Source code for pylops.signalprocessing.FFTND

import logging
import warnings

import numpy as np
import scipy.fft

from pylops.signalprocessing._BaseFFTs import _BaseFFTND, _FFTNorms

logging.basicConfig(format="%(levelname)s: %(message)s", level=logging.WARNING)


class _FFTND_numpy(_BaseFFTND):
    """N-dimensional Fast-Fourier Transform using NumPy"""

    def __init__(
        self,
        dims,
        dirs=(0, 1, 2),
        nffts=None,
        sampling=1.0,
        norm="ortho",
        real=False,
        ifftshift_before=False,
        fftshift_after=False,
        dtype="complex128",
    ):
        super().__init__(
            dims=dims,
            dirs=dirs,
            nffts=nffts,
            sampling=sampling,
            norm=norm,
            real=real,
            ifftshift_before=ifftshift_before,
            fftshift_after=fftshift_after,
            dtype=dtype,
        )
        if self.cdtype != np.complex128:
            warnings.warn(
                f"numpy backend always returns complex128 dtype. To respect the passed dtype, data will be cast to {self.cdtype}."
            )

        self._norm_kwargs = {"norm": None}  # equivalent to "backward" in Numpy/Scipy
        if self.norm is _FFTNorms.ORTHO:
            self._norm_kwargs["norm"] = "ortho"
        elif self.norm is _FFTNorms.NONE:
            self._scale = np.prod(self.nffts)
        elif self.norm is _FFTNorms.ONE_OVER_N:
            self._scale = 1.0 / np.prod(self.nffts)

    def _matvec(self, x):
        x = np.reshape(x, self.dims)
        if self.ifftshift_before.any():
            x = np.fft.ifftshift(x, axes=self.dirs[self.ifftshift_before])
        if not self.clinear:
            x = np.real(x)
        if self.real:
            y = np.fft.rfftn(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
            # Apply scaling to obtain a correct adjoint for this operator
            y = np.swapaxes(y, -1, self.dirs[-1])
            y[..., 1 : 1 + (self.nffts[-1] - 1) // 2] *= np.sqrt(2)
            y = np.swapaxes(y, self.dirs[-1], -1)
        else:
            y = np.fft.fftn(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
        if self.norm is _FFTNorms.ONE_OVER_N:
            y *= self._scale
        y = y.astype(self.cdtype)
        if self.fftshift_after.any():
            y = np.fft.fftshift(y, axes=self.dirs[self.fftshift_after])
        return y.ravel()

    def _rmatvec(self, x):
        x = np.reshape(x, self.dims_fft)
        if self.fftshift_after.any():
            x = np.fft.ifftshift(x, axes=self.dirs[self.fftshift_after])
        if self.real:
            # Apply scaling to obtain a correct adjoint for this operator
            x = x.copy()
            x = np.swapaxes(x, -1, self.dirs[-1])
            x[..., 1 : 1 + (self.nffts[-1] - 1) // 2] /= np.sqrt(2)
            x = np.swapaxes(x, self.dirs[-1], -1)
            y = np.fft.irfftn(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
        else:
            y = np.fft.ifftn(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
        if self.norm is _FFTNorms.NONE:
            y *= self._scale
        for direction, nfft in zip(self.dirs, self.nffts):
            if nfft > self.dims[direction]:
                y = np.take(y, range(self.dims[direction]), axis=direction)
        if self.doifftpad:
            y = np.pad(y, self.ifftpad)
        if not self.clinear:
            y = np.real(y)
        y = y.astype(self.rdtype)
        if self.ifftshift_before.any():
            y = np.fft.fftshift(y, axes=self.dirs[self.ifftshift_before])
        return y.ravel()

    def __truediv__(self, y):
        if self.norm is not _FFTNorms.ORTHO:
            return self._rmatvec(y) / self._scale
        return self._rmatvec(y)


