MarsFM: Shading-Regularized Flow Matching for Martian Relief Estimation

Marius F. R. Juston

Spectral Band	Wavelength Range (nm)	Primary Geological Applications
Blue–Green (BG)	400–600	Detection of frost, ice, and atmospheric scattering phenomena
Panchromatic (RED)	550–850	High-resolution topographic mapping, structural geomorphology analysis, and digital elevation model (DEM) extraction
Near-Infrared (IR)	800–1000	Mineralogical differentiation and sub-pixel spectral unmixing (e.g., integration with CRISM datasets)

Appendix

TABLE III : Spectral band characteristics and primary geological applications

Fig. 2 : Per-pixel radiometrically calibrated distribution of 149,921 valid patches with a patch size of 0.005 degrees over 422 HiRISE measurement readings. These patches are derived from the HiRISE dataset and subsampled using the target-region keyword ”Olympus”. The histograms are unnormalized frequency histograms that provide aggregated information on approximately 46 billion pixels per channel. The red lines in the histograms represent the 2nd and 98th percentiles of the data.

Channel	$N_{\text{pixels}}$	Mean	Std
Near-infrared	$4.5949\text{\times}{10}^{10}$	$0.1965$	$0.0569$
Red	$4.9983\text{\times}{10}^{10}$	$0.1668$	$0.0391$
Blue-green	$4.6060\text{\times}{10}^{10}$	$0.0723$	$0.0211$

Appendix

TABLE IV : Summary statistics of the dataset across spectral channels. P2 and P98 represent the 2% and 98% percentiles. CV denotes the coefficient of variation (

\sigma/\mu

). DRCR denotes dynamic range compression ratio (

(P_{98}-P_{2})/\mu

).

Fig. 3 : Topographic Map of Mars [ 34 ] through the view of Mercator Projection with longitude range of

-180^{\circ}

to

180^{\circ}

and latitude range of approximately

-57^{\circ}

to

57^{\circ}

. The map was generated by the U.S. Geological Survey, and on the map were overlaid all the HiRISE sample locations in blue within that topographical range, and in red is the subsampled dataset used as the scope of the project. The sub-sampled dataset was selected by filtering samples with the rationale description ”Olympus”, rather than a bounding-box selection approach.

Parameter	Example Value	Scientific Representation / Description
OBSERVATION_ID	TRA_000873_2015	Unique identifier for this orbital capture sequence, documenting a transition-phase observation of Becquerel Crater stratigraphy.
PRODUCT_ID	TRA_000873_2015_COLOR	Specific identifier for this data product, indicating it is the synthesized color composite image.
OBSERVATION_START_TIME	2006-10-03T12:51:33.015	Precise Universal Time Coordinated (UTC) timestamp when the instrument began recording the observation.
MAP_PROJECTION_TYPE	EQUIRECTANGULAR	Cartographic mapping method applied. Utilizes a spherical approximation (local radius $R=3393.83$ km) where latitude and longitude grid lines intersect at right angles.
MAP_SCALE	0.25 meters/pixel	Spatial resolution of the projected map. Each pixel represents a 0.25 m $\times$ 0.25 m area on the Martian surface at the center latitude.
CENTER_LATITUDE / LONGITUDE	20.000 ∘ , 180.000 ∘	Planetocentric coordinates defining the center of the map projection.

Appendix

TABLE V : Comprehensive PDS Metadata and Viewing Parameters for HiRISE Observation

Fig. 4 : Full pipeline architecture overview of the Mars HiRISE dataset loader and sampler

Fig. 5 : Full pipeline architecture overview of the Mars HiRISE dataset preprocessing to convert the JPEG2000 images into Cloud-Optimized GeoTIFF

Fig. 7 : HiRISE targeting vocabulary. The word clouds summarize the most prominent terms in observation rationales after stopword filtering. They show that HiRISE text is strongly geologic rather than generic metadata, with repeated concepts such as craters, dunes, layers, polar processes, slopes, deposits, and channels.

