Parametric Level Set Methods for Inverse Problems

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.

💡 Research Summary

This paper introduces a parametric level set (PLS) framework for solving shape‑based inverse problems, where the unknown domain (obstacle) is represented implicitly by a level set function that is expressed as a linear combination of adaptive, compactly‑supported radial basis functions (RBFs). Traditional level set methods treat the level set function as an infinite‑dimensional field discretized on a dense grid. Consequently they suffer from three major drawbacks: (1) a very high dimensional optimization problem (one variable per grid node), (2) the need for periodic re‑initialization to maintain a signed‑distance property, and (3) reliance on explicit regularization terms to counteract ill‑posedness.

The authors overcome these issues by parameterizing the level set as

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