Dirichlet Splatting: Differentiable Rendering for Wave-Based Inverse Problems

2026-09-30

AI Summary

This study addresses the physical mismatch in existing Gaussian splatting methods, which fail to preserve the sidelobe energy and phase inherent to coherent imaging. To overcome this limitation, this work proposes Dirichlet Splatting, which for the first time employs the physically accurate Dirichlet kernel as a differentiable rendering primitive to perfectly align with finite-window discrete Fourier transform (DFT) measurement mechanisms. Furthermore, a dedicated solver is developed that integrates a Dirichlet sliding Frank-Wolfe algorithm, variable projection, and Levenberg-Marquardt correction techniques to achieve efficient optimization. Experimental results demonstrate that the proposed method attains a reflection center error of merely 0.018 bins in terahertz reconstruction, while delivering a 10- to 50-fold computational speedup compared to waveform-level automatic differentiation.

Abstract

Wave-based coherent imaging, including terahertz tomography, synthetic-aperture acoustics, and millimeter-wave radar, forms images by Fourier-processing finite-length signals, with an exact point spread function that is not Gaussian but a Dirichlet kernel: complex-valued, oscillatory, and periodic. However, transplanting 3D Gaussian splatting to coherent sensing fails by construction; Gaussian splats discard the sidelobe energy (10-20% of the total) and the phase that governs coherent interference between reflectors. Our key idea is to replace the learned Gaussian footprint with the physically exact Dirichlet kernel of the finite-window DFT, modulated by a surfel that carries area, normal, and material, so that the rendering primitive matches the measurement physics instead of approximating it. We pair this primitive with a specialized solver, Dirichlet Sliding Frank-Wolfe (DSFW), that combines variable projection, residual dual certificates, and certificate-driven hard replacement of low-utility surfels, with periodic low-resolution coupled Levenberg-Marquardt correction, navigating the rugged loss landscape that breaks generic first-order optimizers. The Dirichlet kernel admits an O(1) closed-form evaluation, so the forward model matches FFT ground truth to machine precision while remaining differentiable end-to-end. On dense terahertz reconstruction, our method recovers reflector centers to 0.018 bin RMSE, 10-50x faster than waveform-level automatic differentiation, where Gaussian splats fail.

Problem

• coherent imaging
• Dirichlet kernel
• 3D Gaussian splatting
• differentiable rendering
• inverse problems

Innovation

• Dirichlet Splatting
• Differentiable Rendering
• Coherent Imaging
• Sliding Frank-Wolfe
• Inverse Problems