinant.
sigma : float, optional
Standard deviation used for the Gaussian kernel, used for the Hessian
matrix.
approximate : bool, optional
If ``True`` and the image is 2D, use a much faster approximate
computation. This argument has no effect on 3D and higher images.
Returns
-------
out : array
The array of the Determinant of Hessians.
References
----------
.. [1] Herbert Bay, Andreas Ess, Tinne Tuytelaars, Luc Van Gool,
"SURF: Speeded Up Robust Features"
ftp://ftp.vision.ee.ethz.ch/publications/articles/eth_biwi_00517.pdf
Notes
-----
For 2D images when ``approximate=True``, the running time of this method
only depends on size of the image. It is independent of `sigma` as one
would expect. The downside is that the result for `sigma` less than `3`
is not accurate, i.e., not similar to the result obtained if someone
computed the Hessian and took its determinant.
"""
image = img_as_float(image)
if image.ndim == 2 and approximate:
integral = integral_image(image)
return np.array(_hessian_matrix_det(integral, sigma))
else: # slower brute-force implementation for nD images
hessian_mat_array = _hessian_matrix_image(hessian_matrix(image, sigma))
return np.linalg.det(hessian_mat_array)
def _image_orthogonal_matrix22_eigvals(M00, M01, M11):
l1 = (M00 + M11) / 2 + np.sqrt(4 * M01 ** 2 + (M00 - M11) ** 2) / 2
l2 = (M00 + M11) / 2 - np.sqrt(4 * M01 ** 2 + (M00 - M11) ** 2) / 2
return l1, l2
def structure_tensor_eigvals(Axx, Axy, Ayy):
"""Compute Eigen values of structure tensor.
Parameters
----------
Axx : ndarray
Element of the structure tensor for each pixel in the input image.
Axy : ndarray
Element of the structure tensor for each pixel in the input image.
Ayy : ndarray
Element of the structure tensor for each pixel in the input image.
Returns
-------
l1 : ndarray
Larger eigen value for each input matrix.
l2 : ndarray
Smaller eigen value for each input matrix.
Examples
--------
>>> from skimage.feature import structure_tensor, structure_tensor_eigvals
>>> square = np.zeros((5, 5))
>>> square[2, 2] = 1
>>> Axx, Axy, Ayy = structure_tensor(square, sigma=0.1)
>>> structure_tensor_eigvals(Axx, Axy, Ayy)[0]
array([[ 0., 0., 0., 0., 0.],
[ 0., 2., 4., 2., 0.],
[ 0., 4., 0., 4., 0.],
[ 0., 2., 4., 2., 0.],
[ 0., 0., 0., 0., 0.]])
"""
return _image_orthogonal_matrix22_eigvals(Axx, Axy, Ayy)
def hessian_matrix_eigvals(H_elems, Hxy=None, Hyy=None, Hxx=None):
"""Compute Eigenvalues of Hessian matrix.
Parameters
----------
H_elems : list of ndarray
The upper-diagonal elements of the Hessian matrix, as returned
by `hessian_matrix`.
Hxy : ndarray, deprecated
Element of the Hessian matrix for each pixel in the input image.
Hyy : ndarray, deprecated
Element of the Hessian matrix for each pixel in the input image.
Hxx : ndarray, deprecated
Element of the Hessian matrix for each pixel in the input image.
Returns
-------
eigs : ndarray
The eigenvalues of the Hessian matrix, in decreasing order. The
eigenvalues are the leading dimension. That is, ``eigs[i, j, k]``
contains the ith-largest eigenvalue at position (j, k).
Examples
--------
>>> from skimage.feature import hessian_matrix, hessian_matrix_eigvals
>>> square = np.zeros((5, 5))
>>> square[2, 2] = 4
>>> H_elems = hessian_matrix(square, sigma=0.1, order='rc')
>>> hessian_matrix_eigvals(H_elems)[0]
array([[ 0., 0., 2., 0., 0.],
[ 0., 1., 0., 1., 0.],
[ 2., 0., -2., 0., 2.],
[ 0., 1., 0., 1., 0.],
[ 0., 0., 2., 0., 0.]])
