PaperlayerView exact revision on arXiv

Source-linked AI summary

PCFlow: Physics-Conditioned Flow Matching for GPR B-Scan Image Synthesis

Zhijie Shen, Chenchen Fu, Xuanhao Chang, Hongtao Bai, Lili He

arXiv:2609.07300v1cs.LG

TL;DR

Connecting physical scene descriptions to radar-response geometry remains a challenge in GPR B-scan synthesis. PCFlow uses a Maxwell-informed physical condition field to guide conditional flow matching. On in-distribution and out-of-distribution tests, it achieves more accurate response geometry than generative baselines while maintaining strong image fidelity.

  • Problem

    Connecting scene-domain physical descriptions to B-scan response geometry remains a key conditioning challenge in GPR forward modeling.

    Source: §I · p. 1

  • Method

    PCFlow builds a dense Maxwell-informed condition field from physical scene cues and response-domain priors to guide conditional flow matching for GPR B-scan synthesis.

    Source: §I · p. 2

  • Results

    Tests on both in-distribution and out-of-distribution sets show more accurate response geometry than GAN, diffusion, ControlNet-style, and text-guided baselines, with strong PSNR, SSIM, and LPIPS image fidelity.

    Source: §I · p. 2

  • Takeaways & Limitations

    The findings support geometry-aware evaluation of GPR synthesis, since moderate PSNR and SSIM can coexist with response-mask IoU close to zero for baselines.

    Source: §VI · p. 9

    The closed-form travel-time prior is tailored to single-pipeline configurations in homogeneous or weakly heterogeneous backgrounds, simplifying real-world subsurface clutter.

    Source: §VI · p. 9

Abstract

from arXiv · show

Ground-penetrating radar (GPR) B-scan image synthesis is important for data augmentation, algorithm validation, and simulation acceleration, yet generating radargrams with both visual realism and physical consistency remains challenging. Existing learning-based generative models often emphasize visual appearance but provide limited control over response geometry. In this paper, we propose PCFlow, a physics-conditioned flow matching framework for fast GPR B-scan image synthesis. The core of PCFlow is a Maxwell-informed dense physical condition field constructed from the parameterized physical model used for electromagnetic simulation, including material properties, target geometry, propagation cues, and response-domain priors. This condition field provides an interpretable interface between physical scene parameters and radar response geometry, and guides conditional flow matching in the VAE latent space toward physically feasible generation paths. We evaluate PCFlow on a gprMax-based buried-pipeline dataset with both in-distribution and out-of-distribution test cases. Experimental results show that PCFlow generates images with more accurate response geometry and high visual fidelity, demonstrating its effectiveness for controllable and physically faithful radar image synthesis.

I. INTRODUCTION … A. Problem Formulation and Overview

PCFlow targets fast, physically faithful GPR B-scan synthesis from parameterized simulation scenes by encoding dense radar-aligned physical conditions and using them to guide latent flow matching.

I. INTRODUCTION

  • GPR simulation provides physically reliable responses but makes large parameter sweeps costly, while learned radargrams can look plausible yet lack control from parameterized scenes over target-response geometry.Fast, physically faithful synthesis supports data augmentation, algorithm validation, simulation acceleration, and learning-based subsurface interpretation.Source: §I · p. 1
  • PCFlow addresses the challenge of bridging scene-domain material and geometry properties with radargram-domain responses through an interpretable, radar-aligned condition.The condition combines physical cues with response priors such as approximate travel-time curves, attenuation patterns, and hyperbola anchors.Source: §I · pp. 1–2
  • PCFlow constructs a deterministic Maxwell-informed dense condition field and uses conditional flow matching in VAE latent space, with Physics-SPADE modulation and classifier-free guidance.The encoder integrates material, geometry, propagation, and response-domain priors; Physics-SPADE and guidance strengthen condition injection and condition-specific generation, especially under distribution shifts.Source: §I · p. 2
  • On in- and out-of-distribution tests, PCFlow achieved more accurate response geometry than GAN, diffusion, ControlNet-style, and text-guided baselines while maintaining strong image fidelity.The gprMax buried-pipeline dataset contains approximately 800 condition–response pairs; image fidelity was assessed with PSNR, SSIM, and LPIPS.Source: §I · p. 2

