Metrics of curves in shape optimization and analysis - Andrea Carlo ...

(a) (b) (c) (d) (e)Figure 3: Segmentation of occluded plane shape.(From Tsai et al. [60] c○ 2001 IEEE. Reproduced with permission).2.2 Shape optimization & active contours2.2.1 A short history of active contoursIn the late 20th century, the most common approach to image segmentationwas a combination of edge detection and contour continuation. With edgedetection [5], small edge elements would be identified by a local analysis ofthe image; then a continuation method would be employed to reconstructfull contours. The methods were suffering from two drawbacks, edge detectionbeing too sensitive to noise and to photometric features (such as, sharp shadows,reflections) that were not related to the physical structure of the image; andcontinuation was in essence a NP-complete algorithm.Active contours were introduced by Kass et al. [26] as a simpler approachto the segmentation problem. The idea is to minimize an energy F (c) (wherethe variable c is a contour i.e. a curve), that contains an edge-based attractionterm and a smoothness term, which becomes large when the curve is irregular.To search for the minimum of F (c), a curve evolution method was derived fromthe calculus of variations. The evolving curve would then be attracted to thefeatures of interest, while mantaining the property of being a closed smoothcontour.An unjustified feature of the model of [26] is that the evolution is dependenton the way the contour is parameterized. Hence Caselles et al. [6], Malladi et al.[33] revisited the idea of [26] in the new form of a geometric evolution, whichcould be implemented by the level set method invented by Osher and Sethian[44]. Thereafter, Kichenassamy et al. [27] Caselles et al. [7] presented the activecontour approach to the edge detection problem in the form of the minimizationof a geometric energy that is a generalization of Euclidean arc length. Again, theauthors derived the gradient descent flow in order to minimize the geometricenergy.In contrast to the edge-based approaches for active contours (mentionedabove), region-based energies for active contours have been proposed in Ronfard[46] Zhu et al. [69] Yezzi et al. [64] Chan and Vese [8]. In these approaches, anenergy is designed to be minimized when the curve partitions the image intostatistically distinct regions. This kind of energy has provided many desirablefeatures; for example, it provides less sensitivity to noise, better ability to capture11