Multi-Level-Set-Based Physics-Driven Neural Network to Solve 3-D Inverse Scattering Problems

1 Introduction

• Existing approaches face challenges with sharp-interface preservation, computational cost, hyperparameter sensitivity, or generalization to unseen scattering scenarios.The introduction also notes that multi-object, multi-material level-set reconstruction remains challenging when target number, material distribution, and topology are unknown.Source: §1 · pp. 2–3
• Voxel-wise contrast representations do not explicitly exploit piecewise homogeneity, although practical scatterers often vary mainly across material interfaces.Level-set methods are motivated as a way to represent boundaries and handle topology changes.Source: §1 · p. 3
• LSPDNN represents unknown scatterers with multiple coordinate-dependent neural level-set functions rather than voxel-wise contrast values.The learned functions implicitly determine material interfaces and object topology.Source: §1 · p. 3
• Its soft-union model separates global object support from local material distribution using component weights and learnable complex-permittivity candidates.The formulation is intended to support multi-object and multi-material reconstruction in one differentiable framework.Source: §1 · p. 3
• Model-consistent TV regularizes soft material-region indicators, encouraging coherent regions while suppressing fragmented material assignments.The penalty is applied to region indicators rather than directly smoothing permittivity candidates.Source: §1 · p. 3
• Adaptive regularization-weight learning reduces reliance on manually selected loss weights by initializing weights after a data-only warm-up and learning them during optimization.The weights act as learnable precision parameters for balancing prior terms.Source: §1 · p. 4

2 Formulation of Inverse Scattering Problems

• The imaging setup illuminates the domain of interest with incident plane waves and measures scattered fields at receivers on a surrounding observation surface.The system is described as a 3-D imaging configuration.Source: §2 · p. 4
• The inverse problem is to reconstruct the unknown contrast distribution from measured scattered fields.The contrast is defined from relative permittivity as χ(r) = ϵr(r)−1.Source: §2 · pp. 4–5
• The objective combines a data-discrepancy term with regularization, and its hyperparameters are optimized during training or iteration.Ldata constrains data discrepancy; R imposes prior constraints on the desired solution.Source: §2 · p. 5

3 Inversion Scheme

• A coordinate-based Fourier-feature MLP predicts residual level sets, while back-propagation updates network weights, gates, and material candidates until the iteration limit.The normalized spatial coordinates are Fourier-embedded and passed through a four-layer MLP with 64 hidden units and Tanh activations.Source: §3 · pp. 5–6

3.1 Multi-Level-Set representation model

• The multi-level-set representation decouples object support from material assignment, combining components by soft union and weighting them with learnable complex permittivity candidates.The soft union forms global support without artificial accumulation in overlapping regions, enabling joint geometry and complex-contrast reconstruction for multi-object problems.Source: §3.1 · pp. 6, 8

3.2 Loss Function

• The loss combines physics-based data fidelity with TV on material-region indicators, support binarization, single-component impurity, and component-activation penalties.Applying TV to qunionpk promotes material-region continuity in multi-permittivity reconstruction.Source: §3.2 · pp. 8–9

3.3 Adaptive Regularization Weight Learning Strategy

• To reduce manual weight tuning, the solver performs a data-only warm-up, initializes regularization weights from that reconstruction state, and then learns them.The paper uses 100 warm-up iterations, during which measured scattered fields alone drive reconstruction.Source: §3.3 · p. 9
• After warm-up, weights are initialized from the current reconstruction so the total regularization contribution targets 5% of data-fidelity loss, equally divided among four terms.The initialization aims to make each weighted regularization term comparable in magnitude to the data-fidelity loss.Source: §3.3 · pp. 9–10
• An auxiliary term prevents learnable weights from collapsing to zero and makes the derivative with respect to log-weight approximately zero at initialization.This starts the weights in a balanced state before they adapt during reconstruction.Source: §3.3 · p. 10
• From iteration Jw onward, log-weights are optimized jointly with contrast-representation parameters and updated through gradients of the total objective.At each iteration, weights are recovered as λi = exp(αi) and substituted into the objective.Source: §3.3 · p. 10

3.4 Training Settings and Evaluation Indicator

• Training uses Adam with initial learning rates of 1 × 10^-2 for shape parameters and 1 × 10^-1 for complex permittivity candidates, halved every 150 epochs for 800 epochs.The solvers were trained on a workstation with 128 GB RAM, a 3.2 GHz i9 CPU, and an NVIDIA GeForce RTX 4090 GPU.Source: §3.4 · p. 10
• Reconstruction discrepancy is quantified by the relative error between the reconstructed relative-permittivity distribution and ground truth.The supplied passage identifies the indicator but does not include its formula.Source: §3.4 · p. 10

4.1 Reconstruction of Complex Digit-like Scatterers

• For digit-like scatterers, LSPDNN preserved curved boundaries and hollow regions and produced more uniform material values closer to ground truth than the baselines.The point-clouds method showed local discontinuities and lost fine boundary details; PDNN showed distorted boundaries, nonuniform materials, and severe loss of hollow or concave regions, especially in the imaginary part.Source: Figure 4 · p. 12, §4.1 · pp. 12–13

