Fourier phase encodes important spatial image structure, but recovering an image without measured spectral magnitude requires additional constraints and leaves absolute intensity ambiguous. We propose a projection-based two-stage framework that combines Fourier-phase and spatial-support constraints with an image-specific neural prior. The first stage alternates constraint enforcement with regularized neural-prior updates, while the second performs phase/support refinement alone with guaranteed convergence. We evaluate two neural-prior implementations on the same 77 microscopy images and compare them with a constraint-only baseline. After 500 final refinement passes, the best-performing variant achieves 31.41 dB pooled PSNR, 35.75 dB mean PSNR, and 0.9531 mean SSIM, improving pooled PSNR by 1.51~dB and reducing pooled MSE by 29.3% relative to the baseline. The results demonstrate the benefit of combining neural guidance with explicit constraint refinement at the evaluated iteration budget, while showing that lower phase residual alone does not guarantee greater reconstruction accuracy.
Figures & tables
Figure 1: Flow diagram of the phase-only reconstruction framework. Starting from the constant magnitude initialization, each iteration applies Fourier-phase replacement, inverse transformation, support cropping, and nonnegativity enforcement. The constrained image is processed by the neural prior and combined with its output through relaxation.
Figure 2: Convergence diagnostics on the paired 77-image subset. Left: mean squared change between successive neural-prior outputs and relaxed estimates during the outer stage. Right: mean squared change between successive estimates during the final phase/support-only refinement.
Method
L
Pooled MSE
Pooled PSNR
Mean PSNR
Mean SSIM
Phase residual
(dB)
(dB)
Phase/support
100
1.785×10−3
27.48
28.21±2.44
0.6973±0.0892
1.79×10−2
Phase/support
500
1.080×10−3
29.66
32.17±5.56
0.8564±0.1010
6.17×10−3
Self-Diff.
0
1.132×10−3
29.46
33.78±7.17
0.9392±0.0583
6.85×10−3
Self-Diff.
300
1.101×10−3
29.58
35.34±10.04
0.9408±0.0628
7.30×10−4
Self-Diff.
500
1.099×10−3
29.59
35.59±10.66
0.9408±0.0631
3.62×10−4
Table 1: Paired comparison on the same 77 images. Mean PSNR and SSIM are reported as mean ± standard deviation across images.
Figure 3: Effect of the final phase/support refinement on SelfDeblur over the paired 77-image subset. Left: pooled and mean per-image PSNR. Right: mean SSIM.
Figure 4: Phase-only reconstruction of a BBBC006 image: (a) initialization scaled by a=50 ; (b) SelfDeblur reconstruction after 100 outer iterations and 500 final phase/support passes; (c) reference. All panels share the display range [0,1] .
Oppenheim and Lim (1981) showed that natural images stay recognizable when reconstructed from their Fourier phase alone, while the magnitude carries little of their identity. We ask whether trained image classifiers reproduce this asymmetry inside their hidden layers, and we test it causally: given two images, we transplant the phase of one onto the magnitude of the other at a chosen layer and record which image the prediction follows. In PRISM2D, GFNet, and ViT-B/16 the prediction follows the phase or sign donor, and deleting all image-specific magnitude barely moves accuracy, so identity rides on phase while image-specific magnitude is largely dispensable to the readout. ResNet-50 at first seems to break the pattern, because transplanting sign after its ReLUs does nothing; a fair intervention before the ReLU reveals a strong latent sign code in the late blocks, and a DC-only control shows the readout consumes a channel-wise spatial average. Controls rule out the trivial case in which magnitude simply stops depending on the image. The architectures therefore share a phase/sign identity code but expose it in different bases, set by rectification and readout geometry, which gives a mechanistic account of the texture--shape gap between CNNs and attention models.
Deep unrolling methods for inverse problems augment traditional reconstruction algorithms with learned priors to improve their performance. These methods have been developed under the implicit assumption that the geometry of the measurement operator does not change between training and deployment; when this is no longer the case, the quality of obtained reconstructions can degrade significantly. In this paper, we argue that this is not an inherent limitation of the deep unrolling paradigm and can be mitigated by a careful choice of the underlying iterative algorithm. We propose a deep unrolling framework for the phase retrieval problem that augments the alternating projections method with a learned denoising step. Our method obtains high-quality reconstructions without the need for retraining in settings where conventional model-based alternatives struggle, such as when the dynamic range of the measurement operator changes significantly at test time.
Electrical & Computer Engineering, University of Washington · NSF-Simons National Institute for Theory and Mathematics in Biology · Department of Computer Science, University of Chicago +3
Phase retrieval - recovering a complex-valued field from intensity measurements - is typically solved using variants of the Gerchberg-Saxton (GS) algorithm, understood as alternating projections between measurement planes. Meanwhile, modern computational imaging increasingly relies on gradient-based optimization and automatic differentiation. Here we show that these two approaches are mathematically identical: the GS magnitude replacement step is exactly a unit gradient descent step on an amplitude least-squares loss. This equivalence enables seamless integration of classical phase retrieval with differentiable physics pipelines. We further identify two complementary probabilistic interpretations of this equivalence: globally, the amplitude loss is the negative log-likelihood under Gaussian amplitude noise; locally, each projection step arises as a Bayesian update with the propagated field as prior. The local view provides qualitative guidance for relaxation in iterative phase retrieval.
Jakob Schroeder, Andreas Döpp
Fakultät für Physik, Ludwig-Maximilian-Universität München, Am Coulombwall 1, 85748 Garching, Germany · Munich Center for Machine Learning (MCML) · Max Planck Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, Garching 85748, Germany