An Efficient Algorithm for Level Set Method Preserving Distance Function

Abstract

The level set method [31] is a popular technique for tracking moving interfaces in several disciplines including computer vision and fluid dynamics. However, despite its high flexibility, the original level set method is limited by two important numerical issues. Firstly, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features s.a. the curvature or the contour normal. Secondly, the level set algorithm is slow because the time step is limited by the standard CFL condition, which is also essential to the numerical stability of the iterative scheme. Recent advances with graph cut methods [4,3] and continuous convex relaxation methods [7,5,16] provide powerful alternatives to the level set method for image processing problems because they are fast, accurate and guaranteed to find the global minimizer independently to the initialization. These recent techniques use binary functions to represent the contour rather than distance functions, which are usually considered for the level set method. However, the binary function cannot provide the distance information, which can be essential for some applications s.a. the surface reconstruction problem from scattered points and the cortex segmentation problem in medical imaging. In this paper, we propose a fast algorithm to preserve distance functions in level set methods. Our algorithm is inspired by recent efficient ℓ 1 optimization techniques, which will provide an efficient and easy to implement algorithm. It is interesting to note that our algorithm is not limited by the CFL condition and it naturally preserves the level set function as a distance function during the evolution, which avoids the classical re-distancing problem in level set methods. We apply the proposed algorithm to carry out image segmentation, where our methods proves to be 5 to 6 times faster than standard distance preserving level set techniques. We also present two applications where preserving a distance function is essential. Nonetheless, our method stays generic and can be applied to any level set methods that require the distance information. 1 Report Documentation Page SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution unlimited SUPPLEMENTARY NOTES ABSTRACTThe level set method [31] is a popular technique for tracking moving interfaces in several disciplines including computer vision and uid dynamics. However, de-spite its high exibility, the original level set method is limited by two important numerical issues. Firstly, the level set method does not implicitly preserve the level set function as a distance function, which is necessary to estimate accurately geometric features s.a. the curvature or the contour normal. Secondly, the level set algorithm is slow because the time step is limited by the standard CFL con-dition, which is also essent...

Some fast projection methods based on Chan-Vese model for image segmentation

“…Most importantly, even if the LSF is initialized as a piecewise constant function, it can be corrected automatically due to the iterative projection computation. Therefore, our proposed methods have both higher computational efficiency and better SDF fidelity than those reported in [30,34,36]. What is worth mentioning here is that our proposed algorithms are quite generic and can be easily extended to all models using VLSM for multiphase image segmentation, motion segmentation, 3D reconstruction etc.…”

“…Using the similar idea, [36] introduced four auxiliary variables and four Lagrangian multipliers to deal with the same constrained optimization problem. The minimization problem is reformulated as following, and here, we name it as completely augmented Lagrangian method (CALM).…”

“…However, the penalty parameter limits the time step for the LSF evolution due to CourantFriedrichs-Lewy (CFL) condition [35] and thus the SDF cannot be preserved unless penalty parameter is very large, which cannot guarantee the stability of numerical computation. In order to avoid CFL condition, researchers in [36] proposed completely augmented Lagrangian method by introducing eight auxiliary variables and four penalty parameters, leading to numerous sub-minimization and sub-maximization problems for every introduced variable. Therefore, the resulting models are very complicated.…”

“…In this paper, we investigate the relationship between the TV regularizer term of Chan-Vese model and the constraint of LSF as a SDF and then propose a new model with fewer auxiliary variables in comparison with [36]. In this case, we can transform the constraint into a very simple algebra equation that can be explicitly implemented via direct projection approach without reinitialization.…”

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