Parameter identification and algorithmic construction of fractal interpolation functions: Applications in digital imaging and visualization

This dissertation examines the theory and applications of fractal interpolation. Its main contribution is the parameter identification, algorithmic construction and applications of fractal interpolation. We focus on the self-affine and piecewise self-affine fractal interpolation functions that are based on the theory of iterated function systems. Specifically, we present two novel methods for parameter identification that are based on minimising the symmetric difference between bounding volumes of appropriately chosen points, achieving lower errors compared to existing methods. We also present a novel method that aims at preserving the fractal dimension of the initial set of points. Beyond these, we introduce a new method for curve fitting using fractal interpolation, allowing a more economical representation compared to existing ones. Moreover, we construct non-tensor product bivariate fractal interpolation surfaces. As far as the applications are concerned, we focus on isosurface triangulation, point cloud modelling, active shape models as well as representation and compression of medical and geographic data; fractal interpolation is used as the core of the proposed methods yielding better results or overcoming limitations of existing methodologies.