Diplomarbeit: Level Sets for Symbolic Reconstruction

Student: Jakob Vogel
Advisor: Prof. Dr. Nassir Navab
Supervision by: Andreas Keil

Abstract

The task of segmenting medical image data often can be carried out by using basic algorithms like region growing, e.g. There are, however, clinical questions which give rise to advanced segmentation problems. In order to solve those problems, more sophisticated algorithmic approaches have to be applied. One of them is called "Level Sets" and provides the programmer with a lot of options to define his segmentation goals. The segmentation is achieved by adjusting region boundaries. These adjustments are driven by user-defined energies and forces. Minimizing those energies leads to partial differential equations which – in the end – have to be solved using finite differences.

The boundaries are represented implicitely, defining them as the zero level set (hence the name) of some function :

(In contrast, "Snakes" are explicitely defined curves or surfaces using splines or other basis functions.) A boundary's movement over time can then be modeled as evolution of the defining level set function :

Resources and Literature

Primary:

• [1] J.A. Sethian. Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, 2nd edition, 1999; download of introduction, chapter 1, and bibliography
• [2] J.A. Sethian's excellent web resource
• [3] D. Adalsteinsson and J.A. Sethian. A fast level set method for propagating interfaces. J. Computational Physics, 118(2):269-277, 1995. doi:10.1006/jcph.1995.1098
• [4] R. Malladi, J.A. Sethian, and B.C. Vemuri. Shape modeling with front propagation: A level set approach. IEEE Trans. Pattern Anal. Machine Intell., 17(2):158-175, 1995. doi:10.1109/34.368173

Documents

• Final presentation