weights.py

import numpy as np
import numpy.typing as npt
import scipy.ndimage as ndimage
import voltools as vt
from typing import Optional, Union
from pytom_tm.io import UnequalSpacingError
from itertools import pairwise


constants = {
    # Dictionary of physical constants required for calculation.
    "c": 299792458,  # m/s
    "el": 1.60217646e-19,  # C
    "h": 6.62606896e-34,  # J*S
    "h_ev": 4.13566733e-15,  # eV*s
    "h_bar": 1.054571628e-34,  # J*s
    "h_bar_ev": 6.58211899e-16,  # eV*s
    "na": 6.02214179e23,  # mol-1
    "re": 2.817940289458e-15,  # m
    "rw": 2.976e-10,  # m
    "me": 9.10938215e-31,  # kg
    "me_ev": 0.510998910e6,  # ev/c^2
    "kb": 1.3806503e-23,  # m^2 kgs^-2 K^-1
    "eps0": 8.854187817620e-12,  # F/m
}


def hwhm_to_sigma(hwhm: float) -> float:
    """Convert half width of half maximum of a Gaussian to sigma by dividing by sqrt(2 * ln(2)).

    Parameters
    ----------
    hwhm: float
        half width of half maximum of Gaussian

    Returns
    -------
    sigma: float
        sigma of Gaussian
    """
    return hwhm / (np.sqrt(2 * np.log(2)))


def sigma_to_hwhm(sigma: float) -> float:
    """Convert sigma to half width of half maximum of a Gaussian by multiplying with sqrt(2 * ln(2)).

    Parameters
    ----------
    sigma: float
        sigma of Gaussian

    Returns
    -------
    hwhm: float
        half width of half maximum of Gaussian
    """
    return sigma * (np.sqrt(2 * np.log(2)))


def wavelength_ev2m(voltage: float) -> float:
    """Calculate wavelength of electrons from voltage.

    Parameters
    ----------
    voltage: float
        voltage of wave in eV

    Returns
    -------
    lambda: float
        wavelength of electrons in m
    """
    h = constants["h"]
    e = constants["el"]
    m = constants["me"]
    c = constants["c"]

    _lambda = h / np.sqrt(e * voltage * m * (e / m * voltage / c**2 + 2))

    return _lambda


def radial_reduced_grid(
    shape: Union[tuple[int, int, int], tuple[int, int]], shape_is_reduced: bool = False
) -> npt.NDArray[float]:
    """Calculates a Fourier space radial reduced grid for the given input shape, with the 0 frequency in the center
    of the output image. Values range from 0 in the center to 1 at Nyquist frequency.

    By default, it is assumed shape belongs to a real space array, which causes the function to return a
    grid with the last dimension reduced, i.e. shape[-1] // 2 + 1 (ideal for creating frequency
    dependent filters). However, setting radial_reduced_grid(..., shape_is_reduced=True) the shape is assumed to
    already be in a reduced form.

    Parameters
    ----------
    shape: Union[tuple[int, int, int], tuple[int, int]]
        2D/3D input shape, usually the .shape attribute of a numpy array
    shape_is_reduced: bool, default False
        whether the shape is already in a reduced fourier format, False by default

    Returns
    ----------
    radial_reduced_grid: npt.NDArray[float]
        fourier space frequency grid, 0 in center, 1 at nyquist
    """
    if len(shape) not in [2, 3]:
        raise ValueError("radial_reduced_grid() only works for 2D or 3D shapes")
    reduced_dim = shape[-1] if shape_is_reduced else shape[-1] // 2 + 1
    if len(shape) == 3:
        x = (
            np.abs(
                np.arange(
                    -shape[0] // 2 + shape[0] % 2, shape[0] // 2 + shape[0] % 2, 1.0
                )
            )
            / (shape[0] // 2)
        )[:, np.newaxis, np.newaxis]
        y = (
            np.abs(
                np.arange(
                    -shape[1] // 2 + shape[1] % 2, shape[1] // 2 + shape[1] % 2, 1.0
                )
            )
            / (shape[1] // 2)
        )[:, np.newaxis]
        z = np.arange(0, reduced_dim, 1.0) / (reduced_dim - 1)
        return np.sqrt(x**2 + y**2 + z**2)
    elif len(shape) == 2:
        x = (
            np.abs(
                np.arange(
                    -shape[0] // 2 + shape[0] % 2, shape[0] // 2 + shape[0] % 2, 1.0
                )
            )
            / (shape[0] // 2)
        )[:, np.newaxis]
        y = np.arange(0, reduced_dim, 1.0) / (reduced_dim - 1)
        return np.sqrt(x**2 + y**2)


