WavePCNet: Physics-Driven Coherent Imaging Network

• WavePCNet is a physics-driven neural framework that integrates Fresnel diffraction constraints and tri-phase propagation to overcome limitations in low-SNR, scattered imaging.
• It employs a Tri-Phase Wavefront Complex-Propagation Reprojection module with momentum memory and FFT-based computations to suppress non-physical artifacts.
• Experimental validation shows that WavePCNet outperforms conventional methods with up to an 18.9% gain in accuracy on challenging passive imaging datasets.

The Wavefront Propagating Compensation Network (WavePCNet) is a physics-driven neural framework designed to enhance passive obscured object detection in coherent imaging scenarios. By directly simulating wavefront propagation, WavePCNet introduces strong inductive constraints grounded in the physics of Fresnel diffraction and incorporates bespoke modules that address the ill-posedness and instability endemic to highly-scattered, low-SNR intensity observations. The architecture systematically fuses a Tri-Phase Wavefront Complex-Propagation Reprojection (TriWCP) module with a momentum memory mechanism and a high-frequency cross-layer compensation stream, consistently outperforming conventional methods on accuracy and robustness benchmarks under challenging physical conditions (Zheng et al., 26 Nov 2025).

1. Physical Foundations and Motivation

WavePCNet is motivated by the physics of coherent light propagation governed by the paraxial Helmholtz equation. In free space or weakly scattering media, the complex optical field $\mathcal U(\boldsymbol\xi)$ evolves from a source plane to an observation plane at distance $z$ via the Fresnel diffraction integral:

$$\mathcal E(\boldsymbol\rho) = \frac{e^{ikz}}{i\lambda z} \iint_{\mathbb R^2} \mathcal U(\boldsymbol\xi) \exp\biggl(\frac{ik}{2z}\|\boldsymbol\rho-\boldsymbol\xi\|^2\biggr) \mathrm d\boldsymbol\xi$$

where $k = 2\pi/\lambda$. Only a spatial-frequency disk $\Omega_z$—the "Fresnel passband"—is physically admissible for propagation, enforcing a strict band-limitedness in the field's Fourier domain. Standard convolutional approaches and real-valued CNNs fail to respect these constraints, often resulting in solutions that violate physical laws, especially under low SNR and multiply scattered illumination.

2. Tri-Phase Wavefront Complex-Propagation Reprojection (TriWCP)

TriWCP is a centerpiece in WavePCNet, embedding the Fresnel propagation operator to project estimated complex fields onto the low-rank, physically admissible manifold $\mathbb S_z$. The approach involves three steps: at each iteration $t$, the gradient of the task loss with respect to the current field is computed and split into physical and non-physical components. A lightweight momentum memory mechanism accumulates the history of gradient residuals orthogonal to $\mathbb S_z$, and a convolutional smoothing kernel $\mathcal K$ suppresses high-frequency, non-physical oscillations.

The update direction is given by

$$\Delta^{(t)}(\mathbf r) = \frac{ \boldsymbol m^{(t)}(\mathbf r) - \int\mathcal K(\mathbf r-\mathbf r')\,\boldsymbol\delta^{(t-1)}(\mathbf r')\,\mathrm d\mathbf r' }{ \sqrt{\boldsymbol v^{(t)}(\mathbf r)} + \varepsilon }$$

with $\boldsymbol m^{(t)}$ and $\boldsymbol v^{(t)}$ the accumulated momentum and energy terms, and $\varepsilon$ a stabilizer. After exactly three such updates, the solution is finally reprojected onto $\mathbb S_z$ using the operator $\mathcal P_{\mathbb S_z}$. This tri-phase design optimally trades off expressiveness and stability, avoiding both underfitting (insufficient physical flexibility) and divergence due to excessive unconstrained updates.

The algorithmic pipeline:

Input: speckle intensity I_w → initial complex field Ψ⁰ Precompute: Fresnel operator 𝒯_z and projector 𝒫𝕊_z for t in 1 to 3 do g ← ∂L/∂Ψ^(t-1) g_phys ← 𝒫𝕊_z[g] δ^(t-1) ← g - g_phys m^(t) ← β₁·m^(t-1) + 𝒜_z^†[δ^(t-1)] v^(t) ← β₂·v^(t-1) + |𝒜_z^†[δ^(t-1)]|² s^(t-1) ← K * δ^(t-1) Δ^(t) ← (m^(t) - s^(t-1)) / (√v^(t) + ε) Ψ^(t) ← Ψ^(t-1) - γ_t · Δ^(t) end for Ψ_final ← 𝒫𝕊_z[ Ψ⁰ - ∑_{t=1}^3 γ_t·Δ^(t) ] Output: complex field Ψ_final → downstream feature maps

FFT-based implementation ensures computational efficiency, with only minor overhead beyond vanilla CNN inference.

