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We are interested in geometric approximation by parameterization oftwo-dimensional multiple-component shapes, in particular when the number ofcomponents is a priori unknown. Starting a standard method based on successiveshape deformations with a one-component initial shape in order to approximate amultiple-component target shape usually leads the deformation flow to make theboundary evolve until it surrounds all the components of the target shape. Thisclassical phenomenon tends to create double points on the boundary of theapproximated shape. In order to improve the approximation of multiple-componentshapes (without any knowledge on the number of components in advance), we usein this paper a piecewise B\'ezier parameterization and we consider twoprocedures called intersecting control polygons detection and flip procedure.The first one allows to prevent potential collisions between two parts of theboundary of the approximated shape, and the second one permits to change itstopology by dividing a one-component shape into a two-component shape. For anexperimental purpose, we include these two processes in a basic geometricalshape optimization algorithm and test it on the classical inverse obstacleproblem. This new approach allows to obtain a numerical approximation of theunknown inclusion, detecting both the topology (i.e. the number of connectedcomponents) and the shape of the obstacle. Several numerical simulations areperformed.

Flip procedure in geometric approximation of multiple-component shapes -- Application to multiple-inclusion detection