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Self-Calibrating Dense Displacement Fields for Reliable Co-Registration of Large Optical Satellite Imagery

Shoukun Sun, Zhe Wang, Sanaz Salati, Jiyin Zhang, Hui Wang, Xiaogang Ma

arXiv:2608.22300v1cs.CV

TL;DR

Optical satellite co-registration methods cover only some of the differences between real image pairs, although sub-pixel alignment matters for multi-temporal and multi-sensor analysis. SCDF estimates dense displacement fields with a self-calibrating, pyramid-based loop. Across 584 constructed-ground-truth problems, it is never more than 1.39× from the best result on any real-pair block, while every baseline is 5.3–783× worse on some block.

  • Problem

    Optical satellite co-registration methods use motion assumptions that cover only some real-pair differences, while sub-pixel alignment is a prerequisite for many multi-temporal and multi-sensor analyses.

    Source: §I · pp. 1–2

  • Method

    SCDF estimates a dense displacement field through a pyramid-level predict–measure–filter loop, with acceptance decisions self-calibrated from each image pair.

    Source: C. The cascade: one loop, repeated per… · p. 4, §VII · p. 13

  • Results

    Across 584 constructed-ground-truth problems, SCDF is never more than 1.39× from the best result on any real-pair block, while every baseline is 5.3–783× worse on some block.

    Source: §VII · p. 13

  • Takeaways & Limitations

    The comparison indicates that worst-case behavior distinguishes methods more than precision on fixtures within their deformation models.

    Source: §VII · p. 13

    Development and evaluation cover only same-optical, same-bandfamily pairs; cross-modal registration remains untested.

    Source: B. Limitations of the method · p. 12

Abstract

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Co-registration underlies nearly every multi-temporal and multi-sensor use of optical satellite imagery, and operational products still carry documented offsets well above the fraction-of-a-pixel scale at which change detection, time series, and data fusion degrade. Real image pairs differ along several axes at once (sensor response, scene content, viewing geometry, resolution, mosaic seams), and the last of these is not a single global motion. Existing tools embed a motion model and constants tuned to their development data; a pair that fits is registered precisely, while one that does not either fails to match or returns a result wrong by tens of pixels with no failure reported. Learned matchers add a GPU requirement and carry no accuracy guarantee outside their training distribution. We present SCDF (self-calibrating displacement fields), a training-free, GPU-free estimator whose motion model is the dense per-pixel displacement field itself, so no scene motion falls outside the model. A single predict--measure--filter loop runs over a resolution pyramid: the accumulated field predicts where each patch of the moving image falls in the reference, RootSIFT matching and a correlation pass measure the displacement there to sub-pixel precision, and filters whose thresholds are all calibrated on the image pair itself decide what survives. One configuration, with no per-dataset tuning, processes full $8192^2$ scenes on a single CPU core. On 584 constructed-ground-truth pairs built from real Sentinel-2, Landsat-8/9, and NAIP imagery, against seven classical baselines and two zero-shot pretrained matchers, SCDF registers every pair with zero failures, reduces the best baseline's real-pair median end-point error from 6.83 to 4.17m, and cuts its 90th percentile from 17.8 to 7.77m.

I. INTRODUCTION · II. RELATED WORK

SCDF targets reliable optical-satellite co-registration with a dense displacement field and image-pair-calibrated decisions. The related work shows how fixed motion models, search ranges, thresholds, and training distributions constrain existing approaches.

I. INTRODUCTION

  • SCDF produces fields on all 584 benchmark cells, ranks top-two on 19 of 20 real-pair blocks, and stays within 1.39× of the best result everywhere.Figure 1 illustrates the real-pair block comparison; the introduction reports the aggregate findings.Source: Figure 1 · p. 1, §I · p. 2
  • Every method reaches sub-pixel accuracy when its model covers a fixture, but uncovered differences can yield no match or fields wrong by tens of pixels.The best-completing baseline finishes 554 of 584 problems, yet on 8 of 20 real-pair blocks its result is 10× or more from the best achieved there.Source: §I · pp. 1–2
  • SCDF combines coarse-to-fine patch matching, sub-pixel phase-correlation re-measurement, lattice densification, and cross-validated interpolation without imposing a global motion model.Its acceptance gates and thresholds are calibrated on the image pair; one default configuration is used across all 584 benchmark cells without per-dataset tuning.Source: §I · p. 2
  • The benchmark uses 584 real-image cells with dense, verified-exact constructed ground truth and scores errors in meters, retaining declined cells as +∞ and comparing per-cell tail quantiles and ratio-to-best.The protocol is designed for varying pixel sizes and catastrophic or declined registrations.Source: §I · p. 2