3. Momentum Memory and Energy Spectral Mechanism

At each update, TriWCP integrates a momentum memory mechanism which accumulates only the gradient components orthogonal to the physical subspace $\mathbb S_z$. Exponential decay factors $(\beta_1, \beta_2)$ modulate the persistence of memory and help reject temporally persistent non-physical perturbations. The momentum memory is mathematically encoded as:

$$\boldsymbol m^{(t)}(\mathbf r) = \sum_{\tau=1}^t \beta_1^{t-\tau} \mathcal A_z^\dagger \left[ \boldsymbol g^{(\tau)} - \mathcal P_{\mathbb S_z}[ \boldsymbol g^{(\tau)} ] \right](\mathbf r)$$

$$\boldsymbol v^{(t)}(\mathbf r) = \sum_{\tau=1}^t \beta_2^{t-\tau} \left| \mathcal A_z^\dagger \left[ \boldsymbol g^{(\tau)} - \mathcal P_{\mathbb S_z}[ \boldsymbol g^{(\tau)} ] \right](\mathbf r) \right|^2$$

where $\mathcal A_z^\dagger$ denotes a generalized adjoint propagator accommodating multi-order scattering. This mechanism substantially improves numerical stability, particularly in the presence of high-frequency noise and unpredictable speckle statistics.

4. High-Frequency Cross-Layer Compensation and Stream Fusion

Following the TriWCP module, the network processes the stabilized complex field through a High-Frequency Cross-Layer Compensation stream. This branch employs multi-dilation band-pass filters and semantics-guided attention to selectively enhance high-spatial-frequency features—such as edges and fine contours—that may be suppressed by the hard propagation constraint of TriWCP. The features from both TriWCP and the compensation branch are fused, yielding a representation that retains physical coherence while restoring fine-grained spatial detail under very weak illumination and severe occlusion.

A plausible implication is that this fusion mechanism underpins the network's high interpretability and robustness, as it corrects for physically necessary but perceptually problematic loss of structure during the initial wavefront-constrained inference.

5. Algorithmic and Computational Details

The TriWCP module is strategically placed early in the WavePCNet backbone, acting on the initial complex field reconstructed from the wall's observed speckle intensity. Key computational components—such as the orthogonal projector $\mathcal P_{\mathbb S_z}$ and adjoint propagator $\mathcal A_z^\dagger$—are implemented via efficient FFT-based multiplication by the optical transfer function (OTF) associated with the Fresnel kernel and passband mask. The smoothing kernel $K$ is instantiated as a $3 \times 3$ Gaussian, and step sizes $(\gamma_1, \gamma_2, \gamma_3)$ are calibrated via cross-validation over $[0.01, 0.1]$.

The tri-phase loop introduces minimal extra computational cost, with only a few FFTs, lightweight convolutions, and pixel-wise algebra per inference pass. This results in real-time performance at approximately 40 FPS and a model size of 26M parameters.

6. Experimental Validation and Performance Metrics

WavePCNet, with TriWCP enabled, was evaluated on four physically collected out-of-distribution datasets targeting passive obscured object detection. Disabling TriWCP—while retaining the cross-layer compensation—resulted in a $\approx 18.9\%$ reduction in Maximum-F score and a $0.114$ increase in MAE on the OOD-DUTS benchmark, highlighting its critical role in physically meaningful reconstruction. The decay factors were set to $(\beta_1 = 0.9, \beta_2 = 0.999)$, and the model outperformed other contemporary approaches by $4-6\%$ absolute gain in F-measure and $0.06-0.08$ reduction in MAE.

The experimental design confirms the necessity of physics-driven constraints for passive, coherent imaging in multiply-scattered settings. WavePCNet's tri-modal pipeline and stability mechanisms collectively enable robust segmentation and localization under extremely weak, ambiguous observation conditions (Zheng et al., 26 Nov 2025).

7. Significance, Limitations, and Outlook

WavePCNet, anchored by TriWCP, establishes a paradigm shift towards embedding hard physics constraints within neural architectures for passive imaging. Its spectral projection and memory-augmented updates directly address the pathologies of conventional models in coherent scatter-dominated regimes. While the architecture achieves real-time performance and state-of-the-art metrics on challenging datasets, its reliance on accurate forward models and precise calibration of propagation operators suggests sensitivity to domain-specific transfer. A plausible implication is that future work may extend this methodology to broader classes of wave-based imaging or adapt the spectral constraint operator for other physical modalities where band-limited propagation and phase coherence are essential.