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Hierarchical Segmentation

During resolution reduction by diffusion, local signal variations smooth out as ''time'' t proceeds. Extrema disappear one after the other (only rarely a new on in created, see [6]), and finally only the strongest signal variations survive. Thus a hierarchical ordering of extrema is induced by the diffusion process.

The images f(x,y;t) obtained from the original data f(x,y;t) obey further a concept of causality [3]: every value tex2html_wrap_inline367 found at a specific scale tex2html_wrap_inline369 can be traced back to the original resolution t=0. The contour sheets tex2html_wrap_inline373 are, except for a finite number of critical values tex2html_wrap_inline375 , two-dimensional submanifolds of tex2html_wrap_inline377 . This follows directly from Sard's theorem. At the critical values tex2html_wrap_inline375 , the topology of the contour sheets changes. These events correspond to the aforementioned disappearance or appearance of local extrema.

From Thom's theory it follows that every such event has qualitatively the same generic form [3]. Since f(x,y;t) is a one-parameter family of functions in t, the only possible way to change is given by the so-called fold catastrophe, generically described through the function

equation38

In figure 2 the contour sheets of this function are displayed near a critical value tex2html_wrap_inline375 . For tex2html_wrap_inline387, we see a branching of the contour-sheet as one descends towards finer resolutions (figure 2.a).

   figure46

Figure 2: The change of the topology of contour sheets near a critical value tex2html_wrap_inline375 . Connected at tex2html_wrap_inline387 (a), the sheet splits into two disjunct sets for tex2html_wrap_inline393 (c).

For tex2html_wrap_inline393 (figure 2.c), the manifold separates into two disjunct pieces, one continuing towards coarser resolutions, and the other one descending in scale space towards finer resolutions. Clearly, the surface spawned from the main surface is orientable, and encircles an area at the finest resolution which is topological equivalent to a disk (or a union of discs if it branches again).

Between critical values, all surfaces are similar and can be continuously deformed into each other. We thus have an onion-like, hierarchical structure of contour sheets, changing topology only at a few critical values tex2html_wrap_inline375 , corresponding to the disappearance of local extrema present in the input data.

Several schemes have been proposed for exploiting the structure of scale space [3, 6, 7]. Most approaches need a very dense sampling of scale space and are therefore computationally expensive. Furthermore, the connection with biological vision systems is not clear. We propose here a single and simple neuronal mechanism for utilizing scale space: the tracing and merging of contour sheets in scale space by neuronal oscillators. Within this ansatz, neurons distributed over scale space at discrete points tex2html_wrap_inline359 can participate in a common slow modulation of their mean firing rate, if they code approximately the same intensity level. By this process, each contour-sheet in scale space is can be marked with a specific "timecode". These patches of synchronization are allowed to merge further, exhibiting a common frequency, if -- in scale space -- no pronounced border exists between them. As we will show, such a process is able to segment and mark data into a few object chunks.

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