mirror of
https://github.com/wassname/scikit-image.git
synced 2026-06-27 22:23:29 +08:00
Juan Nunez-Iglesias
844 lines
26 KiB
Python
844 lines
26 KiB
Python
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, pad
|
|
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
|
|
|
|
|
|
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
|
|
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):
|
|
"""Compute Hessian matrix.
|
|
|
|
The Hessian matrix is defined as::
|
|
|
|
H = [Hxx Hxy]
|
|
[Hxy Hyy]
|
|
|
|
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.
|
|
|
|
Returns
|
|
-------
|
|
Hxx : ndarray
|
|
Element of the Hessian matrix for each pixel in the input image.
|
|
Hxy : ndarray
|
|
Element of the Hessian matrix for each pixel in the input image.
|
|
Hyy : 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
|
|
>>> Hxx, Hxy, Hyy = hessian_matrix(square, sigma=0.1)
|
|
>>> Hxy
|
|
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)
|
|
|
|
gradients = np.gradient(gaussian_filtered)
|
|
axes = range(image.ndim)
|
|
H_elems = [np.gradient(gradients[ax0], axis=ax1)
|
|
for ax0, ax1 in combinations_with_replacement(axes, 2)]
|
|
|
|
if image.ndim == 2:
|
|
# The legacy 2D code followed (x, y) convention, so we swap the axis
|
|
# order to maintain compatibility with old code
|
|
H_elems.reverse()
|
|
return H_elems
|
|
|
|
|
|
def hessian_matrix_det(image, sigma):
|
|
"""Computes the approximate Hessian Determinant over an image.
|
|
|
|
This 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
|
|
Standard deviation used for the Gaussian kernel, used for the Hessian
|
|
matrix.
|
|
|
|
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
|
|
-----
|
|
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 it's
|
|
determinant.
|
|
"""
|
|
|
|
image = img_as_float(image)
|
|
image = integral_image(image)
|
|
return np.array(_hessian_matrix_det(image, sigma))
|
|
|
|
|
|
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(Hxx, Hxy, Hyy):
|
|
"""Compute Eigen values of Hessian matrix.
|
|
|
|
Parameters
|
|
----------
|
|
Hxx : ndarray
|
|
Element of the Hessian matrix for each pixel in the input image.
|
|
Hxy : ndarray
|
|
Element of the Hessian matrix for each pixel in the input image.
|
|
Hyy : ndarray
|
|
Element of the Hessian matrix 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 hessian_matrix, hessian_matrix_eigvals
|
|
>>> square = np.zeros((5, 5))
|
|
>>> square[2, 2] = -1 / 1591.54943092
|
|
>>> Hxx, Hxy, Hyy = hessian_matrix(square, sigma=0.1)
|
|
>>> hessian_matrix_eigvals(Hxx, Hxy, Hyy)[0]
|
|
array([[ 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.]])
|
|
|
|
"""
|
|
|
|
return _image_orthogonal_matrix22_eigvals(Hxx, Hxy, Hyy)
|
|
|
|
|
|
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 = 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([[2, 2],
|
|
[2, 3],
|
|
[3, 2],
|
|
[3, 3]])
|
|
>>> 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
|