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Certified Abstention for Provably-Unidentifiable Index-Phase Recovery in Image Registration

Dinh Tuan Nguyen, Trong Nhan Nguyen, Vu Nguyen Minh Hung, Tien Minh Khoi Nguyen, Jaeyeop Choi, Sudip Mondal, Jae Sung Ahn, Junghwan Oh

arXiv:2610.29272v1cs.CV

TL;DR

The paper asks when an image’s sample-to-position map is recoverable under per-line index shifts, a question that matters because a reported position can otherwise be fabricated. It proves an identifiability boundary and uses certified recovery with abstention; results show both a measured scanning offset and modality-specific limits on abstention and risk control.

  • Problem

    Registration theory has largely left open when a per-line integer index shift is recoverable or provably ambiguous, despite its consequences for geometric correction.

    Source: §I · p. 1

  • Method

    The paper proves acquisition-side limits and uses a certificate to recover identifiable offsets while abstaining on provably ambiguous regions.

    Source: Contributions · p. 2, §VII · p. 11

  • Results

    A deployed laser-ultrasound C-scanner showed a rigid, direction-bimodal 17.3-column index offset, while cross-modal abstention completeness and kept-region risk control remained modality-specific.

    Source: §I · p. 1, C. What transfers across modalities · p. 10

  • Takeaways & Limitations

    In this setting, an origin-coupled reference can restore phase fidelity, while the certificate declines recovery where the offset is provably ambiguous.

    Source: §VII · p. 11, §VIII · p. 12

    The theory excludes continuous intra-block warp, and the certificate assumes exchangeable calibration data while bounding mean kept-region risk rather than per-instance correctness.

    Source: §VII · pp. 11–12

Abstract

from arXiv · show

Image registration underpins measurement across scanning, camera, and microscopy imaging, yet a fundamental question is rarely posed: when is the sample-to-position map recoverable at all, and when does any estimator that reports a position necessarily fabricate one? We show that in reference-perturbed sampled imaging, where samples are localized through a perturbed reference, position fidelity (where samples land) and phase fidelity (which reference index becomes each block's origin) are logically independent: no policy acting on position information alone can guarantee phase fidelity. We derive a tight resolution threshold separating recoverable from provably-unidentifiable registration, and a recovery framework that certifies identifiable regions--recovering them exactly--while abstaining on provably-unidentifiable regions rather than hallucinating structure. On a deployed scanning system we measure a direction-bimodal registration offset consistent with the theory. Against classical and learned restoration baselines, certificate-free offset estimators fabricate structure on the unidentifiable regime, whereas the certified method abstains rather than fabricating. Finally, the certified-abstention mechanism carries across scanning, rolling-shutter, and cryo-electron-microscopy imaging--declining rather than fabricating on the regime it flags--though how completely it abstains and how tightly it bounds kept-region risk stay modality-specific (at extreme SNR the cryo-EM certificate abstains only partially): a structural-versus-statistical boundary in registration reliability.

I. INTRODUCTION

The paper studies per-line integer index shifts that can leave a scanned image sharp while making its sample-to-position map unreliable. It proves when phase is unrecoverable and proposes certified recovery that abstains rather than fabricating structure.

I. INTRODUCTION

  • The paper proves that position fidelity and phase fidelity are independent, so position-only policies cannot guarantee which reference index becomes each block’s origin.Its recovery framework certifies identifiable regions and abstains on provably-unidentifiable ones.Source: Contributions · p. 2
  • An origin-coupled synchronization primitive restores phase fidelity, and an imperfect reference suffices only when δ < ∆y/(2vmax).This is a theoretical necessary-and-sufficient remedy and tight timing threshold.Source: Contributions · p. 2
  • The certificate decodes offsets from metadata, identifies image-texture shifts above a detectability limit, and flags texture-free or periodic regions as ambiguous.A distribution-free conformal wrapper bounds mean kept-region risk by α, not per-instance risk.Source: Contributions · p. 2
  • Hardware timestamps reveal a rigid, direction-bimodal 17.3-column offset on a deployed position-triggered laser-ultrasound C-scanner.The paper identifies this as one empirical instance, not a multi-condition characterization.Source: Contributions · p. 2
  • Certified abstention transfers to rolling-shutter and cryo-EM settings, but abstention completeness and kept-region risk control remain modality-specific.At extreme SNR, the cryo-EM certificate abstains only partially.Source: Contributions · p. 2
  • On the synthetic benchmark, the certified method fabricates on 0% of the provably-unidentifiable band, while certificate-free estimators fabricate on up to 98%.The certified method also recovers identifiable rows exactly.Source: Contributions · p. 2

