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Efficient Continuous DEM Reconstruction under Limited Target-Resolution Supervision

Zekai Shi, Meng Zhang, Haokun Zhang, Bo Zhang

arXiv:2609.29864v1cs.CVcs.LGeess.IV

TL;DR

Paired elevation references may be available only at coarser output resolutions than the finer grids applications require. SCOPE predicts reusable local residual functions from a low-resolution coefficient field, then evaluates them to construct denser terrain grids. It performs strongly across supervised-scale, unseen-scale, and external marine evaluations, with low incremental counted computation at higher output density.

  • Problem

    Paired training references may be available at coarser output resolutions than desired, leaving the question of reconstruction at finer grids without target-scale training labels.

    Source: §1.1 · p. 1, §5.1 · p. 13

  • Method

    SCOPE predicts a low-resolution latent coefficient field whose coefficients parameterize reusable local Fourier residuals, fused over a bicubic base surface.

    Source: §3.1 · pp. 5–6

  • Results

    SCOPE leads six supervised-scale metrics, reduces unseen-scale land RMSE and MAE by approximately 12% relative to bicubic, and lowers external-marine RMSE than EBCF-CDEM and LIIF-MS.

    Source: §6 · p. 16

  • Takeaways & Limitations

    The evaluations support reusable terrain functions as an approach to scale-flexible DEM reconstruction under limited target-resolution supervision.

    Source: §6 · p. 16

    Main quantitative comparisons use the checkpoint-selection validation subset rather than a separate held-out main-study test set.

    Source: §5.6 · p. 15

Abstract

from arXiv · show

High-resolution digital elevation models (DEMs) support Earth observation applications, but paired training references are often available only at coarser output resolutions. Reconstructing finer terrain grids therefore requires both effective transfer beyond the supervised scale and control of dense-query computation. To address this problem, SCOPE learns a continuous terrain representation from coarser-resolution pairs. It predicts a latent coefficient field on the low-resolution grid and reuses local Fourier residual functions through basis evaluation and geometry-guided ensemble fusion. This separates high-dimensional coefficient prediction from output-grid construction. Experiments on geographically distributed land--ocean samples assess supervised reconstruction, unseen-scale inference, cross-domain generalization, and theoretical computation. SCOPE leads the compared methods across six metrics in the main supervised-scale evaluation. At an unseen factor three times the training factor, land reconstruction reduces RMSE and MAE by approximately 12\% relative to bicubic interpolation, with errors close to target-scale fine-tuning. Ninefold output density increases counted multiply--accumulate operations by only about 2\%. Frozen-model validation on held-out external marine regions reduces RMSE relative to the DEM-specific implicit baseline EBCF-CDEM by approximately 19\% under self-downsampling and 2\% with cross-product inputs, while also yielding lower RMSE than LIIF-MS in both settings. These results demonstrate the value of reusable coefficient fields for accurate reconstruction beyond the supervised resolution with low incremental arithmetic cost.

1. Introduction

Paired elevation references may be available only at coarser output resolutions than desired, while fixed-grid learned methods are tied to predefined output grids and continuous querying can repeat costly decoding. SCOPE instead predicts reusable local residual functions from a low-resolution coefficient field, separating coefficient prediction from dense output construction.

1.1. DEM Products and Limited Target-Resolution Supervision

  • Paired DEMs may provide only coarser-resolution training labels than the desired grid, motivating reconstruction beyond interpolation without new target-scale labels.The paper calls learning from paired references at a coarser output resolution than the desired grid limited target-resolution supervision.Source: §1.1 · p. 1

1.2. Limitations of Fixed-Scale DEM Reconstruction

  • Fixed-scale learned grid-to-grid methods target predefined output grids, while interpolation permits arbitrary spacing but does not learn terrain-dependent corrections from paired observations.The paper argues that learned elevation reconstruction and output-grid flexibility must be considered together when desired resolution exceeds available supervision.Source: §1.2 · pp. 1–2

1.3. Continuous Terrain Representation

  • Continuous terrain methods differ in their predicted quantities, but flexible coordinate querying alone does not establish accuracy beyond training scales and can repeat decoding work at each output coordinate.LIIF predicts values, LTE uses query-centered frequency-aware estimation, and EBCF-CDEM predicts elevation bias relative to a nearest-neighbor base.Source: §1.3 · p. 2

