Separable Parameterization of the Sphere using Compactly Supported Interpolatory Basis Functions

D. Schmitter, R. Delgado-Gonzalo, M. Unser

First International Conference on Subdivision, Geometric and Algebraic Methods, Isogeometric Analysis and Refinability in Tuscany (SMART'14), Pontignano, Italian Republic, September 28-October 1, 2014.

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The segmentation of 3D objects with sphere-like topology is a current challenge in the field of biomedical imaging. Examples of such segmentation problems include brain segmentation in 3D MRI images or blob-like cell segmentation in 3D microscopy. Aside from the geometric requirements that the model needs to meet, a fast implementation and the possibility of user-interaction have to be considered. Active contour models (a.k.a. snakes) have proven to be effective for such segmentation challenges [1].

Recently, we have presented a parameterization of the sphere that relies on exponential B-spline bases in order to construct a spherical 3D snake [2, 3]. The bases have minimal support which allows an efficient computational implementation of the model.

This time we show how to construct such spherical snakes using basis functions that are capable of reproducing ellipsoids, while at the same time being interpolatory. This property makes them good candidates for user-interactive applications because the control points of the snake lie directly on its surface. The proposed basis functions still have compact support while respecting the interpolation condition. In order for the snake to remain closed and smooth when deforming, special attention must be paid to the poles of the sphere. Therefore, we need to explicitly impose the necessary interpolation and smoothness conditions directly at the poles. For this purpose we reformulate these conditions with respect to the corresponding control points thus ensuring that the Gaussian curvature remains well defined for every point on the surface. Finally, we will show examples of real medical and biological 3D structures which can be segmented with the sphere snake.

References

1. M. Kass, A. Witkin, D. Terzopoulos, "Snakes: Active Contour Models," International Journal of Computer Vision, vol. 1, no. 4, pp. 321-331, January 1998.

2. R. Delgado-Gonzalo, N. Chenouard, M. Unser, "Spline-Based Deforming Ellipsoids for Interactive 3D Bioimage Segmentation," IEEE Transactions on Image Processing, vol. 22, no. 10, pp. 3926-3940, October 2013.

3. D. Schmitter, R. Delgado-Gonzalo, G. Krueger, M. Unser, "Atlas-Free Brain Segmentation in 3D Proton-Density-Like MRI Images," Proceedings of the Eleventh IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI'14), Beijing, People's Republic of China, April 29-May 2, 2014, pp. 629-632.

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