Hierarchical Curve Reconstruction. Part I: Bifurcation analysis and Recovery of Smooth Curves

1 I n t r o d u c t i o n The problem of curve inference from a brightness image is of fundamental importance for image analysis. Computing a curve representation is generally a difficult task since brightness data provides only uncertain and ambiguous informat ion about curve location. Two sources of uncertainty are curve bifurcations (junctions) and "invisible curves" (e.g. the sides of the Kanisza triangle). Local information is not sufficient to deal with these problems and "global" information has to be used somehow. Methods based on optimization of a cost functional derived according to Bayesian, minimum description length, or energy-based principles [4, 9, 11, 12, 20] introduce global information by simply adding an appropr ia te te rm to the cost functional. These formulations are simple and compact but lead usually to computationally intractable problems. Moreover, it is often difficult or impossible to guarantee that the optimal solution of these cost functionals represents correctly all the desired features, such as junctions and invisible curves [16]. Curve evolution approaches, where the computed curves are defined to be the stat ionary solutions of some differential equation [7, 19], can be computat ionally efficient but usually require some external initialization in order to converge to the desired solution. A way to exploit global information efficiently without the need of external initialization is to use a hierarchy of intermediate representations between the brightness da ta and the final representation. The complexity and spatial extent of the descriptors in these representations increase gradually as one moves up in this hierarchy. Global information is introduced gradually so that computa t ion is always efficient. * Research supported by US Army grant DAAL03-92-G-0115, Center for Intelligent Control Systems