pylops.signalprocessing.FourierRadon2D

class pylops.signalprocessing.FourierRadon2D(taxis, haxis, pxaxis, nfft, flims=None, kind='linear', engine='numpy', num_threads_per_blocks=(32, 32), dtype='float64', name='R')

2D Fourier Radon transform

Apply Radon forward (and adjoint) transform using Fast Fourier Transform to a 2-dimensional array of size \([n_{p_x} \times n_t]\) (and \([n_x \times n_t]\)).

Note that forward and adjoint follow the same convention of the time-space implementation in pylops.signalprocessing.Radon2D.

Parameters:

taxisnumpy.ndarray

Time axis

haxisnumpy.ndarray

Spatial axis

pxaxisnumpy.ndarray

Axis of scanning variable \(p_x\) of parametric curve

nfftint

Number of samples in Fourier transform

flimstuple, optional

Indices of lower and upper limits of Fourier axis to be used in the application of the Radon matrix (when None, use entire axis)

kindstr

Curve to be used for stacking/spreading (linear, parabolic)

enginestr

Engine used for computation (numpy or numba or cuda)

num_threads_per_blockstuple

Number of threads in each block (only when engine=cuda)

dtypestr

Type of elements in input array.

namestr

Name of operator (to be used by pylops.utils.describe.describe)

Attributes:

haxisnumpy.ndarray

Spatial axis (or squared axis when kind='parabolic')

nhint

Number of samples in spatial axis.

ntint

Number of samples in time axis.

dhfloat

Sampling step in spatial axis.

dtfloat

Sampling step in time axis.

pxnumpy.ndarray

Axis of scanning variable \(p_x\) of parametric curve

npxint

Number of samples in \(p_x\) axis.

fnumpy.ndarray

Fourier axis.

nfft2int

Number of samples in positive Fourier axis.

cdtypestr

Complex type associated with dtype.

flimstuple

Indices of lower and upper limits of Fourier axis to be used in

num_blocks_matvectuple

Number of blocks in each dimension for matvec (only when engine=cuda)

num_blocks_rmatvectuple

Number of blocks in each dimension for rmatvec (only when engine=cuda)

dimstuple

Shape of the array after the adjoint, but before flattening.

For example, x_reshaped = (Op.H * y.ravel()).reshape(Op.dims).

dimsdtuple

Shape of the array after the forward, but before flattening.

For example, y_reshaped = (Op * x.ravel()).reshape(Op.dimsd).

shapetuple

Operator shape.

Raises:

ValueError

If engine is neither numpy, numba, nor cuda.

Notes

The FourierRadon2D operator applies the Radon transform in the frequency domain. After transforming a 2-dimensional array of size \([n_x \times n_t]\) into the frequency domain, the following linear transformation is applied to each frequency component in adjoint mode:

\[\begin{split}\begin{bmatrix} m(p_{x,1}, \omega_i) \\ m(p_{x,2}, \omega_i) \\ \vdots \\ m(p_{x,N_p}, \omega_i) \end{bmatrix} = \begin{bmatrix} e^{-j \omega_i p_{x,1} x^l_1} & e^{-j \omega_i p_{x,1} x^l_2} & \ldots & e^{-j \omega_i p_{x,1} x^l_{N_x}} \\ e^{-j \omega_i p_{x,2} x^l_1} & e^{-j \omega_i p_{x,2} x^l_2} & \ldots & e^{-j \omega_i p_{x,2} x^l_{N_x}} \\ \vdots & \vdots & \ddots & \vdots \\ e^{-j \omega_i p_{x,N_p} x^l_1} & e^{-j \omega_i p_{x,N_p} x^l_2} & \ldots & e^{-j \omega_i p_{x,N_p} x^l_{N_x}} \\ \end{bmatrix} \begin{bmatrix} d(x_1, \omega_i) \\ d(x_2, \omega_i) \\ \vdots \\ d(x_{N_x}, \omega_i) \end{bmatrix}\end{split}\]

where \(l=1,2\). Similarly the forward mode is implemented by applying the transpose and complex conjugate of the above matrix to the model transformed to the Fourier domain.

Refer to [1] for more theoretical and implementation details.

[1]

Sacchi, M. “Statistical and Transform Methods for Geophysical Signal Processing”, 2007.

Methods

__init__(taxis, haxis, pxaxis, nfft[, ...])
adjoint()
apply_columns(cols)	Apply subset of columns of operator
cond([uselobpcg])	Condition number of linear operator.
conj()	Complex conjugate operator
div(y[, niter, densesolver])	Solve the linear problem \(\mathbf{y}=\mathbf{A}\mathbf{x}\).
dot(x)	Matrix-matrix or matrix-vector multiplication.
eigs([neigs, symmetric, niter, uselobpcg])	Most significant eigenvalues of linear operator.
matmat(X)	Matrix-matrix multiplication.
matvec(x)	Matrix-vector multiplication.
reset_count()	Reset counters
rmatmat(X)	Matrix-matrix multiplication.
rmatvec(x)	Adjoint matrix-vector multiplication.
todense([backend])	Return dense matrix.
toimag([forw, adj])	Imag operator
toreal([forw, adj])	Real operator
tosparse()	Return sparse matrix.
trace([neval, method, backend])	Trace of linear operator.
transpose()

Examples using pylops.signalprocessing.FourierRadon2D

Fourier Radon Transform

Fourier Radon Transform