2.1. Active Contour Models
Since active contour was introduced to the vision community by Kass et al.
(1988), extensive researches was done on “snakes” or parametric active contour models for boundary detection. The classical approach is based on deforming an initial contour towards the boundary of the object to be detected. The deformation is obtained by trying to minimize a functional designed such that its minimum is
obtained at the boundary of the object. The energy functional is basically composed of two components, one controls the smoothness of the curve and another attracts the curve towards the boundary. However, there are three key difficulties with parametric active contour algorithms. First, the initial contour must be close to the true boundary or it will likely converge to the wrong result. Then, active contours have difficulties progressing into boundary concavities. Finally, energy model is not capable of handling changes in the topology of the evolving contour when direct
implementations are performed. An approach insensitive to initialization and the ability to move into boundary concavities is proposed in [1]. The author present external forces originate from an edge map of the image to provide larger capture range. However, this parametric model cannot handles topology changes as well.
Recently, novel geometric models of active contours were proposed [2] [4].
These models are based on the theory of curve evolution and geometric flows, which has received a large amount of attention in recent years. It allows automatic changes in the topology when implemented using the level-sets based numerical algorithm
[6][7]. Thereby, several objects can be detected simultaneously without previous knowledge of their number in the scene and without using special tracking procedures.
However, because the flow may be slow to converge in practice, a constant term is added to keep the curve moving in the desired direction. Kaleem et al. [10] modify this term based on the gradient flow derived from a weighted area functional, with image dependent weighting factor. Since this flow requires the computation of only first order derivatives, it offers significant computational savings over the weighted length minimizing flow.
Active contour models that rely on the edge-detector or image gradient can detect only objects with edges defined by gradient. In practice, the discrete gradients are bounded and then the stopping function is never zero on the edges, and the curve may pass through the boundary. Chan and Vese detailed a level set implementation of the Mumford-Shah functional [9], which is based on the use of the Heaviside function as an indicator function for the separate phases. The idea is to partition the given image into two homogeneous regions, without a stopping edge-detector. The authors also extend this binary image oriented method to segment images with more than two regions by multiphase level sets [12].
To segment objects in textured background, Paragios and Deriche proposed a region-based energy, where statistical models were used for textured object and background regions [16]. They extended the region model to the mixture of Gaussians for magnitude of Gabor filter responses. The texture segmentation is obtained by unifying region and boundary-based information.
Tracking is another segmentation method by using the segmentation results in
the image frames history. If the contour is initialized with its previous position, contour segmentation approaches become object trackers, and tracking is defined based on motion information to evolve an initial object contour[4][14][17][18][19].
In [4], the evolution equation for contour is obtained by image differences, this tracking algorithm can be applied only when an important degree of similarity among the images and displacements involved are small. G. Tsechpenakis et al. proposed a method handling the appearance of occlusions between different objects [19]. The use of the object motion history and statistical measurements provide information for the extraction of uncertainty regions. In [14], tracking is expressed as detection and tracking of moving objects in image sequences. In the proposed algorithm, a detection step forces a closed curve to converge towards moving areas of an image, while a tracking step evolves the curve to coincide with the exact boundary of the moving object. The tracking step is only an intensity boundary detection algorithm using active contours and implemented using level sets. Since the tracking step relies on the previous frame, the background is assumed to be stationary. The problem addressed in [17] is that object tracking can be treated as two-class discriminate analysis of pixels, where the classes correspond to the object and the background regions. Since his approach compute for each pixel by brute-force search in a circular neighborhood, there are two problems exist even when strong assumption on intensity boundaries.
First, the contour cannot capture the parts of the object near where existing a background region with similar intensities to them. The second, the background around the contour will be classified as the object if there exist some pixels with similar intensities within the object. The classification criterion is extended in [18]. A window of specified fixed size is defined for each pixel around the contour. The contour will move in the direction that can equalize the numbers of pixels within the
window that belong to two classes (object and background). Also, shape priors are used to recover the missing object regions during occlusion. Nevertheless, since the shape prior takes effect only when the occlusions are detected, this approach still suffers the problems as encountered by [17].
Our approach is the incorporation of knowledge about the shape with [2], to control the difficult conditions of the image. However, different from[18], we improve the tracking results of [17] instead of handling occlusions of the objects.
2.2. Shape Priors
In the substantial literature of deformable models, there are three main
mechanisms can be found to constraint the shape of the curve during the evolution of the deformable model:
1. Free-form approaches: These methods do not encode a default shape, but the energy functional imposes smoothness and compactness of the boundary of the surface. They can be seen as general, weak and local shape constraints.
2. Analytical parametric templates: The analytical shape constraints are defined by the distribution of the admissible parameters. These methods are commonly used when some prior information about the geometrical shape is available, which can be encoded using a small number of parameters.
3. Prototype-based constraints: Shapes are represented by the mean shape of a collection of individuals and their statistical variations. These methods require either training or global shape modeling.
T. F. Cootes et al. propose a method that uses point distribution model (PDM) [13] as the prototype-based constraints. It describes the average and characteristic shape variations of a set of training samples, which are given in the form of a set of points on the learning boundaries. In [21], the authors investigate the use of discrete cosine transform (DCT) coefficients in describing object shape. The method starts
with local shape parameterization, then, the shape is converted into an implicit representation using global shape parameters. As can be seen, incorporating prior shape information in a deformable model, requires either training or global shape modeling. Training involves manual interaction to accumulate information on the shape variability of the
same object class. Global modeling can be characterized using only a few parameters, and tend to be much more stable than local properties. The choice of a certain shape representation determines to a great extent the flexibility, processing speed, and amount of user interaction.
Leventon et al. [23] have incorporated statistical shape information into the evolution of geodesic active contours. They compute a prior on shape variation given a set of training instances. Each curve in the training dataset is embedded as the zero level-set of three-dimensional surface, which is a signed distance function. Daniel Cremers et al. [22] propose a closed-form, spline based solution for incorporating invariance with respect to similarity transformations in the variational framework.
Dainiel Cremers and Stefano Soatto [20] integrate prior shape knowledge into level set based segmentation methods and proposed dissimilarity measures for shapes encoded by the signed distance function.
The proposed shape prior is a global shape model using the initial contour as the shape prior but not the training set from the image history, therefore the number of parameters involved can be reduced. It is motivated by modeling the flow field of the shape forces as geodesic active contours [23], incorporating invariant transformation by the pseudo distance measure [22] and alternatively computing the total energy and the shape energy during the evolution of the curve.