II. RELATED WORK AND POSITIONING

Prior registration work estimates offsets or uncertainty, but does not establish when an image’s index origin is recoverable. The paper frames this gap through an acquisition model with block-wise sampling and re-initialization.

II. RELATED WORK AND POSITIONING

  • Existing scanning correction and reconstruction work handles observed offsets but does not model index fidelity or when residual offsets are provably ambiguous.The paper fills this gap with an acquisition-side contract, timing threshold, and abstention certificate.Source: §II · pp. 2–3
  • Registration bounds estimate shift variance, while uncertainty-aware registration, flags low reliability rather than proving that no estimate is recoverable.The paper’s distinction is structural identifiability, not a confidence interval.Source: §II · p. 3
  • Jitter reconstruction treats continuous position perturbations, whereas this paper studies an unknown integer index offset per block.Order tracking, assumes an available angle reference; scan-distortion methods, do not provide an acquisition-side fidelity contract or abstention certificate.Source: §II · p. 3

III. THE GRID-LOCKED BLOCK-SEQUENTIAL MODEL

  • The paper formalizes a grid-locked, block-sequential sampling model and distinguishes the discrete block index origin from signal phase.Its scope is the acquisition and image-registration problem under that model.Source: §III · p. 3

A. Geometry, crossings, and blocks

The model describes grid crossings, block capture, and re-initialization delays. Its fidelity contract separates within-block position, between-block phase, and block assembly.

A. Geometry, crossings, and blocks

  • A static field is sampled along a fast axis at grid crossings spaced by ∆y, with blocks corresponding to traversals at successive slow-axis positions.The analysis uses crossing times bounded by the admissible speed range; crossing positions remain on the absolute grid.Source: A. Geometry, crossings, and blocks · p. 3

B. Count-locked capture with re-initialization

  • Each block captures a fixed count of consecutive crossings after a nonzero re-initialization delay, and asynchronous delays determine which crossing becomes record 0.The paper defines a position-only policy as one based on the encoder trajectory, not the realized delays.Source: B. Count-locked capture with… · p. 3

C. The three-layer fidelity contract

  • The fidelity contract separates consecutive within-block grid positions (L1), a shared index origin across blocks (L2), and correct row and reversal mapping (L3).Spatial fidelity requires all three; L3 is standard bookkeeping and is not analyzed further.Source: C. The three-layer fidelity contract · p. 4

D. The separation

Grid-locked sampling guarantees samples land on the grid, but asynchronous re-initialization can change which crossing becomes each block’s origin. No position-only policy can guarantee consistent phase; origin-coupled synchronization or sufficiently precise reference timing restores it.

D. The separation

  • Grid-locked triggering preserves consecutive grid positions in every block, but that position fidelity does not ensure consistent block origins.The distinction separates within-block spacing from which physical crossing is stored as column 0.Source: §IV · p. 4, D. The separation · p. 4

IV. POSITION–PHASE SEPARATION

  • No position-only policy guarantees consistent block origins: adversarial re-initialization delays can change the true origin while leaving the policy’s position-stream input unchanged.The resulting phase offset can vary across blocks even though every block retains correct grid spacing.Source: §IV · p. 4
  • An origin-coupled synchronization primitive is necessary and sufficient for consistent origins within the count-locked policy class.Withholding the first imaged pulse until re-initialization is complete and the stage reaches the fixed origin crossing sets every block’s offset to zero.Source: §IV · p. 5
  • Nearest-crossing reference timing guarantees consistent origins for all adversaries only when its error is below Δy/(2vmax).The threshold is tight: at or above the half-cell boundary, an adversary can shift the selected origin to the adjacent crossing.Source: §IV · p. 5

A. Subharmonic capture under an exact block count

With an exact block count, re-initialization latency can force captures to straddle a turnaround, reducing batch cadence. Origin-coupled synchronization removes the delay-induced spill.