II. RELATED WORK

  • Area-based methods such as AROSICS, COSI-Corr, autoRIFT, and GeFolki rely on window translation or small smooth flow, fixed search ranges, and product-tuned thresholds.Their assumptions can limit coverage; AROSICS takes more blocks outright than any other baseline in the benchmark.Source: §II · p. 2
  • Sparse pipelines produce one global transform, which cannot represent non-global deformation such as opposing mosaic-seam shifts.The related work describes SIFT with RANSAC as a canonical sparse pipeline and RIFT and SRIF as multimodal alternatives.Source: §II · pp. 2–3
  • Learned matchers lack performance guarantees outside their training distribution and are memory-bound at small working resolutions, making large-scene use operationally burdensome.An 8192^2 scene requires tiled inference, a GPU, and a multi-gigabyte environment.Source: §I · p. 2, §II · p. 3
  • Conventional checkpoint RMSE, CE90, mean endpoint error, and fixed-tolerance matching scores do not directly suit dense fields spanning widely varying pixel sizes and catastrophic failures.A declined hard pair can disappear from its own average, while a single diverged extrapolation can dominate a mean; the paper presents a benchmark for dense optical-scene fields against constructed truth on real pairs.Source: §II · p. 3

III. METHOD … C. The cascade: one loop, repeated per level

SCDF estimates a dense displacement field directly, without a global motion model or thresholds tuned to a dataset. It carries that field through a coarse-to-fine predict–measure–filter loop, using decisions calibrated on the image pair.

A. Problem statement and conventions

  • The deliverable is a dense displacement field on MOV’s pixel grid, in geographic units in MOV’s CRS, with a valid mask and pixel-center anchoring.The field is undefined outside the REF–MOV overlap and on nodata; when CRSs differ, REF positions are converted to MOV’s CRS before differencing.Source: A. Problem statement and conventions · p. 3

B. Design principle: bounding the worst case

  • SCDF avoids global transforms and computes every decision threshold from the image pair, aiming to bound worst-case errors across heterogeneous inputs.The design predicts that it will rarely be best on a particular condition but will avoid sharp error degradation; the supplied passage says Section V tests that prediction.Source: B. Design principle: bounding the… · pp. 3–4

C. The cascade: one loop, repeated per level

  • SCDF repeats a predict–measure–filter loop from the full overlap to 256-px patches, halving patch size at each level and carrying only the accumulated field between levels.It initializes with a zero-displacement point at the overlap center; the coarsest level searches the entire overlap, with no exhaustive global search.Source: C. The cascade: one loop, repeated per… · p. 4
  • At each patch center, SCDF predicts displacement by confidence-weighted moving least squares and places a REF search window using the local GSD ratio.When resolutions differ, the finer side is downsampled to a common grid; the local affine model follows Schaefer et al..Source: C. The cascade: one loop, repeated per… · p. 4
  • RootSIFT correspondences must pass a ratio test, local RANSAC affine, and forward/backward cycle check; accepted inliers become confidence-scored displacement samples.A stored match is overwritten only by a more confident one; its displacement, measured flag, and confidence are recorded in D, F, and C, respectively. RootSIFT,; RANSAC.Source: C. The cascade: one loop, repeated per… · p. 4
  • At every level, SCDF removes gross magnitude outliers and tests each match against a leave-one-out, confidence- and proximity-weighted local-affine prediction, with residuals scaled robustly from that level’s distribution.The affine fit accommodates locally linear displacement gradients; the magnitude gate is the sole absolute bound, preventing a bad measurement from moving far from the prior.Source: C. The cascade: one loop, repeated per… · pp. 4–5
  • SCDF studentizes residuals for prediction leverage, reweights fits once with Tukey biweights, and applies 32-px non-maximum suppression to retain the highest-confidence point per neighborhood.The pipeline is recall-first: it generates many candidates, then removes them using tests calibrated on the image pair.Source: C. The cascade: one loop, repeated per… · p. 5

D. The correlation pass: re-measure and densify … IV. BENCHMARK AND EVALUATION PROTOCOL

SCDF refines and densifies its displacement field with pair-calibrated correlation tests, then cross-validates interpolation and applies a common warp procedure. Its evaluation uses constructed ground truth from real satellite imagery across varied pairs.

