# American Institute of Mathematical Sciences

August  2014, 8(3): 845-863. doi: 10.3934/ipi.2014.8.845

## Active arcs and contours

 1 Department of Mathematics, University of California, Irvine, Irvine, CA 92697, United States

Received  September 2012 Revised  November 2013 Published  August 2014

The level set method [33] is a commonly used framework for image segmentation algorithms. For edge detection and segmentation models, the standard level set method provides a flexible curve representation and implementation. However, one drawback has been in the types of curves that can be represented in this standard method. In the classical level set method, the curve must enclose an open set (i.e. loops or contours without endpoints). Thus the classical framework is limited to locating edge sets without endpoints. Using the curve representation from [37,36], we construct a segmentation and edge detection method which can locate arcs (i.e. curves with free endpoints) as well as standard contours. Within this new framework, the variational segmentation model presented here is able to detect general edge structures and linear objects. This energy is composed of two terms, an edge set regularizer and an edge attractor. Our variational model is related to the Mumford and Shah model [29] for joint segmentation and restoration in terms of an asymptotic limit, and in addition, is both general and flexible in terms of its uses and its applications. Numerical results are given on images with a variety of edge structures.
Citation: Hayden Schaeffer. Active arcs and contours. Inverse Problems & Imaging, 2014, 8 (3) : 845-863. doi: 10.3934/ipi.2014.8.845
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