We investigate the topological structure of an image and the hierarchical relationship between local and global structures provided by spatial gradients at different levels of scale. The gradient field curves link stationary points of the image including a local minimum at infinity, and construct the topological structure of the image. The evolution of the topological structure with respect to scale is analysed using pseudograph representation. The hierarchical relationships among the structures at different scales are expressed as trajectories of the stationary points in the scale space, which we call stationary curves. Each toppoint of the local extremum curve generically has a specific gradient field curve, which we call the antidirectional figure-flow curve. The antidirectional figure-flow curve connects between the toppoint and another local extremum to which the toppoint is subordinate. A point at infinity can also be connected to the toppoints of local minimum curves. These hierarchical relationships among the stationary points are expressed as a tree. As an application, we present the temporal segmentation of an image sequence based on the transition of the hierarchical structure.
['Tomoya Sakai', 'Atsushi Imiya']
Gradient Structure of Image in Scale Space