D. The correlation pass: re-measure and densify

  • The correlation pass re-measures matched points and adds accepted measurements on a 48-pixel lattice, preserving rejected matches at reduced confidence while omitting rejected lattice candidates.The matched point’s displacement predicts its correction; lattice candidates use the neighbors’ MLS interpolant. The pass addresses feature localization limits and gaps at mosaic seams.Source: D. The correlation pass: re-measure… · p. 5, Figure 3 · p. 6
  • Correlation peaks supply sub-pixel residuals, but points are accepted only if they pass response, pair-distribution error, and prior-deviation gates.The response threshold compares the peak with the same reference window matched against eight far-away moving-image windows; accepted responses become confidence values.Source: D. The correlation pass: re-measure… · pp. 5–6, Figure 3 · p. 6
  • After re-measurement and densification, the merged points pass the global and leave-one-out filters again.Feature localization is only 0.3–0.5 pixels on the coarser grid, and cross-resolution pairs scale that error by the resolution ratio.Source: D. The correlation pass: re-measure… · pp. 5–6

E. Fill by cross-validated interpolation

  • The field’s interpolator is selected by five-fold cross-validation on the pair’s control points, choosing among seven candidates without consulting ground truth.Each candidate is scored by mean Euclidean error on held-out displacement predictions; the winner is refit on all points to fill the field.Source: E. Fill by cross-validated interpolation · p. 6

F. Warp

  • Warping is downstream of field estimation: all methods use the same GDAL geolocation-array and gdal.Warp procedure, avoiding warp quality as a comparison confound.Nearest-neighbor resampling is the default, and radiometry is left untouched.Source: F. Warp · p. 6

IV. BENCHMARK AND EVALUATION PROTOCOL

  • The benchmark comprises 584 constructed pairs from Sentinel-2, Landsat-8/9, and NAIP imagery across 60 worldwide sites, with known distortions injected to provide scoring truth.The pairs span six groups, from same-scene pairs to five-year temporal gaps and a 1:16.7 resolution ratio; ten methods produce 5,840 frozen, re-scorable runs.Source: §IV · p. 6

A. Sites and image pairs · B. Constructed ground truth on real pairs · C. The four distortion families

The benchmark injects known displacement fields into real optical imagery, preserving realistic sensor and scene differences while making the added motion exactly scoreable. Its three sections define the sampled pairs, distinguish two ground-truth experiments, and specify four distortion families.

A. Sites and image pairs

  • The 60 sites are globally sampled to avoid monotonous scenes, retain weak-texture cases, and prevent overlap; same-sensor pairs also pass cloud/nodata quality control.Sites are selected using WorldCover 2021 composition; the anti-homogeneity gate requires at least two classes above 2% and limits the largest class to 90%.Source: A. Sites and image pairs · p. 6, Figure 4 · p. 7
  • The benchmark contains 584 constructed pairs in 24 blocks; imagery uses original digital numbers without radiometric normalization, generally in 8192² crops at the finer sensor’s GSD.The Landsat half of leg b4 uses 2731² crops at native 30 m; the pair set includes temporal Sentinel-2 revisits and Sentinel-2 comparisons with Landsat and NAIP.Source: A. Sites and image pairs · pp. 6–7

B. Constructed ground truth on real pairs

  • Real imagery has no per-pixel displacement truth, so the benchmark injects known fields into real scenes, preserving their differences while making the added motion exactly scoreable.The four construction families mirror real-pair differences and do not assume the single motion model used by homography benchmarks.Source: B. Constructed ground truth on real… · p. 7
  • E2 distorts real pairs and scores only the added field, making it closer to deployment; E1 distorts a scene against itself and provides an uncontaminated accuracy floor.E2 spans cross-season to five-year revisits and cross-sensor pairs with 1:3 and 1:16.7 GSD ratios. E1 and E2 are not pooled because they answer different questions.Source: B. Constructed ground truth on real… · p. 7
  • E2 accuracy is floored by pre-existing misalignment excluded from its constructed truth; this floor is shared across methods, while the undocumented b5 floor supports ranking only.Same-orbit Sentinel-2 pairs average 0.2–0.4 px misregistration,, and Landsat Collection-2 is harmonized to below 6 m CE90.Source: B. Constructed ground truth on real… · p. 7

C. The four distortion families

  • A1 changes radiometry, blur, and noise, while A3 resamples onto a 3× coarser grid; both leave geometry unchanged, so their ground truth is zero.A1 uses a γ = 1.6 curve, σ = 2 px Gaussian blur, and 4% band-scale noise; A3 uses nodata-aware area averaging.Source: C. The four distortion families · pp. 7–8
  • A2 applies a homography with corner offsets up to 0.02 N (≈164 px), whereas A4 adds opposing ±49 px seam shifts and a ∼0.4 px elastic residual.A4 is seam-dominated and admits no single global affine or homography solution; the injected displacement field is the ground truth and is verified against each constructed mover.Source: C. The four distortion families · pp. 7–8

D. The scoring rule … V. RESULTS

The section defines how methods are scored across cells, explains the comparison sets and best-method ratios, and previews findings from a frozen 5,840-run matrix.