class _FFTND_scipy(_BaseFFTND):
    """N-dimensional Fast-Fourier Transform using SciPy"""

    def __init__(
        self,
        dims,
        dirs=(0, 1, 2),
        nffts=None,
        sampling=1.0,
        norm="ortho",
        real=False,
        ifftshift_before=False,
        fftshift_after=False,
        dtype="complex128",
    ):
        super().__init__(
            dims=dims,
            dirs=dirs,
            nffts=nffts,
            sampling=sampling,
            norm=norm,
            real=real,
            ifftshift_before=ifftshift_before,
            fftshift_after=fftshift_after,
            dtype=dtype,
        )

        self._norm_kwargs = {"norm": None}  # equivalent to "backward" in Numpy/Scipy
        if self.norm is _FFTNorms.ORTHO:
            self._norm_kwargs["norm"] = "ortho"
        elif self.norm is _FFTNorms.NONE:
            self._scale = np.prod(self.nffts)
        elif self.norm is _FFTNorms.ONE_OVER_N:
            self._scale = 1.0 / np.prod(self.nffts)

    def _matvec(self, x):
        x = np.reshape(x, self.dims)
        if self.ifftshift_before.any():
            x = scipy.fft.ifftshift(x, axes=self.dirs[self.ifftshift_before])
        if not self.clinear:
            x = np.real(x)
        if self.real:
            y = scipy.fft.rfftn(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
            # Apply scaling to obtain a correct adjoint for this operator
            y = np.swapaxes(y, -1, self.dirs[-1])
            y[..., 1 : 1 + (self.nffts[-1] - 1) // 2] *= np.sqrt(2)
            y = np.swapaxes(y, self.dirs[-1], -1)
        else:
            y = scipy.fft.fftn(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
        if self.norm is _FFTNorms.ONE_OVER_N:
            y *= self._scale
        if self.fftshift_after.any():
            y = scipy.fft.fftshift(y, axes=self.dirs[self.fftshift_after])
        return y.ravel()

    def _rmatvec(self, x):
        x = np.reshape(x, self.dims_fft)
        if self.fftshift_after.any():
            x = scipy.fft.ifftshift(x, axes=self.dirs[self.fftshift_after])
        if self.real:
            # Apply scaling to obtain a correct adjoint for this operator
            x = x.copy()
            x = np.swapaxes(x, -1, self.dirs[-1])
            x[..., 1 : 1 + (self.nffts[-1] - 1) // 2] /= np.sqrt(2)
            x = np.swapaxes(x, self.dirs[-1], -1)
            y = scipy.fft.irfftn(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
        else:
            y = scipy.fft.ifftn(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
        if self.norm is _FFTNorms.NONE:
            y *= self._scale
        for direction, nfft in zip(self.dirs, self.nffts):
            if nfft > self.dims[direction]:
                y = np.take(y, range(self.dims[direction]), axis=direction)
        if self.doifftpad:
            y = np.pad(y, self.ifftpad)
        if not self.clinear:
            y = np.real(y)
        if self.ifftshift_before.any():
            y = scipy.fft.fftshift(y, axes=self.dirs[self.ifftshift_before])
        return y.ravel()

    def __truediv__(self, y):
        if self.norm is not _FFTNorms.ORTHO:
            return self._rmatvec(y) / self._scale
        return self._rmatvec(y)


