Lifting Lines and Tone: Image-Space Stylization in Path-Space

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.