D. The scoring rule

  • Errors are measured in meters and aggregated by the median across cells; this avoids MOV pixel scales of 0.6–90 m assigning cells up to 150× different weight.Each cell’s median endpoint error and p90 are reported, with the per-cell p90 values also aggregated by their median.Source: D. The scoring rule · p. 8
  • A cell with no field scores +∞ rather than being dropped; a method is marked FAIL when it declines more than half the cells it covers.The same failure rule applies to the median and median per-cell p90.Source: D. The scoring rule · p. 8

E. Baselines and fairness

  • Seven training-free baselines run at official or published defaults on identical inputs and use the same evaluator as SCDF, which runs one configuration on all 584 cells.RIFT and RIFT2 are faithful ports retaining the authors’ FSC outlier rejection; SRIF is reimplemented, while the non-functional port is excluded from rankings and flagged (†).Source: E. Baselines and fairness · p. 8
  • SuperPoint+LightGlue, and LoFTR are reported zero-shot on E1 only and are not ranked against training-free methods.They require a GPU, per-image normalization, and tiling; their correspondences are densified through the authors’ global homography path, so the comparison would also reflect that adaptation layer.Source: E. Baselines and fairness · p. 8

F. The ratio to the best

  • Ratios to the best method on a block, and their per-cell analogue, require no threshold and cancel the unknown b5 floor, but cannot distinguish universal success from universal failure.The paper therefore reports them alongside absolute errors in meters.Source: F. The ratio to the best · p. 8
  • The per-cell ratio is used only on E2 because seven E1 cells have a best error of exactly zero, making its denominator degenerate.Those E1 fixtures have zero displacement, allowing a method to return an exactly identity field.Source: F. The ratio to the best · p. 8

V. RESULTS

  • Across the frozen 5,840-run matrix, every baseline is a specialist with a cliff, while SCDF has no cliff and, as its design predicts, no peak.The paper says the results establish what each experiment measures before showing these patterns.Source: §V · p. 8

A. What each experiment measures · B. Every baseline is a specialist, and every specialist has a cliff · C. A bounded worst case, and no peak

The experiments separate structural fit from performance on real imagery, revealing that each baseline has a fixture where its assumptions fail. SCDF’s central advantage is bounded performance across heterogeneous blocks, though a few open-water cells expose a serious failure mode.

A. What each experiment measures

  • E1 tests structural assumptions with exact ground truth, whereas E2 adds real content change and native misregistration; SCDF spans 2.4–11.5 m by leg and 3.9–4.4 m by construction on E2.E1’s sub-pixel scores do not rank precision; its large errors indicate model classes without a solution, while E2’s precision-limited scores make rankings meaningful.Source: A. What each experiment measures · pp. 8–9, Table V · p. 9

B. Every baseline is a specialist, and every specialist has a cliff

  • SIFT+RANSAC scores 0.062 m on homographic E1/A2 but 446 m on E1/A4; AROSICS wins six E2 blocks yet is 823 m off on E2/A2.SIFT+RANSAC’s E2/A4 error is 458 m; AROSICS is at least 10× from the best in 8 of 20 E2 blocks, reflecting their homography and translation assumptions.Source: B. Every baseline is a specialist, and… · p. 10
  • GeFolki is the steadiest baseline, with a 5.3× worst block and none at least 10× off, but its E1/A2 median is 0.559 m while its per-cell p90 reaches 1424 m.RIFT and RIFT2 score 48 m and 40 m on E2’s radiometric A1 axis, where their cross-modal descriptors target a different problem.Source: B. Every baseline is a specialist, and… · p. 10
  • On E2’s same-sensor legs, global-model methods score 430–615 m on the A4 seam while dense methods resolve those cells to 2.3–5.2 m; autoRIFT declines 94 of 96 A4 cells on 8192² legs.The seam shifts scene halves oppositely; autoRIFT completes 25 of 25 cells on E2/b4-A4, where its fixed 25 px window fits, and is best there at 8.84 m.Source: B. Every baseline is a specialist, and… · p. 10

C. A bounded worst case, and no peak

  • On E2, SCDF is best in 10 of 20 blocks, top-two in 19, worst-case 1.39× the block winner, and produces fields on all 584 cells.The best baseline’s worst block is 5.3×; AROSICS, SIFT+RANSAC, RIFT, and RIFT2 reach 408×, 554×, and at least 10× off in 13 of 20 blocks, respectively.Source: Table VI · p. 11, C. A bounded worst case, and no peak · p. 11
  • On E1, SCDF wins no block but has the lowest overall median, 0.63 m, and p90, 1.27 m, with a worst-block ratio of 8.0×.Its 0.500 m on A2 remains one-twentieth of a pixel despite SIFT+RANSAC’s exact-homography result of 0.062 m; eight other methods have two to four blocks at least 10× off.Source: C. A bounded worst case, and no peak · p. 10, Table VI · p. 11
  • On E2’s 484 cells, SCDF is within 2× of the best on 89.9% and within 5× on 97.7%, while only 1.2% exceed 10×.GeFolki reaches 58.3% and 84.3% within 2× and 5×, and AROSICS 48.1% and 56.6%; respectively, 10.1% and 41.3% of their cells exceed 10×.Source: Figure 7 · p. 11, C. A bounded worst case, and no peak · p. 11
  • SCDF catastrophically fails on three open-water E2 cells at 15.9–73.5 km, which account for 92.9% of its total E2 error sum.The bounded guarantee is at block level; individual cells can still fail.Source: C. A bounded worst case, and no peak · p. 11