1.4. From Coordinate Querying to Reusable Terrain Functions

  • SCOPE predicts a low-resolution latent coefficient field whose coefficients parameterize reusable local Fourier residual functions, evaluated and fused over a bicubic base surface.This allocates high-dimensional coefficient prediction to the LR grid and uses lightweight local evaluation at output queries.Source: §1.4 · p. 2
  • The evaluation distinguishes demonstrated scale extrapolation from output-grid flexibility and combines supervised-scale accuracy, unseen-scale reconstruction, operation counts, and external cross-domain tests.The external tests separate cross-region transfer under synthetic degradation from transfer using matched inputs from another DEM product.Source: §1.4 · p. 2

2. Data and Study Design

The study combines global terrestrial and background bathymetric products with regional coastal and marine datasets. It constructs paired samples across geographically distributed land–ocean terrain, then uses held-out marine regions for frozen-model external evaluation.

2.1. Data Sources

  • TanDEM-X 30 m supplies the primary terrestrial reference, GEBCO_2024 provides global background bathymetry, and regional products add coastal, reef, shallow-shelf, and open-marine terrain.The regional sources include NOAA Coastal Relief Models, Torres Strait and Bass Strait products, and an EMODnet Caribbean tile used only in fixed 5× evaluation.Source: §2.1 · p. 3

2.2. Study Area and Sample Construction

  • The study samples geographically distributed terrain between 65°N and 65°S, combining land references with marine and coastal samples to cover land–ocean transitions and heterogeneous bathymetry.Preprocessing standardizes products and aligns paired or cross-product rasters on a common grid before patch extraction.Source: §2.2 · p. 3
  • Two sample designs support evaluation: controlled pairs degrade an HR reference to create the LR input, while cross-product pairs align LR and HR samples from different elevation products.The controlled design uses a scale-dependent degradation process; the cross-product design includes co-registration, resampling, valid-region intersection, and patch-level quality control.Source: §2.2 · p. 5

2.3. External Marine Data for Cross-Domain Evaluation

  • External evaluation uses 12 held-out marine reference rasters and 3,825 valid patches, comparing 450→90 m self-downsampled inputs with matched GEBCO_2023 inputs.The reference rasters are held out from training and checkpoint selection, and the two input settings are evaluated separately.Source: Table 1 · p. 4, §2.3 · p. 5

3. Methodology

SCOPE reconstructs a continuous elevation surface by adding locally evaluated residuals to a bicubic base. It predicts reusable coefficients on the low-resolution grid, fuses neighboring residual candidates, and uses sign-adaptive activation and a composite training objective.

3.1. SCOPE Network

  • SCOPE adds an LAE-fused residual to a bicubic base surface, using NAF-predicted coefficients and local basis evaluation to reconstruct elevations at continuous query coordinates.The encoder produces an LR-aligned terrain feature field, which NAF transforms into a latent coefficient field; neighboring coefficient vectors parameterize local residual functions rather than storing discrete elevation residuals.Source: §3.1 · pp. 5–6, §3 · p. 5, Figure 4 · p. 6
  • Reusing the same coefficient field at different query densities yields DEM outputs at different reconstruction scales without changing the encoded feature field or adding a scale-specific output layer.NAF evaluates local residual candidates using relative query displacements from neighboring LR locations.Source: §3.1 · p. 6
  • LAE weights neighboring residual candidates using learned compatibility and a logarithmic bilinear prior, retaining bilinear spatial preference while allowing terrain-dependent adjustment.The learned score estimates each candidate’s compatibility with the query location.Source: §3.1 · pp. 6–7

3.2. Sign-Adaptive Smooth Unit

  • SASU is a parameter-free sign-adaptive activation for the elevation residual decoder, combining positive GELU and negative SiLU branches that preserve responses and gradients for negative features.The two branches share their value and first derivative at the origin, supporting a smooth transition during optimization.Source: §3.2 · p. 7

3.3. Training Objective

  • Training combines reconstruction, gradient, and circular direction losses to preserve elevation consistency, local relief variation, and local orientation.The direction loss uses a mask for HR gradient magnitudes above a per-sample relative threshold and is zero when no valid pixels remain.Source: §3.3 · p. 7

4. Experimental Setup and Results

The experiments assess supervised reconstruction, unseen-scale performance, terrain structure, and external transfer using spatially separated validation regions and frozen external-test checkpoints. The fixed protocol trains with 5× supervision and evaluates specified controls at 15×.