Fig. 8 : Top rationale words and phrases. The script counts filtered unigrams and adjacent token bigrams, making the text distribution auditable rather than relying only on the word-cloud visualization.

Fig. 9 : From words to geological categories. Geographic-feature counts track named Mars feature types such as craters, chasmata, valles, fossae, and montes. Theme counts group rationales into science categories including impact craters, gullies/slopes, aeolian processes, polar/ice/frost, layers/stratigraphy, mineralogy, volcanic processes, tectonics, and landing-site/engineering observations.

Fig. 10 : Observation map colored by primary science theme. Each observation is assigned a primary theme from its matched keyword list and plotted by longitude/latitude. This links the keyword-derived catalog characterization to geography; text is not a conditioning input in the reported model.

Fig. 11 : Theme-specific Mars maps. Small multiples reduce overplotting by drawing one hex-density map per science theme, revealing where different rationale classes cluster spatially.

Fig. 12 : Dominant theme per Mars region. Each spatial bin is colored by the locally dominant primary theme, with opacity scaled by log observation count. This preserves a single-map view while reducing the overplotting problem of point-wise thematic maps.

Fig. 13 : Theme imbalance and latitude dependence. The treemap shows the global class imbalance in rationale-derived science themes, while the stacked latitude-band plot shows that theme frequencies vary by equatorial, mid-latitude, and polar regions. These distributions motivate geographically grouped evaluation and careful sampling for any future text-conditioned extension. Keyword themes describe targeting rationales, not verified geological labels for every pixel.

Fig. 14 : Demonstration of the patch sampling for a single strip inside the HiRISE dataset, the gray box represents the max bounding box that is provided from the metadata, the contour plot represents the region where data is available.

Fig. 15 : Thumbnail plot grid of 16 HiRISE measurements from the ”Olympus” dataset target with angular resolution of 0.005 degrees. Due to the radiometry calibration, the image scale is 0-0.2; to improve visualization, we normalize per-patch pixel intensities to the 2nd and 98th percentiles. From these randomly selected plots, we can clearly see the range in random features available in the currently selected subset of the HiRISE dataset.

Fig. 16 : Feasible center regions for four convex footprints at

p=0.5

. Blue shows the footprint, red the sampled feasible boundary, and green a test square. The exact region is convex but need not be polygonal; a sampled polygon approximates it.

Fig. 17 : Column-locked grid and independent-row placement on one synthetic strip at

\delta=0

. The exported counts are 47 and 58 valid patches, with reported coverage 88.2% and 98.9%. These illustrate the benefit on this geometry; the theorem guarantees maximal count within each fixed row, not maximal covered area.

Fig. 18 : The same comparison at

\delta=0.5

. The export reports 183 versus 204 valid patches and 99.6% versus 100% footprint coverage. “Optimal” in the retained plot refers to independent-row placement under the stated restrictions.

Fig. 19 : Nine synthetic footprints comparing column-locked centers (green) and independent-row centers (red). Each panel retains the original counts and relative gain. The examples support a descriptive yield comparison under the sampled geometries; they do not establish a universal coverage improvement.

Fig. 20 : Comprehensive evaluation of the simple versus optimal patch placement algorithms. Panels (A) and (B) illustrate the spatial distribution and coverage gains on a representative synthetic strip. Panel (C) tracks the mean patch density as a function of the internal patch overlap parameter. Panel (D) presents the distribution of per-strip percentage gains across the synthetic dataset, indicating higher patch counts in the illustrated examples; these are descriptive comparisons, not a statistical significance test.

Fig. 21 : Sensitivity analysis of the valid patch yield with respect to the minimum required overlap threshold (

p

). The optimal independent strip packing algorithm consistently identifies more valid patch centers across all threshold constraints than the simple grid baseline.

Fig. 22 : Impact of the patch overlap parameter (

\delta

) on packing density and footprint coverage. The left panel shows how the total patch count scales with increasing overlap, while the right panel illustrates the corresponding increase in total area coverage, highlighting the optimal algorithm’s superior space utilization.

