Further Reading

Deformable contour models are commonly used in image processing and computer vision, for example for shape description [21], object localization [22], and visual tracking [23].

A good starting point to learn about parametric active contours is [24]. These snakes have undergone significant improvements since their conception, for example see the GVF snake in [7,9]. Region-based parametric snake frameworks have also been reported in [25-27]

The geometric model of active contours was simultaneously proposed by Caselles et al. [1] and Malladi et al. [2]. Geometric snakes are based on the theory of curve evolution in time according to intrinsic geometric measures of the image. They are numerically implemented via level sets, the theory of which can be sought in [15,16].

There has been a number of works based on the geometric snake and level set framework. Siddiqi et al. [14] augmented the performance of the standard geometric snake that minimizes a modified length functional by combining it with a weighted area functional. Xu et al. extended their parametric GVF snake [7] into the generalized GVF snake, the GGVF, in [9]. Later, they also established an equivalence model between parametric and geometric active contours [10] using the GGVF. A geometric GGVF snake enhanced with simple region-based information was presented in [10]. Paragios et al. [28,29] presented a boundary and region unifying geometric snake framework which integrates a region segmentation technique with the geometric snake. In [30], Yezzi et al. developed coupled curve evolution equations and combined them with image statistics for images of a known number of region types, with every pixel contributing to the statistics of the regions inside and outside an evolving curve. Using color edge gradients, Sapiro [6] extended the standard geometric snake for use with color images (also see Fig. 10.6). In [11], Chan et al. described a region-segmentation-based active contour that does not use the geometric snake's gradient flow to halt the curve at object boundaries. Instead, this was modeled as an energy minimization of a Mumford-Shah-based minimal partition problem and implemented via level sets. Their use of a segmented region map is similar to the concept we have explored here.

Level set methods can be computationally expensive. A number of fast implementations for geometric snakes have been proposed. The narrow band technique, initially proposed by Chop [31], only deals with pixels that are close to the evolving zero level set to save computation. Later, Adalsterinsson et al. [32] analyzed and optimized this approach. Sethian [33,34] also proposed the fast marching method to reduce the computations, but it requires the contours to monotonically shrink or expand. Some effort has been expended in combining these two methods. In [35], Paragios et al. showed this combination could be efficient in application to motion tracking. Adaptive mesh techniques [36] can also be used to speed up the convergence of PDEs. More recently, additive operative splitting (AOS) schemes were introduced by Weickert et al. [37] as an unconditionally stable numerical scheme for nonlinear diffusion in image processing. The basic idea is to decompose a multidimensional problem into one-dimensional ones. AOS schemes can be easily applied in implementing level set propagation [38].

The mean shift algorithm is a nonparametric technique for estimation of the density gradient, which was first proposed by Fukunaga et al. [39]. The idea was later generalized by Cheng [40]. The technique was extended to various applications, amongst them color image segmentation, by Comaniciu et al. [12, 13,20].

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