Datasets:

AVSim
/

simulation-package

import math

import numpy as np
import scipy.ndimage as ndi

from .._shared.utils import check_nD, _supported_float_type
from ..feature.util import DescriptorExtractor, FeatureDetector
from .._shared.filters import gaussian
from ..transform import rescale
from ..util import img_as_float
from ._sift import _local_max, _ori_distances, _update_histogram


def _edgeness(hxx, hyy, hxy):
    """Compute edgeness (eq. 18 of Otero et. al. IPOL paper)"""
    trace = hxx + hyy
    determinant = hxx * hyy - hxy * hxy
    return (trace * trace) / determinant


def _sparse_gradient(vol, positions):
    """Gradient of a 3D volume at the provided `positions`.
    For SIFT we only need the gradient at specific positions and do not need
    the gradient at the edge positions, so can just use this simple
    implementation instead of numpy.gradient.
    """
    p0 = positions[..., 0]
    p1 = positions[..., 1]
    p2 = positions[..., 2]
    g0 = vol[p0 + 1, p1, p2] - vol[p0 - 1, p1, p2]
    g0 *= 0.5
    g1 = vol[p0, p1 + 1, p2] - vol[p0, p1 - 1, p2]
    g1 *= 0.5
    g2 = vol[p0, p1, p2 + 1] - vol[p0, p1, p2 - 1]
    g2 *= 0.5
    return g0, g1, g2


def _hessian(d, positions):
    """Compute the non-redundant 3D Hessian terms at the requested positions.
    Source: "Anatomy of the SIFT Method"  p.380 (13)
    """
    p0 = positions[..., 0]
    p1 = positions[..., 1]
    p2 = positions[..., 2]
    two_d0 = 2 * d[p0, p1, p2]
    # 0 = row, 1 = col, 2 = octave
    h00 = d[p0 - 1, p1, p2] + d[p0 + 1, p1, p2] - two_d0
    h11 = d[p0, p1 - 1, p2] + d[p0, p1 + 1, p2] - two_d0
    h22 = d[p0, p1, p2 - 1] + d[p0, p1, p2 + 1] - two_d0
    h01 = 0.25 * (
        d[p0 + 1, p1 + 1, p2]
        - d[p0 - 1, p1 + 1, p2]
        - d[p0 + 1, p1 - 1, p2]
        + d[p0 - 1, p1 - 1, p2]
    )
    h02 = 0.25 * (
        d[p0 + 1, p1, p2 + 1]
        - d[p0 + 1, p1, p2 - 1]
        + d[p0 - 1, p1, p2 - 1]
        - d[p0 - 1, p1, p2 + 1]
    )
    h12 = 0.25 * (
        d[p0, p1 + 1, p2 + 1]
        - d[p0, p1 + 1, p2 - 1]
        + d[p0, p1 - 1, p2 - 1]
        - d[p0, p1 - 1, p2 + 1]
    )
    return (h00, h11, h22, h01, h02, h12)


def _offsets(grad, hess):
    """Compute position refinement offsets from gradient and Hessian.
    This is equivalent to np.linalg.solve(-H, J) where H is the Hessian
    matrix and J is the gradient (Jacobian).
    This analytical solution is adapted from (BSD-licensed) C code by
    Otero et. al (see SIFT docstring References).
    """
    h00, h11, h22, h01, h02, h12 = hess
    g0, g1, g2 = grad
    det = h00 * h11 * h22
    det -= h00 * h12 * h12
    det -= h01 * h01 * h22
    det += 2 * h01 * h02 * h12
    det -= h02 * h02 * h11
    aa = (h11 * h22 - h12 * h12) / det
    ab = (h02 * h12 - h01 * h22) / det
    ac = (h01 * h12 - h02 * h11) / det
    bb = (h00 * h22 - h02 * h02) / det
    bc = (h01 * h02 - h00 * h12) / det
    cc = (h00 * h11 - h01 * h01) / det
    offset0 = -aa * g0 - ab * g1 - ac * g2
    offset1 = -ab * g0 - bb * g1 - bc * g2
    offset2 = -ac * g0 - bc * g1 - cc * g2
    return np.stack((offset0, offset1, offset2), axis=-1)


