Datasets:

AVSim
/

simulation-package

import math

import numpy as np
import scipy.ndimage as ndi
from scipy import spatial

from .._shared.filters import gaussian
from .._shared.utils import _supported_float_type, check_nD
from ..transform import integral_image
from ..util import img_as_float
from ._hessian_det_appx import _hessian_matrix_det
from .peak import peak_local_max

# This basic blob detection algorithm is based on:
# http://www.cs.utah.edu/~jfishbau/advimproc/project1/ (04.04.2013)
# Theory behind: https://en.wikipedia.org/wiki/Blob_detection (04.04.2013)


def _compute_disk_overlap(d, r1, r2):
    """
    Compute fraction of surface overlap between two disks of radii
    ``r1`` and ``r2``, with centers separated by a distance ``d``.
    Parameters
    ----------
    d : float
        Distance between centers.
    r1 : float
        Radius of the first disk.
    r2 : float
        Radius of the second disk.
    Returns
    -------
    fraction: float
        Fraction of area of the overlap between the two disks.
    """

    ratio1 = (d**2 + r1**2 - r2**2) / (2 * d * r1)
    ratio1 = np.clip(ratio1, -1, 1)
    acos1 = math.acos(ratio1)

    ratio2 = (d**2 + r2**2 - r1**2) / (2 * d * r2)
    ratio2 = np.clip(ratio2, -1, 1)
    acos2 = math.acos(ratio2)

    a = -d + r2 + r1
    b = d - r2 + r1
    c = d + r2 - r1
    d = d + r2 + r1
    area = r1**2 * acos1 + r2**2 * acos2 - 0.5 * math.sqrt(abs(a * b * c * d))
    return area / (math.pi * (min(r1, r2) ** 2))


def _compute_sphere_overlap(d, r1, r2):
    """
    Compute volume overlap fraction between two spheres of radii
    ``r1`` and ``r2``, with centers separated by a distance ``d``.
    Parameters
    ----------
    d : float
        Distance between centers.
    r1 : float
        Radius of the first sphere.
    r2 : float
        Radius of the second sphere.
    Returns
    -------
    fraction: float
        Fraction of volume of the overlap between the two spheres.
    Notes
    -----
    See for example http://mathworld.wolfram.com/Sphere-SphereIntersection.html
    for more details.
    """
    vol = (
        math.pi
        / (12 * d)
        * (r1 + r2 - d) ** 2
        * (d**2 + 2 * d * (r1 + r2) - 3 * (r1**2 + r2**2) + 6 * r1 * r2)
    )
    return vol / (4.0 / 3 * math.pi * min(r1, r2) ** 3)


def _blob_overlap(blob1, blob2, *, sigma_dim=1):
    """Finds the overlapping area fraction between two blobs.
    Returns a float representing fraction of overlapped area. Note that 0.0
    is *always* returned for dimension greater than 3.
    Parameters
    ----------
    blob1 : sequence of arrays
        A sequence of ``(row, col, sigma)`` or ``(pln, row, col, sigma)``,
        where ``row, col`` (or ``(pln, row, col)``) are coordinates
        of blob and ``sigma`` is the standard deviation of the Gaussian kernel
        which detected the blob.
    blob2 : sequence of arrays
        A sequence of ``(row, col, sigma)`` or ``(pln, row, col, sigma)``,
        where ``row, col`` (or ``(pln, row, col)``) are coordinates
        of blob and ``sigma`` is the standard deviation of the Gaussian kernel
        which detected the blob.
    sigma_dim : int, optional
        The dimensionality of the sigma value. Can be 1 or the same as the
        dimensionality of the blob space (2 or 3).
    Returns
    -------
    f : float
        Fraction of overlapped area (or volume in 3D).
    """
    ndim = len(blob1) - sigma_dim
    if ndim > 3:
        return 0.0
    root_ndim = math.sqrt(ndim)

