Providing a Visual Understanding of Holography Through Phase Space Representations

Tobias Birnbaum,Peter Schelkens,Tomasz Kozacki

doi:10.3390/app10144766

Tobias Birnbaum, Peter Schelkens + Show 1 more

Open Access

https://doi.org/10.3390/app10144766

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Abstract

Digital holograms are a prime example for signals, which are best understood in phase space—the joint space of spatial coordinates and spatial frequencies. Many characteristics, as well as optical operations can be visualized therein with so called phase space representations (PSRs). However, literature relies often only on symbolic PSRs or on, in practice, visually insufficient PSRs like the Wigner–Ville representation. In this tutorial-style paper, we will showcase the S-method, which is both a PSR that can be calculated directly from any given signal, and that allows for a clear visual interpretation. We will highlight the power of space-frequency analysis in digital holography, explain why this specific PSR is recommended, discuss a broad range of basic operations, and briefly overview several interesting practical questions in digital holography.

Highlights

The phase space of a signal is the joint space of the signals native domain and its Fourier domain.It can consist of time and frequency for temporal signals or spatial positions and spatial frequencies for spatial signals
Shifting the window with respect to the signal and re-evaluating the Fourier transform each time, will provide a representation of excited frequencies over space. This concept of a windowed Fourier transform is known as the short-term Fourier transform (STFT), which is given at a location ξ px ∈ {0, . . . , N − 1}
We provided a motivation of phase space analysis for digital holography (DH) and discussed aspects of it on individual

Summary

Introduction

The phase space of a signal is the joint space of the signals native domain and its Fourier domain.It can consist of time and frequency for temporal signals or spatial positions and spatial frequencies for spatial signals. Phase space representations (PSRs; in signal processing literature the term time-frequency representations is more common) are an essential tool initially proposed to aid phase space analysis, which tries to understand systems whose frequency spectrum changes over the course of the signal by visualizing quasi-instantaneous frequencies as they change. They quickly proved themselves useful in many applications of science and engineering. Shifting the window with respect to the signal and re-evaluating the Fourier transform each time, will provide a (quasi-stationary) representation of excited frequencies over space. All other parameters influencing the quality of the STFT, will affect the visual performance of the

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