Provable and Robust Wavefront Sensing via Self-Reference Interferometry

Overview of our proposed self-reference wavefront sensing method.

Overview of coprime-shift self-reference wavefront sensing

(a): Interference of the incoming wavefront with its spatially shifted copy (i.e. self reference) provides pairwise phase differences among shifted pixels.
(b): Phasor differences measured at multiple shifts are propagated using our proposed graph-based algorithm to recover the global phase profile.
(c): The recovered phase enables applications such as target refocusing and imaging through a diffuser.

Application: Computational Refocusing

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Abstract

Wavefront sensing involves estimating the phase and intensity of light, enabling a wide range of imaging applications, from adaptive optics and astronomy to biomedical imaging. Since conventional image sensors can only measure the spatial intensity distribution, phase retrieval arises as the central problem in wavefront sensing. Conventional interferometric approaches like phase-shifting interferometry (PSI) can recover phase information, but they rely on a stable reference beam that is difficult to realize in practical settings. To overcome this limitation, we propose a novel self-reference framework that relies on interference between shifted copies of the incoming wave; this results in pairwise phase differences between shifted pixels. We formulate an analytical solution for the complete phase retrieval based on the propagation of these differences across a connected graph. Furthermore, we provide a theoretical analysis of optimal measurement patterns, proving that co-prime shifts guarantee a connected graph and bound worst-case error accumulation, yielding a provably robust method. Extensive simulations demonstrate that complete phase profiles can be recovered from as few as eight shifted measurements, outperforming several existing approaches. Finally, we validate our framework using a hardware prototype, demonstrating real experiments for optical phase profile recovery, auto-refocusing, and imaging through scattering media.

Method

We recover an unknown wavefront by interfering it with spatially shifted copies of itself. Four phase-shifted intensity measurements provide pairwise phase differences, which are propagated through a graph to recover the full phase profile relative to a reference pixel.

Complete connectivity with co-prime shifts. Pixels are graph nodes; measured phase differences are edges. Co-prime shifts guarantee every pixel is connected to the reference, enabling complete phase recovery.
Robust shortest-path recovery. Phase estimates propagate from the reference along shortest paths. Equal-length estimates are averaged for robustness, while longer paths are discarded since they accumulate more noise.
Provably optimal sensing. For \(N\) pixels, the consecutive co-prime shifts \(s = \lfloor \sqrt{N/2} \rfloor\) and \(t = s + 1\) minimize the maximum hop distance. For \(N = 512\), the optimal shifts are \(s = 16\) and \(t = 17\).
Controlled error propagation. Phase error increases with hop distance, so minimizing the maximum number of hops bounds the worst-case error and reduces overall accumulated noise.

Single-shift path graph versus co-prime two-shift graph — A single shift creates long propagation paths, while two co-prime shifts add long-range connections and reduce the maximum hop distance.

Hardware measurement pipeline diagram — A 4f optical system uses an SLM to generate spatial shifts, record interference measurements, and directly measure amplitude.

Phase error versus hop distance and hop-distance histograms — Phase error increases with hop distance. The optimal pair \((16, 17)\) minimizes the maximum hop distance and concentrates pixels along shorter recovery paths.

Real Results

Seeing through a scattering media

Seeing through a diffuser. We recover the scattered complex field and computationally propagate it to the diffuser plane for phase correction. Subsequent propagation to the object plane successfully reveals the target, which is otherwise invisible.

Simulated results

We evaluate phase recovery from shifted interferometric measurements in a synthetic setting, comparing against gradient descent (GD-Random), PnP-FISTA, WISH, and a Deep Image Prior (DIP) baseline across random, quadratic, and smooth phase profiles. We report the mean phase error at SNR ≈ 22 dB below, where our method achieves the lowest error for both quadratic and smooth phase profiles.

Phase errors (mean ± std) at 22 dB SNR. Our method achieves the best or second-best results across all phase profiles. (**Best bold**, second-best underlined).
Method	Quadratic phase		Random phase		Smooth phase (peaks)
Method	16	32	16	32	16	32
GD-Random	0.695 ± 0.365	0.470 ± 0.434	0.138 ± 0.148	0.079 ± 0.152	0.791 ± 0.361	0.570 ± 0.404
WISH	0.852 ± 0.299	0.423 ± 0.366	0.854 ± 0.320	0.453 ± 0.377	0.898 ± 0.330	0.425 ± 0.367
PnP-FISTA	0.669 ± 0.341	0.456 ± 0.304	0.735 ± 0.329	0.425 ± 0.264	0.663 ± 0.341	0.479 ± 0.338
DIP	1.486 ± 0.031	1.491 ± 0.051	1.495 ± 0.037	1.447 ± 0.065	1.442 ± 0.063	1.423 ± 0.081
Ours	0.158 ± 0.104	0.099 ± 0.083	0.156 ± 0.099	0.094 ± 0.075	0.155 ± 0.096	0.097 ± 0.071
Ours with LS	0.183 ± 0.082	0.126 ± 0.065	0.158 ± 0.075	0.121 ± 0.055	0.205 ± 0.063	0.159 ± 0.050

Visual comparison of recovered phase against baseline methods — Phase recovery of smooth and random phase profiles (SNR 22 dB, 16 measurements). Our method achieves the lowest error on smooth profiles and remains competitive for random phases.

Phase recovery results across test scenes — Phase recovery performance across SNRs in simulations. Our method demonstrates consistent performance across all phase types, maintaining the lowest error in most settings and highly competitive results otherwise. Notably, least-squares refinement (Ours + LS) significantly improves reconstruction accuracy at low SNRs.

BibTeX

If you find this work useful, please cite it. We will update this citation once the ECCV proceedings version is available.

@article{yismaw2026provable,
  title   = {Provable and Robust Wavefront Sensing via Self-Reference Interferometry},
  author  = {Yismaw, Nebiyou and Saragadam, Vishwanath and Sankaranarayanan, Aswin C and Asif, M Salman},
  journal = {arXiv preprint arXiv:2604.03564},
  year    = {2026}
}