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Holographic Phase Retrieval via Wirtinger Flow: Cartesian Form with Auxiliary Amplitude

Ittetsu Uchiyama, Chihiro Tsutake, Keita Takahashi, Toshiaki Fujii

TL;DR

The paper addresses holographic phase retrieval by enabling simultaneous optimization of phase and an auxiliary amplitude for phase-only holograms. It introduces Wirtinger flow in Cartesian form (WFCF), deriving a complex-parameter representation $c_n=a_n+jb_n$ with $h_n(c_n)=\frac{c_n}{|c_n|}$ and a gradient-based update on $\bar{c}_n$, supported by theoretical analysis of gradient geometry and an amplitude-monotonicity property. Empirical results show that WFCF yields higher PSNR and faster convergence than prior methods, including WFPF, GS, and Kaczmarz, both in simulation and optical display experiments. The work demonstrates that incorporating an auxiliary amplitude and exploiting Cartesian Wirtinger calculus can significantly accelerate holographic phase retrieval, with practical implications for faster, higher-quality holographic displays.

Abstract

We propose a new gradient method for holography, where a phase-only hologram is parameterized by not only the phase but also amplitude. The key idea of our approach is the formulation of a phase-only hologram using an auxiliary amplitude. We optimize the parameters using the so-called Wirtinger flow algorithm in the Cartesian domain, which is a gradient method defined on the basis of the Wirtinger calculus. At the early stage of optimization, each element of the hologram exists inside a complex circle, and it can take a large gradient while diverging from the origin. This characteristic contributes to accelerating the gradient descent. Meanwhile, at the final stage of optimization, each element evolves along a complex circle, similar to previous state-of-the-art gradient methods. The experimental results demonstrate that our method outperforms previous methods, primarily due to the optimization of the amplitude.

Holographic Phase Retrieval via Wirtinger Flow: Cartesian Form with Auxiliary Amplitude

TL;DR

The paper addresses holographic phase retrieval by enabling simultaneous optimization of phase and an auxiliary amplitude for phase-only holograms. It introduces Wirtinger flow in Cartesian form (WFCF), deriving a complex-parameter representation with and a gradient-based update on , supported by theoretical analysis of gradient geometry and an amplitude-monotonicity property. Empirical results show that WFCF yields higher PSNR and faster convergence than prior methods, including WFPF, GS, and Kaczmarz, both in simulation and optical display experiments. The work demonstrates that incorporating an auxiliary amplitude and exploiting Cartesian Wirtinger calculus can significantly accelerate holographic phase retrieval, with practical implications for faster, higher-quality holographic displays.

Abstract

We propose a new gradient method for holography, where a phase-only hologram is parameterized by not only the phase but also amplitude. The key idea of our approach is the formulation of a phase-only hologram using an auxiliary amplitude. We optimize the parameters using the so-called Wirtinger flow algorithm in the Cartesian domain, which is a gradient method defined on the basis of the Wirtinger calculus. At the early stage of optimization, each element of the hologram exists inside a complex circle, and it can take a large gradient while diverging from the origin. This characteristic contributes to accelerating the gradient descent. Meanwhile, at the final stage of optimization, each element evolves along a complex circle, similar to previous state-of-the-art gradient methods. The experimental results demonstrate that our method outperforms previous methods, primarily due to the optimization of the amplitude.
Paper Structure (11 sections, 5 theorems, 36 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 11 sections, 5 theorems, 36 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Wirtinger flow in polar form
  3. Wirtinger flow in Cartesian form
  4. Formulation
  5. Geometrical analysis
  6. Experimental results
  7. Comparisons with previous methods
  8. Numerical support for theoretical results
  9. Conclusion
  10. Proof of Theorem \ref{['t1']}
  11. Proof of Theorem \ref{['t2']}

Key Result

Theorem 1

The phase of the gradient satisfies Equation eq:thm1 holds regardless of the initial value $c_n[0]$ and the distance $z$.

Figures (8)

  • Figure 1: Coordinate system.
  • Figure 2: Geometry of gradients.
  • Figure 3: Displayed images and PSNR values in simulation (Dog, Penguins, and Rock).
  • Figure 4: Displayed images and PSNR values in simulation (Flower, Men, and Baby).
  • Figure 5: Optical implementation of holography.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Definition 1: Wirtinger derivative
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Corollary 1: induced from Theorem \ref{['t1']}
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 3 more