corner.py

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] = -1.0 / 1591.54943092
    >>> Hxx, Hxy, Hyy = hessian_matrix(square, sigma=0.1)
    >>> Hxx
    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.]])

    """

    image = _prepare_grayscale_input_2D(image)

    # Window extent which covers > 99% of the normal distribution.
    window_ext = max(1, np.ceil(3 * sigma))

    ky, kx = np.mgrid[-window_ext:window_ext + 1, -window_ext:window_ext + 1]

    # Second derivative Gaussian kernels.
    gaussian_exp = np.exp(-(kx ** 2 + ky ** 2) / (2 * sigma ** 2))
    kernel_xx = 1 / (2 * np.pi * sigma ** 4) * (kx ** 2 / sigma ** 2 - 1)
    kernel_xx *= gaussian_exp
    kernel_xy = 1 / (2 * np.pi * sigma ** 6) * (kx * ky)
    kernel_xy *= gaussian_exp
    kernel_yy = kernel_xx.transpose()

    # Remove small kernel values.
    eps = np.finfo(kernel_xx.dtype).eps
    kernel_xx[np.abs(kernel_xx) < eps * np.abs(kernel_xx).max()] = 0
    kernel_xy[np.abs(kernel_xy) < eps * np.abs(kernel_xy).max()] = 0
    kernel_yy[np.abs(kernel_yy) < eps * np.abs(kernel_yy).max()] = 0

    Hxx = ndi.convolve(image, kernel_xx, mode=mode, cval=cval)
    Hxy = ndi.convolve(image, kernel_xy, mode=mode, cval=cval)
    Hyy = ndi.convolve(image, kernel_yy, mode=mode, cval=cval)

    return Hxx, Hxy, Hyy


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