Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
On the numerical solution of the three-dimensional inverse obstacle scattering problem
Journal of Computational and Applied Mathematics - Special issue on inverse problems in scattering theory
A fast level set method for propagating interfaces
Journal of Computational Physics
A variational level set approach to multiphase motion
Journal of Computational Physics
Inverse obstacle scattering using reduced data
SIAM Journal on Applied Mathematics
Electrical Impedance Tomography
SIAM Review
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Tikhonov Regularization and Total Least Squares
SIAM Journal on Matrix Analysis and Applications
A PDE-based fast local level set method
Journal of Computational Physics
Computational Methods for Inverse Problems
Computational Methods for Inverse Problems
Journal of Computational Physics
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
An extended level set method for shape and topology optimization
Journal of Computational Physics
Variational B-spline level-set: a linear filtering approach for fast deformable model evolution
IEEE Transactions on Image Processing
IEEE Transactions on Image Processing
A curve evolution approach to object-based tomographic reconstruction
IEEE Transactions on Image Processing
IEEE Transactions on Image Processing
A Regularized Gauss-Newton Trust Region Approach to Imaging in Diffuse Optical Tomography
SIAM Journal on Scientific Computing
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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 in a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, reinitialization, and use of signed distance function. Moreover, we show that from a computational point of view, low order representation of the problem paves the way 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, provide 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.