A. GPR B-Scan Image Synthesis and Data Augmentation

  • Prior GPR synthesis methods often provide limited explicit control over hyperbola position, burial depth, aperture, and attenuation; PCFlow encodes physical cues and response priors into dense radargram-aligned conditions.The related work describes prior approaches as GAN-based, image-to-image, or diffusion-based generation and augmentation methods.Source: A. GPR B-Scan Image Synthesis and Data… · p. 2

B. Conditional Generative Models and Physical Control

  • Conditional flow matching provides continuous probability transport from a simple prior and deterministic velocity-field integration, while PCFlow adds dense Maxwell-informed conditioning, Physics-SPADE, and classifier-free guidance.The condition field and modulation inject scene quantities and hyperbola priors into the velocity field; guidance amplifies condition-specific transport at inference.Source: B. Conditional Generative Models and… · p. 2

A. Problem Formulation and Overview

  • Given gprMax condition S—including permittivity, conductivity, geometry, acquisition, simulation domain, and temporal window—PCFlow models pθ(IB | S) through condition construction and latent-space probability transport.The encoder creates dense radargram-aligned conditions, downsamples them to VAE latent resolution, and guides transport from Gaussian noise toward the target B-scan latent representation.Source: A. Problem Formulation and Overview · pp. 2–3

B. Preliminary: Conditional Flow Matching · C. Maxwell-Informed Condition Encoder

PCFlow learns a conditional latent velocity field along a linear flow path from Gaussian noise to GPR image latents. Its deterministic Maxwell-informed encoder supplies dense physical and response-domain conditions intended to reduce transport ambiguity and guide physically consistent generation.

B. Preliminary: Conditional Flow Matching

C. Maxwell-Informed Condition Encoder

1) From physical scene to response-domain geometry:

PCFlow addresses the mismatch between localized scene geometry and hyperbolic radar responses by encoding an approximate response-domain prior. A simplified propagation model specifies the dominant travel-time geometry, while conditional flow matching learns the remaining radargram details.

From physical scene to response-domain geometry

  • PCFlow makes the nonlinear mapping from scene geometry and medium velocity to radar-response travel time explicit through approximate response-domain priors.Scene masks or material maps alone leave the generator to infer this mapping.Source: 1) From physical scene to… · p. 4
  • A delayed-wavelet integral relates response location to material contrast and propagation time, then reduces to the earliest strong pipeline scattering along a dominant two-way travel-time curve.This first-order geometry prior is paired with learned waveform details, interference, later propagation effects, and higher-order scattering.Source: 1) From physical scene to… · p. 4
  • Encoding the approximate travel-time curve in Cresp reduces the burden of learning the scene-to-response mapping, allowing the conditional flow model to focus on residual radargram appearance.A misplaced dominant curve causes image-domain discrepancy around the main reflection.Source: 1) From physical scene to… · p. 4
  • For a buried pipeline, PCFlow approximates two-way travel time using transmitter and receiver positions, burial depth, and homogeneous-background velocity, then maps it to a hyperbola prior.Apex, attenuation, and target-strength priors derive from the travel-time minimum, propagation loss, and material or impedance contrast.Source: 1) From physical scene to… · p. 4

2) Channel instantiation: · D. Conditional Velocity Network and Sampling · 1) SPADE-based physical modulation:

PCFlow constructs a 26-channel Maxwell-informed condition field and injects it into a latent conditional velocity network at multiple scales, combining physical scene and response priors with SPADE modulation.