A. Subharmonic capture under an exact block count

  • Under a zero-surplus count cap, any delay consuming at least one crossing forces the capture window to continue past turnaround into the next sweep.For a sustained delay fraction β, cadence becomes one completed batch per (1+β) sweeps; as β approaches 1, cadence halves to one batch per two sweeps.Source: A. Subharmonic capture under an exact… · p. 5
  • The spill occurs because the window needs nc captures after opening Ni crossings into the sweep, so the remaining captures occur after turnaround in the opposite direction.This produces a mislabeled direction and a batch duration inflated by turnaround dwell.Source: A. Subharmonic capture under an exact… · p. 5
  • The origin-coupled synchronization primitive sets Ni = 0, keeping all nc captures within one sweep and restoring one batch per sweep.The paper identifies the handshake, rather than surplus count, as the structural cure for both cadence and offset.Source: A. Subharmonic capture under an exact… · p. 5

V. RECOVERABILITY OF BROKEN PHASE FIDELITY

Broken block-index offsets can be recovered exactly from capture timestamps when their resolution is sufficient. The timestamp decoder has a tight half-interval error threshold.

V. RECOVERABILITY OF BROKEN PHASE FIDELITY

  • The recovery framework distinguishes exact metadata recovery from image-only recovery and provably ambiguous cases.The per-block shift bound is attributed to Robinson–Milanfar; the paper’s contribution is the integer/block structure, channel trichotomy, ambiguity guards, and abstention certificate.Source: §V · p. 5
  • Per-record capture timestamps recover each block offset exactly by rounding the record-0 timestamp divided by the nominal inter-crossing interval.Applying the inverse shift then restores consistent block origins, given L1 and L3.Source: §V · p. 5
  • Timestamp rounding recovers every offset for every profile if and only if timestamp error εt is less than Δtnom/2, or Δy/(2v) at sweep speed v.For a speed class, the worst-case threshold is Δy/(2vmax).Source: §V · p. 6

A. Image channel: a registration estimator and its detectability limit

The image channel estimates integer block shifts by matching a block to a grid-consistent neighbour, with a certificate tied to local texture and noise. Its recovery bound has an exact Gaussian-noise counterpart, and the synthetic study places the empirical knee near that threshold.

A. Image channel: a registration estimator and its detectability limit

  • The maximum-likelihood integer-shift estimator recovers all R blocks with probability at least 1 − R(2Kmax+1) exp(−ΛminΔy^2/(8σ^2)) under the stated local model.The guarantee uses the minimum local texture information Λmin across blocks.Source: A. Image channel: a registration… · p. 6
  • With Gaussian noise, the adjacent-competitor error is exactly Φ(−ρi/2), where ρi = ΛiΔy^2/(2σ^2) and Φ is the standard normal CDF.Source: A. Image channel: a registration… · p. 6
  • The synthetic study locates the empirical knee at ρ⋆ to within 3%, confirming the recovery threshold with the exact constant.The estimator’s identifiability certificate flags the regime where image-based shift recovery is not identifiable.Source: A. Image channel: a registration… · p. 6

B. Where the image channel fails: ambiguity

The image channel cannot identify shifts in texture-free or periodic regions, so image-only recovery must abstain there. A shared half-cell resolution condition connects acquisition, metadata, and image-based phase recovery.

B. Where the image channel fails: ambiguity

  • Texture-free bands have a flat likelihood, while p-periodic bands leave the offset ambiguous modulo p, regardless of texture energy.Image-only recovery must abstain in both regimes or defer to metadata or anchors.Source: B. Where the image channel fails:… · p. 6

C. Sub-grid residual

  • For continuous within-grid residuals, classical reconstruction results provide sufficient stability conditions only under bandlimited or shift-invariant-space hypotheses.The paper does not claim an if-and-only-if transfer without verifying those hypotheses.Source: C. Sub-grid residual · p. 6

