A central issue in characterizing neuronal growth patterns is whether their arbors form clusters. Formal definitions of clusters have been elusive, although intuitively they appear to be related to the complexity of branching. Standard notions of complexity have been developed for point sets, but neurons are specialized ``curve-like'' objects. Thus we consider the problem of characterizing the local complexity of a ``curve-like'' measurable set. We propose an index of complexity suitable for defining clusters in such objects, together with an algorithm that produces a complexity map which gives, at each point on the set, precisely this index of complexity. Our index is closely related to the classical notions of fractal dimension, since it consists of determining the rate of growth of the area of a dilated set at a given scale, but it differs in two significant ways. First, the dilation is done normal to the local structure of the set, instead of being done isotropically. Second, the rate of growth of the area of this new set, which we named ``normal complexity'', is taken at a fixed (given) scale instead of around zero. The results will be key in choosing the appropriate representation when integrating local information in low level computer vision. As an application, they lead to the quantification of axonal and dendritic tree growth in neurons (see Fig. ).
B. Dubuc, M. Stryker (University of California at San Francisco), S.W. Zucker
Figure: An example of curve-like sets obtained from neuron data reconstruction illustrating the need of using different representations for integration. (a) neuron reconstruction (b) normal complexity map where white stands for high complexity and black for low complexity. Notice how tedious it is to follow the lines in the dendritic cluster in (a).