Kernel Based Algebraic Curve Fitting

An algebraic curve is defined as the zero set of a multivariate polynomial. We consider the problem of fitting an algebraic curve to a set of vectors given an additional set of vectors labelled as interior or exterior to the curve. The problem of fitting a linear curve in this way is shown to lend itself to a support vector representation, allowing non-linear curves and high dimensional surfaces to be estimated using kernel functions. The approach is attractive due to the stability of solutions obtained, the range of functional forms made possible (including polynomials), and the potential for applying well understood regularisation operators from the theory of Support Vector Machines

Different approaches on the implementation of implicit polynomials in visual tracking /

Visual tracking has emerged as an important component of systems in several application areas including vision-based control, human-computer interfaces, surveillance, agricultural automation, medical imaging and visual reconstruction. The central challenge in visual tracking is to keep track of the pose and location of one or more objects through a sequence of frames. Implicit algebraic 2D curves and 3D surfaces are among the most powerful representations and have proven very useful in many model-based applications in the past two decades. With this approach, objects in 2D images are described by their silhouettes and then represented by 2D implicit polynomial curves. In our work, we tried different approaches in order to efficiently apply the powerful implicit algebraic 2D curve representation to the phenomenon of visual tracking. Through the proposed concepts and algorithms, we tried to reduce the computational burden of fitting algorithms. Besides showing the usage of this representation on boundary data simulations, use of the implicit polynomial as a representative of the target region is also experimented on real videos

Covariant-conics decomposition of quartics for 2D shape recognition and alignment

This paper outlines a geometric parameterization of 2D curves where the parameterization is in terms of geometric invariants and terms that determine an intrinsic coordinate system. Thus, we present a new approach to handle two fundamental problems: single-computation alignment and recognition of 2D shapes under affine transformations. The approach is model-based, and every shape is first fit by an implicit fourth degree (quartic) polynomial. Based on the decomposition of this equation into three covariant conics, we are able to define a unique intrinsic reference system that incorporates usable alignment information contained in the implicit polynomial representation, a complete set of geometric invariants, and thus an associated canonical form for a quartic. This representation permits shape recognition based on 8 affine invariants. This is illustrated in experiments with real data sets. 1