D. One condition, three channels: half-cell resolution

  • Phase pinning requires resolving the phase coordinate to less than half a cell, whether at acquisition, through metadata, or from image texture.The image channel uses a half-column estimate; the trigger and timestamp channels require less than half-cell timing resolution.Source: D. One condition, three channels:… · pp. 6–7
  • If none of the three channels supplies the half-cell margin, phase remains unrecoverable; texture-free and periodic bands are cases where image texture cannot supply it.The half-cell budget is a design heuristic for this problem, not a universal cross-modal budget.Source: B. Where the image channel fails:… · p. 6, D. One condition, three channels:… · p. 7

E. Calibrated abstention certificate

A position-only scan exhibits a direction-bimodal 17.3-column block-origin offset despite on-grid column spacing. The section also describes a calibrated abstention certificate, stochastic recovery conditions, and the limits of the single-system evidence.

E. Calibrated abstention certificate

  • The certificate combines matching-cost margin, texture energy, and a periodicity guard, then uses conformal risk control to bound mean kept-region risk under exchangeability.The guarantee is distribution-free and finite-sample, but applies to average kept-region risk—not each kept block—and does not identify the unrecoverable set exactly.Source: E. Calibrated abstention certificate · p. 7

F. Stochastic re-initialization

  • Under the proposed AR(1) model, sufficiently detectable increments and locally non-periodic texture permit exact recovery up to a global constant; one anchor resolves it.If increments fail with probability q, accumulated error grows as a random walk and anchors every L blocks bound it; texture-free or periodic bands make increments unidentifiable.Source: F. Stochastic re-initialization · p. 7
  • The AR(1) model is motivated by observed offset structure, but its parameters have not been fitted at the operating point and no sharp transition is claimed.The reported same-direction lag-1 autocorrelation in the logged capture is 0.40–0.56.Source: F. Stochastic re-initialization · p. 7

VI. EVIDENCE AND VALIDATION

  • The 17.3-column displacement is a rigid per-direction index-origin shift, while intra-block velocity ripple is a distinct, bracketed warp failure.At submicron pitch, the dominant visible artifact is the bracketed L1 warp; isolating L2 is a scope choice, not a claim that it is the only failure mode.Source: §VI · p. 7

A. The phenomenon on a real position-triggered scanner

  • The offset scales with mean speed through Φ = δv/∆y, with a trigger-to-capture gap acting oppositely by scan direction; the headline displacement is 2Φ = 17.3 columns.The proposed decomposition separates the direction-reversing swing from a global constant, whose cause is not attributed.Source: A. The phenomenon on a real… · p. 8
  • On the logged position-only scan, a nonzero block offset on almost every block is consistent with the predicted failure, but one instance does not prove the theorem.The capture used no origin-coupled reference; its per-block offsets are measured directly from hardware timestamps.Source: A. The phenomenon on a real… · p. 8
  • The image-based recovery shown for the real system is qualitative; separating per-block L2 offset from intra-block L1 warp on the same region remains future work.The figure uses rendered-image proxies that mix L1 and L2 and does not quantify the L2 offset.Source: A. The phenomenon on a real… · p. 8, Figure 1 · p. 8
  • 17.3 columns separate the forward and reverse block-origin means (+6.9 and −10.4) in a 903-block position-only scan, while column spacing remains on-grid.This is the logged real-system index-offset result; within-direction standard deviations are 0.97 and 1.19 columns.Source: A. The phenomenon on a real… · p. 8, Table I · p. 9

B. Synthetic corroboration

In the synthetic benchmark, the certified method recovers identifiable blocks exactly and abstains on the provably unidentifiable band, unlike certificate-free offset estimators that fabricate shifts. Controlled simulations also test the paper’s predicted thresholds, cadence, ambiguity, and conformal coverage.