def create_gaussian_low_pass(
    shape: Union[tuple[int, int, int], tuple[int, int]],
    spacing: float,
    resolution: float,
) -> npt.NDArray[float]:
    """Create a 3D Gaussian low-pass filter with cutoff (or HWHM) that is reduced in fourier space.

    Parameters
    ----------
    shape: Union[tuple[int, int, int], tuple[int, int]]
        shape tuple with x,y or x,y,z dimension
    spacing: float
        voxel size in real space
    resolution: float
        resolution in real space to filter towards

    Returns
    ----------
    output: npt.NDArray[float]
        array containing the filter
    """
    q = radial_reduced_grid(shape)

    # 2 * spacing / resolution is cutoff in fourier space
    # then convert cutoff (hwhm) to sigma for gaussian function
    sigma_fourier = hwhm_to_sigma(2 * spacing / resolution)

    return np.fft.ifftshift(np.exp(-(q**2) / (2 * sigma_fourier**2)), axes=(0, 1))


def create_gaussian_high_pass(
    shape: Union[tuple[int, int, int], tuple[int, int]],
    spacing: float,
    resolution: float,
) -> npt.NDArray[float]:
    """Create a 3D Gaussian high-pass filter with cutoff (or HWHM) that is reduced in fourier space.

    Parameters
    ----------
    shape: Union[tuple[int, int, int], tuple[int, int]]
        shape tuple with x,y or x,y,z dimension
    spacing: float
        voxel size in real space
    resolution: float
        resolution in real space to filter towards

    Returns
    ----------
    output: npt.NDArray[float]
        array containing the filter
    """
    q = radial_reduced_grid(shape)

    # 2 * spacing / resolution is cutoff in fourier space
    # then convert cutoff (hwhm) to sigma for gaussian function
    sigma_fourier = hwhm_to_sigma(2 * spacing / resolution)

    return np.fft.ifftshift(1 - np.exp(-(q**2) / (2 * sigma_fourier**2)), axes=(0, 1))


def create_gaussian_band_pass(
    shape: Union[tuple[int, int, int], tuple[int, int]],
    spacing: float,
    low_pass: Optional[float] = None,
    high_pass: Optional[float] = None,
) -> npt.NDArray[float]:
    """Resolution bands presents the resolution shells where information needs to be maintained. For example the bands
    might be (150A, 40A). For a spacing of 15A (nyquist resolution is 30A) this is a mild low pass filter. However,
    quite some low spatial frequencies will be cut by it.

    Parameters
    ----------
    shape: Union[tuple[int, int, int], tuple[int, int]]
        shape tuple with x,y or x,y,z dimension
    spacing: float
        voxel size in real space
    low_pass: Optional[float], default None
        resolution of low-pass filter
    high_pass: Optional[float], default None
        resolution of high-pass filter

    Returns
    ----------
    output: npt.NDArray[float]
        array containing the band-pass filter
    """
    if high_pass is None and low_pass is None:
        raise ValueError("Either low-pass or high-pass needs to be set for band-pass")

    if high_pass is None:
        return create_gaussian_low_pass(shape, spacing, low_pass)
    elif low_pass is None:
        return create_gaussian_high_pass(shape, spacing, high_pass)
    elif low_pass >= high_pass:
        raise ValueError(
            "Second value of band-pass needs to be a high resolution shell."
        )
    else:
        q = radial_reduced_grid(shape)

