# Research

The main topic of my research is concentrated on geometrical multiresolution methods of image processing. These methods are based on the family of functions, known as the so-called "X-lets". The good characterization of these functions one can find here. The new families of functions introduced by me and used in the area of geometrical multiresolution adaptive methods are presented below. Additionally, on the scheme below, the new methods are presented together with the state-of-the-art methods.## Second order wedgelets

Second order wedgelets were described in my PhD Thesis [2]. For the very first time they were presented on the conference
titled "Decision Support Systems" which took place in Zakopane, Poland in 2003, and published in [1]. The first name of second
order wedgelets was *extended wedgelets*. Starting from the paper [3] up to now they have been called as second
order wedgelets.

Second order wedgelets are defined basing on second order beamlets, since wedgelets are defined with the help of linear beamlets. The second order beamlet is defined as a segment of a second order curve (parabola, ellipse or hyperbola). In a natural way second order wedgelets are generalizations of wedgelets. So, the former ones adapts to location, scale and curvature of an edge.

[1] Lisowska A., *Effective coding of images with the use of geometrical wvelets*,
"Decision Support Systems" conference, Zakopane, Poland, 2003.
pdf (in Polish)

[2] Lisowska A. *Geometrical Wavelets and their Generalizations in Digital Image Coding and Processing*,
PhD Thesis, University of Silesia, Sosnowiec, Poland, 2005.
pdf

[3] Lisowska A., *Second Order Wedgelets in Image Coding*, Proceedings of EUROCON '07 Conference, Warsaw, Poland,
pp. 237-244, 2007.

## Smoothlets

Smoothlets were defined in [1] in 2011. They were also described in [2] and in the monograph [3]. The smoothlet is defined as a continuous function basing on a curvilinear beamlet. Smoothlets are generalizations of second order wedgelets. So, smoothlets adapt to location, scale, curvature and blur of an edge.

[1] Lisowska A., *Smoothlets - Multiscale Functions for Adaptive Representations of Images*, IEEE Transactions on Image
Processing, Vol. 20, No. 7, pp. 1777-1787, 2011. pdf

[2] Lisowska A., *Smoothlet Transform: Theory and Applications*, Advances in Imaging and Electron Physics, Elsevier,
Vol. 178, pp. 97-145, 2013.

[3] Lisowska A., *Geometrical Multiresolution Adaptive Transforms. Theory and Applications*, Springer, 2014.
abstract

## Multiwedgelets

Vectors of wedgelets, called shortly multiwedgelets, were introduced in [1] in 2013. Multiwedgelets were designed to efficiently represent multiedges, that is multiple edges which may cross or cover partially. So far, only the single edge model was taken into account in the literature. That is why multiwedgelets were defined. As a generalization of the single edge model, multiwedgelets adapt to location, scale, curvature and multiplicity of edges.

[1] Lisowska A., *Multiwedgelets in Image Denoising*, Lecture Notes in Electrical Engineering, Springer, Dordrecht,
Vol. 240, pp. 3-11, 2013. abstract

## Multismoothlets

Vectors of smoothlets, called shortly multismoothlets, were described in [2] in 2014. Their short description one can find in poster [1]. Multismoothlets represent multiedges with different level of blur. So, multismoothlets adapt to location, scale, curvature, blur and multiplicity of edges.

[1] Lisowska A., *Multismoothlets*, Compressed Sensing and its Applications, Berlin, Germany, 2013.
pdf

[2] Lisowska A., *Geometrical Multiresolution Adaptive Transforms. Theory and Applications*, Springer, 2014.
abstract