B. Synthetic corroboration

  • Five controlled simulation studies test predicted jitter, cadence, detectability, ambiguity, and conformal-coverage behavior against the theoretical results.The real-system measurements establish that the phenomenon occurs; the synthetic studies test the predictions quantitatively.Source: B. Synthetic corroboration · p. 9
  • Identifiable-band RMSE is 0 for the certified method, versus 43% for rigid registration and column-mean destriping and 84%/33% for learned-template and low-rank PCA.The table defines exact recovery over the identifiable band, fabrication as nonzero wrong shifts on p-periodic lines, and abstention coverage on texture-free/periodic bands.Source: B. Synthetic corroboration · p. 9, Table II · p. 9
  • On the p-periodic band, certificate-free offset estimators fabricate at rates of 0.98, 0.88, 0.70, 0.63, and 0.18, while the certificate and metadata oracle abstain 100%.The low fabrication of destriping and PCA reflects their inactivity, not certified abstention; the certified method still recovers identifiable rows exactly.Source: B. Synthetic corroboration · p. 9, Table II · p. 9

C. What transfers across modalities

The certified-abstention mechanism carries across the tested modalities, but abstention completeness and kept-region risk control remain modality-specific. The discussion frames phase recovery around paying a half-cell margin in one channel and identifies limits on the theory and certificate.

C. What transfers across modalities

  • Across rolling-shutter, cryo-EM, and scanning data, certificate-free estimators fabricate on flagged unidentifiable regimes while the certified method declines; abstention completeness and kept-region risk vary by modality.Scanning abstains on essentially all unidentifiable rows and meets its conformal target; rolling-shutter risk floors at 0.17–0.46 versus α=0.10, while extreme-SNR cryo-EM flags 0.39 of an approximately 0.79 unidentifiable regime.Source: C. What transfers across modalities · pp. 9–10
  • Synthetic studies show exact image decoding on textured bands, full certificate abstention on texture-free and periodic bands, and kept-region error at or below α across 20 splits.The metadata channel remains exact across these bands.Source: Figure 4 · p. 10
  • On the synthetic benchmark, certificate-free restorers fabricate on the periodic band, whereas image+cert abstains (1.00) and exactly recovers the identifiable band.Low fabrication by destriping and PCA reflects inertness, not certified abstention.Source: Figure 5 · p. 11

VII. DISCUSSION

  • The half-cell margin can be paid at triggering, through metadata, or through image texture; texture-free and periodic bands lack the image-texture route, so the certificate abstains.A fixed trigger-to-capture gap gives Φ = δv/∆y cells, with the operating rule δv < ∆y/2.Source: §VII · p. 11
  • Tightening the block count to the imaged-column count induces subharmonic capture cadence under any nonzero re-initialization latency; only the origin-coupled handshake fixes offset and cadence.Source: §VII · p. 11
  • The deployed position-triggered C-scanner exhibits nonzero per-block offsets, but a rate-matched logged capture is still needed to quantify the inter-block phase offset separately from intra-block warp.Source: §VII · p. 11
  • The theory addresses inter-block phase, not intra-block warp; the certificate assumes exchangeable calibration data and does not identify the unrecoverable set exactly.The sampler class includes several scanning and line-imaging modalities; reproducibility materials regenerate synthetic and July results, but June raw logs were not retained.Source: C. What transfers across modalities · p. 10, §VII · pp. 11–12
  • Abstention soundness is structural across modalities, but completeness and risk tightness are statistical; the paper does not claim a universal cross-modal law or a direct remedy for generative-model confabulation.The proposed mechanism relies on a computable identifiability structure, which the discussion says generative-model confabulation lacks.Source: §VII · p. 12

VIII. CONCLUSION

The conclusion separates sample-position fidelity from block-origin phase fidelity and gives conditions for prevention or recovery. It also reports a measured scanner offset and provides a reproducibility package with a stated exception.

VIII. CONCLUSION

  • Absolute-position triggering guarantees intra-block position but not inter-block phase; an origin-coupled primitive restores phase, while an imperfect reference must resolve half a grid cell at the fastest speed.Source: §VIII · p. 12
  • A deployed laser-ultrasound C-scanner shows a rigid 17.3-column bidirectional index offset; metadata, sufficiently informative image texture, or a sub-half-cell reference provide recovery or prevention channels.Texture-free and periodic regions remain ambiguous, where the certificate abstains rather than fabricates.Source: §VIII · p. 12

DATA AVAILABILITY

  • The released package regenerates every figure and reported July and synthetic number from committed files, but the June investigation record's raw logs were not retained.Source: DATA AVAILABILITY · p. 12
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