# Local Symmetries and Segmentaton of Shapes

Wednesday, November 15, 2000 - 11:00am - 12:00pm

Keller 3-180

Jayant Shah (Northeastern University)

The set of local symmetry axes is found by analyzing the level curves of a function which is the solution of an elliptic PDE. These level curves may be thought of as successive smoothings of the shape boundary. A point on a level curve is a point of local symmetry if the level curve is symmetric about the gradient vector at that point upto second order. The local symmetry axes may also be described as the ridges and the valleys of the graph of this function. The rationale underlying this approach is that if a shape has certain symmetries, the solution of the PDE ought to reflect these symmetries.

The set of local symmetry axes includes loci which are analogous to the more commonly used medial axes. If a 2D shape is viewed as a collection of ribbons glued together, then the local symmetry axis of each ribbon along its length may be viewed as its medial axis. Alternatively, if the shape is viewed as a distorted circle, distorted by protrusions and indentations, then the local symmetry axis of each protrusion along its length is its medial axis.

There are two main advantages of this approach. It is possible to calculate the necessary properties of the level curves from the differential properties of the function without having to locate the level curves themselves, and the use of an elliptic PDE makes it unnecessary to presmooth the shape boundary. Moreover, it is straightforward to extend the definition of local symmetry to higher dimensions.

Unlike the shape skeletons found by the Blum transform or by decomposing the shape into a set of ribbons, the set of local symmetries is usually not connected. One can obtain a connected set by extending the local symmetry axes to join up with the nearby axes. However, it is more natural to use the local symmetry axes to segment the shape boundary, thus preserving all shape information. The segmentation obtained in this way has the struture of a graph.

The set of local symmetry axes includes loci which are analogous to the more commonly used medial axes. If a 2D shape is viewed as a collection of ribbons glued together, then the local symmetry axis of each ribbon along its length may be viewed as its medial axis. Alternatively, if the shape is viewed as a distorted circle, distorted by protrusions and indentations, then the local symmetry axis of each protrusion along its length is its medial axis.

There are two main advantages of this approach. It is possible to calculate the necessary properties of the level curves from the differential properties of the function without having to locate the level curves themselves, and the use of an elliptic PDE makes it unnecessary to presmooth the shape boundary. Moreover, it is straightforward to extend the definition of local symmetry to higher dimensions.

Unlike the shape skeletons found by the Blum transform or by decomposing the shape into a set of ribbons, the set of local symmetries is usually not connected. One can obtain a connected set by extending the local symmetry axes to join up with the nearby axes. However, it is more natural to use the local symmetry axes to segment the shape boundary, thus preserving all shape information. The segmentation obtained in this way has the struture of a graph.