4.2 Reconstruction of Multi-Object Scatterers

• In multi-object Case 1, LSPDNN recovered the ring-shaped support and two isolated components, achieving 0.33% relative error and 285 s runtime versus PDNN’s 3.30% and 453 s.The reconstructions showed clearer boundaries and more uniform material values consistent with ground truth.Source: Figure 5 · p. 13, §4.2 · p. 15

4.1 Reconstruction of Complex Digit-like Scatterers

• For digit-like targets, LSPDNN reduced average relative error by 89.4% versus point-clouds and 92.1% versus PDNN.Table 1 reports relative error and inference time comparisons.Source: Table 1 · p. 14, §4.1 · p. 14
• The proposed solver’s relative errors were 0.09% for digit “2” (versus 4.90% for point-clouds and 7.24% for PDNN), and 0.43%, 0.24%, and 0.76% for digits “8”, “9”, and “7”, respectively.Source: §4.1 · p. 13
• LSPDNN reduced average runtime versus PDNN from 913.5 s to 594.8 s, a reduction of about 34.9%.The point-clouds method achieved inference below 1 s.Source: Table 1 · p. 14, §4.1 · p. 14

4.2 Reconstruction of Multi-Object Scatterers

• The point-clouds baseline performed poorly on the tested multi-object configurations, which were outside its digit-like training distribution.Its point-set representation and Chamfer-loss supervision do not explicitly enforce piecewise-homogeneous materials or separation of disconnected components, leading to biased material values, merged nearby objects, or missed small ones.Source: §4.2 · p. 14
• In multi-object Case 2, PDNN roughly located the dominant object but lost boundary details, produced nonuniform permittivity, and significantly underestimated material contrast in the imaginary part.The supplied passage ends before reporting LSPDNN’s Case 2 result.Source: §4.2 · p. 15
• Across three multi-object cases, average relative errors were 5.75% (point-clouds), 3.57% (PDNN), and 1.21% (LSPDNN), while LSPDNN cut PDNN runtime 34.3% (457.0 s to 300.3 s).LSPDNN also preserves separated components and spatially uniform material distributions.Source: Table 2 · p. 15, §4.2 · pp. 15–16

4.3 Resolution Capability for Closely Spaced Scatterers

• In all three closely spaced cases, LSPDNN clearly separates the two cubic scatterers without artificial bridging, even at the most challenging center-to-center distance dcc = 0.5λ.Source: §4.3 · p. 17

4.4 Ablation Study on the Loss Function

• Removing LTV, LBin, LGini, or LAct respectively worsens material continuity, support boundaries, material-assignment sharpness, or support compactness.Without LAct, the complex digit-like support shows overestimated relative permittivity and less clearly defined boundaries.Source: §4.4 · p. 18
• The full model had the lowest relative reconstruction errors in both cases (0.09% in Case 1; 0.66% in Case 2); removing LAct or LTV raised Case 1 error to 2.15% or 1.84%, respectively.In Case 2, removing LBin produced the largest degradation, yielding an error of 1.35%.Source: §4.4 · p. 18

4.5 Noise Robustness Analysis

• The proposed LSPDNN solver shows good noise robustness: at SNR = 20 dB, reconstructions remain close to ground truth for all tested targets; at 10 dB, quality decreases, especially for complex or multi-material targets, but main object supports and dominant structural features persist.Source: §4.5 · p. 20
• The noise-robustness analysis examines inverse-scattering reconstruction under noisy measurement conditions.Source: §4.5 · p. 20

4.6 Experimental Validation

• For TwoCubes, LSPDNN recovers two separated cubic scatterers with well-preserved locations and supports, while producing fewer background artifacts than PDNN.Slight boundary distortions remain in the reconstruction.Source: §4.6 · p. 21
• For TwoSpheres, LSPDNN reconstructs compact, geometrically consistent adjacent spheres, preserving their contact region and relative permittivity despite slight surface roughness.The experimental reconstruction is shown in Figure 10.Source: Figure 10 · p. 20, §4.6 · p. 21
• Using only the measured PP channel, LSPDNN outperforms PDNN in artifact suppression and geometric fidelity on sharp-edge and smooth-contact dielectric targets.The results support its applicability to practical 3-D inverse-scattering imaging.Source: §4.6 · p. 21

• LSPDNN combines neural level-set parameterization and a soft-union multi-material contrast model to represent piecewise-homogeneous scatterers with multiple objects and material regions.Source: §5 · p. 22
• Compared with point-clouds and PDNN, LSPDNN better preserves sharp boundaries across the reported reconstruction results.Source: §5 · p. 22
• The solver reconstructs complex geometries, distinguishes closely spaced targets, and recovers multiple material regions with improved accuracy.Source: §5 · p. 22
• Noise tests show stable reconstruction under degraded measurements, and experimental results support reliable recovery of target geometry and material distribution from measured scattered fields.Source: §5 · p. 22
• Future work will target computational efficiency, automatic model-complexity selection, and broader experimental validation under practical measurement uncertainties and model conditions.Source: §5 · p. 22