Attribute	Standard RDR	DTM Ortho
Sample Bits	16	8
Effective Max DN	1023 (10-bit)	255 (8-bit)
Typical Scaling Factor	$\sim 1.24\times 10^{-4}$	$\sim 8.49\times 10^{-5}$
Typical Offset	$\sim 0.033$	$\sim 0.069$

Appendix

TABLE VI : Example encoded-image attributes in the inspected RDR and DTM-ortho products. Read bit depth, scaling, and offset from each label; these examples are not universal product constants.

Fig. 23 : Per-pixel radiometrically calibrated distribution of valid patches with a patch size of 0.018 degrees over 200 HiRISE DTM measurement readings. These patches are derived from the HiRISE DTM dataset and subsampled by filtering samples with a bounding box of

[-120\degree,-30\degree,150\degree,30\degree]

. The histograms are unnormalized frequency histograms that provide aggregated information on approximately 25 billion pixels per channel. The red lines in the histograms represent the 2nd and 98th percentiles of the data.

Fig. 24 : Per-pixel elevation data of valid patches with a patch size of 0.018 degrees over 200 HiRISE DTM measurement readings. These patches are derived from the HiRISE DTM dataset and subsampled by filtering samples within a bounding box of

[-120\degree,-30\degree,150\degree,30\degree]

. The histograms are semi-log-scaled, unnormalized frequency histograms that provide aggregated information on approximately 1 billion pixels per channel (there are fewer pixels due to resolution subsampling). The red lines in the histograms represent the 2nd and 98th percentiles of the data.

Channel	$N_{\text{pixels}}$	Mean	Std
Elevation	$1.6092\text{\times}{10}^{09}$	$-2,070.4332$	$1,152.9997$
Left red	$2.5751\text{\times}{10}^{10}$	$0.1134$	$0.0292$
Right red	$2.6228\text{\times}{10}^{10}$	$0.1176$	$0.0313$

Appendix

TABLE VII : Summary statistics of the dataset across spectral channels. P2 and P98 represent the 2% and 98% percentiles. CV denotes the coefficient of variation (

\sigma/\mu

). DRCR denotes dynamic range compression ratio (

(P_{98}-P_{2})/\mu

).

Fig. 25 : Per-pixel detrended absolute value elevation data of valid patches with a patch size of 0.018 degrees over 200 HiRISE DTM measurement readings. These patches are derived from the HiRISE DTM dataset and subsampled by filtering samples with a bounding box of

[-120\degree,-30\degree,150\degree,30\degree]

. The histograms are semi-log-scaled, unnormalized frequency histograms that provide aggregated information on approximately 1 billion pixels per channel (there are fewer pixels due to resolution subsampling). The red lines in the histograms represent the 2nd and 98th percentiles of the data.

Channel	$N_{\text{pixels}}$	Mean	Std
Elevation	$1.6092\text{\times}{10}^{09}$	$9.1183$	$12.1752$

Appendix

TABLE VIII : Summary statistics of the absolute detrended residual

|Z-\mathrm{plane}|

in meters. P2 and P98 represent the 2% and 98% percentiles. CV denotes the coefficient of variation (

\sigma/\mu

). DRCR denotes dynamic range compression ratio (

(P_{98}-P_{2})/\mu

).

Fig. 26 : Signed-log encoding and its clipping cost. (a) Transfer functions using the illustrative 98th-percentile scale

S_{\mathrm{ref}}\approx 45.9

m. (b) Encoded residual-magnitude densities. (c) Derivative gain relative to linear scaling by the observed maximum (

\sim 730.3

m), not a spatial-resolution gain. (d) Meter-valued error from saturating extreme residuals. The complete study is retained to expose the tradeoff between central dynamic range and tail information loss.

Fig. 27 : Reference-scale selection under an average clipping budget. The sweep relates numerical gain

B(S)

to meter-valued mean absolute clipping error

C(S)

. The marked scale is approximately 76.8 m for a 0.1 m budget. The constraint limits an average over the inspected residual sample; it does not ensure small errors on individual cliffs or establish an instrument-precision threshold.