        # 2 * spacing / resolution is cutoff in fourier space
        # then convert cutoff (hwhm) to sigma for gaussian function
        sigma_high_pass = hwhm_to_sigma(2 * spacing / high_pass)
        sigma_low_pass = hwhm_to_sigma(2 * spacing / low_pass)

        return np.fft.ifftshift(
            (1 - np.exp(-(q**2) / (2 * sigma_high_pass**2)))
            * np.exp(-(q**2) / (2 * sigma_low_pass**2)),
            axes=(0, 1),
        )


def create_wedge(
    shape: tuple[int, int, int],
    tilt_angles: list[float, ...],
    voxel_size: float,
    cut_off_radius: float = 1.0,
    angles_in_degrees: bool = True,
    low_pass: Optional[float] = None,
    high_pass: Optional[float] = None,
    tilt_weighting: bool = False,
    accumulated_dose_per_tilt: Optional[list[float, ...]] = None,
    ctf_params_per_tilt: Optional[list[dict]] = None,
) -> npt.NDArray[float]:
    """This function returns a wedge volume that is either symmetric or asymmetric depending on wedge angle input.

    Parameters
    ----------
    shape: tuple[int, int, int]
        real space shape of volume to which it needs to be applied
    tilt_angles: list[float, ...]
        tilt angles used for reconstructing the tomogram
    voxel_size: float
        voxel size is needed for the calculation of various filters
    cut_off_radius: float, default 1.
        cutoff as a fraction of nyquist, i.e. 1.0 means all the way to nyquist
    angles_in_degrees: bool, default True
        whether angles are in degrees or radians units
    low_pass: Optional[float], default None
        low pass filter resolution in A
    high_pass: Optional[float], default None
        high pass filter resolution in A
    tilt_weighting: bool, default False
        apply tilt weighting
    accumulated_dose_per_tilt: Optional[list[float, ...]], default None
        accumulated dose for each tilt for dose weighting
    ctf_params_per_tilt: Optional[list[dict]], default None
        ctf parameters for each tilt (see _create_tilt_weighted_wedge() for dict specification)

    Returns
    -------
    wedge: npt.NDArray[float]
        wedge volume that is a reduced fourier space object in z, i.e. shape[2] // 2 + 1
    """
    if not isinstance(tilt_angles, list) or len(tilt_angles) < 2:
        raise ValueError("Wedge generation needs at least a list of two tilt angles.")

    if voxel_size <= 0.0:
        raise ValueError(
            "Voxel size in create wedge is smaller or equal to 0, which is an invalid voxel spacing."
        )

    if cut_off_radius > 1:
        print(
            "Warning: wedge cutoff needs to be defined as a fraction of nyquist 0 < c <= 1. Setting value to 1.0."
        )
        cut_off_radius = 1.0
    elif cut_off_radius <= 0:
        raise ValueError("Invalid wedge cutoff: needs to be larger than 0")

    if angles_in_degrees:
        tilt_angles_rad = [np.deg2rad(w) for w in tilt_angles]
    else:
        tilt_angles_rad = tilt_angles

    if tilt_weighting:
        if ctf_params_per_tilt is not None and len(ctf_params_per_tilt) == 1:
            # with only single defocus, copy the params for each tilt
            ctf_params_per_tilt = ctf_params_per_tilt * len(tilt_angles_rad)
        wedge = _create_tilt_weighted_wedge(
            shape,
            tilt_angles_rad,
            cut_off_radius,
            voxel_size,
            accumulated_dose_per_tilt=accumulated_dose_per_tilt,
            ctf_params_per_tilt=ctf_params_per_tilt,
        ).astype(np.float32)
    else:
        wedge_angles = (
            np.pi / 2 - np.abs(min(tilt_angles_rad)),
            np.pi / 2 - np.abs(max(tilt_angles_rad)),
        )
        if np.round(wedge_angles[0], 2) == np.round(wedge_angles[1], 2):
            wedge = _create_symmetric_wedge(
                shape, wedge_angles[0], cut_off_radius
            ).astype(np.float32)
        else:
            wedge = _create_asymmetric_wedge(
                shape, (wedge_angles[0], wedge_angles[1]), cut_off_radius
            ).astype(np.float32)
        if ctf_params_per_tilt is not None:
            # - take ctf params from approx. middle tilt as those are most accurate
            # - in case list has 1 element we take that
            ctf_params = ctf_params_per_tilt[len(ctf_params_per_tilt) // 2]
            wedge *= create_ctf(
                shape,
                voxel_size * 1e-10,
                **ctf_params,
            )

    if not (low_pass is None and high_pass is None):
        return wedge * create_gaussian_band_pass(
            shape, voxel_size, low_pass, high_pass
        ).astype(np.float32)
    else:
        return wedge


def _create_symmetric_wedge(
    shape: tuple[int, int, int], wedge_angle: float, cut_off_radius: float
) -> npt.NDArray[float]:
    """This function returns a symmetric wedge object. Function should not be imported, user should call
    create_wedge().

