pylops.signalprocessing.FFTND

pylops.signalprocessing.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')[source]

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 numpy.fft.fftn (or numpy.fft.rfftn for real models) in forward mode, and to numpy.fft.ifftn (or numpy.fft.irfftn for real models) in adjoint mode, or their CuPy equivalents. Alternatively, when the SciPy engine is chosen, the overloads are of scipy.fft.fftn (or scipy.fft.rfftn for real models) in forward mode, and to scipy.fft.ifftn (or scipy.fft.irfftn for real models) in adjoint mode.

When using real=True, the result of the forward is also multiplied by \(\sqrt{2}\) for all frequency bins except zero and Nyquist along the last direction of dirs, and the input of the adjoint is multiplied by \(1 / \sqrt{2}\) 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 : tuple

Number of samples for each dimension

dirs : tuple or int, optional

Direction(s) along which FFTND is applied

nffts : tuple or 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 : tuple or float, optional

Sampling steps for each direction. When supplied a single value, it is used for all 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 \(1/\sqrt{N_F}\),

where \(N_F\) 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 \(1/N_F\). 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 : 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 : tuple or 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 : tuple or 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 : str, optional

Engine used for fft computation (numpy or scipy).

dtype : 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.

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 applies the N-dimensional forward Fourier transform to a multi-dimensional array. Considering an N-dimensional signal \(d(x_1, \ldots, x_N)\). The FFTND in forward mode is:

\[D(k_1, \ldots, k_N) = \mathscr{F} (d) = \frac{1}{\sqrt{N_F}} \int \int d(x_1, \ldots, x_N) e^{-j2\pi k_1 x_1} \cdots e^{-j 2 \pi k_N x_N} dx_1 \cdots dx_N\]

Similarly, the three-dimensional inverse Fourier transform is applied to the Fourier spectrum \(D(k_z, k_y, k_x)\) in adjoint mode:

\[d(x_1, \ldots, x_N) = \mathscr{F}^{-1} (D) = \frac{1}{\sqrt{N_F}} \int \int D(k_1, \ldots, k_N) e^{-j2\pi k_1 x_1} \cdots e^{-j 2 \pi k_N x_N} dk_1 \cdots dk_N\]

where \(N_F\) 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.

Attributes:
dims_fft : tuple

Shape of the array after the forward, but before linearization. E.g. y_reshaped = (Op * x.ravel()).reshape(Op.dims_fft).

fs : tuple

Each element of the tuple corresponds to the Discrete Fourier Transform sample frequencies along the respective direction given by dirs.

real : bool

When True, uses rfftn/irfftn

rdtype : bool

Expected input type to the forward

cdtype : bool

Output type of the forward. Complex equivalent to rdtype.

shape : tuple

Operator shape

clinear : bool

Operator is complex-linear. Is false when either real=True or when dtype is not a complex type.

explicit : bool

Operator contains a matrix that can be solved explicitly (True) or not (False)

Examples using pylops.signalprocessing.FFTND