D. What the components buy · E. Computational cost · VI. DISCUSSION

SCDF’s measurement stack gains accuracy mainly through the interaction of its stages, while its unablated skeleton preserves registration across ablations. Its single-core dense-field runtime is comparable to other dense estimators, though slower than sparse pipelines.

D. What the components buy

  • Removing all three measurement stages raises typical-cell error 17.1% and per-cell-tail error 41.0%, with a 14.7% superadditive joint effect; each alone costs at most 3.5%.The reported ratios are geometric means of per-cell variant/full-stack error on per-cell median and p90; values above 1 indicate accuracy loss.Source: D. What the components buy · pp. 11–12, Table VII · p. 12
  • Without the stack, A1 radiometric cells are 1.53×/1.83× worse (median/p90), versus 1.004× on A2 homography cells; tails worsen more than medians on every split.These ratios compare variant error with full-stack error.Source: D. What the components buy · p. 12
  • Every ablation variant, including the all-off corner, produced a displacement field on all 60 cells, so no knockout changed whether a cell registered.The passage attributes the bounded worst case to the non-ablated skeleton rather than to any removable stage.Source: D. What the components buy · p. 12

VI. DISCUSSION

  • The unablated skeleton comprises the recall-first policy, leave-one-out filter, null-calibrated response gate, and confidence weighting; its per-window gate matched an oracle-tuned fixed threshold without tuning.The gate was scored against dense ground truth on one cell’s densification lattice; ROC and operating-point results are in Supplement S-V.Source: D. What the components buy · p. 12

E. Computational cost

  • SCDF’s median runtime is 963 s per 8192^2 scene (67 Mpx), on par with GeFolki (992 s) and below RIFT2 (2174 s).The runtime comparison uses one CPU core per method.Source: E. Computational cost · p. 12, Table VIII · p. 12, D. What the components buy · p. 12
  • SIFT+RANSAC takes 29 s and autoRIFT 73 s on the same scenes; SCDF’s cost is dominated by K=8 null correlations and the dense correlation lattice.SCDF cost is flat across full-size constructions and parallelizes over patches, yielding ∼5× lower wall-clock time with 8 workers and bit-identical results.Source: E. Computational cost · p. 12

A. Guidance for practice … S-I. METHOD IMPLEMENTATION DETAILS AND CONSTANTS

SCDF is intended for uninspected, heterogeneous image pairs, while certified regimes can favor specialized methods. The paper also qualifies its results by open-water failures, runtime and scope limits, and constructed-ground-truth accuracy bounds.

A. Guidance for practice

  • For certified regimes, specialists can be preferable; SCDF targets uninspected inputs, where specialists may fail silently and its gates calibrate on each pair.AROSICS is ∼6× faster and as precise on guaranteed same-sensor translational pairs, SIFT+RANSAC reaches 0.062 m on a globally projective pair, and autoRIFT performs best when its window fits known small displacement.Source: A. Guidance for practice · p. 12

B. Limitations of the method

  • On the worst open-water site, only 42 of 433 points verify, and downweighting cannot recover a field without a reliable anchor.The authors identify reverting toward the zero-displacement seed after pair-level verification collapses as future work; under their +∞ scoring rule, a declared non-answer scores the same as a catastrophic field.Source: B. Limitations of the method · p. 12
  • Runtime is unoptimized at ∼16 minutes per 8192^2 scene on one core, and evaluation covers only same-optical, same-band-family pairs.Cross-modal registration is untested, though the matcher is a plug point; the authors make no claims beyond the tested regime.Source: B. Limitations of the method · p. 12

C. Limitations of the benchmark, and disclosures

  • Constructed-ground-truth accuracy is bounded by each leg’s native floor; b5 supports rankings but no absolute accuracy claim, while SRIF is excluded and autoRIFT A4 exceeds its design envelope.The b1–b3 floor is approximately 3–5 m, b4 is documented only as a bound (< 6 m CE90), and single low-texture cells can swing 2–7× under sub-half-pixel perturbations.Source: C. Limitations of the benchmark, and… · p. 13

