Dirichlet Splatting: Differentiable Rendering for Wave-Based Inverse Problems

We present Dirichlet Splatting , a physically grounded differentiable rendering method for inverse problems with wave-based coherent sensors. Unlike optical cameras, these sensors observe a scene in a band-limited Fourier space whose continuous point response is complex-valued and oscillatory. Our key observation is that the point spread function of a reflector in this Fourier space is the Dirichlet kernel , not a Gaussian. We instantiate each splat as a Dirichlet kernel in closed form, model each primitive as a planar surfel rendered through a surfel-aware Dirichlet splat whose complex amplitude is set by wave propagation physics, and introduce a sliding Frank-Wolfe adaptation tailored to the rugged, non-convex landscape that the Dirichlet oscillation induces. On terahertz tomography, Dirichlet Splatting recovers reflector geometry with an order of magnitude fewer parameters, faster convergence, and higher accuracy than 3D Gaussian Splatting.