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Item A Bidirectional Formulation for Walk on Spheres(The Eurographics Association and John Wiley & Sons Ltd., 2022) Qi, Yang; Seyb, Dario; Bitterli, Benedikt; Jarosz, Wojciech; Ghosh, Abhijeet; Wei, Li-YiNumerically solving partial differential equations (PDEs) is central to many applications in computer graphics and scientific modeling. Conventional methods for solving PDEs often need to discretize the space first, making them less efficient for complex geometry. Unlike conventional methods, the walk on spheres (WoS) algorithm recently introduced to graphics is a grid-free Monte Carlo method that can provide numerical solutions of Poisson equations without discretizing space. We draw analogies between WoS and classical rendering algorithms, and find that the WoS algorithm is conceptually equivalent to forward path tracing. Inspired by similar approaches in light transport, we propose a novel WoS reformulation that operates in the reverse direction, starting at source points and estimating the Green's function at ''sensor'' points. Implementations of this algorithm show improvement over classical WoS in solving Poisson equation with sparse sources. Our approach opens exciting avenues for future algorithms for PDE estimation which, analogous to light transport, connect WoS walks starting from sensors and sources and combine different strategies for robust solution algorithms in all cases.Item Fourier Analysis of Correlated Monte Carlo Importance Sampling(© 2020 Eurographics ‐ The European Association for Computer Graphics and John Wiley & Sons Ltd, 2020) Singh, Gurprit; Subr, Kartic; Coeurjolly, David; Ostromoukhov, Victor; Jarosz, Wojciech; Benes, Bedrich and Hauser, HelwigFourier analysis is gaining popularity in image synthesis as a tool for the analysis of error in Monte Carlo (MC) integration. Still, existing tools are only able to analyse convergence under simplifying assumptions (such as randomized shifts) which are not applied in practice during rendering. We reformulate the expressions for bias and variance of sampling‐based integrators to unify non‐uniform sample distributions [importance sampling (IS)] as well as correlations between samples while respecting finite sampling domains. Our unified formulation hints at fundamental limitations of Fourier‐based tools in performing variance analysis for MC integration. At the same time, it reveals that, when combined with correlated sampling, IS can impact convergence rate by introducing or inhibiting discontinuities in the integrand. We demonstrate that the convergence of multiple importance sampling (MIS) is determined by the strategy which converges slowest and propose several simple approaches to overcome this limitation. We show that smoothing light boundaries (as commonly done in production to reduce variance) can improve (M)IS convergence (at a cost of introducing a small amount of bias) since it removes discontinuities within the integration domain. We also propose practical integrand‐ and sample‐mirroring approaches which cancel the impact of boundary discontinuities on the convergence rate of estimators.Item Temporally Sliced Photon Primitives for Time-of-flight Rendering(The Eurographics Association and John Wiley & Sons Ltd., 2022) Liu, Yang; Jiao, Shaojie; Jarosz, Wojciech; Ghosh, Abhijeet; Wei, Li-YiWe derive a class of new Monte Carlo estimators for volumetric time-of-flight rendering, generalizing prior work on transient photon points and beams. Conceptually, our method starts with any steady-state photon primitive – like a photon plane, parallelepiped, or parallelotope – and slices it with a temporal wavefront, producing a primitive of one dimension lower. We show how different unbiased temporally sliced primitives arise by analytically integrating any four dimensions within a novel extended spatio-temporal path space formulation. The differences between these estimators reduce to the determinant of a 4×4 Jacobian matrix, with columns dictated by the chosen dimensions. We then show how to combine the relative strengths of different sliced primitives using multiple importance sampling. Finally, we implement several of the new estimators enabled by our theory and compare them to each other as well as previous techniques.Item A Wave-optics BSDF for Correlated Scatterers(The Eurographics Association and John Wiley & Sons Ltd., 2025) Yang, Ruomai; Kim, Juhyeon; Pediredla, Adithya; Jarosz, Wojciech; Wang, Beibei; Wilkie, AlexanderWe present a wave-optics-based BSDF for simulating the corona effect observed when viewing strong light sources through materials such as certain fabrics or glass surfaces with condensation. These visual phenomena arise from the interference of diffraction patterns caused by correlated, disordered arrangements of droplets or pores. Our method leverages the pair correlation function (PCF) to decouple the spatial relationships between scatterers from the diffraction behavior of individual scatterers. This two-level decomposition allows us to derive a physically based BSDF that provides explicit control over both scatterer shape and spatial correlation. We also introduce a practical importance sampling strategy for integrating our BSDF within a Monte Carlo renderer. Our simulation results and real-world comparisons demonstrate that the method can reliably reproduce the characteristics of the corona effects in various real-world diffractive materials.