    # we divide coordinates by sigma * sqrt(ndim) to rescale space to isotropy,
    # giving spheres of radius = 1 or < 1.
    if blob1[-1] == blob2[-1] == 0:
        return 0.0
    elif blob1[-1] > blob2[-1]:
        max_sigma = blob1[-sigma_dim:]
        r1 = 1
        r2 = blob2[-1] / blob1[-1]
    else:
        max_sigma = blob2[-sigma_dim:]
        r2 = 1
        r1 = blob1[-1] / blob2[-1]
    pos1 = blob1[:ndim] / (max_sigma * root_ndim)
    pos2 = blob2[:ndim] / (max_sigma * root_ndim)

    d = np.sqrt(np.sum((pos2 - pos1) ** 2))
    if d > r1 + r2:  # centers farther than sum of radii, so no overlap
        return 0.0

    # one blob is inside the other
    if d <= abs(r1 - r2):
        return 1.0

    if ndim == 2:
        return _compute_disk_overlap(d, r1, r2)

    else:  # ndim=3 http://mathworld.wolfram.com/Sphere-SphereIntersection.html
        return _compute_sphere_overlap(d, r1, r2)


def _prune_blobs(blobs_array, overlap, *, sigma_dim=1):
    """Eliminated blobs with area overlap.
    Parameters
    ----------
    blobs_array : ndarray
        A 2d array with each row representing 3 (or 4) values,
        ``(row, col, sigma)`` or ``(pln, row, col, sigma)`` in 3D,
        where ``(row, col)`` (``(pln, row, col)``) are coordinates of the blob
        and ``sigma`` is the standard deviation of the Gaussian kernel which
        detected the blob.
        This array must not have a dimension of size 0.
    overlap : float
        A value between 0 and 1. If the fraction of area overlapping for 2
        blobs is greater than `overlap` the smaller blob is eliminated.
    sigma_dim : int, optional
        The number of columns in ``blobs_array`` corresponding to sigmas rather
        than positions.
    Returns
    -------
    A : ndarray
        `array` with overlapping blobs removed.
    """
    sigma = blobs_array[:, -sigma_dim:].max()
    distance = 2 * sigma * math.sqrt(blobs_array.shape[1] - sigma_dim)
    tree = spatial.cKDTree(blobs_array[:, :-sigma_dim])
    pairs = np.array(list(tree.query_pairs(distance)))
    if len(pairs) == 0:
        return blobs_array
    else:
        for i, j in pairs:
            blob1, blob2 = blobs_array[i], blobs_array[j]
            if _blob_overlap(blob1, blob2, sigma_dim=sigma_dim) > overlap:
                # note: this test works even in the anisotropic case because
                # all sigmas increase together.
                if blob1[-1] > blob2[-1]:
                    blob2[-1] = 0
                else:
                    blob1[-1] = 0

    return np.stack([b for b in blobs_array if b[-1] > 0])


def _format_exclude_border(img_ndim, exclude_border):
    """Format an ``exclude_border`` argument as a tuple of ints for calling
    ``peak_local_max``.
    """
    if isinstance(exclude_border, tuple):
        if len(exclude_border) != img_ndim:
            raise ValueError(
                "`exclude_border` should have the same length as the "
                "dimensionality of the image."
            )
        for exclude in exclude_border:
            if not isinstance(exclude, int):
                raise ValueError(
                    "exclude border, when expressed as a tuple, must only "
                    "contain ints."
                )
        return exclude_border + (0,)
    elif isinstance(exclude_border, int):
        return (exclude_border,) * img_ndim + (0,)
    elif exclude_border is True:
        raise ValueError("exclude_border cannot be True")
    elif exclude_border is False:
        return (0,) * (img_ndim + 1)
    else:
        raise ValueError(f'Unsupported value ({exclude_border}) for exclude_border')


