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import math
from itertools import combinations_with_replacement
import numpy as np
from scipy import ndimage as ndi
from scipy import spatial, stats
from .._shared.filters import gaussian
from .._shared.utils import _supported_float_type, safe_as_int, warn
from ..transform import integral_image
from ..util import img_as_float
from ._hessian_det_appx import _hessian_matrix_det
from .corner_cy import _corner_fast, _corner_moravec, _corner_orientations
from .peak import peak_local_max
from .util import _prepare_grayscale_input_2D, _prepare_grayscale_input_nD
def _compute_derivatives(image, mode='constant', cval=0):
"""Compute derivatives in axis directions 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
-------
derivatives : list of ndarray
Derivatives in each axis direction.
"""
derivatives = [
ndi.sobel(image, axis=i, mode=mode, cval=cval) for i in range(image.ndim)
]
return derivatives
def structure_tensor(image, sigma=1, mode='constant', cval=0, order='rc'):
"""Compute structure tensor using sum of squared differences.
The (2-dimensional) structure tensor A is defined as::
A = [Arr Arc]
[Arc Acc]
which is approximated by the weighted sum of squared differences in a local
window around each pixel in the image. This formula can be extended to a
larger number of dimensions (see [1]_).
Parameters
----------
image : ndarray
Input image.
sigma : float or array-like of float, optional
Standard deviation used for the Gaussian kernel, which is used as a
weighting function for the local summation of squared differences.
If sigma is an iterable, its length must be equal to `image.ndim` and
each element is used for the Gaussian kernel applied along its
respective axis.
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 : {'rc', 'xy'}, optional
NOTE: 'xy' is only an option for 2D images, higher dimensions must
always use 'rc' order. This parameter allows for the use of reverse or
forward order of the image axes in gradient computation. 'rc' indicates
the use of the first axis initially (Arr, Arc, Acc), whilst 'xy'
indicates the usage of the last axis initially (Axx, Axy, Ayy).
Returns
-------
A_elems : list of ndarray
Upper-diagonal elements 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
>>> Arr, Arc, Acc = structure_tensor(square, sigma=0.1, order='rc')
>>> Acc
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.]])
See also
--------
structure_tensor_eigenvalues
References
----------
.. [1] https://en.wikipedia.org/wiki/Structure_tensor
"""
if order == 'xy' and image.ndim > 2:
raise ValueError('Only "rc" order is supported for dim > 2.')
if order not in ['rc', 'xy']:
raise ValueError(f'order {order} is invalid. Must be either "rc" or "xy"')
if not np.isscalar(sigma):
sigma = tuple(sigma)
if len(sigma) != image.ndim:
raise ValueError('sigma must have as many elements as image ' 'has axes')
image = _prepare_grayscale_input_nD(image)
derivatives = _compute_derivatives(image, mode=mode, cval=cval)
if order == 'xy':
derivatives = reversed(derivatives)
# structure tensor
A_elems = [
gaussian(der0 * der1, sigma=sigma, mode=mode, cval=cval)
for der0, der1 in combinations_with_replacement(derivatives, 2)
]
return A_elems
def _hessian_matrix_with_gaussian(image, sigma=1, mode='reflect', cval=0, order='rc'):
"""Compute the Hessian via convolutions with Gaussian derivatives.
In 2D, 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 r- and c-directions.
The implementation here also supports n-dimensional data.
Parameters
----------
image : ndarray
Input image.
sigma : float or sequence of float, optional
Standard deviation used for the Gaussian kernel, which sets the
amount of smoothing in terms of pixel-distances. It is
advised to not choose a sigma much less than 1.0, otherwise
aliasing artifacts may occur.
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 : {'rc', 'xy'}, optional
This parameter allows for the use of reverse or forward order of
the image axes in gradient computation. 'rc' indicates the use of
the first axis initially (Hrr, Hrc, Hcc), whilst 'xy' indicates the
usage of the last axis initially (Hxx, Hxy, Hyy)
Returns
-------
H_elems : list of ndarray
Upper-diagonal elements of the hessian matrix for each pixel in the
input image. In 2D, this will be a three element list containing [Hrr,
Hrc, Hcc]. In nD, the list will contain ``(n**2 + n) / 2`` arrays.
