A Second Order Shape Optimization Approach for Image Segmentation

Abstract

The problem of segmentation of a given image using the active contour technique is considered. An abstract calculus to find appropriate speed functions for active contour models in image segmentation or related problems based on variational principles is presented. The speed method from shape sensitivity analysis is used to derive speed functions which correspond to gradient or Newton-type directions for the underlying optimization problem. The Newton-type speed function is found by solving an elliptic problem on the current active contour in every time step. Numerical experiments comparing the classical gradient method with Newton's method are presented.

Discrete gradient flows for shape optimization and applications

“…On the other hand level set-based approaches in shape optimization often handle these quantities explicitly in the time stepping scheme, hence in a less stable manner [7,17].…”

“…We should remark that the choice of threshold is a subtle issue. Hintermüller and Ring report in [17] that a small value suffices in general. Our experiments also show that = 0.1 is well-behaved for a 100 × 100 image if we take unit interpixel distance, thereby giving rise to a computational domain [0, 100] 2 .…”

“…This scalar product has been obtained in [17] by taking the second order shape derivative into account, thereby resulting in a Newton-type flow.…”

13. A Combined Shape-Newton and Topology Optimization Technique in Real-Time Image Segmentation

“…In connection with velocity fields F coming from shape sensitivity analysis [5,18], as for instance F being the negative shape gradient of J or the corresponding Newton direction [9], it is a versatile tool for shape optimization including image segmentation via (1). While the merging or splitting of existing contours can be handled easily by level-set based shape optimization algorithms [2,9], the creation of new components is in general hard (if not impossible) to accomplish by using classical shape sensitivity concepts only. As a remedy we propose the blending of shape sensitivity information with topological sensitivity.…”

“…In [9] the first and second order Eulerian semi-derivatives of the function in (20) were computed. We only provide the formulas and refer to [9] for details. The first order Eulerian derivative is given by…”