2) Channel instantiation:

  • The condition field has 26 channels, combining normalized radargram coordinates with scene-domain material, propagation, discontinuity, and pipeline-geometry cues.These cues describe wave velocity, attenuation, phase accumulation, reflection-related discontinuities, and target shape.Source: 2) Channel instantiation · p. 4
  • Its six response-domain channels encode a hyperbola heatmap, apex, travel-time curve and residual, curve attenuation, and target-strength prior.Derived from an approximate two-way travel-time model, they anchor the dominant B-scan response with coarse physical cues.Source: 2) Channel instantiation · p. 4
  • Although instantiated for buried pipelines, the condition-construction principle can extend to other GPR tasks by selecting task-specific physical priors and response mechanisms.Examples include cavity detection, layered-medium identification, and multi-target recognition.Source: 2) Channel instantiation · p. 4
  • Channel-aware downsampling average-pools smooth dense fields and max-pools sparse or localized fields to preserve continuous trends and sharp structures.This average–max strategy is used when resizing the condition field to the VAE latent resolution.Source: 2) Channel instantiation · pp. 4–5

D. Conditional Velocity Network and Sampling

  • The velocity field uses a UNet, with sinusoidal timestep embeddings,, encoder–bottleneck–decoder skips, residual blocks, and low-resolution self-attention.Two residual blocks are used at each resolution level before downsampling or upsampling; the timestep projection is added in each block.Source: D. Conditional Velocity Network and… · p. 5

1) SPADE-based physical modulation:

  • A condition pyramid feeds resized fields to residual blocks, while SPADE injects spatially varying physical guidance into intermediate features at multiple resolutions.This repeated location-dependent guidance addresses the limits of input-only conditioning for amplitude, polarity, and hyperbolic reflection structures tied to antenna position and travel time.Source: 1) SPADE-based physical modulation · p. 5
  • PCFlow (PL) predicts modulation parameters from all condition channels, enabling flexible cross-channel interactions and often strong fidelity near the training distribution.PCFlow (GR), or Physics-SPADE, separately encodes physics-related channel groups and fuses them to predict modulation parameters, aiming to preserve geometry and response-prior information more robustly under distribution shifts.Source: 1) SPADE-based physical modulation · p. 5

2) Classifier-free guided sampling: … C. Implementation Details

PCFlow uses classifier-free guidance to steer latent sampling toward the encoded simulation condition. The experiments describe a gprMax pipeline dataset, baseline comparisons, and implementation settings.

2) Classifier-free guided sampling:

  • Classifier-free guidance trains conditional and unconditional velocity estimates through random condition dropping, then combines them with a guidance scale s ≥ 1 during latent sampling.Sampling starts from z0 ∼ N(0, I), integrates the guided velocity from t = 0 to t = 1, and decodes the final latent with DVAE.Source: 2) Classifier-free guided sampling · p. 5

A. Dataset and Splits

  • The gprMax dataset contains approximately 800 paired configuration files and direct-wave-removed B-scans spanning four soil types and steel or PVC pipes filled with air or water.Pipe radii, lateral positions, and burial depths vary to produce different response geometries and reflection strengths.Source: A. Dataset and Splits · p. 5
  • The split uses dry sand, wet sand, and wet soil for ID tests and holds out high-conductivity dry clay for OOD testing, with group-aware scene splitting to avoid leakage.It contains 443 training, 88 validation, 70 ID test, and 199 OOD test samples.Source: A. Dataset and Splits · p. 5

B. Baselines

  • Baselines share the same split and validation-based hyperparameter selection; Pix2Pix takes the same dense physical condition field as PCFlow.Conditional DDPM uses encoded physical attributes, while its physics-conditioned variant adds a hyperbola-prior-weighted reconstruction constraint.Source: B. Baselines · pp. 5–6
  • Other controllable diffusion baselines include DDBM’s spatial prior image and a ControlNet-style model that maps the dense condition field to a single-channel control image.DDBM’s prior includes approximate hyperbola shape, apex location, and material-soil attributes.Source: B. Baselines · p. 6
  • Stable Diffusion 2.1 baselines use prompt templates from physical parameters and include zero-shot, UNet LoRA, joint UNet-text-encoder LoRA, and full UNet fine-tuning variants.The variants include LoRA tuning and DataDream-style joint LoRA.Source: B. Baselines · p. 6