"""
image = img_as_float(image)
float_dtype = _supported_float_type(image.dtype)
image = image.astype(float_dtype, copy=False)
if image.ndim > 2 and order == "xy":
raise ValueError("order='xy' is only supported for 2D images.")
if order not in ["rc", "xy"]:
raise ValueError(f"unrecognized order: {order}")
if np.isscalar(sigma):
sigma = (sigma,) * image.ndim
# This function uses `scipy.ndimage.gaussian_filter` with the order
# argument to compute convolutions. For example, specifying
# ``order=[1, 0]`` would apply convolution with a first-order derivative of
# the Gaussian along the first axis and simple Gaussian smoothing along the
# second.
# For small sigma, the SciPy Gaussian filter suffers from aliasing and edge
# artifacts, given that the filter will approximate a sinc or sinc
# derivative which only goes to 0 very slowly (order 1/n**2). Thus, we use
# a much larger truncate value to reduce any edge artifacts.
truncate = 8 if all(s > 1 for s in sigma) else 100
sq1_2 = 1 / math.sqrt(2)
sigma_scaled = tuple(sq1_2 * s for s in sigma)
common_kwargs = dict(sigma=sigma_scaled, mode=mode, cval=cval, truncate=truncate)
gaussian_ = functools.partial(ndi.gaussian_filter, **common_kwargs)
# Apply two successive first order Gaussian derivative operations, as
# detailed in:
# https://dsp.stackexchange.com/questions/78280/are-scipy-second-order-gaussian-derivatives-correct
# 1.) First order along one axis while smoothing (order=0) along the other
ndim = image.ndim
# orders in 2D = ([1, 0], [0, 1])
# in 3D = ([1, 0, 0], [0, 1, 0], [0, 0, 1])
# etc.
orders = tuple([0] * d + [1] + [0] * (ndim - d - 1) for d in range(ndim))
gradients = [gaussian_(image, order=orders[d]) for d in range(ndim)]
# 2.) apply the derivative along another axis as well
axes = range(ndim)
if order == 'xy':
axes = reversed(axes)
H_elems = [
gaussian_(gradients[ax0], order=orders[ax1])
for ax0, ax1 in combinations_with_replacement(axes, 2)
]
return H_elems
def hessian_matrix(
image, sigma=1, mode='constant', cval=0, order='rc', use_gaussian_derivatives=None
):
r"""Compute the Hessian matrix.
In 2D, 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 r- and c-directions.
The implementation here also supports n-dimensional data.
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 : {'rc', 'xy'}, optional
For 2D images, this parameter allows for the use of reverse or forward
order of the image axes in gradient computation. 'rc' indicates the use
of the first axis initially (Hrr, Hrc, Hcc), whilst 'xy' indicates the
usage of the last axis initially (Hxx, Hxy, Hyy). Images with higher
dimension must always use 'rc' order.
use_gaussian_derivatives : bool, optional
Indicates whether the Hessian is computed by convolving with Gaussian
derivatives, or by a simple finite-difference operation.
Returns
-------
H_elems : list of ndarray
Upper-diagonal elements of the hessian matrix for each pixel in the
input image. In 2D, this will be a three element list containing [Hrr,
Hrc, Hcc]. In nD, the list will contain ``(n**2 + n) / 2`` arrays.
Notes
-----
The distributive property of derivatives and convolutions allows us to
restate the derivative of an image, I, smoothed with a Gaussian kernel, G,
as the convolution of the image with the derivative of G.
.. math::
\frac{\partial }{\partial x_i}(I * G) =
I * \left( \frac{\partial }{\partial x_i} G \right)
When ``use_gaussian_derivatives`` is ``True``, this property is used to
compute the second order derivatives that make up the Hessian matrix.
When ``use_gaussian_derivatives`` is ``False``, simple finite differences
on a Gaussian-smoothed image are used instead.
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',
... use_gaussian_derivatives=False)
>>> 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)
float_dtype = _supported_float_type(image.dtype)
image = image.astype(float_dtype, copy=False)
if image.ndim > 2 and order == "xy":
raise ValueError("order='xy' is only supported for 2D images.")
if order not in ["rc", "xy"]:
raise ValueError(f"unrecognized order: {order}")
if use_gaussian_derivatives is None:
use_gaussian_derivatives = False
warn(
"use_gaussian_derivatives currently defaults to False, but will "
"change to True in a future version. Please specify this "
"argument explicitly to maintain the current behavior",
category=FutureWarning,
stacklevel=2,
)
if use_gaussian_derivatives:
return _hessian_matrix_with_gaussian(
image, sigma=sigma, mode=mode, cval=cval, order=order
)
gaussian_filtered = gaussian(image, sigma=sigma, mode=mode, cval=cval)
gradients = np.gradient(gaussian_filtered)
axes = range(image.ndim)
if order == 'xy':
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_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.
Parameters
----------
image : ndarray
The image over which to compute the Hessian Determinant.
sigma : float, optional
Standard deviation of 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)
float_dtype = _supported_float_type(image.dtype)
image = image.astype(float_dtype, copy=False)
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 = _symmetric_image(
hessian_matrix(image, sigma, use_gaussian_derivatives=False)
)
return np.linalg.det(hessian_mat_array)
def _symmetric_compute_eigenvalues(S_elems):
"""Compute eigenvalues from the upper-diagonal entries of a symmetric
matrix.