$S_{ref}$	Saturation Rate (Data lost to tails) (%)	Irreducible Clipping Error (MACE) (m)	Base Expansion Factor
98% (45.9)	1.9871%	0.39506	22.92
Optimal (76.4m)	0.3986%	0.0987	13.69

Appendix

TABLE IX : Study of the

S_{ref}

parameters based on the different scale using the constrained Pareto optimization routine or using the data’s 98% range.

Fig. 28 : Pipeline and criteria for filtering the dataset elements to ensure the dataset can serve as a training dataset for a neural network; the valid components are cached in a manifest file for faster access down the line. The first filter, the near-flat slope , removed patches where the average residual error from the detrended patch was close to zero, representing featureless DTM. The tile-merge seam aims to remove CCD seam artifacts and merged DTM, but it produces non-continuous transition edges along the seams, which throw neural networks off. TIN (Triangulated irregular network) flat plateau represents the potential faceted and manually interpolated areas, which are results of terrain that was very bland (low contrast) or deeply shadowed with low contrast and low signal, which causes the areas to only generally approximate the shape of the terrain [ 43 ] . Finally, low-coverage is an optional non-zero data percentage filter, if not handled as a parameter of the HiRISE patch sampler; see Appendix B .

Fig. 29 : Automated Seam Artifact Detection Pipeline. The proposed architecture identifies unnatural mosaicking boundaries through a three-stage process. Stage 1 computes unblurred step-gradients and calculates a statistical distribution shift (Cohen’s

d

) between adjacent terrain blocks for both the orthoimage and DTM. Stage 2 aggregates these pixel-level signals using multi-angle directional convolutions gated by the statistical shift, producing a base anomaly heatmap. Stage 3 suppresses false positives from natural topography (e.g., craters and dunes) by applying four geometric penalty multipliers: Span Ratio, Sparsity, Piecewise Linearity (via Hough Transform), and Spatial Isolation. The composite score ranks seam candidates; this diagram does not establish detection accuracy.

Fig. 30 : Diagnostic visualization grid for automated seam artifact detection. Each row represents an evaluated scan region comprising eight analytical panels: (1) Ortho+line : RGB orthomosaic with the best-scoring candidate line superimposed; (2) DTM+line : Digital Terrain Model (DTM) with candidate line; (3) Ortho grad : Sharp gradient magnitude utilizing 1D difference kernels to emphasize exact pixel step edges; (4) Seam response : Heatmap of the directional seam scores; (5) Cohen’s

d

: Heatmap representing localized distribution shifts between terrain separated by the candidate line; (6) Span (CCs) : Connected components map of the core signal, utilized to evaluate structural span and filter isolated anomalies; (7) Linearity : Probabilistic Hough transform lines overlaid on a sparsity hot-mask to quantify piecewise linearity; and (8) Per-Angle : A polar plot illustrating the maximum response as a function of orientation angle

\theta

. Textual annotations (left) detail the calculated structural multipliers, composite score, span ratio, sparsity, raw seam score, Cohen’s

d

, and optimal angle

\theta

, classifying the sample as HIGH or LOW by the heuristic threshold. These labels are not calibrated probabilities.

Fig. 31 : Topographic Map of Mars [ 34 ] through the view of Mercator Projection with longitude range of

-180^{\circ}

to

180^{\circ}

and latitude range of approximately

-57^{\circ}

to

57^{\circ}

. The map was generated by the U.S. Geological Survey, and on the map were overlaid all the HiRISE sample locations in blue within that topographical range, and in red is the subsampled dataset used as the scope of the project. The sub-sampled dataset was selected by filtering samples with a bounding-box of

[-120^{\circ},-30^{\circ},150^{\circ},30^{\circ}]

.

