Image analysis has in the last decade experienced a revolution via the development of new tools for the representation and analysis of local image features. At the heart of these developments is the construction of suitable local representations of structure, via decompositions in a set of localized functions. The chosen decomposition then forms the setting for further analysis and/or estimation methods. In particular, compression of a given representation ensures that most decomposition coefficients are of negligible magnitude, and this often simplifies the analysis considerably.
A major issue in designing local analysis methods is the choice of a suitable decomposition, and this is in many cases related to the appropriate localisation of an image. To design â€œoptimalâ€ or well-localised functions, a localisation operator is usually constructed. The eigenfunctions of this operator give a full set of well-localised orthogonal analysis functions. Using more than one function for local estimation has many advantages, and whilst the estimation bias is still controlled, reduced variance estimators may be constructed.
The problem of developing optimal functions in two dimensions requires defining a genuinely two dimensional localisation operator, associated in the spatial domain with the Cartesian distance metric. We discuss the choice of operator and determine the eigensystem of a radial operator. The usage of the eigenfunctions to estimate given local image characteristics is outlined, and we demonstrate the improved performance of our proposed estimation method.
This is joint work with Georgios Metikas to appear in IEEE Transactions on Signal Processing next year.