"""
if Hxy is not None:
if Hxx is None:
Hxx = H_elems
H_elems = [Hxx, Hxy, Hyy]
warn('The API of `hessian_matrix_eigvals` has changed. Use a list of '
'elements instead of separate arguments. The old version of the '
'API will be removed in version 0.16.')
if len(H_elems) == 3: # Use fast Cython code for 2D
eigvals = np.array(_image_orthogonal_matrix22_eigvals(*H_elems))
else:
matrices = _hessian_matrix_image(H_elems)
# eigvalsh returns eigenvalues in increasing order. We want decreasing
eigvals = np.linalg.eigvalsh(matrices)[..., ::-1]
leading_axes = tuple(range(eigvals.ndim - 1))
eigvals = np.transpose(eigvals, (eigvals.ndim - 1,) + leading_axes)
return eigvals
def shape_index(image, sigma=1, mode='constant', cval=0):
"""Compute the shape index.
The shape index, as defined by Koenderink & van Doorn [1]_, is a
single valued measure of local curvature, assuming the image as a 3D plane
with intensities representing heights.
It is derived from the eigen values of the Hessian, and its
value ranges from -1 to 1 (and is undefined (=NaN) in *flat* regions),
with following ranges representing following shapes:
.. table:: Ranges of the shape index and corresponding shapes.
=================== =============
Interval (s in ...) Shape
=================== =============
[ -1, -7/8) Spherical cup
[-7/8, -5/8) Through
[-5/8, -3/8) Rut
[-3/8, -1/8) Saddle rut
[-1/8, +1/8) Saddle
[+1/8, +3/8) Saddle ridge
[+3/8, +5/8) Ridge
[+5/8, +7/8) Dome
[+7/8, +1] Spherical cap
=================== =============
Parameters
----------
image : ndarray
Input image.
sigma : float, optional
Standard deviation used for the Gaussian kernel, which is used for
smoothing the input data before Hessian eigen value calculation.
mode : {'constant', 'reflect', 'wrap', 'nearest', 'mirror'}, optional
How to handle values outside the image borders
cval : float, optional
Used in conjunction with mode 'constant', the value outside
the image boundaries.
Returns
-------
s : ndarray
Shape index
References
----------
.. [1] Koenderink, J. J. & van Doorn, A. J.,
"Surface shape and curvature scales",
Image and Vision Computing, 1992, 10, 557-564.
:DOI:`10.1016/0262-8856(92)90076-F`
Examples
--------
>>> from skimage.feature import shape_index
>>> square = np.zeros((5, 5))
>>> square[2, 2] = 4
>>> s = shape_index(square, sigma=0.1)
>>> s
array([[ nan, nan, -0.5, nan, nan],
[ nan, -0. , nan, -0. , nan],
[-0.5, nan, -1. , nan, -0.5],
[ nan, -0. , nan, -0. , nan],
[ nan, nan, -0.5, nan, nan]])
"""
H = hessian_matrix(image, sigma=sigma, mode=mode, cval=cval, order='rc')
l1, l2 = hessian_matrix_eigvals(H)
return (2.0 / np.pi) * np.arctan((l2 + l1) / (l2 - l1))
def corner_kitchen_rosenfeld(image, mode='constant', cval=0):
"""Compute Kitchen and Rosenfeld corner measure response image.
The corner measure is calculated as follows::
(imxx * imy**2 + imyy * imx**2 - 2 * imxy * imx * imy)
/ (imx**2 + imy**2)
Where imx and imy are the first and imxx, imxy, imyy the second
derivatives.
Parameters
----------
image : ndarray
Input image.
mode : {'constant', 'reflect', 'wrap', 'nearest', 'mirror'}, optional
How to handle values outside the image borders.
cval : float, optional
Used in conjunction with mode 'constant', the value outside
the image boundaries.
Returns
-------
response : ndarray
Kitchen and Rosenfeld response image.
"""
imx, imy = _compute_derivatives(image, mode=mode, cval=cval)
imxx, imxy = _compute_derivatives(imx, mode=mode, cval=cval)
imyx, imyy = _compute_derivatives(imy, mode=mode, cval=cval)
numerator = (imxx * imy ** 2 + imyy * imx ** 2 - 2 * imxy * imx * imy)
denominator = (imx ** 2 + imy ** 2)
response = np.zeros_like(image, dtype=np.double)
mask = denominator != 0
response[mask] = numerator[mask] / denominator[mask]
return response
def corner_harris(image, method='k', k=0.05, eps=1e-6, sigma=1):
"""Compute Harris corner measure response image.