C. Implementation Details

  • Implementation uses a pretrained SDXL VAE, a four-channel latent UNet, 0.1 condition dropout, and a Heun sampler with 50 steps and guidance scale 2.5.Models train for 10k iterations with batch size 8 and learning rate 1 × 10^-4; the EMA checkpoint is used for evaluation.Source: C. Implementation Details · p. 6

D. Evaluation Metrics and Results · 1) Metrics: · 2) Main results:

PCFlow is evaluated for both image fidelity and physical response consistency, with results showing that visual quality alone does not establish physically meaningful radar responses.

1) Metrics:

  • Image fidelity is measured with PSNR, SSIM, and LPIPS, while geometry-aware metrics assess localization, depth, curve consistency, and aperture shape of the dominant early hyperbola.The geometry metrics include apex and ridge-curve errors, opening error, and response-mask IoU; apex and curve errors are in pixels and opening errors in 10^-3 units.Source: 1) Metrics · p. 6
  • Geometry-aware errors compare ground-truth and generated response masks, ridge curves, and apex locations after robust response extraction.The reported metrics include response-mask IoU, with higher IoU indicating better response-region alignment.Source: 1) Metrics · p. 6

2) Main results:

  • PCFlow achieves substantially lower apex and curve errors and higher mask IoU than general generative baselines on both ID and OOD test sets.Baselines often produce near-zero mask IoU; qualitative comparisons show PCFlow's physically consistent responses, with PCFlow (GR) showing better OOD geometry under high-conductivity shifts and PCFlow (PL) strong visual fidelity.Source: 2) Main results · p. 7, 1) Metrics · p. 7, Figure 3 · p. 7
  • PCFlow achieves strong image-level quality, especially on the ID test set.The reported image-fidelity evaluation covers both ID and OOD test sets.Source: 2) Main results · p. 7, Table IV · p. 7
  • Some baselines attain moderate PSNR or SSIM yet fail to produce physically meaningful response regions, so conventional image metrics alone are insufficient for evaluation.The comparison of image-level and geometry-aware metrics reveals this mismatch.Source: 2) Main results · p. 8
  • Training visualizations show dominant response geometry emerging early, while later iterations refine secondary returns and radargram appearance.Weak target responses appear at 0.5K iterations, the dominant hyperbola is clear by 1K, and secondary reflection branches and multi-return structures form around 2K–2.5K.Source: 2) Main results · p. 8, Figure 4 · p. 8

V. ABLATION ANALYSIS · VI. DISCUSSION · VII. CONCLUSION

The ablations identify response-domain priors in PCFlow’s Maxwell-informed condition field as central to preserving OOD response geometry, while the discussion distinguishes physical reliability from visual fidelity and outlines the framework’s scope. The conclusion summarizes PCFlow’s conditioned latent flow-matching approach and its future directions.

V. ABLATION ANALYSIS

  • Removing response-domain priors increases OOD curve error from 3.57 to 106.02 and drops Mask IoU from 0.74 to 0, demonstrating their importance for geometry consistency.The ablation analysis identifies the Maxwell-informed condition field as the main source of PCFlow’s physical consistency.Source: §V · p. 9
  • Response-prior-only retains competitive OOD geometry but weaker image fidelity, whereas material-only and geometry-only fail to localize or guide hyperbolas effectively.The ablations compare grouped and plain modulation with variants retaining different scene-domain channels and response priors; Fig. 5 illustrates the OOD geometry differences.Source: §V · pp. 8–9, Figure 5 · p. 9
  • PCFlow (PL) gives the strongest image fidelity, PCFlow (GR) the best OOD curve consistency, and removing SPADE weakens both geometry and image metrics.These results indicate that the condition field, especially response-domain priors, matters more than any single architectural choice.Source: §V · p. 9

