Computer graphics, visualization, geometric modeling and processing, non-photorealistic rendering, and computational topology.
Vector and tensor field design
Vector field topology
Vector fields appear in a broad range of scientific and engineering domains. Visualization based on vector field topology has gained wide acceptance by researchers and practitioners due to its ability to provide an integral view of important flow features such as fixed points and interactions among them (separatrices). My collaborators and I have developed efficient techniques to extract periodic orbits and incorporate it into vector field topology in the form of a directed graph. Moreover, we have recognized the instability associated with trajectory-based vector field topology and defined a more robust form a vector field topology based on Morse decomposition from dynamical systems.
Asymmetric tensor field visualization
Past research on tensor field visualization has focused on symmetric tensors. Inspired by the pioneering work of Zheng and Pang, my collaborators and I have studied the topology of 2D asymmetric tensors and introduced the notions of eigenvalue and eigenvector manifolds. In addition, we applied our analysis to the study of fluids, which led to a number of interesting observations of data that were not available when using vector field visualization only. We have further enhanced this approach by combining glyph packing with streamline placement for asymmetric tensor visualization.
Surface parameterization has a wide range of applications in computer graphics. The quality of a parameterization largely depends on the amount of the distortion and seams in the parameterization. Most past methods produce a large number of relatively small charts; while having low distortion, this often comes to the cost of having a large amount of seams, over which the signal becomes discontinuous. My collaborators and I developed a rather different approach in which the surface is segmented into meaning parts with relatively simple geometry. This has led to a small number of large patches, thus reducing the amount of seams without significantly increasing the distortion. We also devised a new stretch metric and a new patch unfolding technique by using scaffolding triangles.
Painterly rendering of videos
Artists use different means of stylization to control the focus on different objects in the scene. This allows them to portray complex meaning and achieve certain artistic effects. Most prior work on painterly rendering of videos, however, uses only a single painting style, with fixed global parameters, irrespective of objects and their layout in the images. This often leads to inadequate artistic control. Moreover, brush stroke orientation is typically assumed to follow an everywhere continuous directional field. We produce a video painting system that accounts for the spatial support of objects in the images or videos, and uses this information to specify style parameters and stroke orientation for painterly rendering. Since objects occupy distinct image locations and move relatively smoothly from one video frame to another, our object-based painterly rendering approach is characterized by style parameters that coherently vary in space and time. Space-time-varying style parameters enable more artistic freedom, such as emphasis/de-emphasis, increase or decrease of contrast, exaggeration or abstraction of different objects in the scene in a temporally coherent fashion.
There has been a recent realization by the geometry community that rotational symmetries on a surface can be used to understand the properties of the surface as well as to guide surface parameterization and remeshing. My collaborators and I have formulated N-way rotational symmetries (N-RoSy) in terms of N-th order tensor and provided some basic but important mathematical analysis on N-RoSy fields. We have also discovered a way to transform an N-RoSy field into a vector field, thus allowing the user to design N-RoSy fields by reusing vector field design techniques. Applications of our approaches include geometry remeshing and non-photorealistic rendering. Moreover, we have developed an efficient visualization techniques for N-RoSy fields as well as a technique that automatically generates a 4- or 6-RoSy field from the curvature tensor in the surface. Such a field is then used for remeshing and pattern synthesis.
Visualizing surfaces of large amount of folding and interior is a challenging task. My collaborator and I developed a visibility measure that enables us to distinguish the easy-to-see regions of a surface (exterior) from the hard-to-see regions (creases, interiors). We then apply the visibility information to guide mesh simplification, which allows us to better maintain the appearance of the model as simplification is performed mostly to low-visibility-regions in the model. Later, my collaborators and I have developed a surface segmentation framework in which a surface can be segmented into layers by applying graph cut to the surface visibility function. Once the segmentation is achieved, it can be used to enhance the visualization of the surface, as each layer is rendered differently.
Procedural street modeling
Modeling a city requires the ability to generate a plausible street network. Past procedural techniques in generating street networks lack global control, as the network is often generated incrementally using grammars. Moreover, if the user is not satisfied with the result, the whole network must be regenerated. My collaborators and I recognize the link between a street network and the hyperstreamlines in a tensor field. Consequently, we adapted our tensor field design system to the street modeling, greatly reducing the amount of the work and increasing the reusability of the generated networks.