This corner detector uses information from the auto-correlation matrix A::
A = [(imx**2) (imx*imy)] = [Axx Axy]
[(imx*imy) (imy**2)] [Axy Ayy]
Where imx and imy are first derivatives, averaged with a gaussian filter.
The corner measure is then defined as::
det(A) - k * trace(A)**2
or::
2 * det(A) / (trace(A) + eps)
Parameters
----------
image : ndarray
Input image.
method : {'k', 'eps'}, optional
Method to compute the response image from the auto-correlation matrix.
k : float, optional
Sensitivity factor to separate corners from edges, typically in range
`[0, 0.2]`. Small values of k result in detection of sharp corners.
eps : float, optional
Normalisation factor (Noble's corner measure).
sigma : float, optional
Standard deviation used for the Gaussian kernel, which is used as
weighting function for the auto-correlation matrix.
Returns
-------
response : ndarray
Harris response image.
References
----------
.. [1] http://kiwi.cs.dal.ca/~dparks/CornerDetection/harris.htm
.. [2] http://en.wikipedia.org/wiki/Corner_detection
Examples
--------
>>> from skimage.feature import corner_harris, corner_peaks
>>> square = np.zeros([10, 10])
>>> square[2:8, 2:8] = 1
>>> square.astype(int)
array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])
>>> corner_peaks(corner_harris(square), min_distance=1)
array([[2, 2],
[2, 7],
[7, 2],
[7, 7]])
"""
Axx, Axy, Ayy = structure_tensor(image, sigma)
# determinant
detA = Axx * Ayy - Axy ** 2
# trace
traceA = Axx + Ayy
if method == 'k':
response = detA - k * traceA ** 2
else:
response = 2 * detA / (traceA + eps)
return response
def corner_shi_tomasi(image, sigma=1):
"""Compute Shi-Tomasi (Kanade-Tomasi) corner measure response image.
This corner detector uses information from the auto-correlation matrix A::
A = [(imx**2) (imx*imy)] = [Axx Axy]
[(imx*imy) (imy**2)] [Axy Ayy]
Where imx and imy are first derivatives, averaged with a gaussian filter.
The corner measure is then defined as the smaller eigenvalue of A::
((Axx + Ayy) - sqrt((Axx - Ayy)**2 + 4 * Axy**2)) / 2
Parameters
----------
image : ndarray
Input image.
sigma : float, optional
Standard deviation used for the Gaussian kernel, which is used as
weighting function for the auto-correlation matrix.
Returns
-------
response : ndarray
Shi-Tomasi response image.
References
----------
.. [1] http://kiwi.cs.dal.ca/~dparks/CornerDetection/harris.htm
.. [2] http://en.wikipedia.org/wiki/Corner_detection
Examples
--------
>>> from skimage.feature import corner_shi_tomasi, corner_peaks
>>> square = np.zeros([10, 10])
>>> square[2:8, 2:8] = 1
>>> square.astype(int)
array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])
>>> corner_peaks(corner_shi_tomasi(square), min_distance=1)
array([[2, 2],
[2, 7],
[7, 2],
[7, 7]])
"""
Axx, Axy, Ayy = structure_tensor(image, sigma)
# minimum eigenvalue of A
response = ((Axx + Ayy) - np.sqrt((Axx - Ayy) ** 2 + 4 * Axy ** 2)) / 2
return response
def corner_foerstner(image, sigma=1):
"""Compute Foerstner corner measure response image.
This corner detector uses information from the auto-correlation matrix A::
A = [(imx**2) (imx*imy)] = [Axx Axy]
[(imx*imy) (imy**2)] [Axy Ayy]
Where imx and imy are first derivatives, averaged with a gaussian filter.
The corner measure is then defined as::
w = det(A) / trace(A) (size of error ellipse)
q = 4 * det(A) / trace(A)**2 (roundness of error ellipse)
Parameters
----------
image : ndarray
Input image.
sigma : float, optional
Standard deviation used for the Gaussian kernel, which is used as
weighting function for the auto-correlation matrix.
Returns
-------
w : ndarray
Error ellipse sizes.
q : ndarray
Roundness of error ellipse.