def blob_dog(
    image,
    min_sigma=1,
    max_sigma=50,
    sigma_ratio=1.6,
    threshold=0.5,
    overlap=0.5,
    *,
    threshold_rel=None,
    exclude_border=False,
):
    r"""Finds blobs in the given grayscale image.
    Blobs are found using the Difference of Gaussian (DoG) method [1]_, [2]_.
    For each blob found, the method returns its coordinates and the standard
    deviation of the Gaussian kernel that detected the blob.
    Parameters
    ----------
    image : ndarray
        Input grayscale image, blobs are assumed to be light on dark
        background (white on black).
    min_sigma : scalar or sequence of scalars, optional
        Minimum standard deviation for Gaussian kernel. Keep this value low to
        detect smaller blobs. The standard deviation of the Gaussian kernel
        is given either as a sequence for each axis, or as a single number, in
        which case it is equal for all axes.
    max_sigma : scalar or sequence of scalars, optional
        The maximum standard deviation for Gaussian kernel. Keep this high to
        detect larger blobs. The standard deviation of the Gaussian kernel
        is given either as a sequence for each axis, or as a single number, in
        which case it is equal for all axes.
    sigma_ratio : float, optional
        The ratio between the standard deviation of Gaussian Kernels used for
        computing the Difference of Gaussians
    threshold : float or None, optional
        The absolute lower bound for scale space maxima. Local maxima smaller
        than `threshold` are ignored. Reduce this to detect blobs with lower
        intensities. If `threshold_rel` is also specified, whichever threshold
        is larger will be used. If None, `threshold_rel` is used instead.
    overlap : float, optional
        A value between 0 and 1. If the area of two blobs overlaps by a
        fraction greater than `threshold`, the smaller blob is eliminated.
    threshold_rel : float or None, optional
        Minimum intensity of peaks, calculated as
        ``max(dog_space) * threshold_rel``, where ``dog_space`` refers to the
        stack of Difference-of-Gaussian (DoG) images computed internally. This
        should have a value between 0 and 1. If None, `threshold` is used
        instead.
    exclude_border : tuple of ints, int, or False, optional
        If tuple of ints, the length of the tuple must match the input array's
        dimensionality.  Each element of the tuple will exclude peaks from
        within `exclude_border`-pixels of the border of the image along that
        dimension.
        If nonzero int, `exclude_border` excludes peaks from within
        `exclude_border`-pixels of the border of the image.
        If zero or False, peaks are identified regardless of their
        distance from the border.
    Returns
    -------
    A : (n, image.ndim + sigma) ndarray
        A 2d array with each row representing 2 coordinate values for a 2D
        image, or 3 coordinate values for a 3D image, plus the sigma(s) used.
        When a single sigma is passed, outputs are:
        ``(r, c, sigma)`` or ``(p, r, c, sigma)`` where ``(r, c)`` or
        ``(p, r, c)`` are coordinates of the blob and ``sigma`` is the standard
        deviation of the Gaussian kernel which detected the blob. When an
        anisotropic gaussian is used (sigmas per dimension), the detected sigma
        is returned for each dimension.
    See also
    --------
    skimage.filters.difference_of_gaussians
    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Blob_detection#The_difference_of_Gaussians_approach
    .. [2] Lowe, D. G. "Distinctive Image Features from Scale-Invariant
        Keypoints." International Journal of Computer Vision 60, 91–110 (2004).
        https://www.cs.ubc.ca/~lowe/papers/ijcv04.pdf
        :DOI:`10.1023/B:VISI.0000029664.99615.94`
    Examples
    --------
    >>> from skimage import data, feature
    >>> coins = data.coins()
    >>> feature.blob_dog(coins, threshold=.05, min_sigma=10, max_sigma=40)
    array([[128., 155.,  10.],
           [198., 155.,  10.],
           [124., 338.,  10.],
           [127., 102.,  10.],
           [193., 281.,  10.],
           [126., 208.,  10.],
           [267., 115.,  10.],
           [197., 102.,  10.],
           [198., 215.,  10.],
           [123., 279.,  10.],
           [126.,  46.,  10.],
           [259., 247.,  10.],
           [196.,  43.,  10.],
           [ 54., 276.,  10.],
           [267., 358.,  10.],
           [ 58., 100.,  10.],
           [259., 305.,  10.],
           [185., 347.,  16.],
           [261., 174.,  16.],
           [ 46., 336.,  16.],
           [ 54., 217.,  10.],
           [ 55., 157.,  10.],
           [ 57.,  41.,  10.],
           [260.,  47.,  16.]])
    Notes
    -----
    The radius of each blob is approximately :math:`\sqrt{2}\sigma` for
    a 2-D image and :math:`\sqrt{3}\sigma` for a 3-D image.
    """
    image = img_as_float(image)
    float_dtype = _supported_float_type(image.dtype)
    image = image.astype(float_dtype, copy=False)