Parameters
----------
S_elems : list of ndarray
The upper-diagonal elements of the matrix, as returned by
`hessian_matrix` or `structure_tensor`.
Returns
-------
eigs : ndarray
The eigenvalues of the 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).
"""
if len(S_elems) == 3: # Fast explicit formulas for 2D.
M00, M01, M11 = S_elems
eigs = np.empty((2, *M00.shape), M00.dtype)
eigs[:] = (M00 + M11) / 2
hsqrtdet = np.sqrt(M01**2 + ((M00 - M11) / 2) ** 2)
eigs[0] += hsqrtdet
eigs[1] -= hsqrtdet
return eigs
else:
matrices = _symmetric_image(S_elems)
# eigvalsh returns eigenvalues in increasing order. We want decreasing
eigs = np.linalg.eigvalsh(matrices)[..., ::-1]
leading_axes = tuple(range(eigs.ndim - 1))
return np.transpose(eigs, (eigs.ndim - 1,) + leading_axes)
def _symmetric_image(S_elems):
"""Convert the upper-diagonal elements of a matrix to the full
symmetric matrix.
Parameters
----------
S_elems : list of array
The upper-diagonal elements of the matrix, as returned by
`hessian_matrix` or `structure_tensor`.
Returns
-------
image : array
An array of shape ``(M, N[, ...], image.ndim, image.ndim)``,
containing the matrix corresponding to each coordinate.
"""
image = S_elems[0]
symmetric_image = np.zeros(
image.shape + (image.ndim, image.ndim), dtype=S_elems[0].dtype
)
for idx, (row, col) in enumerate(
combinations_with_replacement(range(image.ndim), 2)
):
symmetric_image[..., row, col] = S_elems[idx]
symmetric_image[..., col, row] = S_elems[idx]
return symmetric_image
def structure_tensor_eigenvalues(A_elems):
"""Compute eigenvalues of structure tensor.
Parameters
----------
A_elems : list of ndarray
The upper-diagonal elements of the structure tensor, as returned
by `structure_tensor`.
Returns
-------
ndarray
The eigenvalues of the structure tensor, in decreasing order. The
eigenvalues are the leading dimension. That is, the coordinate
[i, j, k] corresponds to the ith-largest eigenvalue at position (j, k).
Examples
--------
>>> from skimage.feature import structure_tensor
>>> from skimage.feature import structure_tensor_eigenvalues
>>> square = np.zeros((5, 5))
>>> square[2, 2] = 1
>>> A_elems = structure_tensor(square, sigma=0.1, order='rc')
>>> structure_tensor_eigenvalues(A_elems)[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.]])
See also
--------
structure_tensor
"""
return _symmetric_compute_eigenvalues(A_elems)
def hessian_matrix_eigvals(H_elems):
"""Compute eigenvalues of Hessian matrix.
Parameters
----------
H_elems : list of ndarray
The upper-diagonal elements of the Hessian matrix, as returned
by `hessian_matrix`.
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',
... use_gaussian_derivatives=False)
>>> 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.]])
"""
return _symmetric_compute_eigenvalues(H_elems)
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 eigenvalues 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 : (M, N) 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',
use_gaussian_derivatives=False,
)
l1, l2 = hessian_matrix_eigvals(H)
# don't warn on divide by 0 as occurs in the docstring example
with np.errstate(divide='ignore', invalid='ignore'):
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 : (M, N) 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.
References
----------
.. [1] Kitchen, L., & Rosenfeld, A. (1982). Gray-level corner detection.
Pattern recognition letters, 1(2), 95-102.
:DOI:`10.1016/0167-8655(82)90020-4`
"""
float_dtype = _supported_float_type(image.dtype)
image = image.astype(float_dtype, copy=False)
imy, imx = _compute_derivatives(image, mode=mode, cval=cval)
imxy, imxx = _compute_derivatives(imx, mode=mode, cval=cval)
imyy, imyx = _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=float_dtype)
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 : (M, N) 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] https://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]])
"""
Arr, Arc, Acc = structure_tensor(image, sigma, order='rc')
# determinant
detA = Arr * Acc - Arc**2
# trace
traceA = Arr + Acc
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 : (M, N) 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] https://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]])
"""
Arr, Arc, Acc = structure_tensor(image, sigma, order='rc')
# minimum eigenvalue of A
response = ((Arr + Acc) - np.sqrt((Arr - Acc) ** 2 + 4 * Arc**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 : (M, N) 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] Förstner, W., & Gülch, E. (1987, June). A fast operator for
detection and precise location of distinct points, corners and
centres of circular features. In Proc. ISPRS intercommission
conference on fast processing of photogrammetric data (pp. 281-305).