References
----------
.. [1] http://www.ipb.uni-bonn.de/uploads/tx_ikgpublication/foerstner87.fast.pdf
.. [2] http://en.wikipedia.org/wiki/Corner_detection
Examples
--------
>>> from skimage.feature import corner_foerstner, corner_peaks
>>> square = np.zeros([10, 10])
>>> square[2:8, 2:8] = 1
>>> square.astype(int)
array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 1, 1, 1, 1, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])
>>> w, q = corner_foerstner(square)
>>> accuracy_thresh = 0.5
>>> roundness_thresh = 0.3
>>> foerstner = (q > roundness_thresh) * (w > accuracy_thresh) * w
>>> corner_peaks(foerstner, min_distance=1)
array([[2, 2],
[2, 7],
[7, 2],
[7, 7]])
"""
Axx, Axy, Ayy = structure_tensor(image, sigma)
# determinant
detA = Axx * Ayy - Axy ** 2
# trace
traceA = Axx + Ayy
w = np.zeros_like(image, dtype=np.double)
q = np.zeros_like(image, dtype=np.double)
mask = traceA != 0
w[mask] = detA[mask] / traceA[mask]
q[mask] = 4 * detA[mask] / traceA[mask] ** 2
return w, q
def corner_fast(image, n=12, threshold=0.15):
"""Extract FAST corners for a given image.
Parameters
----------
image : 2D ndarray
Input image.
n : int
Minimum number of consecutive pixels out of 16 pixels on the circle
that should all be either brighter or darker w.r.t testpixel.
A point c on the circle is darker w.r.t test pixel p if
`Ic < Ip - threshold` and brighter if `Ic > Ip + threshold`. Also
stands for the n in `FAST-n` corner detector.
threshold : float
Threshold used in deciding whether the pixels on the circle are
brighter, darker or similar w.r.t. the test pixel. Decrease the
threshold when more corners are desired and vice-versa.
Returns
-------
response : ndarray
FAST corner response image.
References
----------
.. [1] Edward Rosten and Tom Drummond
"Machine Learning for high-speed corner detection",
http://www.edwardrosten.com/work/rosten_2006_machine.pdf
.. [2] Wikipedia, "Features from accelerated segment test",
https://en.wikipedia.org/wiki/Features_from_accelerated_segment_test
Examples
--------
>>> from skimage.feature import corner_fast, corner_peaks
>>> square = np.zeros((12, 12))
>>> square[3:9, 3:9] = 1
>>> square.astype(int)
array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])
>>> corner_peaks(corner_fast(square, 9), min_distance=1)
array([[3, 3],
[3, 8],
[8, 3],
[8, 8]])
"""
image = _prepare_grayscale_input_2D(image)
image = np.ascontiguousarray(image)
response = _corner_fast(image, n, threshold)
return response
def corner_subpix(image, corners, window_size=11, alpha=0.99):
"""Determine subpixel position of corners.
A statistical test decides whether the corner is defined as the
intersection of two edges or a single peak. Depending on the classification
result, the subpixel corner location is determined based on the local
covariance of the grey-values. If the significance level for either
statistical test is not sufficient, the corner cannot be classified, and
the output subpixel position is set to NaN.
Parameters
----------
image : ndarray
Input image.
corners : (N, 2) ndarray
Corner coordinates `(row, col)`.
window_size : int, optional
Search window size for subpixel estimation.
alpha : float, optional
Significance level for corner classification.
Returns
-------
positions : (N, 2) ndarray
Subpixel corner positions. NaN for "not classified" corners.