    # if both min and max sigma are scalar, function returns only one sigma
    scalar_sigma = np.isscalar(max_sigma) and np.isscalar(min_sigma)

    # Gaussian filter requires that sequence-type sigmas have same
    # dimensionality as image. This broadcasts scalar kernels
    if np.isscalar(max_sigma):
        max_sigma = np.full(image.ndim, max_sigma, dtype=float_dtype)
    if np.isscalar(min_sigma):
        min_sigma = np.full(image.ndim, min_sigma, dtype=float_dtype)

    # Convert sequence types to array
    min_sigma = np.asarray(min_sigma, dtype=float_dtype)
    max_sigma = np.asarray(max_sigma, dtype=float_dtype)

    if sigma_ratio <= 1.0:
        raise ValueError('sigma_ratio must be > 1.0')

    # k such that min_sigma*(sigma_ratio**k) > max_sigma
    k = int(np.mean(np.log(max_sigma / min_sigma) / np.log(sigma_ratio) + 1))

    # a geometric progression of standard deviations for gaussian kernels
    sigma_list = np.array([min_sigma * (sigma_ratio**i) for i in range(k + 1)])

    # computing difference between two successive Gaussian blurred images
    # to obtain an approximation of the scale invariant Laplacian of the
    # Gaussian operator
    dog_image_cube = np.empty(image.shape + (k,), dtype=float_dtype)
    gaussian_previous = gaussian(image, sigma=sigma_list[0], mode='reflect')
    for i, s in enumerate(sigma_list[1:]):
        gaussian_current = gaussian(image, sigma=s, mode='reflect')
        dog_image_cube[..., i] = gaussian_previous - gaussian_current
        gaussian_previous = gaussian_current

    # normalization factor for consistency in DoG magnitude
    sf = 1 / (sigma_ratio - 1)
    dog_image_cube *= sf

    exclude_border = _format_exclude_border(image.ndim, exclude_border)
    local_maxima = peak_local_max(
        dog_image_cube,
        threshold_abs=threshold,
        threshold_rel=threshold_rel,
        exclude_border=exclude_border,
        footprint=np.ones((3,) * (image.ndim + 1)),
    )

    # Catch no peaks
    if local_maxima.size == 0:
        return np.empty((0, image.ndim + (1 if scalar_sigma else image.ndim)))

    # Convert local_maxima to float64
    lm = local_maxima.astype(float_dtype)

    # translate final column of lm, which contains the index of the
    # sigma that produced the maximum intensity value, into the sigma
    sigmas_of_peaks = sigma_list[local_maxima[:, -1]]

    if scalar_sigma:
        # select one sigma column, keeping dimension
        sigmas_of_peaks = sigmas_of_peaks[:, 0:1]

    # Remove sigma index and replace with sigmas
    lm = np.hstack([lm[:, :-1], sigmas_of_peaks])

    sigma_dim = sigmas_of_peaks.shape[1]

    return _prune_blobs(lm, overlap, sigma_dim=sigma_dim)


