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

The paper introduces a parametric level‑set (PLS) framework for solving inverse problems that involve reconstructing unknown obstacles or inclusions. Traditional level‑set methods represent the interface implicitly on a full computational grid and evolve it via Hamilton‑Jacobi‑type partial differential equations. While powerful, these approaches suffer from high dimensionality, the need for periodic re‑initialisation to maintain a signed‑distance function, and difficulties in incorporating regularisation in a systematic way.

To overcome these drawbacks, the authors propose to parameterise the level‑set function φ(x) as a finite linear combination of compactly supported radial basis functions (RBFs): \