    Parameters
    ----------
    shape: tuple[int, int, int]
        real space shape of volume to which it needs to be applied
    wedge_angle: float
        angle describing symmetric wedge in radians
    cut_off_radius: float
        cutoff as a fraction of nyquist, i.e. 1.0 means all the way to nyquist

    Returns
    ----------
    wedge: npt.NDArray[float]
        wedge volume that is a reduced fourier space object in z, i.e. shape[2] // 2 + 1
    """
    x = (
        np.abs(
            np.arange(-shape[0] // 2 + shape[0] % 2, shape[0] // 2 + shape[0] % 2, 1.0)
        )
        / (shape[0] // 2)
    )[:, np.newaxis]
    z = np.arange(0, shape[2] // 2 + 1, 1.0) / (shape[2] // 2)

    # calculate the wedge mask with smooth edges
    wedge_2d = x - np.tan(wedge_angle) * z
    limit = (wedge_2d.max() - wedge_2d.min()) / (2 * min(shape[0], shape[2]) // 2)
    wedge_2d[wedge_2d > limit] = limit
    wedge_2d[wedge_2d < -limit] = -limit
    wedge_2d = (wedge_2d - wedge_2d.min()) / (wedge_2d.max() - wedge_2d.min())
    wedge_2d[shape[0] // 2 + 1, 0] = 1  # ensure that the zero frequency point equals 1

    # duplicate in x
    wedge = np.tile(wedge_2d[:, np.newaxis, :], (1, shape[1], 1))

    wedge[radial_reduced_grid(shape) > cut_off_radius] = 0

    # fourier shift to origin
    return np.fft.ifftshift(wedge, axes=(0, 1))


def _create_asymmetric_wedge(
    shape: tuple[int, int, int],
    wedge_angles: tuple[float, float],
    cut_off_radius: float,
) -> npt.NDArray[float]:
    """This function returns an asymmetric wedge object. Function should not be imported, user should call create_wedge().

    Parameters
    ----------
    shape: tuple[int, int, int]
        real space shape of volume to which it needs to be applied
    wedge_angles: tuple[float, float]
        two angles describing asymmetric missing wedge in radians
    cut_off_radius: float
        cutoff as a fraction of nyquist, i.e. 1.0 means all the way to nyquist

    Returns
    ----------
    wedge: npt.NDArray[float]
        wedge volume that is a reduced fourier space object in z, i.e. shape[2] // 2 + 1
    """
    x = (
        np.abs(
            np.arange(-shape[0] // 2 + shape[0] % 2, shape[0] // 2 + shape[0] % 2, 1.0)
        )
        / (shape[0] // 2)
    )[:, np.newaxis]
    z = np.arange(0, shape[2] // 2 + 1, 1.0) / (shape[2] // 2)

    # calculate wedge for first angle
    wedge_section = x - np.tan(wedge_angles[0]) * z
    limit = (wedge_section.max() - wedge_section.min()) / (
        2 * min(shape[0], shape[2]) // 2
    )
    wedge_section[wedge_section > limit] = limit
    wedge_section[wedge_section < -limit] = -limit
    wedge_section = (wedge_section - wedge_section.min()) / (
        wedge_section.max() - wedge_section.min()
    )

    # set top of the wedge
    wedge_2d = wedge_section.copy()

    # calculate wedge for second angle
    wedge_section = x - np.tan(wedge_angles[1]) * z
    limit = (wedge_section.max() - wedge_section.min()) / (
        2 * min(shape[0], shape[2]) // 2
    )
    wedge_section[wedge_section > limit] = limit
    wedge_section[wedge_section < -limit] = -limit
    wedge_section = (wedge_section - wedge_section.min()) / (
        wedge_section.max() - wedge_section.min()
    )

    # set bottom of wedge and put 0 frequency to 1
    wedge_2d[shape[0] // 2 + 1 :] = wedge_section[shape[0] // 2 + 1 :]
    wedge_2d[shape[0] // 2 + 1, 0] = 1