class SIFT(FeatureDetector, DescriptorExtractor):
    """SIFT feature detection and descriptor extraction.
    Parameters
    ----------
    upsampling : int, optional
        Prior to the feature detection the image is upscaled by a factor
        of 1 (no upscaling), 2 or 4. Method: Bi-cubic interpolation.
    n_octaves : int, optional
        Maximum number of octaves. With every octave the image size is
        halved and the sigma doubled. The number of octaves will be
        reduced as needed to keep at least 12 pixels along each dimension
        at the smallest scale.
    n_scales : int, optional
        Maximum number of scales in every octave.
    sigma_min : float, optional
        The blur level of the seed image. If upsampling is enabled
        sigma_min is scaled by factor 1/upsampling
    sigma_in : float, optional
        The assumed blur level of the input image.
    c_dog : float, optional
        Threshold to discard low contrast extrema in the DoG. It's final
        value is dependent on n_scales by the relation:
        final_c_dog = (2^(1/n_scales)-1) / (2^(1/3)-1) * c_dog
    c_edge : float, optional
        Threshold to discard extrema that lie in edges. If H is the
        Hessian of an extremum, its "edgeness" is described by
        tr(H)²/det(H). If the edgeness is higher than
        (c_edge + 1)²/c_edge, the extremum is discarded.
    n_bins : int, optional
        Number of bins in the histogram that describes the gradient
        orientations around keypoint.
    lambda_ori : float, optional
        The window used to find the reference orientation of a keypoint
        has a width of 6 * lambda_ori * sigma and is weighted by a
        standard deviation of 2 * lambda_ori * sigma.
    c_max : float, optional
        The threshold at which a secondary peak in the orientation
        histogram is accepted as orientation
    lambda_descr : float, optional
        The window used to define the descriptor of a keypoint has a width
        of 2 * lambda_descr * sigma * (n_hist+1)/n_hist and is weighted by
        a standard deviation of lambda_descr * sigma.
    n_hist : int, optional
        The window used to define the descriptor of a keypoint consists of
        n_hist * n_hist histograms.
    n_ori : int, optional
        The number of bins in the histograms of the descriptor patch.
    Attributes
    ----------
    delta_min : float
        The sampling distance of the first octave. It's final value is
        1/upsampling.
    float_dtype : type
        The datatype of the image.
    scalespace_sigmas : (n_octaves, n_scales + 3) array
        The sigma value of all scales in all octaves.
    keypoints : (N, 2) array
        Keypoint coordinates as ``(row, col)``.
    positions : (N, 2) array
        Subpixel-precision keypoint coordinates as ``(row, col)``.
    sigmas : (N,) array
        The corresponding sigma (blur) value of a keypoint.
    scales : (N,) array
        The corresponding scale of a keypoint.
    orientations : (N,) array
        The orientations of the gradient around every keypoint.
    octaves : (N,) array
        The corresponding octave of a keypoint.
    descriptors : (N, n_hist*n_hist*n_ori) array
        The descriptors of a keypoint.
    Notes
    -----
    The SIFT algorithm was developed by David Lowe [1]_, [2]_ and later
    patented by the University of British Columbia. Since the patent expired in
    2020 it's free to use. The implementation here closely follows the
    detailed description in [3]_, including use of the same default parameters.
    References
    ----------
    .. [1] D.G. Lowe. "Object recognition from local scale-invariant
           features", Proceedings of the Seventh IEEE International
           Conference on Computer Vision, 1999, vol.2, pp. 1150-1157.
           :DOI:`10.1109/ICCV.1999.790410`
    .. [2] D.G. Lowe. "Distinctive Image Features from Scale-Invariant
           Keypoints", International Journal of Computer Vision, 2004,
           vol. 60, pp. 91–110.
           :DOI:`10.1023/B:VISI.0000029664.99615.94`
    .. [3] I. R. Otero and M. Delbracio. "Anatomy of the SIFT Method",
           Image Processing On Line, 4 (2014), pp. 370–396.
           :DOI:`10.5201/ipol.2014.82`
    Examples
    --------
    >>> from skimage.feature import SIFT, match_descriptors
    >>> from skimage.data import camera
    >>> from skimage.transform import rotate
    >>> img1 = camera()
    >>> img2 = rotate(camera(), 90)
    >>> detector_extractor1 = SIFT()
    >>> detector_extractor2 = SIFT()
    >>> detector_extractor1.detect_and_extract(img1)
    >>> detector_extractor2.detect_and_extract(img2)
    >>> matches = match_descriptors(detector_extractor1.descriptors,
    ...                             detector_extractor2.descriptors,
    ...                             max_ratio=0.6)
    >>> matches[10:15]
    array([[ 10, 412],
           [ 11, 417],
           [ 12, 407],
           [ 13, 411],
           [ 14, 406]])
    >>> detector_extractor1.keypoints[matches[10:15, 0]]
    array([[ 95, 214],
           [ 97, 211],
           [ 97, 218],
           [102, 215],
           [104, 218]])
    >>> detector_extractor2.keypoints[matches[10:15, 1]]
    array([[297,  95],
           [301,  97],
           [294,  97],
           [297, 102],
           [293, 104]])
    """

