  In this paper, a parametric level set method for reconstruction of obstacles
in general inverse problems is considered. General evolution equations for the
reconstruction of unknown obstacles are derived in terms of the underlying
level set parameters. We show that using the appropriate form of parameterizing
the level set function results a significantly lower dimensional problem, which
bypasses many difficulties with traditional level set methods, such as
regularization, re-initialization and use of signed distance function.
Moreover, we show that from a computational point of view, low order
representation of the problem paves the path for easier use of Newton and
quasi-Newton methods. Specifically for the purposes of this paper, we
parameterize the level set function in terms of adaptive compactly supported
radial basis functions, which used in the proposed manner provides flexibility
in presenting a larger class of shapes with fewer terms. Also they provide a
"narrow-banding" advantage which can further reduce the number of active
unknowns at each step of the evolution. The performance of the proposed
approach is examined in three examples of inverse problems, i.e., electrical
resistance tomography, X-ray computed tomography and diffuse optical
tomography.