Fig. 32 : Full DTM pipeline architecture overview of the Mars DTM HiRISE dataset loader and sampler

Fig. 33 : Diagnostic visualization of the smooth diffusion infilling process prior to VAE optimization. Each row illustrates a single sample containing missing or invalid data, processed across five comparative panels: (1) Confidence Mask : A binary representation delineating valid observations from void regions; (2) Masked Ortho : The original RGB orthomosaic with invalid regions explicitly masked; (3) Smooth Fill Ortho : The resulting orthomosaic after applying a Gaussian Markov Random Field (GMRF) conditional solve, which smoothly diffuses boundary pixels to infill the voids; (4) Masked DTM : The corresponding Digital Terrain Model (DTM) with missing elevation data isolated; and (5) Smooth Fill DTM : The elevation map reconstructed via the identical GMRF sparse linear solve. The fill reduces zero-padding discontinuities, but the original mask still identifies unobserved values and the panels retain visible failure cases.

Fig. 35 : Intermediate fields for the pixel-space objectives. Each row shows an archived sample: (a) input orthoimage; (b) stereo-reference DTM; (c) gradient magnitude at the processed grid scale; (d) gradient magnitude after

4\times

pooling; and (e) log-magnitude Fourier spectrum. Pooling exposes broader spatial variation, while the spectrum summarizes frequency content without feature location. The optional FFL weights large spectral discrepancies rather than exclusively high frequencies; it is disabled in the reported run. These diagnostic fields document the operators, not measured improvements from each loss.

Fig. 36 : Huber loss diagnostic. Columns, left to right: (1) real orthoimage (percentile-stretched, masked); (2) GT DTM (terrain colormap,

[-1,1]\!\to\![0,1]

); (3) predicted DTM; (4) signed error

d-\hat{d}

on a symmetric RdBu_r colormap, scale

\pm v_{\max}

reported in the column title; (5) absolute error

|d-\hat{d}|

on a magma colormap, displayed up to

2\delta

; (6) per-pixel Huber loss

\rho_{\delta}(\cdot)

, displayed up to

\delta^{2}

. The

L_{2}

-to-

L_{1}

transition is visible by comparing columns (5) and (6): large residuals enter the linear branch of the penalty. Display saturation is separate from the loss, which continues to grow linearly.

Fig. 37 : Laplacian diagnostic. Columns show orthoimage, reference relief, reference Laplacian, predicted relief, predicted Laplacian, and absolute Laplacian error. Shared display limits permit spatial comparison of the derivative fields. Interpreting red/blue as bowl/peak requires checking the stencil sign; the standard positive-neighbor Laplacian is positive for a bowl. These are derivatives of the represented relief field, not independently calibrated physical curvatures.

Fig. 38 : Ordinal ranking diagnostic. Columns: (1) orthoimage; (2) GT DTM; (3) predicted DTM; (4) pair-sample panel — 250 randomly-drawn pairs overlaid on a dimmed ortho. Green segments mark pairs whose GT ordering is unambiguous (

|\Delta\hat{d}|>\tau

) and whose prediction agrees ; red segments mark ordering violations ; grey segments are ambiguous pairs (

|\Delta\hat{d}|\leq\tau

) which are excluded from the loss. The per-row caption reports the ordered/total pair count and the violation rate; (5) local violation rate on a coarse

32\!\times\!32

grid (magma colormap). Hot spots reveal where the model systematically predicts the wrong relative ordering — typically crater rims and shadowed regions — and are directly actionable for dataset reweighting or active-learning sample selection. Legend at the foot of the figure.

Fig. 39 : Distribution of fitted illumination parameters. (a) Unit directions after the upper-hemisphere correction; color encodes fitted directional gain and marker size ambient offset. (b) Polar view of the direction distribution. (c) Gain versus offset with a fitted trend. The hemisphere support is imposed by the estimator, and direction coverage is not proof of unbiased recovery. Positive gain–offset association can arise from exposure or the fitting procedure as well as atmospheric effects [ 30 , 79 ] . Independent metadata comparison is needed to validate the fitted directions and physical interpretation.

Fig. 40 : Sensitivity of image-loss components to direction perturbation. The mean Pearson, SSIM, and combined losses increase over the plotted angular range. Bands are labeled 95% intervals in the archived export; the resampling unit and procedure must accompany the experiment. Pearson varies more strongly than SSIM in this plot. The comparison characterizes the two components under fixed geometry; it does not prove that both are necessary for reconstruction accuracy.

