Characterization of subdomain geometry and cardinality

Characterize the shapes and determine the number of subdomains induced when the random field is projected onto a pixel domain, including cases involving occlusion, perspective, reflection, or refraction.

Background

The method represents a pixel as a sum of contributions from subdomains on which the projected random field is constant. The variance correction depends on the areas of these subdomains, but the proposed Monte Carlo sketch estimates the relevant statistics without explicitly enumerating the subdomains or knowing their geometry. The paper therefore leaves the shapes and number of these subdomains unspecified, particularly for complex three-dimensional light transport involving occlusion, perspective, reflection, and refraction.

References

We also do not know the shape of the subdomains $\pixelSubdomains$. It would depend on $\world$. In particular, if $\world operates in 3D, it might involve effects like occlusion and perspective or even reflection and refraction, which will result in a complex partition of the pixel domain $\pixelDomain$ into arbitrarily shaped subdomains $\pixelSubdomains$.

— Tabula Rasa: Monte Carlo estimation of unit-variance noise with controlled spatio-temporal correlation  (2610.11653 - Ritschel et al., 8 Oct 2026) in Section 3, subsection “Sub-pixel decomposition”