    def __init__(
        self,
        upsampling=2,
        n_octaves=8,
        n_scales=3,
        sigma_min=1.6,
        sigma_in=0.5,
        c_dog=0.04 / 3,
        c_edge=10,
        n_bins=36,
        lambda_ori=1.5,
        c_max=0.8,
        lambda_descr=6,
        n_hist=4,
        n_ori=8,
    ):
        if upsampling in [1, 2, 4]:
            self.upsampling = upsampling
        else:
            raise ValueError("upsampling must be 1, 2 or 4")
        self.n_octaves = n_octaves
        self.n_scales = n_scales
        self.sigma_min = sigma_min / upsampling
        self.sigma_in = sigma_in
        self.c_dog = (2 ** (1 / n_scales) - 1) / (2 ** (1 / 3) - 1) * c_dog
        self.c_edge = c_edge
        self.n_bins = n_bins
        self.lambda_ori = lambda_ori
        self.c_max = c_max
        self.lambda_descr = lambda_descr
        self.n_hist = n_hist
        self.n_ori = n_ori
        self.delta_min = 1 / upsampling
        self.float_dtype = None
        self.scalespace_sigmas = None
        self.keypoints = None
        self.positions = None
        self.sigmas = None
        self.scales = None
        self.orientations = None
        self.octaves = None
        self.descriptors = None

    @property
    def deltas(self):
        """The sampling distances of all octaves"""
        deltas = self.delta_min * np.power(
            2, np.arange(self.n_octaves), dtype=self.float_dtype
        )
        return deltas

    def _set_number_of_octaves(self, image_shape):
        size_min = 12  # minimum size of last octave
        s0 = min(image_shape) * self.upsampling
        max_octaves = int(math.log2(s0 / size_min) + 1)
        if max_octaves < self.n_octaves:
            self.n_octaves = max_octaves

    def _create_scalespace(self, image):
        """Source: "Anatomy of the SIFT Method" Alg. 1
        Construction of the scalespace by gradually blurring (scales) and
        downscaling (octaves) the image.
        """
        scalespace = []
        if self.upsampling > 1:
            image = rescale(image, self.upsampling, order=1)

        # smooth to sigma_min, assuming sigma_in
        image = gaussian(
            image,
            sigma=self.upsampling * math.sqrt(self.sigma_min**2 - self.sigma_in**2),
            mode='reflect',
        )

        # Eq. 10:  sigmas.shape = (n_octaves, n_scales + 3).
        # The three extra scales are:
        #    One for the differences needed for DoG and two auxiliary
        #    images (one at either end) for peak_local_max with exclude
        #    border = True (see Fig. 5)
        # The smoothing doubles after n_scales steps.
        tmp = np.power(2, np.arange(self.n_scales + 3) / self.n_scales)
        tmp *= self.sigma_min
        # all sigmas for the gaussian scalespace
        sigmas = self.deltas[:, np.newaxis] / self.deltas[0] * tmp[np.newaxis, :]
        self.scalespace_sigmas = sigmas

