Multidimensional factorization through helical mapping

TL;DR

This paper introduces a helical mapping approach for multidimensional spectral factorization, enabling efficient analysis by converting multidimensional data into one dimension, with applications demonstrated in wave propagation and helioseismology.

Contribution

It presents a novel helical mapping method for multidimensional spectral factorization, linking PDE solutions with cepstral analysis and demonstrating its effectiveness in practical applications.

Findings

Helical mapping converges to semi-causal solutions in multidimensional spectral factorization.

The method effectively decouples PDE solutions in physical systems.

Applications include wave propagation and helioseismology analysis.

Abstract

This paper proposes a new perspective on the problem of multidimensional spectral factorization, through helical mapping: $d$-dimensional ($d$D) data arrays are vectorized, processed by $1$D cepstral analysis and then remapped onto the original space. Partial differential equations (PDEs) are the basic framework to describe the evolution of physical phenomena. We observe that the minimum phase helical solution asymptotically converges to the $d$D semi-causal solution, and allows to decouple the two solutions arising from PDEs describing physical systems. We prove this equivalence in the theoretical framework of cepstral analysis, and we also illustrate the validity of helical factorization through a $2$D wave propagation example and a $3$D application to helioseismology.