pylops.signalprocessing.FFTND¶
- pylops.signalprocessing.FFTND(dims, axes=(-3, -2, -1), nffts=None, sampling=1.0, norm='ortho', real=False, ifftshift_before=False, fftshift_after=False, engine='scipy', dtype='complex128', name='F')[source]¶
N-dimensional Fast-Fourier Transform.
Apply N-dimensional Fast-Fourier Transform (FFT) to any n
axes
of a multi-dimensional array.Using the default NumPy engine, the FFT operator is an overload to either the NumPy
numpy.fft.fftn
(ornumpy.fft.rfftn
for real models) in forward mode, and tonumpy.fft.ifftn
(ornumpy.fft.irfftn
for real models) in adjoint mode, or their CuPy equivalents. Alternatively, when the SciPy engine is chosen, the overloads are ofscipy.fft.fftn
(orscipy.fft.rfftn
for real models) in forward mode, and toscipy.fft.ifftn
(orscipy.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 lastaxes
, 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 axis inaxes
at positive frequencies only as values at negative frequencies are simply their complex conjugates.- Parameters
- dims
tuple
Number of samples for each dimension
- axes
int
, optional New in version 2.0.0.
Axes (or axis) along which FFTND is applied
- nffts
tuple
orint
, optional Number of samples in Fourier Transform for each axis in
axes
. In case only one dimension needs to be specified, useNone
for the other dimension in the tuple. An axis withNone
will usedims[axis]
asnfft
. When supplying a tuple, the length must agree with that ofaxes
. When a single value is passed, it will be used for allaxes`. As such the default is equivalent to ``nffts=(None, ..., None)
.- sampling
tuple
orfloat
, optional Sampling steps for each direction. When supplied a single value, it is used for all directions. Unlike
nffts
, anyNone
will not be converted to the default value.- norm{“ortho”, “none”, “1/n”}, optional
New in version 1.17.0.
“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
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 ofaxes
)- ifftshift_before
tuple
orbool
, optional New in version 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
tuple
orbool
, optional New in version 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
str
, optional New in version 1.17.0.
Engine used for fft computation (
numpy
orscipy
).- 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 returningdtype
different thancomplex128
. As such, when using the NumPy backend, arrays will be force-cast to types corresponding to the supplieddtype
. The SciPy backend supports all precisions natively. Under both backends, when a realdtype
is supplied, a real result will be enforced on the result of thermatvec
and the input of thematvec
.- name
str
, optional New in version 2.0.0.
Name of operator (to be used by
pylops.utils.describe.describe
)
- dims
- Raises
- ValueError
If
nffts
orsampling
are not either a single value or tuple with the same dimensionaxes
.If
norm
is not one of “ortho”, “none”, or “1/n”.
- NotImplementedError
If
engine
is neithernumpy
, norscipy
.
Notes
The FFTND operator (using
norm="ortho"
) 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:\[D(k_1, \ldots, k_N) = \mathscr{F} (d) = \frac{1}{\sqrt{N_F}} \int\limits_{-\infty}^\infty \cdots \int\limits_{-\infty}^\infty d(x_1, \ldots, x_N) e^{-j2\pi k_1 x_1} \cdots e^{-j 2 \pi k_N x_N} \,\mathrm{d}x_1 \cdots \mathrm{d}x_N\]Similarly, the three-dimensional inverse Fourier transform is applied to the Fourier spectrum \(D(k_z, k_y, k_x)\) in adjoint mode:
\[d(x_1, \ldots, x_N) = \mathscr{F}^{-1} (D) = \frac{1}{\sqrt{N_F}} \int\limits_{-\infty}^\infty \cdots \int\limits_{-\infty}^\infty D(k_1, \ldots, k_N) e^{-j2\pi k_1 x_1} \cdots e^{-j 2 \pi k_N x_N} \,\mathrm{d}k_1 \cdots \mathrm{d}k_N\]where \(N_F\) 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 (usingnorm="ortho"
) is a special operator in that the adjoint is also the inverse of the forward mode. For other norms, this does not hold (seenorm
help). However, for any norm, the N-dimensional Fourier transform is Hermitian for real input signals.- Attributes
- dimsd
tuple
Shape of the array after the forward, but before linearization.
For example,
y_reshaped = (Op * x.ravel()).reshape(Op.dimsd)
.- fs
tuple
Each element of the tuple corresponds to the Discrete Fourier Transform sample frequencies along the respective direction given by
axes
.- real
bool
When
True
, usesrfftn
/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
New in version 1.17.0.
Operator is complex-linear. Is false when either
real=True
or whendtype
is not a complex type.- explicit
bool
Operator contains a matrix that can be solved explicitly (
True
) or not (False
)
- dimsd