        # Eq. 7: Gaussian smoothing depends on difference with previous sigma
        #        gaussian_sigmas.shape = (n_octaves, n_scales + 2)
        var_diff = np.diff(sigmas * sigmas, axis=1)
        gaussian_sigmas = np.sqrt(var_diff) / self.deltas[:, np.newaxis]

        # one octave is represented by a 3D image with depth (n_scales+x)
        for o in range(self.n_octaves):
            # Temporarily put scales axis first so octave[i] is C-contiguous
            # (this makes Gaussian filtering faster).
            octave = np.empty(
                (self.n_scales + 3,) + image.shape, dtype=self.float_dtype, order='C'
            )
            octave[0] = image
            for s in range(1, self.n_scales + 3):
                # blur new scale assuming sigma of the last one
                gaussian(
                    octave[s - 1],
                    sigma=gaussian_sigmas[o, s - 1],
                    mode='reflect',
                    out=octave[s],
                )
            # move scales to last axis as expected by other methods
            scalespace.append(np.moveaxis(octave, 0, -1))
            if o < self.n_octaves - 1:
                # downscale the image by taking every second pixel
                image = octave[self.n_scales][::2, ::2]
        return scalespace

    def _inrange(self, a, dim):
        return (
            (a[:, 0] > 0)
            & (a[:, 0] < dim[0] - 1)
            & (a[:, 1] > 0)
            & (a[:, 1] < dim[1] - 1)
        )

    def _find_localize_evaluate(self, dogspace, img_shape):
        """Source: "Anatomy of the SIFT Method" Alg. 4-9
        1) first find all extrema of a (3, 3, 3) neighborhood
        2) use second order Taylor development to refine the positions to
           sub-pixel precision
        3) filter out extrema that have low contrast and lie on edges or close
           to the image borders
        """
        extrema_pos = []
        extrema_scales = []
        extrema_sigmas = []
        threshold = self.c_dog * 0.8
        for o, (octave, delta) in enumerate(zip(dogspace, self.deltas)):
            # find extrema
            keys = _local_max(np.ascontiguousarray(octave), threshold)
            if keys.size == 0:
                extrema_pos.append(np.empty((0, 2)))
                continue

            # localize extrema
            oshape = octave.shape
            refinement_iterations = 5
            offset_max = 0.6
            for i in range(refinement_iterations):
                if i > 0:
                    # exclude any keys that have moved out of bounds
                    keys = keys[self._inrange(keys, oshape), :]

                # Jacobian and Hessian of all extrema
                grad = _sparse_gradient(octave, keys)
                hess = _hessian(octave, keys)

                # solve for offset of the extremum
                off = _offsets(grad, hess)
                if i == refinement_iterations - 1:
                    break
                # offset is too big and an increase would not bring us out of
                # bounds
                wrong_position_pos = np.logical_and(
                    off > offset_max, keys + 1 < tuple([a - 1 for a in oshape])
                )
                wrong_position_neg = np.logical_and(off < -offset_max, keys - 1 > 0)
                if not np.any(np.logical_or(wrong_position_neg, wrong_position_pos)):
                    break
                keys[wrong_position_pos] += 1
                keys[wrong_position_neg] -= 1

            # mask for all extrema that have been localized successfully
            finished = np.all(np.abs(off) < offset_max, axis=1)
            keys = keys[finished]
            off = off[finished]
            grad = [g[finished] for g in grad]

            # value of extremum in octave
            vals = octave[keys[:, 0], keys[:, 1], keys[:, 2]]
            # values at interpolated point
            w = vals
            for i in range(3):
                w += 0.5 * grad[i] * off[:, i]

            h00, h11, h01 = hess[0][finished], hess[1][finished], hess[3][finished]

            sigmaratio = self.scalespace_sigmas[0, 1] / self.scalespace_sigmas[0, 0]

