Biomedical Imaging GroupSTI
English only   BIG > Publications > Active Contours

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 All BibTeX References

Exponential B-Splines and the Design of Active Contours and Surfaces for Biomedical Image Analysis

R. Delgado-Gonzalo, M. Unser

Proceedings of the Workshop on New Trends in Subdivision and Related Applications (NTSRA'12), Milano, Italian Republic, September 4-7, 2012, pp. 22.

Snakes are effective tools for image segmentation. Within a 2D image, a snake is a 1D curve that evolves from an initial position, which is usually specified by a user, toward the boundary of an object. Within a 3D image, a snake is represented by a 2D surface. In the literature, these methods are also known as active contours or active surfaces.

Research has been fruitful in this area, and many snake variants have emerged. Among them, we are interested in the spline-based kind, where the curve or the surface is described continuously by some coefficients (a.k.a. control points) using basis functions. These snakes have become popular because it is possible for the user to interact with them, not only when specifying their initial position, but also during the segmentation process. This is often achieved by allowing the user to specify anchor points the curve or surface should go through.

Our interest is to characterize the spline-like integer-shift-invariant bases involved in the design of this kind of snakes. We prove that any compact-support function that reproduces a subspace of the exponential polynomials can be expressed as the convolution of an exponential B-spline with a compact-support distribution. As a direct consequence of this factorization theorem, we show that the minimal-support basis functions of that subspace are linear combinations of derivatives of exponential B-splines. These minimal-support basis functions form a natural multiscale hierarchy, which we utilize to design fast multiresolution algorithms and subdivision schemes for the representation of closed geometric curves and surfaces. This makes them attractive from a computational point of view.

Finally, we illustrate our scheme by building efficient active contours and surfaces capable of exactly reproducing ellipses in 2D and ellipsoids in 3D irrespective of their position and orientation.

AUTHOR="Delgado-Gonzalo, R. and Unser, M.",
TITLE="Exponential \mbox{{B}-Splines} and the Design of Active Contours
        and Surfaces for Biomedical Image Analysis",
BOOKTITLE="Proceedings of the Workshop on New Trends in Subdivision and
        Related Applications ({NTSRA'12})",
address="Milano, Italian Republic",
month="September 4-7,",

© 2012 BICOCCA. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from BICOCCA.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.