Source code for pylops.signalprocessing.FFT2D

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 _FFT2D_numpy(_BaseFFTND):
    """Two dimensional Fast-Fourier Transform using NumPy"""

    def __init__(
        self,
        dims,
        dirs=(0, 1),
        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 casted to {self.cdtype}."
            )

        # checks
        if self.ndim < 2:
            raise ValueError("FFT2D requires at least two input dimensions")
        if self.ndirs != 2:
            raise ValueError("FFT2D must be applied along exactly two dimensions")

        self.f1, self.f2 = self.fs
        del self.fs

        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.rfft2(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.fft2(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.irfft2(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
        else:
            y = np.fft.ifft2(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
        if self.norm is _FFTNorms.NONE:
            y *= self._scale
        y = np.take(y, range(self.dims[self.dirs[0]]), axis=self.dirs[0])
        y = np.take(y, range(self.dims[self.dirs[1]]), axis=self.dirs[1])
        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 _FFT2D_scipy(_BaseFFTND):
    """Two dimensional Fast-Fourier Transform using SciPy"""

    def __init__(
        self,
        dims,
        dirs=(0, 1),
        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,
        )

        # checks
        if self.ndim < 2:
            raise ValueError("FFT2D requires at least two input dimensions")
        if self.ndirs != 2:
            raise ValueError("FFT2D must be applied along exactly two dimensions")

        self.f1, self.f2 = self.fs
        del self.fs

        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.sqrt(np.prod(self.nffts))
        elif self.norm is _FFTNorms.ONE_OVER_N:
            self._scale = np.sqrt(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.rfft2(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.fft2(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.irfft2(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
        else:
            y = scipy.fft.ifft2(x, s=self.nffts, axes=self.dirs, **self._norm_kwargs)
        if self.norm is _FFTNorms.NONE:
            y *= self._scale
        y = np.take(y, range(self.dims[self.dirs[0]]), axis=self.dirs[0])
        y = np.take(y, range(self.dims[self.dirs[1]]), axis=self.dirs[1])
        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 / self._scale
        return self._rmatvec(y)


[docs]def FFT2D( dims, dirs=(0, 1), nffts=None, sampling=1.0, norm="ortho", real=False, ifftshift_before=False, fftshift_after=False, dtype="complex128", engine="numpy", ): r"""Two dimensional Fast-Fourier Transform. Apply two dimensional Fast-Fourier Transform (FFT) to any pair of 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.fft2` (or :py:func:`numpy.fft.rfft2` for real models) in forward mode, and to :py:func:`numpy.fft.ifft2` (or :py:func:`numpy.fft.irfft2` 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.fft2` (or :py:func:`scipy.fft.rfft2` for real models) in forward mode, and to :py:func:`scipy.fft.ifft2` (or :py:func:`scipy.fft.irfft2` 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, 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`, optional Pair of directions along which FFT2D 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 both directions. Unlike ``nffts``, ``None``s will not be converted to the default value. norm : `{"ortho", "none", "1/n"}`, optional * "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 that 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 FFT2D operator is applied (last element of ``dirs``) ifftshift_before : :obj:`tuple` or :obj:`bool`, optional 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 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 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``s different than ``complex128``. As such, when using the NumPy backend, arrays will be force-casted 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. E.g. ``y_reshaped = (Op * x.ravel()).reshape(Op.dims_fft)``. f1 : :obj:`numpy.ndarray` Discrete Fourier Transform sample frequencies along ``dir[0]`` f2 : :obj:`numpy.ndarray` Discrete Fourier Transform sample frequencies along ``dir[1]`` real : :obj:`bool` When True, uses ``rfft2``/``irfft2`` 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` 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) Raises ------ ValueError If ``dims`` has less than two elements. If ``dirs`` does not have exactly two elements. If ``nffts`` or ``sampling`` are not either a single value or a tuple with two elements. If ``norm`` is not one of "ortho", "none", or "1/n". NotImplementedError If ``engine`` is neither ``numpy``, nor ``scipy``. Notes ----- The FFT2D operator (using `norm="ortho"`) applies the two-dimensional forward Fourier transform to a signal :math:`d(y, x)` in forward mode: .. math:: D(k_y, k_x) = \mathscr{F} (d) = \frac{1}{\sqrt{N_F}} \int \int d(y, x) e^{-j2\pi k_yy} e^{-j2\pi k_xx} dy dx Similarly, the two-dimensional inverse Fourier transform is applied to the Fourier spectrum :math:`D(k_y, k_x)` in adjoint mode: .. math:: d(y,x) = \mathscr{F}^{-1} (D) = \frac{1}{\sqrt{N_F}} \int \int D(k_y, k_x) e^{j2\pi k_yy} e^{j2\pi k_xx} dk_y dk_x 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 FFT2D 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 2D Fourier transform is Hermitian for real input signals. """ if engine == "numpy": f = _FFT2D_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 = _FFT2D_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