VI. DISCUSSION

  • Moderate baseline PSNR and SSIM can coexist with response-mask IoU near zero, while PL favors in-distribution sharpness and GR stabilizes geometry in high-conductivity OOD scenarios.Grouped physical modulation is beneficial when attenuation, material contrast, and response priors vary jointly outside the training distribution.Source: §VI · p. 9
  • Response-domain priors provide soft radargram-aligned travel-time and coarse-amplitude anchors, while the velocity network learns scattering, multipath interference, phase changes, and antenna coupling.The condition field guides latent-space transport toward physically consistent manifolds without analytically replacing full-wave simulation or prescribing the final radargram.Source: §VI · p. 9
  • The closed-form travel-time prior assumes single pipelines in homogeneous or weakly heterogeneous media; extensions include multiple travel-time curves, numerical fields, layered or cluttered scenes, and real GPR data.Ray-based tracking or numerically estimated travel-time fields could replace the hyperbolic approximation for complex stratified or laterally varying media.Source: §VI · pp. 9–10

VII. CONCLUSION

  • PCFlow combines scene cues and response priors in an interpretable dense condition field, using SPADE modulation and classifier-free guidance to condition VAE-latent flow matching.Experiments on gprMax data show improved response-geometry consistency without sacrificing visual fidelity, particularly in high-conductivity OOD scenarios; future work includes layered media, cluttered multi-target scenes, real measurements, and adaptive guidance.Source: §VII · p. 10

APPENDIX A SUPPLEMENTARY DERIVATIONS FOR THE MAXWELL-INFORMED ENCODER · A. Maxwell-Related Propagation Quantities · B. From Born Scattering to a Delayed-Wavelet B-Scan Model

Appendix A supplements the Maxwell-informed condition encoder with propagation-quantity and channel motivations, then derives the delayed-wavelet B-scan model from a first-order Born representation. It specifies the approximations connecting electromagnetic contrast and two-way travel time to radargram response localization.

APPENDIX A SUPPLEMENTARY DERIVATIONS FOR THE MAXWELL-INFORMED ENCODER

  • Appendix A provides derivation paths and channel-level implementation details for the Maxwell-informed encoder, complementing definitions already given in the main text.It refers to the delayed-wavelet response model, single-pipeline reduction, shift-based travel-time error relation, and approximate pipeline two-way travel-time curve.Source: Appendix A · p. 12

A. Maxwell-Related Propagation Quantities

  • For non-magnetic media, relative permittivity controls wave velocity and phase accumulation, conductivity controls propagation loss, and larger permittivity shifts target responses later.The appendix assumes μ ≈ μ0.Source: A. Maxwell-Related Propagation… · p. 12
  • The derived material quantities motivate material, velocity, attenuation, phase, impedance-edge, and response-strength channels in the condition field.The appendix identifies wave impedance as an additional propagation quantity.Source: A. Maxwell-Related Propagation… · p. 12

