Morse decomposition provides a numerically stable topological representation of vector fields that is crucial for their rigorous interpretation. However, Morse decomposition is not unique, and its granularity directly impacts its computational cost. In this paper, we propose an automatic refinement scheme to construct the Morse Connection Graph (MCG) of a given vector field in a hierarchical fashion. Our framework allows a Morse set to be refined through a local update of the flow combinatorialization graph, as well as the connection regions between Morse sets. The computation is fast because the most expensive computation is concentrated on a small portion of the domain. Furthermore, the present work allows the generation of a topologically consistent hierarchy of MCGs, which cannot be obtained using a global method. The classification of the extracted Morse sets is a crucial step for the construction of the MCG, for which the Poincare´ index is inadequate. We make use of an upper bound for the Conley index, provided by the Betti numbers of an index pair for a translation along the flow, to classify the Morse sets. This upper bound is sufficiently accurate for Morse set classification and provides supportive information for the automatic refinement process. An improved visualization technique for MCG is developed to incorporate the Conley indices. Finally, we apply the proposed techniques to a number of synthetic and realworld simulation data to demonstrate their utility.

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. In this paper, we propose 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.