Fig. 41 : Effect of illumination perturbations on one patch. Columns show the observed image, surrogate render, signed z-residual, residual density, and a distribution summary. Rows use the fitted reference direction, a

+45^{\circ}

azimuth shift, and a

+45^{\circ}

elevation shift. The baseline is not independently measured solar ground truth. The retained RMSE/MAE annotations quantify image residuals; they are not terrain errors or proof of a unique inverse.

Fig. 42 : Retained negative-curvature case. This archived tangent-plane contour accompanies the sample classified as a saddle. The reported negative eigenvalue,

-6.21\times 10^{-3}

, cannot be dismissed as numerical error merely because it is small. A reliable classification requires stationarity checks, finite-difference or autodifferentiation consistency, and sensitivity to numerical tolerances. The example qualifies the broader local-curvature result.

Fig. 43 : Local tangent-plane diagnostics for 794 patches. Top row: Hessian-eigenvalue empirical CDFs, tolerance-based classification (793 minima, one saddle), and log condition numbers for positive-curvature cases. Bottom row: gradient norms, projections of the dominant eigenvector on the azimuth/elevation basis, and angular differences between fitted reference directions and empirical optima. The original export reports a 99.9% minimum classification with bootstrap interval

[99.6,100.0]\%

, median gradient norm near

1.5\times 10^{-3}

, and condition-number mode near 1.8. These describe the inspected sample and numerical criteria. The unit quarter-arc in the projection panel is a basis identity. Angular disagreement, with a mode near

10^{\circ}

, compares two fitting objectives, not an estimate against independent solar ground truth.

Fig. 44 : Mean illumination-loss landscape and uncertainty of the mean. The left panel uses a Lambert azimuthal equal-area projection centered on the fitted reference direction. Its displayed loss scale spans approximately 0.744–1.014. The right panel’s SEM scale spans approximately 0.007625–0.008825. The smooth average basin characterizes the sampled terrains and directions; SEM measures uncertainty of an average, not scene-to-scene ambiguity. Neither panel guarantees a unique per-scene optimum or global optimizer convergence.

Fig. 45 : Exploratory embedding and parameter probes. The archived embedding is colored by angular error (left) and standardized gain/offset variation (right). The displayed angular probe has

R^{2}=0.917

; nuisance probes have

R^{2}=-0.034

and

-0.115

. These scores summarize the specified probes, not a general proof of statistical independence. Reproduction requires the input feature construction and terrain-disjoint probe protocol; an embedding of parameters and an embedding of residual features are different experiments.

Fig. 46 : Intermediate shading and comparison fields. The complete diagnostic grid retains the input orthoimage, reference DTM, normal field, two render computations, and the z-normalized render/image/residual views. Reference normals are used for this inspection; prediction normals supply the training objective. The absolute z-residual is a diagnostic image, not the local SSIM map itself. Bright residuals indicate spatial disagreement that may arise from terrain, reflectance, shadows, registration, or illumination; global normalization removes only positive affine exposure changes under the stated assumptions.

Loss	Supervises	Scale	Noise-robust?	Section
$\mathcal{L}_{\text{FM}}$	latent velocity	all latent	partial	D-B
$\mathcal{L}_{\text{huber}}$	absolute depth	values	reduced tail influence	D-C
$\mathcal{L}_{\text{norm}}$	local slope	mid–high	partial	D-D
$\mathcal{L}_{\text{grad}}$	slope multi-scale	low–high	offset-invariant	D-E
$\mathcal{L}_{\text{lap}}$	curvature	high	partial	D-F
$\mathcal{L}_{\text{FFL}}$	frequency bands	adaptive bands	no	D-G

Appendix

TABLE X : Role of each loss term. “Scale” denotes the spatial frequency each term primarily supervises.