B. From Born Scattering to a Delayed-Wavelet B-Scan Model

  • The derivation starts from a first-order Born scattering representation in which the background Green function and electromagnetic contrast describe the scattered response.The representation is stated for a transmitter and receiver pair in the frequency domain.Source: B. From Born Scattering to a… · p. 12
  • Under a high-frequency approximation, the Green-function amplitude collects geometric spreading, attenuation, antenna response, and polarization effects, while phase follows transmitter-to-scatterer-to-receiver travel time.Substituting the approximate Green function into the Born representation yields a kernel governed by this two-way travel time.Source: B. From Born Scattering to a… · p. 12
  • For each acquisition position, source spectrum, antenna response, amplitude, and polarization terms are collected into H_a, with effective contrast χ(r) and two-way travel time τ_a(r).Source: B. From Born Scattering to a… · p. 12
  • Factoring H_a(r, ω) into source spectrum Ψ(ω) and slowly varying W_a(r) under a narrowband or weak-dispersion approximation yields a delayed time-domain wavelet plus residual terms.The inverse Fourier transform gives the time-domain wavelet ψ; ρ_a represents residual terms outside the first-order approximation.Source: B. From Born Scattering to a… · p. 12
  • Mapping acquisition position to horizontal B-scan coordinate and time to vertical coordinate yields a model in which radargram response localization depends on electromagnetic contrast and two-way travel time.Source: B. From Born Scattering to a… · p. 12

C. Single-Pipeline Response and Response-Domain Priors · APPENDIX B TRAVEL-TIME ERROR LOWER BOUND

The single-pipeline priors approximate the dominant response with a localized scatterer, using travel-time geometry and coarse path-length-dependent attenuation while leaving residual wave effects to the conditional network. Appendix B derives a local travel-time error lower bound under wavelet and residual-size conditions, explaining why a plausible-looking B-scan may still be physically inconsistent.

C. Single-Pipeline Response and Response-Domain Priors

  • A localized effective scatterer models the dominant pipeline response with a two-way travel-time curve, effective amplitude, and residual full-wave effects.This reduces the delayed-wavelet integral to the single-response model in (8).Source: C. Single-Pipeline Response and… · p. 12
  • The two-way travel-time prior follows from transmitter and receiver path lengths and effective background velocity, yielding a hyperbola with apex arrival time 2d/vbg for monostatic or near-monostatic scans.Source: C. Single-Pipeline Response and… · p. 12
  • A compact path-length-dependent attenuation and spreading term motivates the response attenuation prior, while the velocity network learns polarity, interference, antenna effects, finite-size scattering, and other residual details.The prior provides only coarse amplitude guidance.Source: C. Single-Pipeline Response and… · p. 12

APPENDIX B TRAVEL-TIME ERROR LOWER BOUND

  • Ignoring residuals, the response mismatch compares amplitude-scaled wavelets shifted by predicted and true travel times; for ψ∈H1(R) with ∥ψ′∥2>0, small shifts admit a positive local L2 bound.Source: Appendix B · p. 13
  • When the effective amplitude is bounded below on the main response support, the bound applies there, but residual full-wave effects weaken it in proportion to their norm unless they overwhelm the dominant shifted-wavelet component.Source: Appendix B · p. 13
  • A visually plausible B-scan can remain physically inconsistent if its dominant response curve is misplaced.Source: Appendix B · p. 13

APPENDIX C RICKER WAVELET CASE

The appendix verifies that Ricker-like GPR pulses satisfy the local shift-identifiability assumption: near the main lobe, temporal misalignment produces nonzero quadratic response error.

  • Ricker-like pulses have a nonzero, square-integrable derivative, so local temporal misalignment around the main lobe induces nonzero quadratic response error.Source: Appendix C · p. 13

APPENDIX D CONDITION CHANNEL SPECIFICATION

Appendix D specifies implementation details of the 26-channel Maxwell-informed condition field, including scalar normalization and pooling choices that preserve material trends and response anchors at latent resolution.

Appendix D Condition Channel Specification

  • Scalar physical fields are clipped to predefined ranges and linearly normalized to [0, 1].Source: Appendix D · p. 13
  • Response-domain channels are constructed from travel-time prior (11); Twin denotes the temporal window of the B-scan.Source: Appendix D · p. 13
  • Smooth dense fields use average pooling, while sparse or localized structures use max pooling to preserve material trends and sharp response anchors at VAE latent resolution.The 26 channels are summarized in Table VII.Source: Appendix D · p. 13

Related papers

Loading 2609.07300v1…