https://cseweb.ucsd.edu/classes/sp02/cse252/foerstner/foerstner.pdf
.. [2] https://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]])
"""
Arr, Arc, Acc = structure_tensor(image, sigma, order='rc')
# determinant
detA = Arr * Acc - Arc**2
# trace
traceA = Arr + Acc
w = np.zeros_like(image, dtype=detA.dtype)
q = np.zeros_like(w)
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 : (M, N) ndarray
Input image.
n : int, optional
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, optional
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] Rosten, E., & Drummond, T. (2006, May). Machine learning for
high-speed corner detection. In European conference on computer
vision (pp. 430-443). Springer, Berlin, Heidelberg.
:DOI:`10.1007/11744023_34`
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 : (M, N) ndarray
Input image.
corners : (K, 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 : (K, 2) ndarray
Subpixel corner positions. NaN for "not classified" corners.
References
----------
.. [1] Förstner, W., & Gülch, E. (1987, June). A fast operator for
detection and precise location of distinct points, corners and
centres of circular features. In Proc. ISPRS intercommission
conference on fast processing of photogrammetric data (pp. 281-305).
https://cseweb.ucsd.edu/classes/sp02/cse252/foerstner/foerstner.pdf
.. [2] https://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
float_dtype = _supported_float_type(image.dtype)
image = image.astype(float_dtype, copy=False)
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=float_dtype)
N_edge = np.zeros((2, 2), dtype=float_dtype)
b_dot = np.zeros((2,), dtype=float_dtype)
b_edge = np.zeros((2,), dtype=float_dtype)
# 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=float_dtype)
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]
winy, winx = _compute_derivatives(window, mode='constant', cval=0)
# compute gradient squares 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=None,
exclude_border=True,
indices=True,
num_peaks=np.inf,
footprint=None,
labels=None,
*,
num_peaks_per_label=np.inf,
p_norm=np.inf,
):
"""Find peaks 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
----------
image : (M, N) ndarray
Input image.
min_distance : int, optional
The minimal allowed distance separating peaks.
* : *
See :py:meth:`skimage.feature.peak_local_max`.
p_norm : float
Which Minkowski p-norm to use. Should be in the range [1, inf].
A finite large p may cause a ValueError if overflow can occur.
``inf`` corresponds to the Chebyshev distance and 2 to the
Euclidean distance.
Returns
-------
output : ndarray or ndarray of bools
* If `indices = True` : (row, column, ...) coordinates of peaks.
* If `indices = False` : Boolean array shaped like `image`, with peaks
represented by True values.
See also
--------
skimage.feature.peak_local_max
Notes
-----
.. versionchanged:: 0.18
The default value of `threshold_rel` has changed to None, which
corresponds to letting `skimage.feature.peak_local_max` decide on the
default. This is equivalent to `threshold_rel=0`.
The `num_peaks` limit is applied before suppression of connected peaks.
To limit the number of peaks after suppression, set `num_peaks=np.inf` and
post-process the output of this function.
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]])
"""
if np.isinf(num_peaks):
num_peaks = None
# Get the coordinates of the detected peaks
coords = peak_local_max(
image,
min_distance=min_distance,
threshold_abs=threshold_abs,
threshold_rel=threshold_rel,
exclude_border=exclude_border,
num_peaks=np.inf,
footprint=footprint,
labels=labels,
num_peaks_per_label=num_peaks_per_label,
)
if len(coords):
# Use KDtree to find the peaks that are too close to each other
tree = spatial.cKDTree(coords)
rejected_peaks_indices = set()
for idx, point in enumerate(coords):
if idx not in rejected_peaks_indices:
candidates = tree.query_ball_point(point, r=min_distance, p=p_norm)
candidates.remove(idx)
rejected_peaks_indices.update(candidates)
# Remove the peaks that are too close to each other
coords = np.delete(coords, tuple(rejected_peaks_indices), axis=0)[:num_peaks]
if indices:
return coords
peaks = np.zeros_like(image, dtype=bool)
peaks[tuple(coords.T)] = True
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 : (M, N) ndarray
Input image.
window_size : int, optional
Window size.
Returns
-------
response : ndarray
Moravec response image.
References
----------
.. [1] https://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]])
"""
image = img_as_float(image)
float_dtype = _supported_float_type(image.dtype)
image = image.astype(float_dtype, copy=False)
return _corner_moravec(np.ascontiguousarray(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 : (M, N) array
Input grayscale image.
corners : (K, 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 : (K, 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.])
"""
image = _prepare_grayscale_input_2D(image)
return _corner_orientations(image, corners, mask)
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