    # duplicate in x
    wedge = np.tile(wedge_2d[:, np.newaxis, :], (1, shape[1], 1))

    wedge[radial_reduced_grid(shape) > cut_off_radius] = 0

    # fourier shift to origin
    return np.fft.ifftshift(wedge, axes=(0, 1))


def _create_tilt_weighted_wedge(
    shape: tuple[int, int, int],
    tilt_angles: list[float, ...],
    cut_off_radius: float,
    pixel_size_angstrom: float,
    accumulated_dose_per_tilt: Optional[list[float, ...]] = None,
    ctf_params_per_tilt: Optional[list[dict]] = None,
) -> npt.NDArray[float]:
    """
    The following B-factor heuristic is used (as mentioned in the M paper, and introduced in RELION 1.4):
        "The B factor is increased by 4Å2 per 1e− Å−2 of exposure, and each tilt is weighted as cos θ."

    Relation between B-factor and the sigma of a gaussian:

        B = 8 * pi ** 2 * sigma_motion ** 2

    i.e. sigma_motion = sqrt( B / (8 * pi ** 2)). Belonging to a Gaussian blur:

        exp( -2 * pi ** 2 * sigma_motion ** 2 * q ** 2 )

    Parameters
    ----------
    shape: tuple[int, int, int]
        shape of volume to model the wedge for
    tilt_angles: list[float, ...]
        tilt angles is a list of angle in radian units
    cut_off_radius: float
        cut off for the mask as a fraction of nyquist, value between 0 and 1
    pixel_size_angstrom: float
        the pixel size as a value in Å
    accumulated_dose_per_tilt: list[float, ...], default None
        the accumulated dose in e− Å−2
    ctf_params_per_tilt: list[dict, ...], default None
        the ctf parameters per tilt angle, list of dicts where each dict has the following keys:
        - 'defocus'; in um
        - 'amplitude'; fraction of amplitude contrast between 0 and 1
        - 'voltage'; in keV
        - 'cs'; spherical abberation in mm
        - 'phase_shift_deg'; phase shift for phase plates in deg

    Returns
    -------
    wedge: npt.NDArray[float]
        structured wedge mask in fourier reduced form, i.e. output shape is (shape[0], shape[1], shape[2] // 2 + 1)
    """
    if accumulated_dose_per_tilt is not None and len(accumulated_dose_per_tilt) != len(
        tilt_angles
    ):
        raise ValueError(
            "in _create_tilt_weighted_wedge the list of accumulated dose per tilt does not have the same "
            "length as the tilt angle list!"
        )
    if ctf_params_per_tilt is not None and len(ctf_params_per_tilt) != len(tilt_angles):
        raise ValueError(
            "in _create_tilt_weighted_wedge the list of CTF parameters per tilt does not have the same "
            "length as the tilt angle list!"
        )
    if not all([shape[0] == s for s in shape[1:]]):
        raise UnequalSpacingError(
            "Input shape for structured wedge needs to be a square box. "
            "Otherwise the frequencies in fourier space are not equal across dimensions."
        )

    image_size = shape[0]  # assign to size variable as all dimensions are equal size
    tilt = np.zeros(shape)
    q_grid = radial_reduced_grid(shape)
    tilt_weighted_wedge = np.zeros((image_size, image_size, image_size // 2 + 1))

    # create ramp weights to correct tilt summation for overlap
    tilt_increment = min([abs(x - y) for x, y in pairwise(tilt_angles)])
    # Crowther freq. determines till what point adjacent tilts overlap in Fourier space
    overlap_frequency = 1 / (tilt_increment * image_size)
    freq_1d = (
        np.abs(
            np.arange(
                -image_size // 2 + image_size % 2, image_size // 2 + image_size % 2, 1.0
            )
        )
        / (image_size // 2)
        * 0.5
    )  # multiply with .5 for nyquist frequency
    ramp_filter = freq_1d / overlap_frequency
    ramp_filter[ramp_filter > 1] = 1  # linear increase up to overlap frequency