[docs]def FFTND( dims, dirs=(0, 1, 2), nffts=None, sampling=1.0, norm="ortho", real=False, ifftshift_before=False, fftshift_after=False, dtype="complex128", engine="scipy", ): r"""N-dimensional Fast-Fourier Transform. Apply N-dimensional Fast-Fourier Transform (FFT) to any n axes of a multi-dimensional array depending on the choice of ``dirs``. Using the default NumPy engine, the FFT operator is an overload to either the NumPy :py:func:`numpy.fft.fftn` (or :py:func:`numpy.fft.rfftn` for real models) in forward mode, and to :py:func:`numpy.fft.ifftn` (or :py:func:`numpy.fft.irfftn` for real models) in adjoint mode, or their CuPy equivalents. Alternatively, when the SciPy engine is chosen, the overloads are of :py:func:`scipy.fft.fftn` (or :py:func:`scipy.fft.rfftn` for real models) in forward mode, and to :py:func:`scipy.fft.ifftn` (or :py:func:`scipy.fft.irfftn` for real models) in adjoint mode. When using ``real=True``, the result of the forward is also multiplied by :math:`\sqrt{2}` for all frequency bins except zero and Nyquist along the last direction of ``dirs``, and the input of the adjoint is multiplied by :math:`1 / \sqrt{2}` for the same frequencies. For a real valued input signal, it is advised to use the flag ``real=True`` as it stores the values of the Fourier transform of the last direction at positive frequencies only as values at negative frequencies are simply their complex conjugates. Parameters ---------- dims : :obj:`tuple` Number of samples for each dimension dirs : :obj:`tuple` or :obj:`int`, optional Direction(s) along which FFTND is applied nffts : :obj:`tuple` or :obj:`int`, optional Number of samples in Fourier Transform for each direction. In case only one dimension needs to be specified, use ``None`` for the other dimension in the tuple. The direction with None will use ``dims[dir]`` as ``nfft``. When supplying a tuple, the order must agree with that of ``dirs``. When a single value is passed, it will be used for both directions. As such the default is equivalent to ``nffts=(None,..., None)``. sampling : :obj:`tuple` or :obj:`float`, optional Sampling steps for each direction. When supplied a single value, it is used for all directions. Unlike ``nffts``, any ``None`` will not be converted to the default value. norm : `{"ortho", "none", "1/n"}`, optional .. versionadded:: 1.17.0 - "ortho": Scales forward and adjoint FFT transforms with :math:`1/\sqrt{N_F}`, where :math:`N_F` is the number of samples in the Fourier domain given by product of all elements of ``nffts``. - "none": Does not scale the forward or the adjoint FFT transforms. - "1/n": Scales both the forward and adjoint FFT transforms by :math:`1/N_F`. .. note:: For "none" and "1/n", the operator is not unitary, that is, the adjoint is not the inverse. To invert the operator, simply use ``Op \ y``. real : :obj:`bool`, optional Model to which fft is applied has real numbers (``True``) or not (``False``). Used to enforce that the output of adjoint of a real model is real. Note that the real FFT is applied only to the first dimension to which the FFTND operator is applied (last element of ``dirs``) ifftshift_before : :obj:`tuple` or :obj:`bool`, optional .. versionadded:: 1.17.0 Apply ifftshift (``True``) or not (``False``) to model vector (before FFT). Consider using this option when the model vector's respective axis is symmetric with respect to the zero value sample. This will shift the zero value sample to coincide with the zero index sample. With such an arrangement, FFT will not introduce a sample-dependent phase-shift when compared to the continuous Fourier Transform. When passing a single value, the shift will the same for every direction. Pass a tuple to specify which dimensions are shifted. fftshift_after : :obj:`tuple` or :obj:`bool`, optional .. versionadded:: 1.17.0 Apply fftshift (``True``) or not (``False``) to data vector (after FFT). Consider using this option when you require frequencies to be arranged naturally, from negative to positive. When