pylops.signalprocessing.FFT2D

class pylops.signalprocessing.FFT2D(dims, dirs=(0, 1), nffts=(None, None), sampling=(1.0, 1.0), dtype='complex128')[source]

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. Note that the FFT2D operator is a simple overload to the numpy numpy.fft.fft2 in forward mode and to the numpy numpy.fft.ifft2 in adjoint mode, however scaling is taken into account differently to guarantee that the operator is passing the dot-test.

Parameters:
dims : tuple

Number of samples for each dimension

dirs : tuple, optional

Pair of directions along which FFT2D is applied

nffts : tuple, optional

Number of samples in Fourier Transform for each direction (same as input if nffts=(None, None))

sampling : tuple, optional

Sampling steps dy and dx

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. Nevertheless, the provided dtype will be enforced on the vector returned by the rmatvec method.
Raises:
ValueError

If dims has less than two elements, and if dirs, nffts, or sampling has more or less than two elements.

Notes

The FFT2D operator applies the two-dimensional forward Fourier transform to a signal \(d(y,x)\) in forward mode:

\[D(k_y, k_x) = \mathscr{F} (d) = \int \int d(y,x) e^{-j2\pi k_yy} e^{-j2\pi k_xx} dy dx\]

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

\[d(y,x) = \mathscr{F}^{-1} (D) = \int \int D(k_y, k_x) e^{j2\pi k_yy} e^{j2\pi k_xx} dk_y dk_x\]

Both operators are effectively discretized and solved by a fast iterative algorithm known as Fast Fourier Transform.

Attributes:
shape : tuple

Operator shape

explicit : bool

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

Methods

__init__(self, dims[, dirs, nffts, …]) Initialize this LinearOperator.
adjoint(self) Hermitian adjoint.
apply_columns(self, cols) Apply subset of columns of operator
cond(self, **kwargs_eig) Condition number of linear operator.
conj(self) Complex conjugate operator
div(self, y[, niter]) Solve the linear problem \(\mathbf{y}=\mathbf{A}\mathbf{x}\).
dot(self, x) Matrix-matrix or matrix-vector multiplication.
eigs(self[, neigs, symmetric, niter]) Most significant eigenvalues of linear operator.
matmat(self, X) Matrix-matrix multiplication.
matvec(self, x) Matrix-vector multiplication.
rmatmat(self, X) Adjoint matrix-matrix multiplication.
rmatvec(self, x) Adjoint matrix-vector multiplication.
todense(self) Return dense matrix.
transpose(self) Transpose this linear operator.