4.1. Experimental Design

  • Evaluation covers supervised-scale reconstruction, reconstruction beyond the supervised scale, terrain-structure preservation, and external transfer, with external marine data reserved for frozen-model testing.Training and validation regions are spatially separated before patch extraction, and checkpoints are fixed before external testing.Source: §4.1 · p. 7, §4 · p. 7
  • The default protocol uses 5× supervision from nominal 450 m inputs and 90 m labels, then evaluates the fixed-5× model at 15× on the same spatial footprint.The 15× evaluation uses the land portion of the fixed-5× evaluation data; SCOPE-15D and SCOPE-15FT are controls supplied with 30 m target-resolution supervision.Source: §4.1 · pp. 7–8
  • SCOPE’s full configuration uses an LR-aligned 192-dimensional feature field, 96 basis frequencies, four-candidate LAE fusion, residual scaling, post-refinement, and SASU activation.Optimization uses approximately 230,000 LR–HR patch pairs; the objective weights elevation, gradient, and aspect losses at 1.0, 0.05, and 0.01.Source: §4.1 · p. 8
  • External evaluation compares frozen SCOPE-5F and baseline checkpoints on the same 3,825 valid reference patches per input setting, reporting per-patch physical-domain RMSE and MAE means.Comparisons retain bicubic interpolation and include unseen-scale inference, direct 15× training, and 5× training followed by 15× fine-tuning.Source: §4.1 · p. 8

4.2. Ablation Study on Key Components

At the supervised 5× scale, SCOPE leads the compared methods across six metrics; ablations show that its base surface, local fusion, refinement, terrain-aware supervision, and activation contribute to balanced reconstruction.

4.2. Ablation Study on Key Components

  • Removing the bicubic base raises RMSE from 4.150 to 4.682 and lowers Corr. from 0.659 to 0.521; bilinear and nearest bases are also slightly weaker than bicubic.Replacing LAE with bilinear fusion causes a small but consistent decline.Source: §4.2 · p. 8, Table 2 · p. 9
  • Removing post-refinement increases slope and aspect errors, while removing aspect-related loss causes the strongest degradation among loss variants, especially on those terrain metrics.The L1-only variant remains competitive in elevation accuracy but has weaker terrain-structure metrics.Source: §4.2 · p. 8
  • SASU is the most effective tested activation overall: SiLU is close, whereas GELU degrades elevation accuracy and terrain-related metrics more clearly.The ablations collectively support the bicubic base, adaptive local fusion, refinement, terrain-aware supervision, and SASU as contributors to final performance.Source: §4.2 · p. 8

4.3. Quantitative Comparison with Baselines

  • At 5×, SCOPE ranks first across all six metrics, reducing RMSE, MAE, and slope error by 6.04%, 5.28%, and 7.16% against the strongest non-SCOPE results.Aspect error falls by 1.21%, while PSNR and Corr. increase by 1.32 dB and 0.019; SCOPE also reduces counted GMACs by 79.77% versus LIIF and 77.62% versus LTE.Source: §4.3 · pp. 8–9

4.4. Fixed-Scale Visual Analysis

Fixed-scale visual comparisons examine land and marine/coastal reconstructions, their error patterns and profiles, and whether methods preserve directional terrain structure. SCOPE is reported to retain fine-scale relief and directional continuity more faithfully than the compared alternatives.

4.4. Fixed-Scale Visual Analysis

  • Representative land and marine patches are compared at 5× against the HR reference, LR input, interpolation, fixed-scale networks, and continuous reconstruction baselines.The complete predefined evaluation set, rather than only the visual examples, is used for quantitative metrics.Source: §4.4 · p. 9, §4.4.1 · p. 9

4.4.1. Visual Reconstruction Comparison

  • SCOPE better preserves land ridge–valley detail and marine/coastal transitions, while other methods smooth or blur fine relief; EBCF-CDEM shows nearest-neighbor block artifacts in the marine example.The land and marine panels use shared elevation ranges, with enlarged regions showing local structures.Source: §4.4.1 · p. 9, Figure 5 · p. 9, Figure 6 · p. 10

