Dirichlet splatting improves wave-based 3D imaging accuracy and speed
Dirichlet Splatting: Differentiable Rendering for Wave-Based Inverse Problems
Graphics
Summary
Imaging methods that use waves, like terahertz or radar scans, rely on mathematical functions to form pictures. The common use of smooth Gaussian shapes to represent points in 3D space loses important parts of the signal and phase information needed for clear images. The authors replace these with a precise mathematical function called the Dirichlet kernel, which better matches how waves behave. This leads to faster and more accurate recovery of object locations in 3D imaging than previous methods.
What this means in practice
- •For terahertz imaging teams: Improve 3D terahertz image reconstructions with higher accuracy and faster processing using Dirichlet splatting methods.
- •For radar system engineers: Enhance millimeter-wave radar imagery by deploying accurate wave-based rendering primitives that capture phase and sidelobe effects.
Authors
Xingyu Chen, Wuqiong Zhao, Xinyu Zhang, Tzu-Mao Li
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.