    # generate 2d weights along the tilt axis
    ramp_weighting = np.tile(ramp_filter[:, np.newaxis], (1, image_size))

    for i, alpha in enumerate(tilt_angles):
        if ctf_params_per_tilt is not None:
            ctf = np.fft.fftshift(
                create_ctf(
                    (image_size,) * 2,
                    pixel_size_angstrom * 1e-10,
                    **ctf_params_per_tilt[i],
                ),
                axes=0,
            )
            tilt[:, :, image_size // 2] = (
                np.concatenate(
                    (  # duplicate and flip the CTF around the 0 frequency; then concatenate to make it non-reduced
                        np.flip(ctf[:, 1 : 1 + image_size - ctf.shape[1]], axis=1),
                        ctf,
                    ),
                    axis=1,
                )
                * ramp_weighting
            )
        else:
            tilt[:, :, image_size // 2] = ramp_weighting

        # rotate the image weights to the tilt angle
        rotated = np.flip(
            vt.transform(
                tilt,
                rotation=(0, alpha, 0),
                rotation_units="rad",
                rotation_order="rxyz",
                center=(image_size // 2,) * 3,
                interpolation="filt_bspline",
                device="cpu",
            )[:, :, : image_size // 2 + 1],  # crop back z-axis to reduced Fourier form
            axis=2,
        )

        # weight with exposure and tilt dampening
        if accumulated_dose_per_tilt is not None:
            q_squared = (q_grid / (2 * pixel_size_angstrom)) ** 2
            sigma_motion = np.sqrt(accumulated_dose_per_tilt[i] * 4 / (8 * np.pi**2))
            weighted_tilt = (
                rotated
                * np.cos(alpha)  # apply tilt-dependent weighting
                * np.exp(
                    -2 * np.pi**2 * sigma_motion**2 * q_squared
                )  # apply dose-weighting
            )
        else:
            weighted_tilt = (
                rotated * np.cos(alpha)  # apply tilt-dependent weighting
            )

        tilt_weighted_wedge += weighted_tilt

    tilt_weighted_wedge[q_grid > cut_off_radius] = 0

    return np.fft.ifftshift(tilt_weighted_wedge, axes=(0, 1))


def create_ctf(
    shape: Union[tuple[int, int, int], tuple[int, int]],
    pixel_size: float,
    defocus: float,
    amplitude_contrast: float,
    voltage: float,
    spherical_aberration: float,
    cut_after_first_zero: bool = False,
    flip_phase: bool = False,
    phase_shift_deg: float = 0.0,
) -> npt.NDArray[float]:
    """Create a CTF in a 3D volume in reduced format.

    Parameters
    ----------
    shape: Union[tuple[int, int, int], tuple[int, int]]
        dimensions of volume to create ctf in
    pixel_size: float
        pixel size for ctf in m
    defocus: float
        defocus for ctf in m
    amplitude_contrast: float
        the fraction of amplitude contrast in the ctf
    voltage: float
        acceleration voltage of the microscope in eV
    spherical_aberration: float
        spherical aberration in m
    cut_after_first_zero: bool, default False
        whether to cut ctf after first zero crossing
    flip_phase: bool, default False
        make ctf fully positive/negative to imitate ctf correction by phase flipping
    phase_shift_deg: float, default .0
        additional phase shift to model phase plates, similar to
        `https://github.com/dtegunov/tom_deconv` except the ctf defintion in tom
        produces the inverse curve of what we have here

    Returns
    -------
    ctf: npt.NDArray[float]
        CTF in 3D
    """
    k = radial_reduced_grid(shape) / (2 * pixel_size)  # frequencies in fourier space

    _lambda = wavelength_ev2m(voltage)

    # phase contrast transfer
    chi = (
        np.pi * _lambda * defocus * k**2
        - 0.5 * np.pi * spherical_aberration * _lambda**3 * k**4
    )
    # amplitude contrast term
    tan_term = np.arctan(amplitude_contrast / np.sqrt(1 - amplitude_contrast**2))

    # determine the ctf
    ctf = -np.sin(chi + tan_term + np.deg2rad(phase_shift_deg))

    if cut_after_first_zero:  # find frequency to cut first zero

        def chi_1d(q):
            return (
                np.pi * _lambda * defocus * q**2
                - 0.5 * np.pi * spherical_aberration * _lambda**3 * q**4
            )

        def ctf_1d(q):
            return -np.sin(chi_1d(q) + tan_term)