4.4.2. Elevation Error Distribution and Profile Analysis

  • Bicubic reduces block artifacts but leaves boundary errors; SCOPE has fewer high-error responses at complex transitions and maintains similar or better profile fidelity than EDSR and SwinIR with stronger geomorphological consistency.Profiles reveal smoothed extremes in bicubic and underestimated or shifted peak–valley amplitudes in LIIF.Source: §4.4.2 · pp. 10–11, Figure 7 · p. 10, Figure 8 · p. 10

4.4.3. Directional Terrain Structure Analysis

  • SCOPE preserves fine-scale directional variations with spatial coherence and fewer artificial discontinuities, closer to HR aspect patterns than the compared methods.Figures show aspect patterns for representative land ridges and valleys and marine/coastal transitions.Source: §4.4.3 · p. 11, Figure 10 · p. 11, Figure 11 · p. 11

4.5. Cross-Domain Evaluation

Cross-domain evaluation separates matched-product supervision from frozen-model tests on external marine regions. SCOPE-Direct leads elevation-error metrics in the matched experiment, while the frozen SCOPE-5F checkpoint performs strongly under self-downsampling and more variably with cross-product inputs.

4.5.1. Matched-Product Supervision and Transfer

  • The matched LR–HR experiment compares paired-data SCOPE-Direct with synthetic-data SCOPE-Transfer, separately from the frozen external-marine test.Figures 12–13 visualize land and marine/coastal reconstructions for SCOPE-Direct.Source: §4.5.1 · pp. 11–12, Table 4 · p. 12, Figure 12 · p. 12
  • SCOPE-Direct has the lowest RMSE and MAE, whereas SCOPE-Transfer leads grid-space slope error; bicubic leads aspect error, PSNR, and Corr.SCOPE-Transfer RMSE and MAE are 8.546 and 6.223, versus 8.452 and 6.145 for bicubic.Source: §4.5.1 · p. 11, Table 4 · p. 12

4.5.2. Frozen-Model Evaluation on External Marine Regions

  • Under self-downsampling, frozen SCOPE-5F leads all six metrics and reduces RMSE by 19.40% versus EBCF-CDEM; with matched GEBCO_2023 inputs, its RMSE is 2.08% lower than EBCF-CDEM.In the cross-product setting, bicubic leads RMSE, MAE, and Slope and has the highest PSNR; SCOPE-5F’s RMSE and MAE are 3.69% and 3.22% higher than bicubic.Source: §4.5.2 · p. 12, Figure 13 · p. 12

4.6. Reconstruction Beyond the Supervised Scale

The scale-transfer experiment tests whether a model trained only at 5× can reconstruct at an unseen 15× scale without target-scale supervision or retraining. SCOPE-5F improves on key baselines while requiring little additional counted arithmetic as output density increases.

4.6. Reconstruction Beyond the Supervised Scale

  • The unseen 15× test queries the fixed-5× model on the same spatial support, increasing output-grid density ninefold without 15× supervision or retraining.This is presented as a stricter test than querying within or near the supervised scale range.Source: §4.6 · p. 12
  • At unseen 15× on land, SCOPE-5F RMSE is 6.753 and MAE is 4.609, reductions of 21.66% and 22.00% versus LIIF-MS and 34.21% and 36.12% versus EBCF-CDEM.Against bicubic, SCOPE-5F improves RMSE, MAE, slope error, PSNR, and Corr.; aspect errors are 71.534 for SCOPE-5F and 71.386 for bicubic.Source: §4.6 · p. 12
  • The controlled variants perform closely, with SCOPE-15FT’s RMSE 0.08% below SCOPE-5F; SCOPE-5F also reduces counted GMACs by 97.23% versus LIIF-MS and 69.09% versus EBCF-CDEM.At 15×, EBCF-CDEM uses 1.450M parameters and 84.799 GMACs.Source: §4.6 · pp. 12–13

5. Discussion

SCOPE reuses a learned coefficient field to reconstruct unseen output scales, while local basis evaluation and geometry-guided fusion limit repeated query-side computation. The discussion reports useful land scale transfer and computational savings, but bounds claims about cross-product generalization and geographic and measurement scope.

