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 novel segment-based light transport framework that uses segments as the basic unit of light transport. Unlike vertex-based formulations, our segment-based formulation naturally accommodates the disconnected subpaths encountered in photon density estimation and path filtering methods, and opens the door to a wide range of new rendering methods that consider segments as a sampling primitive. To facilitate the development of segment-based rendering methods, we introduce several segment sampling techniques and estimation strategies, including a highly-performant recursive estimator. One of our key contributions is a general-purpose segment sampling framework based on marginal multiple importance sampling (MMIS). To demonstrate the practicality of our sampling framework, we show how it allows us to easily implement a robust bidirectional path filtering method — challenging under a vertex-based formulation — achieving superior filtering efficiency and convergence compared to state-of-the-art approaches.

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.

Multiple importance sampling (MIS) is a powerful tool to combine different sampling techniques in a provably good manner. MIS requires that the techniques’ probability density functions (PDFs) are readily evaluable point-wise. However, this requirement may not be satisfied when (some of) those PDFs are marginals, i.e., integrals of other PDFs. We generalize MIS to combine samples from such marginal PDFs. The key idea is to consider each marginalization domain as a continuous space of sampling techniques with readily evaluable (conditional) PDFs. We stochastically select techniques from these spaces and combine the samples drawn from them into an unbiased estimator. Prior work has dealt with the special cases of multiple classical techniques or a single marginal one. Our formulation can handle mixtures of those.

Feature lines visualize the shape and structure of 3D objects, and are an essential component of many non-photorealistic rendering styles. Existing feature line rendering methods, however, are only able to render feature lines in limited contexts, such as on immediately visible surfaces or in specular reflections. We present a novel, path-based method for feature line rendering that allows for the accurate rendering of feature lines in the presence of complex physical phenomena such as glossy reflection, depth-of-field, and dispersion. Our key insight is that feature lines can be modeled as view-dependent light sources. These light sources can be sampled as a part of ordinary paths , and seamlessly integrate into existing physically-based rendering methods. We illustrate the effectiveness of our method in several real-world rendering scenarios with a variety of different physical phenomena.

Abstract Markov chain Monte Carlo (MCMC) sampling is a powerful approach to generate samples from an arbitrary distribution. The application to light transport simulation allows us to efficiently handle complex light transport such as highly occluded scenes. Since light transport paths in MCMC methods are sampled according to the path contributions over the sampling domain covering the whole image, bright pixels receive more samples than dark pixels to represent differences in the brightness. This variation in the number of samples per pixel is a fundamental property of MCMC methods. This property often leads to uneven convergence over the image, which is a notorious and fundamental issue of any MCMC method to date. We present a novel stratification method of MCMC light transport methods. Our stratification method, for the first time, breaks the fundamental limitation that the number of samples per pixel is uncontrollable. Our method guarantees that every pixel receives a specified number of samples by running a single Markov chain per pixel. We rely on the fact that different MCMC processes should converge to the same result when the sampling domain and the integrand are the same. We thus subdivide an image into multiple overlapping tiles associated with each pixel, run an independent MCMC process in each of them, and then align all of the tiles such that overlapping regions match. This can be formulated as an optimization problem similar to the reconstruction step for gradient‐domain rendering. Further, our method can exploit the coherency of integrands among neighboring pixels via coherent Markov chains and replica exchange. Images rendered with our method exhibit much more predictable convergence compared to existing MCMC methods.

Multiple importance sampling (MIS) is a provably good way to combine a finite set of sampling techniques to reduce variance in Monte Carlo integral estimation. However, there exist integration problems for which a continuum of sampling techniques is available. To handle such cases we establish a continuous MIS (CMIS) formulation as a generalization of MIS to uncountably infinite sets of techniques. Our formulation is equipped with a base estimator that is coupled with a provably optimal balance heuristic and a practical stochastic MIS (SMIS) estimator that makes CMIS accessible to a broad range of problems. To illustrate the effectiveness and utility of our framework, we apply it to three different light transport applications, showing improved performance over the prior state-of-the-art techniques.