Fig. 47 : Archived flow-matching and loss architecture. The complete diagram preserves the latent encoding, interpolant, velocity prediction, endpoint decoding, eight objective branches, and weighted optimization. VAE parameters are frozen while the decoder remains differentiable with respect to its input. FFL is optional and has zero reported weight. The revised Equation ( 28 ) includes source augmentation and logit-normal time sampling. Geometric losses compare VAE-decoded fields. The archived run shades normalized decoded relief; the current code first inverts the signed logarithm. The recovered run configuration governs the reported results, while the diagram retains historical schematic annotations.

Metric	Direction	Value
Differential geometry diagnostics
Normal angular error ( $\degree$ )	$\downarrow$	$0.380\pm 0.206$
Slope RMSE ( $\degree$ )	$\downarrow$	$0.489\pm 0.303$
Curvature RMSE ( $\times 10^{-3}$ )	$\downarrow$	$2.93\pm 1.79$
Depth-Boundary F-score	$\uparrow$	$0.075\pm 0.147$
Structural, image, and spectral diagnostics

Appendix

TABLE XI : Recovered MarsFM test export from run 88c8 , checkpoint 9476: mean

\pm

sample standard deviation over 1012 rank-zero patch records, with 16 Euler steps and one realization. Relief errors follow affine alignment in normalized signed-log space. The default positive-reference mask restricts scoring support. Angular values are grid-coordinate diagnostics, not physical-slope errors. Numerical rounding follows the recovered CSV.

Fig. 48 : Prediction triptych, validation step 9827. Left: HiRISE RED input; center: predicted relief; right: stereo-derived reference. The common color scale reveals broad terrain correspondence, amplitude differences, and smoothing. The historical meter annotation represents normalized relief in this export, not physical height. Exporter: plot_prediction_triptych .

Fig. 49 : Directional cross-sections, step 9827. Solid reference, dashed prediction, and shaded difference along four central slices. Broad residual regions remain in addition to errors at sharp transitions. The retained elevation-unit labels require export verification. Exporter: plot_cross_sections .

Fig. 50 : RGB normal fields, step 9827. Prediction (left) and reference (right), with channels encoding

(n_{x},n_{y},n_{z})

. The prediction contains less fine-scale variation. Angles refer to the operator’s coordinate convention until metric conversion is verified. Exporter: plot_normal_maps .

Fig. 51 : Slope diagnostic, step 9827. Input image, reference slope, and absolute slope difference. The image shows spatially distributed disagreement; the retained degree labels describe the exported operator, not independently verified physical slopes. Exporter: plot_slope_error_map .

Fig. 52 : Archived shading comparison, step 9827. Input orthoimage, predicted-relief render, reference-relief render, and absolute z-normalized residual. The last panel is labeled as a loss view in the original asset, but it is not itself an SSIM map. Reference rendering provides a comparison under the same surrogate; it is not an accuracy bound. Exporter: plot_lunar_lambert_comparison .

Fig. 53 : Radial power spectra, step 9827. Solid reference, dashed prediction, and dotted

f^{-2.5}

comparator. The prediction has less power over much of the plotted range and a small excess at the extreme tail. The original frequency-axis units require correction from the FFT metadata before making physical-scale claims. Exporter: plot_radial_psd_curves .

Fig. 54 : Edge and Laplacian diagnostics, step 9827. Blue: reference-only edges; red: prediction-only; purple: overlap. The other panels show the predicted Laplacian and its absolute disagreement with the reference. Unmatched edges require investigation and are not automatically evidence of super-resolution. Exporter: plot_geomorphometric_analysis .

Fig. 55 : Absolute residual and marginal means, step 9827. Main panel: pixelwise prediction–reference error. Right and bottom: row and column means. The summary values are retained, but the meter labels conflict with the recorded normalized-relief export. Exporter: plot_error_heatmap .

Fig. 56 : Prediction–reference hexbin scatter, step 9827. The figure reports

R^{2}=0.2832

and shows a compressed predicted range relative to the dashed identity line. Verify axis units and alignment upon export. Exporter: plot_elevation_scatter .

