Figure 1: A subdivision surface character (3D model courtesy of Peter Eriksson) is animated and rendered using Modo ray-tracing engine. Three poses are shown in the middle: our screen-space mean curvature (depicted on the top with concave regions in red, convex regions in blue) is used to modulate a basic shading (bottom-left) yielding an exaggerated appearance (bottom-right). The image on the right illustrates another shading modulation based on our offset measurement τ .
Introduction
Surface curvature is a measure commonly employed in Computer Graphics for a vast range of applications: for modeling purposes of course, but also to drive texture generation, or to produce exaggerated or stylized shading results (see Figure 1).
Curvature is a differential property expressed at each surface point. When working with meshes, it can be efficiently computed per vertex using GPU-based techniques [Griffin et al. 2011]. However, this approach becomes impractical when dealing with production quality scenes that involve various representations (e.g., subdivision surfaces, implicit and procedural geometry, displacement and bump ma ps) and/or deformable objects. This severely limits its use in real-time editing (as in sculpting) or for shading animated scenes.
Screen-Space Curvature
Our approach may be applied either in ray-tracing or compositing contexts. For each sample (ray or pixel), we start by collecting neighbor samples in screen-space that are closer than a threshold distance t. We also reject neighbors that are farther than a threshold relative depth d, to avoid collecting samples across occluding contours. We then recover a normal vector and a 3D position for each neighbor sample. The resulting local point cloud is fit with an algebraic sphere where each sample is weighted by a compactly-supported function parametrized by t.
Our fitting is performed using the method of Mellado et al. [2012], which has the benefit of characterizing the resulting sphere with geometrically-meaningful parameters. This provides us not only with a mean curvature estimate κ, but also with an offset τ that identifies local relief and a smooth reconstructed normal vector η.With this approach, t controls the scale of the local reconstruction. As shown in Figure 2, our approach produces an accurate approximation of object-space mean curvature, which is due to the local nature of differential properties. There is a notable difference though: farther objects exhibit less details than closer ones with the screen-space approach, which provides for automatic simplification and reduced aliasing artifacts.
Figure 2: Comparison between mean curvature estimated in object- (left) and screen-space (right), shown at 2 different scales.
An alternative to our screen-space curvature would be to apply 2D derivative filters on normals [Vergne et al. 2009]. However, this approach neglects 3D positions, and raises issues nearby occluding contours, whereas our fitting remains well-defined.
Applications
We have implemented SSC in Modo on top of its ray-tracing engine, and in CUDA for post-processing, as shown in the video. An example animation is depicted in Figure 1, where we visualize our mean curvature estimate κ with a color code, along with a simple curvature-based shading result. Our method is applied at each frame with a negligible performance overhead compared to rendering (for both pixel- and ray-based implementations), and it exhibits natural temporal coherence.
In future work, we plan to use SSC during sculpting to enhance small (yet important) surface details, and for non-photorealistic rendering to guide stylization. In both cases, we believe that not only κ but also τ and η will prove to be valuable control parameters.
This work has been presented as a Siggraph Talk (2013) in Anaheim (follow this link to get publication details and slides).
