Poisson Summation Formulae Associated with the Special Affine Fourier Transform and Offset Hilbert Transform

This paper investigates the generalized pattern of Poisson summation formulae from the special affine Fourier transform (SAFT) and offset Hilbert transform (OHT) points of view. Several novel summation formulae are derived accordingly. Firstly, the relationship between SAFT (or OHT) and Fourier transform (FT) is obtained. Then, the generalized Poisson sum formulae are obtained based on above relationships. The novel results can be regarded as the generalizations of the classical results in several transform domains such as FT, fractional Fourier transform, and the linear canonical transform

Sampling theorems in the OLCT and the OLCHT domains by polar coordinates

The sampling theorem for the offset linear canonical transform (OLCT) of bandlimited functions in polar coordinates is an important mathematical tool in many fields of signal processing and medical imaging. This paper investigates two sampling theorems for interpolating \Omega bandlimited and highest frequency bandlimited functions f(r,{\theta}) in the OLCT and the offset linear canonical Hankel transform (OLCHT) domains by polar coordinates. Based on the classical Stark's interpolation formulas, we derive the sampling theorems for \Omega bandlimited functions f(r,{\theta}) in the OLCT and the OLCHT domains, respectively. The first interpolation formula is concise and applicable. Due to the consistency of the OLCHT order, the second interpolation formula is superior to the first interpolation formula in computational complexity.Comment: 24 page

Special Affine Stockwell Transform Theory, Uncertainty Principles and Applications

In this paper, we study the convolution structure in the special affine Fourier transform domain to combine the advantages of the well known special affine Fourier and Stockwell transforms into a novel integral transform coined as special affine Stockwell transform and investigate the associated constant Q property in the joint time frequency domain. The preliminary analysis encompasses the derivation of the fundamental properties, Rayleighs energy theorem, inversion formula and range theorem. Besides, we also derive a direct relationship between the recently introduced special affine scaled Wigner distribution and the proposed SAST. Further, we establish Heisenbergs uncertainty principle, logarithmic uncertainty principle and Nazarovs uncertainty principle associated with the proposed SAST. Towards the culmination of this paper, some potential applications with simulation are presented.Comment: arXiv admin note: text overlap with arXiv:2010.01972 by other author

5D seismic data completion and denoising using a novel class of tensor decompositions

We have developed a novel strategy for simultaneous interpolation and denoising of prestack seismic data. Most seismic surveys fail to cover all possible source-receiver combinations, leading to missing data especially in the midpoint-offset domain. This undersampling can complicate certain data processing steps such as amplitude-variation-with-offset analysis and migration. Data interpolation can mitigate the impact of missing traces. We considered the prestack data as a 5D multidimensional array or otherwise referred to as a 5D tensor. Using synthetic data sets, we first found that prestack data can be well approximated by a low-rank tensor under a recently proposed framework for tensor singular value decomposition (tSVD). Under this low-rank assumption, we proposed a complexity-penalized algorithm for the recovery of missing traces and data denoising. In this algorithm, the complexity regularization was controlled by tuning a single regularization parameter using a statistical test. We tested the performance of the proposed algorithm on synthetic and real data to show that missing data can be reliably recovered under heavy downsampling. In addition, we demonstrated that compressibility, i.e., approximation of the data by a low-rank tensor, of seismic data under tSVD depended on the velocity model complexity and shot and receiver spacing. We further found that compressibility correlated with the recovery of missing data because high compressibility implied good recovery and vice versa.National Science Foundation (U.S.). Graduate Research Fellowship (Grant DGE-0806676)National Science Foundation (U.S.). Division of Computing and Communication Foundations (Grant NSF-1319653