def blob_log(
    image,
    min_sigma=1,
    max_sigma=50,
    num_sigma=10,
    threshold=0.2,
    overlap=0.5,
    log_scale=False,
    *,
    threshold_rel=None,
    exclude_border=False,
):
    r"""Finds blobs in the given grayscale image.
    Blobs are found using the Laplacian of Gaussian (LoG) method [1]_.
    For each blob found, the method returns its coordinates and the standard
    deviation of the Gaussian kernel that detected the blob.
    Parameters
    ----------
    image : ndarray
        Input grayscale image, blobs are assumed to be light on dark
        background (white on black).
    min_sigma : scalar or sequence of scalars, optional
        Minimum standard deviation for Gaussian kernel. Keep this value low to
        detect smaller blobs. The standard deviation of the Gaussian kernel
        is given either as a sequence for each axis, or as a single number, in
        which case it is equal for all axes.
    max_sigma : scalar or sequence of scalars, optional
        The maximum standard deviation for Gaussian kernel. Keep this high to
        detect larger blobs. The standard deviation of the Gaussian kernel
        is given either as a sequence for each axis, or as a single number, in
        which case it is equal for all axes.
    num_sigma : int, optional
        The number of evenly spaced values for standard deviation of the
        Gaussian kernel to consider on the closed interval
        ``[min_sigma, max_sigma]``.
    threshold : float or None, optional
        The absolute lower bound for scale space maxima. Local maxima smaller
        than `threshold` are ignored. Reduce this to detect blobs with lower
        intensities. If `threshold_rel` is also specified, whichever threshold
        is larger will be used. If None, `threshold_rel` is used instead.
    overlap : float, optional
        A value between 0 and 1. If the area of two blobs overlaps by a
        fraction greater than `threshold`, the smaller blob is eliminated.
    log_scale : bool, optional
        If set intermediate values of standard deviations are interpolated
        using a logarithmic scale to the base `10`. If not, linear
        interpolation is used.
    threshold_rel : float or None, optional
        Minimum intensity of peaks, calculated as
        ``max(log_space) * threshold_rel``, where ``log_space`` refers to the
        stack of Laplacian-of-Gaussian (LoG) images computed internally. This
        should have a value between 0 and 1. If None, `threshold` is used
        instead.
    exclude_border : tuple of ints, int, or False, optional
        If tuple of ints, the length of the tuple must match the input array's
        dimensionality.  Each element of the tuple will exclude peaks from
        within `exclude_border`-pixels of the border of the image along that
        dimension.
        If nonzero int, `exclude_border` excludes peaks from within
        `exclude_border`-pixels of the border of the image.
        If zero or False, peaks are identified regardless of their
        distance from the border.
    Returns
    -------
    A : (n, image.ndim + sigma) ndarray
        A 2d array with each row representing 2 coordinate values for a 2D
        image, or 3 coordinate values for a 3D image, plus the sigma(s) used.
        When a single sigma is passed, outputs are:
        ``(r, c, sigma)`` or ``(p, r, c, sigma)`` where ``(r, c)`` or
        ``(p, r, c)`` are coordinates of the blob and ``sigma`` is the standard
        deviation of the Gaussian kernel which detected the blob. When an
        anisotropic gaussian is used (sigmas per dimension), the detected sigma
        is returned for each dimension.
    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Blob_detection#The_Laplacian_of_Gaussian
    Examples
    --------
    >>> from skimage import data, feature, exposure
    >>> img = data.coins()
    >>> img = exposure.equalize_hist(img)  # improves detection
    >>> feature.blob_log(img, threshold = .3)
    array([[124.        , 336.        ,  11.88888889],
           [198.        , 155.        ,  11.88888889],
           [194.        , 213.        ,  17.33333333],
           [121.        , 272.        ,  17.33333333],
           [263.        , 244.        ,  17.33333333],
           [194.        , 276.        ,  17.33333333],
           [266.        , 115.        ,  11.88888889],
           [128.        , 154.        ,  11.88888889],
           [260.        , 174.        ,  17.33333333],
           [198.        , 103.        ,  11.88888889],
           [126.        , 208.        ,  11.88888889],
           [127.        , 102.        ,  11.88888889],
           [263.        , 302.        ,  17.33333333],
           [197.        ,  44.        ,  11.88888889],
           [185.        , 344.        ,  17.33333333],
           [126.        ,  46.        ,  11.88888889],
           [113.        , 323.        ,   1.        ]])
    Notes
    -----
    The radius of each blob is approximately :math:`\sqrt{2}\sigma` for
    a 2-D image and :math:`\sqrt{3}\sigma` for a 3-D image.
    """
    image = img_as_float(image)
    float_dtype = _supported_float_type(image.dtype)
    image = image.astype(float_dtype, copy=False)