VII. CONCLUSION

  • Across 584 constructed-ground-truth problems, SCDF’s distance-from-best is at most 1.39× on any real-pair block, while every baseline is 5.3–783× off or declines cells.On non-global deformation, global-model methods can be hundreds of meters wrong where dense fields resolve to a few meters; the authors propose reporting meter-based order statistics that retain declined cells alongside distance-from-best.Source: §VII · p. 13
  • SCDF’s acceptance gates self-calibrate under one default configuration, but the authors identify collapsed verification as a key failure case for future fallback work.Other proposed directions include learned matchers behind the same gates, manual-checkpoint evaluation on undistorted cross-sensor pairs, true multimodal extension, and runtime engineering.Source: §VII · p. 13
  • The benchmark data and scripts are archived, and the SCDF implementation is available publicly.The benchmark can be regenerated end to end from public Planetary Computer collections using the archived deterministic construction scripts.Source: §VII · p. 13

Supplementary Material for:

  • The supplement collects implementation constants, baseline configurations, audit disclosures, data-pipeline details, component studies, and zero-shot matcher details.Its section, table, and figure numbers carry an “S” prefix; references to main-text sections point back to the main paper.Source: Supplementary Material for · p. 15

A. Internal constants · B. Leave-one-out filter: two refinements

SCDF uses one fixed set of internal constants across all 584 benchmark cells, while two refinements make its leave-one-out local-affine filter account for prediction leverage and reduce neighbor-corrupted fits.

A. Internal constants

  • All internal pipeline constants are fixed, not exposed as user configuration, and identical across the 584 benchmark cells.Table S1 lists the constants, but the supplied passage does not provide their values.Source: A. Internal constants · p. 15, Table S1 · p. 15

B. Leave-one-out filter: two refinements

  • The leave-one-out filter predicts each match’s displacement from its 12 nearest other matches using the MLS prior’s distance- and confidence-weighted affine model, then tests the residual.The filter evaluates the residual against the test in main-text Eq. (4).Source: B. Leave-one-out filter: two refinements · p. 16
  • The leave-one-out prediction is linear in neighbor values and is evaluated using normalized weights, centered offsets, and a moment matrix.These quantities define the closed-form prediction described across the supplied passages.Source: B. Leave-one-out filter: two refinements · p. 16
  • Because hull-edge and sparse-area matches have higher leverage than interior matches, scaling residuals by leverage prevents an interior-calibrated threshold from discarding valid extrapolative matches.The passages describe edge or sparse-area leverage as several times larger than interior leverage.Source: B. Leave-one-out filter: two refinements · p. 16
  • When a neighborhood’s affine system is degenerate because its neighbors are collinear, the filter falls back to the weighted mean.The fallback uses g² = 0.Source: B. Leave-one-out filter: two refinements · p. 16
  • A first-pass outlier can inflate neighboring residuals, and matches across a displacement discontinuity can corrupt each other’s fits.One robust reweighting pass is used to address both effects.Source: B. Leave-one-out filter: two refinements · p. 16
  • The filter applies one Tukey-biweight reweighting pass based on first-pass residuals, recomputes the fits, and retains above-threshold matches only if they remain inconsistent with the cleaned neighborhood.The biweight uses c = 4.685 and a robust residual scale of 1.4826 MAD.Source: B. Leave-one-out filter: two refinements · p. 16

C. Correlation-pass formalism

The correlation pass scores candidate shifts between same-extent windows on a coarse grid, then uses pair-calibrated gates to accept or reject measurements. Its gates separately assess lock uniqueness, window similarity, and deviation from the predicted displacement.

C. Correlation-pass formalism

  • Mean-removed, Hann-tapered windows are correlated via the correlation theorem to score every candidate shift at once in coarse pixels.The finer window is area-downsampled to the coarse grid, and the taper suppresses the DFT's wrap-around seam.Source: C. Correlation-pass formalism · p. 16
  • The residual peak is localized to 1/32 px by evaluating Eq. (S1) on an upsampled grid near the integer peak using a matrix-multiply DFT.Only the small neighborhood is evaluated; the windows themselves are never upsampled.Source: C. Correlation-pass formalism · p. 16
  • The response ρ tests whether the shift lock is unique, while normalized mismatch ϵ tests whether the aligned windows look alike; the pass gates these axes separately.Open water can look alike without locking to a unique shift, while a sharp cross-sensor lock can be unique despite radiometric differences.Source: C. Correlation-pass formalism · p. 16
  • Measurements must pass response, error, and prior-deviation gates; the error bound uses a batch quantile clipped to 0.6–0.97, and the displacement limit is 4 coarse px when re-measuring and 3 when densifying.The response gate compares against K=8 unrelated batch partners and is skipped for batches with fewer than 16 windows; error quantiles are 0.5 when re-measuring and 0.4 when densifying.Source: C. Correlation-pass formalism · p. 16
  • A measurement failing any gate is rejected rather than clipped, and both data-driven thresholds are computed only after the full batch is measured.For gate-failure downweighting, confidence is scaled by the failed point's response relative to the accepted windows' median response.Source: C. Correlation-pass formalism · p. 16

S-II. BASELINE CONFIGURATIONS AND AUDIT DISCLOSURES … A. Formalism, equations, and parameters

The paper documents baseline configurations and failure modes, then specifies four controlled image constructions with verified ground-truth displacement fields. These details define the benchmark protocol and the conditions under which particular baselines fail.

