Image Reconstruction from Phase with Untrained Neural Priors

• Fourier phase carries substantial spatial image information but does not determine absolute intensity.Source: §1 · p. 1
• Classical reconstruction and identifiability studies use additional assumptions, including known spatial support.Source: §1 · p. 1
• On 77 microscopy images, the proposed two-stage framework combines phase/support constraints with neural-prior updates and improves reconstruction quality over constraint-only iterations at the evaluated budget.Lower phase residual alone does not guarantee greater image-domain accuracy.Source: §1 · p. 1

• Earlier phase-reconstruction work includes structural analysis by Oppenheim and Lim, reconstruction conditions by Hayes et al., and iterative restoration by Levi and Stark [3].Source: §2 · p. 1
• Known-phase and known-support image sets are described as closed and convex, supporting successive orthogonal projections onto the phase and support constraints.The phase-only intersection remains large because positive rescalings share the same phase.Source: §2 · p. 1
• DIP and SelfDeblur are image-specific priors; here the neural block serves as an intermediate prior for Fourier-phase reconstruction, not standalone deblurring.Self-Diffusion is also evaluated as an alternative untrained prior under the same reconstruction constraints.Source: §2 · pp. 1–2
• Projection-based recovery builds on Youla and Webb, Sezan and Stark, Levi and Stark, and convolution-based recovery by C¸ etin et al..C¸ etin et al. also considered zero-phase FIR filter design.Source: §2 · p. 2

3.1. Successive Projections onto Convex Sets

• Each projection cycle retains the current Fourier magnitude while imposing measured phase, then crops to the known support and enforces nonnegativity; absolute intensity remains unrecoverable from phase alone.The implementation zero-pads to a 2M×2N canvas for Fourier processing before top-left support cropping.Source: §3.1 · p. 2

3.2. Neural-prior stage

• The neural-prior stage supplies the constrained image to an untrained neural block, using either SelfDeblur or Self-Diffusion to generate the next estimate.Network weights are initialized anew for each image, without pretrained weights or parameter transfer.Source: §3.2 · p. 2

3.3. Proof of Convergence

• Final phase/support-only passes cannot increase error to a correctly scaled reference, and the complete procedure is globally convergent to the constraint intersection.These passes retain signed values and impose no clipping to [0, 1].Source: §3.3 · p. 3

4. EXPERIMENTAL PROTOCOL

• BBBC006v1 microscopy evaluation uses channel w1 at focal plane z = 16, with one deterministic 256 × 256 center crop per field of view generating a 512 × 512 exact Fourier-phase observation.Integer intensities are divided by 4095 without image-specific contrast normalization, and no measurement noise is added.Source: §4 · p. 3
• The figures show outer-stage and final-stage update changes, SelfDeblur quality across refinement passes, and a three-panel phase-only reconstruction example.The example contrasts the initialization scaled by a = 50, the SelfDeblur reconstruction after 100 outer iterations and 500 refinement passes, and the reference, all displayed over [0, 1].Source: Figure 2 · p. 3, Figure 3 · p. 3, Figure 4 · p. 4
• Both priors receive 10,000 inner training iterations per image before final phase/support refinement, with 100 outer iterations and checkpoints through 500 refinement passes.The paired sample contains 77 images selected from those successfully reconstructed by both methods; settings are fixed across images.Source: §4 · pp. 3–4
• Quality is measured by MSE, PSNR, and SSIM, while reporting also includes runtime and relative phase residual.SSIM uses an 11 × 11 Gaussian window with standard deviation 1.5; there is no clipping, alignment, or reference-fitted intensity rescaling.Source: §4 · p. 4

5. RESULTS AND DISCUSSION

• Before final refinement, Self-Diffusion has higher mean PSNR and SSIM than SelfDeblur, but SelfDeblur already has higher pooled PSNR despite its larger phase residual.The respective mean PSNR and SSIM values are 33.78 dB and 0.9392 for Self-Diffusion versus 31.71 dB and 0.8925 for SelfDeblur; pooled PSNR is 30.06 versus 29.46 dB.Source: §5 · p. 4
• 31.41 dB pooled PSNR, 35.75 dB mean PSNR, and 0.9531 mean SSIM are reached by SelfDeblur after 500 final refinement passes.Self-Diffusion reaches 29.59 dB pooled PSNR and 0.9408 mean SSIM, while its mean PSNR reaches 35.59 dB.Source: §5 · p. 4
• At the final checkpoint, SelfDeblur has stronger overall image-domain metrics and is faster than Self-Diffusion: 353.4 s/image versus 541.4 s/image.Self-Diffusion's mean cropped phase residual falls from 6.85 × 10^-3 to 3.62 × 10^-4, compared with SelfDeblur's decrease from 1.48 × 10^-2 to 1.53 × 10^-3.Source: §5 · p. 4
• Self-Diffusion generally makes smaller updates during both stages, whereas SelfDeblur's larger final-stage updates correspond to greater image-domain quality improvement.The comparison shows that smaller ∆MSE during refinement does not by itself indicate a better reconstruction.Source: §5 · p. 4

6. CONCLUSION

• The proposed two-stage algorithm combines Fourier-phase and spatial constraints with an image-specific neural prior that initializes globally convergent constraint-only refinement.SelfDeblur and Self-Diffusion serve as alternative neural components in the framework.Source: §6 · p. 4
• Experiments on 77 microscopy images show that incorporating a neural prior can improve reconstruction quality over constraint-only iterations at the evaluated iteration budget.Source: §6 · p. 4
• Future work will address noisy or incomplete phase measurements, uncertain support, and adaptive refinement strategies.Source: §6 · p. 4