            # filter for contrast, edgeness and borders
            contrast_threshold = self.c_dog
            contrast_filter = np.abs(w) > contrast_threshold

            edge_threshold = np.square(self.c_edge + 1) / self.c_edge
            edge_response = _edgeness(
                h00[contrast_filter], h11[contrast_filter], h01[contrast_filter]
            )
            edge_filter = np.abs(edge_response) <= edge_threshold

            keys = keys[contrast_filter][edge_filter]
            off = off[contrast_filter][edge_filter]
            yx = ((keys[:, :2] + off[:, :2]) * delta).astype(self.float_dtype)

            sigmas = self.scalespace_sigmas[o, keys[:, 2]] * np.power(
                sigmaratio, off[:, 2]
            )
            border_filter = np.all(
                np.logical_and(
                    (yx - sigmas[:, np.newaxis]) > 0.0,
                    (yx + sigmas[:, np.newaxis]) < img_shape,
                ),
                axis=1,
            )
            extrema_pos.append(yx[border_filter])
            extrema_scales.append(keys[border_filter, 2])
            extrema_sigmas.append(sigmas[border_filter])

        octave_indices = np.concatenate(
            [np.full(len(p), i) for i, p in enumerate(extrema_pos)]
        )

        if len(octave_indices) == 0:
            raise RuntimeError(
                "SIFT found no features. Try passing in an image containing "
                "greater intensity contrasts between adjacent pixels."
            )

        extrema_pos = np.concatenate(extrema_pos)
        extrema_scales = np.concatenate(extrema_scales)
        extrema_sigmas = np.concatenate(extrema_sigmas)
        return extrema_pos, extrema_scales, extrema_sigmas, octave_indices

    def _fit(self, h):
        """Refine the position of the peak by fitting it to a parabola"""
        return (h[0] - h[2]) / (2 * (h[0] + h[2] - 2 * h[1]))

    def _compute_orientation(
        self, positions_oct, scales_oct, sigmas_oct, octaves, gaussian_scalespace
    ):
        """Source: "Anatomy of the SIFT Method" Alg. 11
        Calculates the orientation of the gradient around every keypoint
        """
        gradient_space = []
        # list for keypoints that have more than one reference orientation
        keypoint_indices = []
        keypoint_angles = []
        keypoint_octave = []
        orientations = np.zeros_like(sigmas_oct, dtype=self.float_dtype)
        key_count = 0
        for o, (octave, delta) in enumerate(zip(gaussian_scalespace, self.deltas)):
            gradient_space.append(np.gradient(octave))

            in_oct = octaves == o
            if not np.any(in_oct):
                continue
            positions = positions_oct[in_oct]
            scales = scales_oct[in_oct]
            sigmas = sigmas_oct[in_oct]

            oshape = octave.shape[:2]
            # convert to octave's dimensions
            yx = positions / delta
            sigma = sigmas / delta

            # dimensions of the patch
            radius = 3 * self.lambda_ori * sigma
            p_min = np.maximum(0, yx - radius[:, np.newaxis] + 0.5).astype(int)
            p_max = np.minimum(
                yx + radius[:, np.newaxis] + 0.5, (oshape[0] - 1, oshape[1] - 1)
            ).astype(int)
            # orientation histogram
            hist = np.empty(self.n_bins, dtype=self.float_dtype)
            avg_kernel = np.full((3,), 1 / 3, dtype=self.float_dtype)
            for k in range(len(yx)):
                hist[:] = 0

                # use the patch coordinates to get the gradient and then
                # normalize them
                r, c = np.meshgrid(
                    np.arange(p_min[k, 0], p_max[k, 0] + 1),
                    np.arange(p_min[k, 1], p_max[k, 1] + 1),
                    indexing='ij',
                    sparse=True,
                )
                gradient_row = gradient_space[o][0][r, c, scales[k]]
                gradient_col = gradient_space[o][1][r, c, scales[k]]
                r = r.astype(self.float_dtype, copy=False)
                c = c.astype(self.float_dtype, copy=False)
                r -= yx[k, 0]
                c -= yx[k, 1]

                # gradient magnitude and angles
                magnitude = np.sqrt(np.square(gradient_row) + np.square(gradient_col))
                theta = np.mod(np.arctan2(gradient_col, gradient_row), 2 * np.pi)

                # more weight to center values
                kernel = np.exp(
                    np.divide(r * r + c * c, -2 * (self.lambda_ori * sigma[k]) ** 2)
                )

                # fill the histogram
                bins = np.floor(
                    (theta / (2 * np.pi) * self.n_bins + 0.5) % self.n_bins
                ).astype(int)
                np.add.at(hist, bins, kernel * magnitude)

