Eric Tosan

Fractal modeling using free form techniques

IFS models have become a powerful tool for the construction of fractal sets. They offer a straightforward way to generate complex, self‐similar figures as attractors. However, they present the fundamental disadvantage of providing little control over the global form of the attractor. In contrast, free form techniques give a high control over smooth man‐made objects with the use of a set of control points.

Fractal coding of shapes based on a projected IFS model

This paper addresses the problem of approximation of natural complex shapes. Using MPEG-7 terminology, this problem can be considered as the search of a descriptor for a shape feature. This shape can be defined either as a frontier between image regions or a natural curve. For this purpose, an original descriptor which combines iterated function system (IFS) model and the notion of free form curves is proposed. A set of control points allows one to define the IFS model in a barycentric space. This generalization adds a real flexibility to fractal approximation techniques enriching the set of contractive operators which are candidate to model the self-similarity. This new descriptor named projected IFS model allows the reconstruction of a shape using a projection via the control points. It is adapted to the representation of both smooth shapes (man-made objects, body,..) and fractal shapes (mountain, cloud, tree,..). Results on synthetic shapes and a real mountain shape are presented.

A FRACTAL APPROXIMATION OF CURVES

This paper deals with the approximation of rough curves using a fractal model. The approximation criterion is based on a curve parametrization and the fractal model is a projected iterated function system (IFS) model. This model unifies the IFS model and a classical model used in computer graphics (free form representation with control points). To formulate the approximation problem, we introduce a family of curves based on our model. Then, the approximation problem has a nonlinear fitting formulation. We have tested this fractal approximation method on both smooth and rough synthetic curves. A shape extracted from a natural scene has also been approximated. The obtained results are very satisfying from both quantitative and visual point of view.

Design and reconstruction of fractal surfaces

A method for the design and reconstruction of rough surfaces is introduced. A fractal model based on projected IFS attractors allows the definition of free form fractal shapes controlled with a set of points. Ibis flexible model has good fitting properties for recovering surfaces. The approximation is formulated as a non-linear fitting problem and resolved using a modified LEVENBERG-MARQUARDT minimisation method. The main applications are shape design, shape reconstruction and geometric data compression.

Fractal Inverse Problem: Approximation Formulation and Differential Methods

An analytical approach to fractal inverse problem is presented in this paper. We recall the construction of an Hilbert space with address functions, that constitute a general framework for fractal modeling, since it includes IFS made fractals. A large scale of applications is shown, ranging from scalar function approximation to image compression.