not applying fftshift after FFT, frequencies are arranged from zero to largest positive, and then from negative Nyquist to the frequency bin before zero. When passing a single value, the shift will the same for every direction. Pass a tuple to specify which dimensions are shifted. engine : :obj:`str`, optional .. versionadded:: 1.17.0 Engine used for fft computation (``numpy`` or ``scipy``). dtype : :obj:`str`, optional Type of elements in input array. Note that the ``dtype`` of the operator is the corresponding complex type even when a real type is provided. In addition, note that the NumPy backend does not support returning ``dtype`` different than ``complex128``. As such, when using the NumPy backend, arrays will be force-cast to types corresponding to the supplied ``dtype``. The SciPy backend supports all precisions natively. Under both backends, when a real ``dtype`` is supplied, a real result will be enforced on the result of the ``rmatvec`` and the input of the ``matvec``. Attributes ---------- dims_fft : :obj:`tuple` Shape of the array after the forward, but before linearization. For example, ``y_reshaped = (Op * x.ravel()).reshape(Op.dims_fft)``. fs : :obj:`tuple` Each element of the tuple corresponds to the Discrete Fourier Transform sample frequencies along the respective direction given by ``dirs``. real : :obj:`bool` When ``True``, uses ``rfftn``/``irfftn`` rdtype : :obj:`bool` Expected input type to the forward cdtype : :obj:`bool` Output type of the forward. Complex equivalent to ``rdtype``. shape : :obj:`tuple` Operator shape clinear : :obj:`bool` .. versionadded:: 1.17.0 Operator is complex-linear. Is false when either ``real=True`` or when ``dtype`` is not a complex type. explicit : :obj:`bool` Operator contains a matrix that can be solved explicitly (``True``) or not (``False``) See Also -------- FFT: One-dimensional FFT FFT2D: Two-dimensional FFT Raises ------ ValueError - If ``nffts`` or ``sampling`` are not either a single value or tuple with the same dimension ``dirs``. - If ``norm`` is not one of "ortho", "none", or "1/n". NotImplementedError If ``engine`` is neither ``numpy``, nor ``scipy``. Notes ----- The FFTND operator (using ``norm="ortho"``) applies the N-dimensional forward Fourier transform to a multi-dimensional array. Considering an N-dimensional signal :math:`d(x_1, \ldots, x_N)`. The FFTND in forward mode is: .. math:: D(k_1, \ldots, k_N) = \mathscr{F} (d) = \frac{1}{\sqrt{N_F}} \int\limits_{-\infty}^\infty \cdots \int\limits_{-\infty}^\infty d(x_1, \ldots, x_N) e^{-j2\pi k_1 x_1} \cdots e^{-j 2 \pi k_N x_N} \,\mathrm{d}x_1 \cdots \mathrm{d}x_N Similarly, the three-dimensional inverse Fourier transform is applied to the Fourier spectrum :math:`D(k_z, k_y, k_x)` in adjoint mode: .. math:: d(x_1, \ldots, x_N) = \mathscr{F}^{-1} (D) = \frac{1}{\sqrt{N_F}} \int\limits_{-\infty}^\infty \cdots \int\limits_{-\infty}^\infty D(k_1, \ldots, k_N) e^{-j2\pi k_1 x_1} \cdots e^{-j 2 \pi k_N x_N} \,\mathrm{d}k_1 \cdots \mathrm{d}k_N where :math:`N_F` is the number of samples in the Fourier domain given by the product of the element of ``nffts``. Both operators are effectively discretized and solved by a fast iterative algorithm known as Fast Fourier Transform. Note that the FFTND operator (using ``norm="ortho"``) is a special operator in that the adjoint is also the inverse of the forward mode. For other norms, this does not hold (see ``norm`` help). However, for any norm, the N-dimensional Fourier transform is Hermitian for real input signals. """ if engine == "numpy": f = _FFTND_numpy( dims=dims, dirs=dirs, nffts=nffts, sampling=sampling, norm=norm, real=real, ifftshift_before=ifftshift_before, fftshift_after=fftshift_after, dtype=dtype, ) elif engine == "scipy": f = _FFTND_scipy( dims=dims, dirs=dirs, nffts=nffts, sampling=sampling, norm=norm, real=real, ifftshift_before=ifftshift_before, fftshift_after=fftshift_after, dtype=dtype, ) else: raise NotImplementedError("engine must be numpy or scipy") return f