                # smooth the histogram and find the maximum
                hist = np.concatenate((hist[-6:], hist, hist[:6]))
                for _ in range(6):  # number of smoothings
                    hist = np.convolve(hist, avg_kernel, mode='same')
                hist = hist[6:-6]
                max_filter = ndi.maximum_filter(hist, [3], mode='wrap')

                # if an angle is in 80% percent range of the maximum, a
                # new keypoint is created for it
                maxima = np.nonzero(
                    np.logical_and(
                        hist >= (self.c_max * np.max(hist)), max_filter == hist
                    )
                )

                # save the angles
                for c, m in enumerate(maxima[0]):
                    neigh = np.arange(m - 1, m + 2) % len(hist)
                    # use neighbors to fit a parabola, to get more accurate
                    # result
                    ori = (m + self._fit(hist[neigh]) + 0.5) * 2 * np.pi / self.n_bins
                    if ori > np.pi:
                        ori -= 2 * np.pi
                    if c == 0:
                        orientations[key_count] = ori
                    else:
                        keypoint_indices.append(key_count)
                        keypoint_angles.append(ori)
                        keypoint_octave.append(o)
                key_count += 1
        self.positions = np.concatenate(
            (positions_oct, positions_oct[keypoint_indices])
        )
        self.scales = np.concatenate((scales_oct, scales_oct[keypoint_indices]))
        self.sigmas = np.concatenate((sigmas_oct, sigmas_oct[keypoint_indices]))
        self.orientations = np.concatenate((orientations, keypoint_angles))
        self.octaves = np.concatenate((octaves, keypoint_octave))
        # return the gradient_space to reuse it to find the descriptor
        return gradient_space

    def _rotate(self, row, col, angle):
        c = math.cos(angle)
        s = math.sin(angle)
        rot_row = c * row + s * col
        rot_col = -s * row + c * col
        return rot_row, rot_col

    def _compute_descriptor(self, gradient_space):
        """Source: "Anatomy of the SIFT Method" Alg. 12
        Calculates the descriptor for every keypoint
        """
        n_key = len(self.scales)
        self.descriptors = np.empty(
            (n_key, self.n_hist**2 * self.n_ori), dtype=np.uint8
        )

        # indices of the histograms
        hists = np.arange(1, self.n_hist + 1, dtype=self.float_dtype)
        # indices of the bins
        bins = np.arange(1, self.n_ori + 1, dtype=self.float_dtype)

        key_numbers = np.arange(n_key)
        for o, (gradient, delta) in enumerate(zip(gradient_space, self.deltas)):
            in_oct = self.octaves == o
            if not np.any(in_oct):
                continue
            positions = self.positions[in_oct]
            scales = self.scales[in_oct]
            sigmas = self.sigmas[in_oct]
            orientations = self.orientations[in_oct]
            numbers = key_numbers[in_oct]

            dim = gradient[0].shape[:2]
            center_pos = positions / delta
            sigma = sigmas / delta

            # dimensions of the patch
            radius = self.lambda_descr * (1 + 1 / self.n_hist) * sigma
            radius_patch = math.sqrt(2) * radius
            p_min = np.asarray(
                np.maximum(0, center_pos - radius_patch[:, np.newaxis] + 0.5), dtype=int
            )
            p_max = np.asarray(
                np.minimum(
                    center_pos + radius_patch[:, np.newaxis] + 0.5,
                    (dim[0] - 1, dim[1] - 1),
                ),
                dtype=int,
            )

            for k in range(len(p_max)):
                rad_k = float(radius[k])
                ori = float(orientations[k])
                histograms = np.zeros(
                    (self.n_hist, self.n_hist, self.n_ori), dtype=self.float_dtype
                )
                # the patch
                r, c = np.meshgrid(
                    np.arange(p_min[k, 0], p_max[k, 0]),
                    np.arange(p_min[k, 1], p_max[k, 1]),
                    indexing='ij',
                    sparse=True,
                )
                # normalized coordinates
                r_norm = np.subtract(r, center_pos[k, 0], dtype=self.float_dtype)
                c_norm = np.subtract(c, center_pos[k, 1], dtype=self.float_dtype)
                r_norm, c_norm = self._rotate(r_norm, c_norm, ori)