References
----------
.. [1] http://www.ipb.uni-bonn.de/uploads/tx_ikgpublication/\
foerstner87.fast.pdf
.. [2] http://en.wikipedia.org/wiki/Corner_detection
Examples
--------
>>> from skimage.feature import corner_harris, corner_peaks, corner_subpix
>>> img = np.zeros((10, 10))
>>> img[:5, :5] = 1
>>> img[5:, 5:] = 1
>>> img.astype(int)
array([[1, 1, 1, 1, 1, 0, 0, 0, 0, 0],
[1, 1, 1, 1, 1, 0, 0, 0, 0, 0],
[1, 1, 1, 1, 1, 0, 0, 0, 0, 0],
[1, 1, 1, 1, 1, 0, 0, 0, 0, 0],
[1, 1, 1, 1, 1, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 1, 1, 1, 1],
[0, 0, 0, 0, 0, 1, 1, 1, 1, 1],
[0, 0, 0, 0, 0, 1, 1, 1, 1, 1],
[0, 0, 0, 0, 0, 1, 1, 1, 1, 1],
[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]])
>>> coords = corner_peaks(corner_harris(img), min_distance=2)
>>> coords_subpix = corner_subpix(img, coords, window_size=7)
>>> coords_subpix
array([[ 4.5, 4.5]])
"""
# window extent in one direction
wext = (window_size - 1) // 2
image = np.pad(image, pad_width=wext, mode='constant', constant_values=0)
# add pad width, make sure to not modify the input values in-place
corners = safe_as_int(corners + wext)
# normal equation arrays
N_dot = np.zeros((2, 2), dtype=np.double)
N_edge = np.zeros((2, 2), dtype=np.double)
b_dot = np.zeros((2, ), dtype=np.double)
b_edge = np.zeros((2, ), dtype=np.double)
# critical statistical test values
redundancy = window_size ** 2 - 2
t_crit_dot = stats.f.isf(1 - alpha, redundancy, redundancy)
t_crit_edge = stats.f.isf(alpha, redundancy, redundancy)
# coordinates of pixels within window
y, x = np.mgrid[- wext:wext + 1, - wext:wext + 1]
corners_subpix = np.zeros_like(corners, dtype=np.double)
for i, (y0, x0) in enumerate(corners):
# crop window around corner + border for sobel operator
miny = y0 - wext - 1
maxy = y0 + wext + 2
minx = x0 - wext - 1
maxx = x0 + wext + 2
window = image[miny:maxy, minx:maxx]
winx, winy = _compute_derivatives(window, mode='constant', cval=0)
# compute gradient suares and remove border
winx_winx = (winx * winx)[1:-1, 1:-1]
winx_winy = (winx * winy)[1:-1, 1:-1]
winy_winy = (winy * winy)[1:-1, 1:-1]
# sum of squared differences (mean instead of gaussian filter)
Axx = np.sum(winx_winx)
Axy = np.sum(winx_winy)
Ayy = np.sum(winy_winy)
# sum of squared differences weighted with coordinates
# (mean instead of gaussian filter)
bxx_x = np.sum(winx_winx * x)
bxx_y = np.sum(winx_winx * y)
bxy_x = np.sum(winx_winy * x)
bxy_y = np.sum(winx_winy * y)
byy_x = np.sum(winy_winy * x)
byy_y = np.sum(winy_winy * y)
# normal equations for subpixel position
N_dot[0, 0] = Axx
N_dot[0, 1] = N_dot[1, 0] = - Axy
N_dot[1, 1] = Ayy
N_edge[0, 0] = Ayy
N_edge[0, 1] = N_edge[1, 0] = Axy
N_edge[1, 1] = Axx
b_dot[:] = bxx_y - bxy_x, byy_x - bxy_y
b_edge[:] = byy_y + bxy_x, bxx_x + bxy_y
# estimated positions
try:
est_dot = np.linalg.solve(N_dot, b_dot)
est_edge = np.linalg.solve(N_edge, b_edge)
except np.linalg.LinAlgError:
# if image is constant the system is singular
corners_subpix[i, :] = np.nan, np.nan
continue
# residuals
ry_dot = y - est_dot[0]
rx_dot = x - est_dot[1]
ry_edge = y - est_edge[0]
rx_edge = x - est_edge[1]
# squared residuals
rxx_dot = rx_dot * rx_dot
rxy_dot = rx_dot * ry_dot
ryy_dot = ry_dot * ry_dot
rxx_edge = rx_edge * rx_edge
rxy_edge = rx_edge * ry_edge
ryy_edge = ry_edge * ry_edge
# determine corner class (dot or edge)
# variance for different models
var_dot = np.sum(winx_winx * ryy_dot - 2 * winx_winy * rxy_dot
+ winy_winy * rxx_dot)
var_edge = np.sum(winy_winy * ryy_edge + 2 * winx_winy * rxy_edge
+ winx_winx * rxx_edge)
# test value (F-distributed)
if var_dot < np.spacing(1) and var_edge < np.spacing(1):
t = np.nan
elif var_dot == 0:
t = np.inf
else:
t = var_edge / var_dot
# 1 for edge, -1 for dot, 0 for "not classified"
corner_class = int(t < t_crit_edge) - int(t > t_crit_dot)
if corner_class == -1:
corners_subpix[i, :] = y0 + est_dot[0], x0 + est_dot[1]
elif corner_class == 0:
corners_subpix[i, :] = np.nan, np.nan
elif corner_class == 1:
corners_subpix[i, :] = y0 + est_edge[0], x0 + est_edge[1]
# subtract pad width
corners_subpix -= wext
return corners_subpix
def corner_peaks(image, min_distance=1, threshold_abs=None, threshold_rel=0.1,
exclude_border=True, indices=True, num_peaks=np.inf,
footprint=None, labels=None):
"""Find corners in corner measure response image.
