Edge aware anisotropic diffusion for 3D scalar data on regular lattices

We present a novel anisotropic diffusion model targeted for 3D scalar field data. Our model preserves material boundaries as well as fine tubular structures while noise is smoothed out. One of the major novelties is the use of the directional second derivative to define material boundaries instead of the gradient magnitude for thresholding. This results in a diffusion model that has much lower sensitivity to the diffusion parameter and smoothes material boundaries consistently compared to gradient magnitude based techniques. We analyze the stability and convergence of the proposed diffusion and demonstrate its denoising capabilities for both analytic and real data. We also discuss applications in the context of volume rendering. We extend our algorithm to non-Cartesian lattices such as Body Centric Cubic (BCC). The key to such an extension is a method to estimate derivatives reliably. Therefore, we present a general framework to estimate derivatives on arbitrary regular lattices. With this framework a user can design filters with compact support and specify a polynomial order of accuracy.