[Paper Review] Non-iterative complex wave-field reconstruction based on Kramers-Kronig relations

Cheng Shen, An Pan+2 more|arXiv (Cornell University)|May 11, 2020

Digital Holography and Microscopy44 references76 citations

TL;DR

This paper proposes a non-iterative, parameter-free method for complex wave-field reconstruction using Kramers-Kronig (KK) relations, leveraging the mathematical analogy between amplitude-modulated pupil measurements and off-axis holography. By exploiting the analyticity of band-limited signals, it recovers phase from just two intensity measurements without prior sample assumptions, achieving high-fidelity reconstructions comparable to gold-standard FPM with significantly reduced data and computational cost.

ABSTRACT

A new computational imaging method to reconstruct the complex wave-field is reported. Due to the existence of zero frequency component, the measured signal by amplitude modulation of pupil has a spectrum similar to the one of off-axis hologram. The mathematical analogy between them is established in this paper. Based on this observation and analyticity of band-limited signal under any diffraction-limited system, an algorithm from Kramers-Kronig (KK) relations is utilized to recover the phase information only from the intensity patterns. From the sensing side, only two measurements are required at least. From the reconstruction algorithm side, our method is iteration-free and parameter-free, also without any assumption on sample characteristics. It owns several advantages over existing phase imaging methods and could provide a unique perspective to understand current computational imaging methods.

Motivation & Objective

• To develop a fast, non-iterative method for reconstructing complex wave-fields from intensity-only measurements.
• To overcome the limitations of existing phase imaging techniques that require multiple measurements, iterative algorithms, or sample priors.
• To establish a mathematical link between amplitude-modulated pupil measurements and off-axis holography to enable phase recovery via Kramers-Kronig relations.
• To demonstrate that the DC component's position in the Fourier domain enables phase retrieval without a reference wave, unlike conventional holography.
• To validate the method experimentally on biological samples with performance comparable to full-field FPM using only two measurements.

Proposed method

• Utilizes a 4f imaging system with pupil plane amplitude modulation via an iris diaphragm or SLM to create two distinct frequency band measurements.
• Exploits the Fourier transform's frequency-shifting property to show that two measurements with shifted pupil masks produce spectra analogous to off-axis holograms.
• Models the measured intensity as a sum of self- and cross-interference terms in the Fourier domain, where the cross-term contains the complex field information.
• Applies the Kramers-Kronig relations to the band-limited, analytic signal in the Fourier domain to recover the complex wave-field from intensity-only data.
• Employs a two-step reconstruction: first, the spectrum is decomposed into a delta function and a shifted sample spectrum; second, KK relations recover the complex field from the intensity spectrum.
• Uses the analyticity of band-limited signals under diffraction-limited systems to ensure the KK relations are mathematically valid and stable for phase recovery.

Experimental results

Research questions

• RQ1 Can the phase of a complex wave-field be reconstructed from just two intensity measurements without iterative algorithms or sample priors?
• RQ2 Is there a fundamental mathematical equivalence between amplitude-modulated pupil measurements and off-axis holography that enables phase retrieval via KK relations?
• RQ3 How does the position of the DC component in the Fourier domain affect the feasibility and stability of phase recovery using KK relations?
• RQ4 Can this method outperform existing non-interferometric phase imaging techniques like DPC and FPM in terms of speed, accuracy, and robustness to strong samples?
• RQ5 What is the role of the pupil’s spectral offset in enabling non-iterative, parameter-free phase retrieval in coherent imaging systems?

Key findings

• The method achieves complex wave-field reconstruction from only two intensity measurements, significantly reducing data acquisition time and volume compared to FPM.
• The reconstruction is iteration-free and parameter-free, with no assumptions on sample characteristics, making it robust for strong or non-weak phase objects.
• In experimental validation on plant cells, the KKSAI method achieved a normalized cross-correlation (NCC) of 0.9410 for amplitude and 0.8052 for phase in ROI1, and 0.9341 and 0.7832 in ROI2, respectively, closely matching the gold-standard FPM reconstruction.
• Digital refocusing using the angular spectrum method yielded best focus distances of -49 µm and -30 µm for the two ROIs, matching the ground truth, confirming accurate phase recovery.
• Quantitative comparison showed that KKSAI outperformed both DPC and FPM in phase reconstruction, with NCC values of 0.8052 and 0.7832 (vs. 0.7198 and 0.6801 for DPC) and FSIM scores of 0.9769 and 0.9727 (vs. 0.9613 and 0.9575 for DPC).
• The method avoids the twin-image problem common in in-line holography and does not require a reference arm, unlike off-axis holography, while still enabling high space-bandwidth product imaging.