Reconstructing Thin Shapes from Boundary Electrical Measurements with Level Sets

We present a new technique for reconstructing thin shapes of fixed thickness from boundary electrical measurements. The main application which we have in mind is crack detection, but the approach is not limited to that. We introduce an extension of the level set technique for modelling thin shapes. Two level set functions are employed in order to achieve this: the first one models the location and form of the thin shape, and the second one models the length and connectivity of it. A gradient based method is derived in order to define evolution laws for these two level set functions which minimize the least squares data misfit. This evolution of the two level set functions corresponds to an evolution of the thin shape or crack during the reconstruction. We present numerical experiments which demonstrate that our technique is able to recover disconnected thin shapes or cracks from noisy simulated boundary data.