Fourier Analysis and Holographic Representations of 1D and 2D Signals

TL;DR

This paper explores Fourier analysis and holographic transforms for signal representation, introducing new methods like CHFT and demonstrating their potential in image and 1D signal processing.

Contribution

It introduces the Continuous Holographic Fourier Transform (CHFT) and a progressive transmission scheme based on DHFT, expanding holographic signal representation techniques.

Findings

Demonstrated the effectiveness of DHFT for image representation

Proposed and discussed theoretical aspects of CHFT for 1D signals

Presented experimental results validating the methods

Abstract

In this paper, we focus on Fourier analysis and holographic transforms for signal representation. For instance, in the case of image processing, the holographic representation has the property that an arbitrary portion of the transformed image enables reconstruction of the whole image with details missing. We focus on holographic representation defined through the Fourier Transforms. Thus, We firstly review some results in Fourier transform and Fourier series. Next, we review the Discrete Holographic Fourier Transform (DHFT) for image representation. Then, we describe the contributions of our work. We show a simple scheme for progressive transmission based on the DHFT. Next, we propose the Continuous Holographic Fourier Transform (CHFT) and discuss some theoretical aspects of it for 1D signals. Finally, some testes are presented in the experimental results