Many non-photorealistic rendering (NPR) styles, such as feature lines and hatching, are defined by image-space structure inherited from hand-drawn media. While recent path-space formulations like the stylized rendering equation (SRE) enable stylization to interact naturally with light transport, they provide no mechanism for enforcing image-space consistency. We present a conceptual framework for lifting image-space stylizations into path-space in a principled, SRE-compatible manner. Our key observation is that image-space consistency can be achieved by establishing geometrically-driven mappings from image-space to path-space. We demonstrate this through two complementary stylizations: feature lines and tone. For feature line rendering, we introduce a conditional lifting based on partial path-space variation, where a geometric path parametrization is combined with parallel transport to preserve image-space structure under distribution effects. This enables a curvature-aware, stochastic, geometry-driven formulation of line detection that generalizes prior ray-based methods. For tone-based styles, such as hatching and halftone, we propose a canonical lifting anchored to a material-independent mapping, motivated by stationary-phase arguments from wave optics. Its locally invertible approximation enables evaluation of image-space tone fields at arbitrary path vertices while preserving image-space structure under complex light transport. Both methods are implemented as ordinary style functions under the SRE and work with existing estimation and sampling strategies; demonstrating how image-space structure can be preserved within path-space rendering, enabling a broader class of expressive, physically-grounded NPR styles.

We propose a generalization of the rendering equation that captures both the realistic light transport of physically-based rendering (PBR) and a subset of non-photorealistic rendering (NPR) stylizations in a principled manner. The proposed formulation is based on the key observation that both classical transport and certain NPR stylizations can be modeled as a function of expectation. Given this observation, we generalize the recursive integrals of the rendering equation to recursive functions of expectation. As estimating functions of expectation can be challenging, especially recursive ones, we provide a toolkit for unbiased and biased estimation comprising prior work, general strategies, and a novel build-your-own strategy for constructing more complex unbiased estimators from simpler unbiased estimators. We then use this toolkit to construct a complete estimator for the proposed recursive formulation, and implement a sampling algorithm that is both conceptually simple and leverages many of the components of an ordinary path tracer. To demonstrate the practicality of the proposed method we showcase how it captures several existing stylizations like color mapping, cel shading, and cross-hatching, fuses NPR and PBR visuals, and allows us to explore visuals that were previously challenging under existing formulations.

Given a partition of a graph into connected components, the membership oracle asserts whether any two vertices of the graph lie in the same component or not. We prove that for $n\ge k\ge 2$, learning the components of an $n$-vertex hidden graph with $k$ components requires at least $(k-1)n-\binom k2$ membership queries. Our result improves on the best known information-theoretic bound of $\Omega(n\log k)$ queries, and exactly matches the query complexity of the algorithm introduced by [Reyzin and Srivastava, 2007] for this problem. Additionally, we introduce an oracle that can learn the number of components of $G$ in asymptotically fewer queries than learning the full partition, thus answering another question posed by the same authors. Lastly, we introduce a more applicable version of this oracle, and prove asymptotically tight bounds of $\widetilde\Theta(m)$ queries for both learning and verifying an $m$-edge hidden graph $G$ using it.

Abstract Monte Carlo integration is a technique for numerically estimating a definite integral by stochastically sampling its integrand. These samples can be averaged to make an improved estimate, and the progressive estimates form a sequence that converges to the integral value on the limit. Unfortunately, the sequence of Monte Carlo estimates converges at a rate of O( ), where n denotes the sample count, effectively slowing down as more samples are drawn. To overcome this, we can apply sequence transformation, which transforms one converging sequence into another with the goal of accelerating the rate of convergence. However, analytically finding such a transformation for Monte Carlo estimates can be challenging, due to both the stochastic nature of the sequence, and the complexity of the integrand. In this paper, we propose to leverage neural networks to learn sequence transformations that improve the convergence of the progressive estimates of Monte Carlo integration. We demonstrate the effectiveness of our method on several canonical 1D integration problems as well as applications in light transport simulation.