A. Configurations

  • Baselines used official or published defaults on identical inputs and evaluator, with single-CPU execution enforced and verified at 99.7% single-core occupancy.SIFT+RANSAC used 8,000 features and a 5 px reprojection threshold; AROSICS used adaptive defaults, and autoRIFT used its official chip settings and 25 px search limit.Source: A. Configurations · p. 17

B. Disclosure: SRIF port is non-functional

  • The SRIF reimplementation produced invalid geometry, including a 7,259 px translation from three inliers on a near-identity cell, so its results are excluded from rankings and aggregates.The disclosure attributes the defect to the port, noting that the paper omits at least one parameter that had to be guessed; SRIF rows remain flagged † for completeness.Source: B. Disclosure: SRIF port is… · p. 17

C. Disclosure: autoRIFT’s A4 failure is its design envelope

  • autoRIFT’s fixed 25 px search window explains most A4 failures: the seam displacement exceeds the window, while 27312 crops with a ±16 px seam complete 25/25 cells.The paper leaves search_limit at its published default; on 81922 legs, autoRIFT completes 3/25 E1/b1 and 2/96 E2 A4 cells.Source: C. Disclosure: autoRIFT’s A4 failure… · p. 17

A. Formalism, equations, and parameters

  • All four constructions define a pixel-space displacement field on a mover grid, with valid pixels requiring an in-bounds bilinear footprint free of nodata samples.Distortions affect raw samples while preserving dtype and nodata; magnitudes are fixed fractions of N and random draws use a fixed seed shared across sites.Source: A. Formalism, equations, and parameters · p. 17
  • A1 applies radiometric and point-spread changes with zero ground-truth displacement, using band-specific tone remapping, blur, and re-noising.Its tone exponent is γ = 1.6, gains are sampled from U(0.80, 1.20), biases from U(−0.06, 0.06), and Gaussian blur uses σ = 2 px with nodata-aware normalized convolution.Source: A. Formalism, equations, and parameters · pp. 17–18
  • A2 applies a global homography by mapping the four scene corners to independently offset positions, with τ = 0.02N (about 164 px at 81922).The same inverse homography generates both the mover image and its ground-truth field.Source: A. Formalism, equations, and parameters · p. 18
  • A3 changes grid resolution through nodata-aware area averaging and uses the exact realized scale factor, making its zero-displacement truth exact.Using the nominal factor 3 instead would create a linear truth ramp reaching about 0.33 MOV pixels at the far edge; the frozen data use the corrected exact factor.Source: A. Formalism, equations, and parameters · p. 18
  • A4 combines a mosaic seam with a weak elastic residual, shifting image halves oppositely by ±49 px at 81922 across a smooth transition.The residual is generated from 24 px per-axis noise on a 16×-downsampled grid and smoothed to about 0.41 px effective amplitude at 81922; construction truth was verified by reconstruction with 0.0 error.Source: A. Formalism, equations, and parameters · p. 18

S-IV. SITE SAMPLING AND DATA PIPELINE · S-V. COMPONENT STUDIES: GATE ROC AND FACTORIAL ABLATION · A. Response-gate ROC

The response gate achieves a ground-truth-tuned floor’s operating point without tuning, while the data pipeline samples geographically separated, compositionally diverse sites and applies explicit imagery-pair quality rules. The supplied passages do not report factorial-ablation findings.

