Generic Events for the Gradient Squared with Application to Multi-Scale Segmentation

In the Gaussian scale-space formalism, image features are often deened as loci of diierential invariants. A typical behavior of these is topological stability in open intervals of the scale axis. However, it is generic that the feature topology changes at speciic scales in so-called catastrophe events. In this paper, we show that the generic Gaussian scale-space catastrophe events for the gradient magnitude squared, Li Li , are the fold catastrophe and the cusp catastrophe. These results are applied to a scale-space formulation of segmentation with catchment basins/watersheds. The common problem of over-segmentation when segmenting with catchment basins of the gradient magnitude is solved by the multi-scale formulation. The necessary linking of segments across scale is based naturally on the catastrophe analysis for Li Li. Veriied segmentation results on 3D medical images are presented.