                # select coordinates and gradient values within the patch
                inside = np.maximum(np.abs(r_norm), np.abs(c_norm)) < rad_k
                r_norm, c_norm = r_norm[inside], c_norm[inside]
                r_idx, c_idx = np.nonzero(inside)
                r = r[r_idx, 0]
                c = c[0, c_idx]
                gradient_row = gradient[0][r, c, scales[k]]
                gradient_col = gradient[1][r, c, scales[k]]
                # compute the (relative) gradient orientation
                theta = np.arctan2(gradient_col, gradient_row) - ori
                lam_sig = self.lambda_descr * float(sigma[k])
                # Gaussian weighted kernel magnitude
                kernel = np.exp((r_norm * r_norm + c_norm * c_norm) / (-2 * lam_sig**2))
                magnitude = (
                    np.sqrt(gradient_row * gradient_row + gradient_col * gradient_col)
                    * kernel
                )

                lam_sig_ratio = 2 * lam_sig / self.n_hist
                rc_bins = (hists - (1 + self.n_hist) / 2) * lam_sig_ratio
                rc_bin_spacing = lam_sig_ratio
                ori_bins = (2 * np.pi * bins) / self.n_ori

                # distances to the histograms and bins
                dist_r = np.abs(np.subtract.outer(rc_bins, r_norm))
                dist_c = np.abs(np.subtract.outer(rc_bins, c_norm))

                # the orientation histograms/bins that get the contribution
                near_t, near_t_val = _ori_distances(ori_bins, theta)

                # create the histogram
                _update_histogram(
                    histograms,
                    near_t,
                    near_t_val,
                    magnitude,
                    dist_r,
                    dist_c,
                    rc_bin_spacing,
                )

                # convert the histograms to a 1d descriptor
                histograms = histograms.reshape(-1)
                # saturate the descriptor
                histograms = np.minimum(histograms, 0.2 * np.linalg.norm(histograms))
                # normalize the descriptor
                descriptor = (512 * histograms) / np.linalg.norm(histograms)
                # quantize the descriptor
                descriptor = np.minimum(np.floor(descriptor), 255)
                self.descriptors[numbers[k], :] = descriptor

    def _preprocess(self, image):
        check_nD(image, 2)
        image = img_as_float(image)
        self.float_dtype = _supported_float_type(image.dtype)
        image = image.astype(self.float_dtype, copy=False)

        self._set_number_of_octaves(image.shape)
        return image

    def detect(self, image):
        """Detect the keypoints.
        Parameters
        ----------
        image : 2D array
            Input image.
        """
        image = self._preprocess(image)

        gaussian_scalespace = self._create_scalespace(image)

        dog_scalespace = [np.diff(layer, axis=2) for layer in gaussian_scalespace]

        positions, scales, sigmas, octaves = self._find_localize_evaluate(
            dog_scalespace, image.shape
        )

        self._compute_orientation(
            positions, scales, sigmas, octaves, gaussian_scalespace
        )

        self.keypoints = self.positions.round().astype(int)

    def extract(self, image):
        """Extract the descriptors for all keypoints in the image.
        Parameters
        ----------
        image : 2D array
            Input image.
        """
        image = self._preprocess(image)

        gaussian_scalespace = self._create_scalespace(image)

        gradient_space = [np.gradient(octave) for octave in gaussian_scalespace]

        self._compute_descriptor(gradient_space)

    def detect_and_extract(self, image):
        """Detect the keypoints and extract their descriptors.
        Parameters
        ----------
        image : 2D array
            Input image.
        """
        image = self._preprocess(image)

        gaussian_scalespace = self._create_scalespace(image)

        dog_scalespace = [np.diff(layer, axis=2) for layer in gaussian_scalespace]

        positions, scales, sigmas, octaves = self._find_localize_evaluate(
            dog_scalespace, image.shape
        )

        gradient_space = self._compute_orientation(
            positions, scales, sigmas, octaves, gaussian_scalespace
        )

        self._compute_descriptor(gradient_space)

        self.keypoints = self.positions.round().astype(int)