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One solution is to add variable parameters to rigid geometric models. Solina and Bajcsy [10] use superquadrics as a generic shape model. A restricted solution space can be obtained by specifying the range of admissible shape parameters. This model accommodates several shape variations including tapering and bending.

An alternative approach is to constrain the deformation of a flexible shape model to a subspace of the total configuration space. Ideally, this subspace should be learned from examples. This is the approach adopted by Cootes et al. [11] who present a

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method to produce a 2-D generic shape model (point distribution model") represented by a set of landmarks. Baumberg and Hogg [12] extend their method by (1) describing models as 2-D spline curves, and (2) acquiring training shape instances automatically from images and therefore automating the entire procedure of model construction.

In this paper, we present a method for automatically producing a generic 3-D shape model, representing a given class of objects, directly from sequences of 2- D images, and demonstrates its applications to 3-D shape recovery, tracking and recognition. For experimental purposes, the objects we are interested in are vehicles, although the method can also be applied to other classes of object. The generic model is composed of a mean shape and a orthonormal basis extracted by analysing the variances of shapes in a set of training shapes obtained using the method proposed in [5]. Each training shape is first represented by a shape vector which consists of the set of control points of a B-spline surface. The shape vectors are aligned by scaling and translation. Principle component analysis is used to obtain a set of unit eigenvectors describing the most significant variances in the shape. These eigenvectors then form a orthonormal basis defining the space of admissible shapes. The produced model is flexible in the variances of shape, and is also restricted to a specific class of objects. Applications in 3-D shape recovery, tracking and recognition can then be easily achieved within one procedure by fitting the model to the object in each image in order to estimate the pose parameters of the model, and the shape parameters which in turn are used to identify the object.

2 Acquisition of Generic Model

2.1 3-D shape recovery

In [5], a method for recovering the 3-D shapes of objects from 2-D image sequences is proposed, assuming the object is rigid and mirror symmetrical, and moves on a ground plane. With this method, instances of various shapes in a generic class of shapes can be obtained. Figure 1 shows some results from this work. The recovered shape model is a closed surface with two poles which is represented discretely by P ?Q sample points r(p; q) on the surface

r(p; q) =

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