This differs from `skimage.feature.peak_local_max` in that it suppresses
multiple connected peaks with the same accumulator value.
Parameters
----------
* : *
See :py:meth:`skimage.feature.peak_local_max`.
Examples
--------
>>> from skimage.feature import peak_local_max
>>> response = np.zeros((5, 5))
>>> response[2:4, 2:4] = 1
>>> response
array([[ 0., 0., 0., 0., 0.],
[ 0., 0., 0., 0., 0.],
[ 0., 0., 1., 1., 0.],
[ 0., 0., 1., 1., 0.],
[ 0., 0., 0., 0., 0.]])
>>> peak_local_max(response)
array([[3, 3],
[3, 2],
[2, 3],
[2, 2]])
>>> corner_peaks(response)
array([[2, 2]])
"""
peaks = peak_local_max(image, min_distance=min_distance,
threshold_abs=threshold_abs,
threshold_rel=threshold_rel,
exclude_border=exclude_border,
indices=False, num_peaks=num_peaks,
footprint=footprint, labels=labels)
if min_distance > 0:
coords = np.transpose(peaks.nonzero())
for r, c in coords:
if peaks[r, c]:
peaks[r - min_distance:r + min_distance + 1,
c - min_distance:c + min_distance + 1] = False
peaks[r, c] = True
if indices is True:
return np.transpose(peaks.nonzero())
else:
return peaks
def corner_moravec(image, window_size=1):
"""Compute Moravec corner measure response image.
This is one of the simplest corner detectors and is comparatively fast but
has several limitations (e.g. not rotation invariant).
Parameters
----------
image : ndarray
Input image.
window_size : int, optional
Window size.
Returns
-------
response : ndarray
Moravec response image.
References
----------
.. [1] http://kiwi.cs.dal.ca/~dparks/CornerDetection/moravec.htm
.. [2] http://en.wikipedia.org/wiki/Corner_detection
Examples
--------
>>> from skimage.feature import corner_moravec
>>> square = np.zeros([7, 7])
>>> square[3, 3] = 1
>>> square.astype(int)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
>>> corner_moravec(square).astype(int)
array([[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 1, 2, 1, 0, 0],
[0, 0, 1, 1, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0]])
"""
return _corner_moravec(image, window_size)
def corner_orientations(image, corners, mask):
"""Compute the orientation of corners.
The orientation of corners is computed using the first order central moment
i.e. the center of mass approach. The corner orientation is the angle of
the vector from the corner coordinate to the intensity centroid in the
local neighborhood around the corner calculated using first order central
moment.
Parameters
----------
image : 2D array
Input grayscale image.
corners : (N, 2) array
Corner coordinates as ``(row, col)``.
mask : 2D array
Mask defining the local neighborhood of the corner used for the
calculation of the central moment.
Returns
-------
orientations : (N, 1) array
Orientations of corners in the range [-pi, pi].
References
----------
.. [1] Ethan Rublee, Vincent Rabaud, Kurt Konolige and Gary Bradski
"ORB : An efficient alternative to SIFT and SURF"
http://www.vision.cs.chubu.ac.jp/CV-R/pdf/Rublee_iccv2011.pdf
.. [2] Paul L. Rosin, "Measuring Corner Properties"
http://users.cs.cf.ac.uk/Paul.Rosin/corner2.pdf
Examples
--------
>>> from skimage.morphology import octagon
>>> from skimage.feature import (corner_fast, corner_peaks,
... corner_orientations)
>>> square = np.zeros((12, 12))
>>> square[3:9, 3:9] = 1
>>> square.astype(int)
array([[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]])
>>> corners = corner_peaks(corner_fast(square, 9), min_distance=1)
>>> corners
array([[3, 3],
[3, 8],
[8, 3],
[8, 8]])
>>> orientations = corner_orientations(square, corners, octagon(3, 2))
>>> np.rad2deg(orientations)
array([ 45., 135., -45., -135.])
"""
return _corner_orientations(image, corners, mask)