    # if both min and max sigma are scalar, function returns only one sigma
    scalar_sigma = True if np.isscalar(max_sigma) and np.isscalar(min_sigma) else False

    # Gaussian filter requires that sequence-type sigmas have same
    # dimensionality as image. This broadcasts scalar kernels
    if np.isscalar(max_sigma):
        max_sigma = np.full(image.ndim, max_sigma, dtype=float_dtype)
    if np.isscalar(min_sigma):
        min_sigma = np.full(image.ndim, min_sigma, dtype=float_dtype)

    # Convert sequence types to array
    min_sigma = np.asarray(min_sigma, dtype=float_dtype)
    max_sigma = np.asarray(max_sigma, dtype=float_dtype)

    if log_scale:
        start = np.log10(min_sigma)
        stop = np.log10(max_sigma)
        sigma_list = np.logspace(start, stop, num_sigma)
    else:
        sigma_list = np.linspace(min_sigma, max_sigma, num_sigma)

    # computing gaussian laplace
    image_cube = np.empty(image.shape + (len(sigma_list),), dtype=float_dtype)
    for i, s in enumerate(sigma_list):
        # average s**2 provides scale invariance
        image_cube[..., i] = -ndi.gaussian_laplace(image, s) * np.mean(s) ** 2

    exclude_border = _format_exclude_border(image.ndim, exclude_border)
    local_maxima = peak_local_max(
        image_cube,
        threshold_abs=threshold,
        threshold_rel=threshold_rel,
        exclude_border=exclude_border,
        footprint=np.ones((3,) * (image.ndim + 1)),
    )

    # Catch no peaks
    if local_maxima.size == 0:
        return np.empty((0, image.ndim + (1 if scalar_sigma else image.ndim)))

    # Convert local_maxima to float64
    lm = local_maxima.astype(float_dtype)

    # translate final column of lm, which contains the index of the
    # sigma that produced the maximum intensity value, into the sigma
    sigmas_of_peaks = sigma_list[local_maxima[:, -1]]

    if scalar_sigma:
        # select one sigma column, keeping dimension
        sigmas_of_peaks = sigmas_of_peaks[:, 0:1]

    # Remove sigma index and replace with sigmas
    lm = np.hstack([lm[:, :-1], sigmas_of_peaks])

    sigma_dim = sigmas_of_peaks.shape[1]

    return _prune_blobs(lm, overlap, sigma_dim=sigma_dim)