Fig. 57 : Four archived metric distributions. Panels: RMSE, AbsRel,

\delta_{1}

, and normal angular error, with per-patch points and reported summary values. Ratio metrics require clarification for signed relief; the panel is not evidence of boundary-F-score bimodality. Exporter: plot_metric_distributions .

Fig. 58 : Decoded flow evolution, step 9827. Five states at

t=0,0.25,0.5,0.75,1

followed by the reference DTM. The changing appearance illustrates the image-to-relief transition, not a direct measurement of latent linearity. Exporter: plot_flow_evolution .

Fig. 59 : Archived step-count versus RMSE study. The horizontal scale is linear, and one step has the lowest RMSE. The original “zero-shot” annotation is inconsistent with the reported fine-tuned model and must not be read as a zero-shot experiment. The figure is preserved pending regeneration with corrected terminology, NFE convention, and band definition. Exporter: plot_timestep_ablation .

Fig. 60 : Validation metrics during one training run. RMSE, photo-consistency, and

\delta_{1}

trajectories, with checkpoint 9476 marked. These are validation metrics, not eight component-loss curves. Their improvement does not isolate a curriculum or individual objective contribution. Exporter: plot_convergence_curves .

Fig. 61 : Learned blend trajectory.

L

remains near its 0.5 initialization and is approximately 0.490 at the selected checkpoint. The trace shows modest optimization change; it does not establish a physical reflectance measurement or a global conditioning guarantee.

Fig. 62 : Sampling variability,

N=8

, step 9827. Input orthoimage, ensemble-mean relief, and pixelwise variance. The original title uses “epistemic” terminology, but fixed-weight sample variance alone does not establish that interpretation or calibration. The recorded normalized-relief pipeline does not justify the displayed physical variance units. Exporter: plot_uncertainty_map .

Euler steps	1	2	4	8	10	20
RMSE	$0.0935\pm 0.0819$	$0.0944\pm 0.0834$	$0.0938\pm 0.0819$	$0.0948\pm 0.0834$	$0.0957\pm 0.0849$	$0.0951\pm 0.0837$
Normal error ( ∘ )	$0.363\pm 0.187$	$0.364\pm 0.187$	$0.366\pm 0.189$	$0.368\pm 0.191$	$0.368\pm 0.191$	$0.369\pm 0.192$
Slope RMSE ( ∘ )	$0.480\pm 0.293$	$0.485\pm 0.302$	$0.482\pm 0.296$	$0.483\pm 0.294$	$0.477\pm 0.288$	$0.478\pm 0.283$
Curvature RMSE ( $\times 10^{3}$ )	$2.76\pm 1.80$	$2.80\pm 1.87$	$2.79\pm 1.84$	$2.78\pm 1.80$	$2.75\pm 1.75$	$2.75\pm 1.69$
Boundary F-score	$0.072\pm 0.135$	$0.083\pm 0.153$	$0.085\pm 0.154$	$0.079\pm 0.148$	$0.086\pm 0.157$	$0.085\pm 0.160$
Topographic MS-SSIM	$0.552\pm 0.163$	$0.547\pm 0.163$	$0.546\pm 0.164$	$0.542\pm 0.159$	$0.540\pm 0.160$	$0.538\pm 0.160$

Steps	1	2	4	8	10	20
RMSE	0.0935	0.0944	0.0938	0.0948	0.0957	0.0951
$\delta_{1}$ (%)	23.56	23.18	23.39	23.12	22.92	22.90

MarsFM: Shading-Regularized Flow Matching for Martian Relief Estimation

Abstract

Figures & tables

Appendix

Remote Sensing Image Understanding

Benchmarking stereo reconstruction for 3D printable Martian terrain models

MARTIAN: A Rendering Framework for Aerial Mars Imagery from HiRISE Orbital Data

MarsRecon: Self-Supervised and Multimodal Surface Representations for Mars

Abstract

Remote Sensing Image Understanding

Explore similar work

Benchmarking stereo reconstruction for 3D printable Martian terrain models

MARTIAN: A Rendering Framework for Aerial Mars Imagery from HiRISE Orbital Data

MarsRecon: Self-Supervised and Multimodal Surface Representations for Mars