5.1. Crossing the Supervision Boundary: From Known Scales to Unseen Reconstruction

  • Land reconstruction remains effective at 30 m when trained with 90 m labels from nominal 450 m inputs, with direct 15× training and fine-tuning yielding close results.This supports reconstruction beyond the supervised output scale without establishing that fixed-scale training is intrinsically superior.Source: §5.1 · p. 13

5.2. Implications for Earth Observation Terrain Products

  • External marine tests distinguish regional generalization under self-downsampling from real-product transfer, where metric-specific rankings do not establish a uniform cross-product advantage.Acquisition source, vertical reference, interpolation history, and terrain distribution may affect matched-product reconstruction; aggregate results do not isolate their contributions.Source: §5.2 · p. 13

5.3. Why Lightweight Local Attentive Ensemble Is Sufficient

  • LAE gives small, consistent improvements over Transformer-style fusion across terrain-related metrics, weighting four local residual candidates using their relative displacements.The query-stage task is compact local compatibility weighting, not long-range interaction learning.Source: §5.3 · pp. 13, 15, Table 7 · p. 15

5.4. E

  • SASU lowers reported RMSE from 4.404 to 4.150 and grid-space slope error from 14.939 to 12.051 versus GELU, while results are nearly identical to SiLU.The authors retain SASU as the default but do not claim a substantial advantage over SiLU.Source: §5.4 · p. 15

5.5. Computational Scaling and Inference E

  • From 5× to 15×, counted GMACs increase by 0.515 for SCOPE, versus 819.528 for LIIF and 73.646 for EBCF-CDEM.SCOPE still has a cost linear in the number of output queries; its LR-side work dominates at the evaluated sizes, so ninefold output density produces only a small total increase.Source: §5.5 · p. 15
  • Figure 15’s trajectories show limited arithmetic growth but increasing SCOPE latency with output density; EBCF-CDEM and EDSR are faster at several plotted factors.The comparison does not establish SCOPE as universally fastest, and 20× and 30× extend cost analysis only, not reference-based accuracy evaluation.Source: §5.5 · p. 15, Figure 15 · p. 15

5.6. Limitations and Extensions toward Latent Geomorphological Prior Learning

  • Evaluation scope is bounded by selected regions and sampling, excludes polar and high-latitude terrain outside 65°N–65°S, and does not imply complete tested coverage from product extents or tile boundaries.Slope-related metrics use unit pixel spacing and assess grid-space structure rather than physical slope errors at 90 m or 30 m horizontal spacing; fine-scale reconstruction remains an inference, not newly observed topographic measurements.Source: §5.6 · p. 16
  • A proposed extension is latent geomorphological prior learning with cross-scale consistency, terrain geometry, semantics, and multi-source geospatial priors.The discussion distinguishes this direction from directly adopting codebook- or diffusion-based image super-resolution.Source: §5.6 · p. 16

6. Conclusion

SCOPE uses a reusable coefficient field to reconstruct denser DEM grids from coarser-resolution paired training data. It performs strongly at supervised and unseen scales, with low incremental counted computation and favorable results in held-out marine validation.

6. Conclusion

  • Under fixed 5× supervision, SCOPE ranks first across six metrics; at unseen 15× on land, RMSE and MAE are approximately 12% lower than bicubic, with RMSE within 0.08% of target-scale fine-tuning.The unseen 15× evaluation uses a factor three times the training factor.Source: §6 · p. 16
  • Ninefold output density increases counted computation by approximately 2%, supporting reconstruction beyond supervised resolution with low incremental arithmetic cost.The approach learns a reusable coefficient field from available pairs and reconstructs the denser grid through local basis evaluation and geometry-guided ensemble fusion.Source: §6 · p. 16
  • On 3,825 held-out external marine patches, SCOPE reduces RMSE versus EBCF-CDEM by approximately 19% under self-downsampling and 2% with matched GEBCO_2023 inputs, and has lower RMSE than LIIF-MS in both settings.It leads all six metrics under self-downsampling and the learned models in RMSE, MAE, Slope, and PSNR under matched cross-product inputs.Source: §6 · p. 16

Competing Interests

The authors report no relevant financial or non-financial interests to disclose.

Competing Interests

Data Availability

The analyzed datasets are publicly available from their original providers.

Data Availability

  • The datasets analyzed are publicly available from their original providers, with names and access information given in the manuscript.Source: Data Availability · p. 16
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