def blob_doh(
    image,
    min_sigma=1,
    max_sigma=30,
    num_sigma=10,
    threshold=0.01,
    overlap=0.5,
    log_scale=False,
    *,
    threshold_rel=None,
):
    """Finds blobs in the given grayscale image.
    Blobs are found using the Determinant of Hessian method [1]_. For each blob
    found, the method returns its coordinates and the standard deviation
    of the Gaussian Kernel used for the Hessian matrix whose determinant
    detected the blob. Determinant of Hessians is approximated using [2]_.
    Parameters
    ----------
    image : 2D ndarray
        Input grayscale image. Blobs can either be light on dark or vice versa.
    min_sigma : float, optional
        The minimum standard deviation for Gaussian Kernel used to compute
        Hessian matrix. Keep this value low to detect smaller blobs.
        The standard deviation of the Gaussian kernel is given either as a
        sequence for each axis, or as a single number, in which case it is
        equal for all axes.
    max_sigma : float, optional
        The maximum standard deviation for Gaussian Kernel used to compute
        Hessian matrix. Keep this value high to detect larger blobs.
        The standard deviation of the Gaussian kernel is given either as a
        sequence for each axis, or as a single number, in which case it is
        equal for all axes.
    num_sigma : int, optional
        The number of evenly spaced values for standard deviation of the
        Gaussian kernel to consider on the closed interval
        ``[min_sigma, max_sigma]``.
    threshold : float or None, optional
        The absolute lower bound for scale space maxima. Local maxima smaller
        than `threshold` are ignored. Reduce this to detect blobs with lower
        intensities. If `threshold_rel` is also specified, whichever threshold
        is larger will be used. If None, `threshold_rel` is used instead.
    overlap : float, optional
        A value between 0 and 1. If the area of two blobs overlaps by a
        fraction greater than `threshold`, the smaller blob is eliminated.
    log_scale : bool, optional
        If set intermediate values of standard deviations are interpolated
        using a logarithmic scale to the base `10`. If not, linear
        interpolation is used.
    threshold_rel : float or None, optional
        Minimum intensity of peaks, calculated as
        ``max(doh_space) * threshold_rel``, where ``doh_space`` refers to the
        stack of Determinant-of-Hessian (DoH) images computed internally. This
        should have a value between 0 and 1. If None, `threshold` is used
        instead.
    Returns
    -------
    A : (n, 3) ndarray
        A 2d array with each row representing 3 values, ``(y,x,sigma)``
        where ``(y,x)`` are coordinates of the blob and ``sigma`` is the
        standard deviation of the Gaussian kernel of the Hessian Matrix whose
        determinant detected the blob.
    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Blob_detection#The_determinant_of_the_Hessian
    .. [2] 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
    Examples
    --------
    >>> from skimage import data, feature
    >>> img = data.coins()
    >>> feature.blob_doh(img)
    array([[197.        , 153.        ,  20.33333333],
           [124.        , 336.        ,  20.33333333],
           [126.        , 153.        ,  20.33333333],
           [195.        , 100.        ,  23.55555556],
           [192.        , 212.        ,  23.55555556],
           [121.        , 271.        ,  30.        ],
           [126.        , 101.        ,  20.33333333],
           [193.        , 275.        ,  23.55555556],
           [123.        , 205.        ,  20.33333333],
           [270.        , 363.        ,  30.        ],
           [265.        , 113.        ,  23.55555556],
           [262.        , 243.        ,  23.55555556],
           [185.        , 348.        ,  30.        ],
           [156.        , 302.        ,  30.        ],
           [123.        ,  44.        ,  23.55555556],
           [260.        , 173.        ,  30.        ],
           [197.        ,  44.        ,  20.33333333]])
    Notes
    -----
    The radius of each blob is approximately `sigma`.
    Computation of Determinant of Hessians is independent of the standard
    deviation. Therefore detecting larger blobs won't take more time. In
    methods line :py:meth:`blob_dog` and :py:meth:`blob_log` the computation
    of Gaussians for larger `sigma` takes more time. The downside is that
    this method can't be used for detecting blobs of radius less than `3px`
    due to the box filters used in the approximation of Hessian Determinant.
    """
    check_nD(image, 2)

    image = img_as_float(image)
    float_dtype = _supported_float_type(image.dtype)
    image = image.astype(float_dtype, copy=False)

    image = integral_image(image)

    if log_scale:
        start, stop = math.log(min_sigma, 10), math.log(max_sigma, 10)
        sigma_list = np.logspace(start, stop, num_sigma)
    else:
        sigma_list = np.linspace(min_sigma, max_sigma, num_sigma)

    image_cube = np.empty(shape=image.shape + (len(sigma_list),), dtype=float_dtype)
    for j, s in enumerate(sigma_list):
        image_cube[..., j] = _hessian_matrix_det(image, s)

    local_maxima = peak_local_max(
        image_cube,
        threshold_abs=threshold,
        threshold_rel=threshold_rel,
        exclude_border=False,
        footprint=np.ones((3,) * image_cube.ndim),
    )

    # Catch no peaks
    if local_maxima.size == 0:
        return np.empty((0, 3))
    # Convert local_maxima to float64
    lm = local_maxima.astype(np.float64)
    # Convert the last index to its corresponding scale value
    lm[:, -1] = sigma_list[local_maxima[:, -1]]
    return _prune_blobs(lm, overlap)