        # sample 1d ctf and get indices of zero crossing
        k_range = np.arange(max(k.shape)) / max(k.shape) / (2 * pixel_size)
        values = ctf_1d(k_range)
        zero_crossings = np.where(np.diff(np.sign(values)))[0]

        # for overfocus the first crossing needs to be skipped, for example see: Yonekura et al. 2006 JSB
        k_cutoff = (
            k_range[zero_crossings[0]] if defocus > 0 else k_range[zero_crossings[1]]
        )

        # filter the ctf with the cutoff frequency
        ctf[k > k_cutoff] = 0

    if flip_phase:  # take absolute, ensures matching contrast
        ctf = np.abs(ctf)
    else:  # multiply the ctf with -1 if we have overfocus, this allows the user to
        # always match the contrast of the input template with the contrast of the
        # tomogram: if the tomogram is black the reference should be black.
        ctf *= -1 if defocus > 0 else 1

    return np.fft.ifftshift(ctf, axes=(0, 1) if len(shape) == 3 else 0)


def radial_average(
    weights: npt.NDArray[float],
) -> tuple[npt.NDArray[float], npt.NDArray[float]]:
    """This calculates the radial average of a reduced fourier space function.

    Parameters
    ----------
    weights: npt.NDArray[float]
        3D array to be radially averaged: in fourier reduced form and with the origin in the corner.

    Returns
    -------
    (q, mean): tuple[npt.NDArray[float], npt.NDArray[float]]
        A tuple of two 1d numpy arrays. Their length equals half of largest input dimension.
    """
    if len(weights.shape) not in [2, 3]:
        raise ValueError("Radial average calculation only works for 2d/3d arrays")

    # get the number of sampling points from the largest fourier dimension,
    # unless the reduced dimensions is already the largest one
    sampling_points = max(max(weights.shape[:-1]) // 2 + 1, weights.shape[-1])

    q = np.arange(sampling_points)
    q_grid = np.floor(
        # convert to radial indices in the fourier power spectrum, 0.5 is added to obtain the correct ring
        radial_reduced_grid(weights.shape, shape_is_reduced=True)
        * (sampling_points - 1)
        + 0.5
    ).astype(int)
    mean = ndimage.mean(
        np.fft.fftshift(weights, axes=(0, 1) if len(weights.shape) == 3 else 0),
        labels=q_grid,
        index=q,
    )

    return q, mean


def power_spectrum_profile(image: npt.NDArray[float]) -> npt.NDArray[float]:
    """Calculate the power spectrum for a real space array and then find the profile (radial average).

    Parameters
    ----------
    image: npt.NDArray[float]
        2D/3D array in real space for which the power spectrum profile needs to be calculated

    Returns
    -------
    profile: npt.NDArray[float]
        A 1D numpy array
    """
    if len(image.shape) not in [2, 3]:
        raise ValueError(
            "Power spectrum profile calculation only works for 2d/3d arrays."
        )

    _, power_profile = radial_average(np.abs(np.fft.rfftn(image)) ** 2)

    return power_profile


def profile_to_weighting(
    profile: npt.NDArray[float], shape: Union[tuple[int, int], tuple[int, int, int]]
) -> npt.NDArray[float]:
    """Calculate a radial weighing (filter) from a spectrum profile.

    Parameters
    ----------
    profile: npt.NDArray[float]
        power spectrum profile (or other 1d profile) to transform in a fourier space filter
    shape: Union[tuple[int, int], tuple[int, int, int]]
        2D/3D array shape in real space for which the fourier reduced weights are calculated

    Returns
    -------
    weighting: npt.NDArray[float]
        Reduced Fourier space weighting for shape
    """
    if len(profile.shape) != 1:
        raise ValueError("Profile passed to profile_to_weighting is not 1-dimensional.")
    if len(shape) not in [2, 3]:
        raise ValueError("Shape passed to profile_to_weighting needs to be 2D/3D.")

    q_grid = radial_reduced_grid(shape)

    weights = ndimage.map_coordinates(
        profile, q_grid.flatten()[np.newaxis, :] * profile.shape[0], order=1
    ).reshape(q_grid.shape)

    weights[q_grid > 1] = 0

    return np.fft.ifftshift(weights, axes=(0, 1) if len(shape) == 3 else 0)