S-IV. SITE SAMPLING AND DATA PIPELINE

  • Sites are drawn globally with area weighting and accepted only when no WorldCover class exceeds 90% and at least two main classes exceed 2%.The sampling targets at least 10 sites per class, but built-up is under-covered at 4–5 sites because it never dominates an 82-km window.Source: S-IV. SITE SAMPLING AND DATA PIPELINE · p. 18
  • Accepted sites are separated by at least one footprint side, preventing crop overlap, and per-site class fractions are stored in the released site polygons.Probe footprints are approximately 82 km for Sentinel-2/Landsat and 5 km for NAIP, decimated to approximately 100 m.Source: S-IV. SITE SAMPLING AND DATA PIPELINE · p. 18
  • Imagery comes from Microsoft Planetary Computer STAC collections, with scene cloud below 10% and crop-level invalid fractions at or below 20% for Sentinel-2.Sentinel-2 temporal pairs must share an MGRS tile and relative orbit; Landsat uses its QA band for quality control, while NAIP requires all-zero nodata exclusion.Source: S-IV. SITE SAMPLING AND DATA PIPELINE · p. 18
  • Sentinel-2/NAIP pairs use same-year NAIP tiles from 2022 or 2023 and carry 0.7–2.1 years of landscape change (median 1.9 years) in addition to the modality gap.Same-year tiles avoid cross-vintage seams; the selected NAIP year maximizes tile coverage over each site within 2017–2024.Source: S-IV. SITE SAMPLING AND DATA PIPELINE · p. 18

A. Response-gate ROC

  • Ground-truth windows are labeled good at EPE ≤0.5 coarse px or junk above 1.5 only for scoring; the gate itself sees none of those labels.The evaluation compares true- and false-positive rates against the ROC curve of an absolute response floor.Source: A. Response-gate ROC · p. 18, Figure S1 · p. 19
  • A single absolute response floor is unsuitable because the per-window standard ranges from 0.14–1.13, holding smooth windows to a higher bar than textured ones.The gate’s operating point is compared with the ROC curve from sweeping an absolute floor across its full range.Source: A. Response-gate ROC · p. 18, Figure S1 · p. 19
  • 71% of good windows are accepted and 6% of junk windows are accepted (TPR 0.71 / FPR 0.06), matching the frontier of a ground-truth-tuned response floor without tuning.Parity, not dominance, is the claim; Fig. S1 shows the operating point among 342 measurable windows from one cell.Source: A. Response-gate ROC · p. 18, Figure S1 · p. 19

B. Factorial ablation of the measurement stack · S-VI. ZERO-SHOT DEEP-LEARNING BASELINES

The factorial ablation shows that the measurement stages jointly improve precision, especially where matches are imprecise, while reliability changes little. The zero-shot baseline section specifies the pretrained models and the adaptations used to run them on the benchmark imagery.

B. Factorial ablation of the measurement stack

  • Eight variants were tested on the same stratified 60-cell subset, and all registered every cell at the same median coverage (0.947).Cells recur across legs and constructions, so paired Wilcoxon p values are descriptive strength, not literal error rates.Source: B. Factorial ablation of the… · p. 19
  • Removing any one stage costs at most 3.5% on the median; removing all three costs 17.1% on the median and 41.0% on p90.Every two-stage knockout falls between those outcomes, and no single stage carries precision alone.Source: Table S2 · p. 19, B. Factorial ablation of the… · p. 19
  • The three stage main effects are positive on both responses (p ≤0.010), and their joint effect exceeds the sum of their single-stage effects by 14.7% (p=1.7 × 10^-9).All three interactions are positive: each stage is worth more when the others are absent, so the stages act as substitutes rather than additive contributors.Source: B. Factorial ablation of the… · p. 19, Table S3 · p. 20
  • Without the stack, A1 loses most (1.53× median, 1.83× p90), while the 1:3 cross-resolution leg b4 is most damaged on median (1.29×; 1.50× on p90).A2 pays nothing (1.004); switching re-measurement off helps there (V1 on A2 is 0.887 median, 0.874 p90), offsetting A1’s loss in the pooled V1 result.Source: B. Factorial ablation of the… · p. 19, Table S4 · p. 20
  • Under the 0.90 sub-pixel-fraction diagnostic, clean-rate is 0.717 for every variant except V4 (0.683) and V7 (0.667); no single-stage knockout flips any cell.Worst single-cell degradation rises with knockout depth: 1.57× (V1), 2.38× (V4), and 3.49× (V7); V0 beats V7 on 46 of 60 cells.Source: B. Factorial ablation of the… · p. 20

S-VI. ZERO-SHOT DEEP-LEARNING BASELINES

  • SuperPoint+LightGlue and LoFTR were run strictly zero-shot with published pretrained weights, default inference settings, and no fine-tuning or domain adaptation.LoFTR used outdoor weights via kornia 0.8.3.Source: S-VI. ZERO-SHOT DEEP-LEARNING BASELINES · p. 20
  • To adapt the imagery, both models received grayscale 8-bit inputs made with a 2–98th-percentile stretch, then corresponding REF and MOV tiles were matched and aggregated.LightGlue used 2048^2 tiles with 256 px overlap; LoFTR used 1024^2 tiles with 128 px overlap; tile-local matches were densified using the same global-homography estimation as SIFT+RANSAC.Source: S-VI. ZERO-SHOT DEEP-LEARNING BASELINES · p. 20

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