Agnieszka Lisowska

Second Order Wedgelets in Image Coding

In these days efficient image coding plays a very important role. There are well known and recognized theories concerning this topic such as wavelets or new and recently developed ones as geometrical wavelets. The latter one, thanks to the possibility of catching discontinuities in different locations, scales and orientations better reflects the Human Visual System than classical wavelets. In the paper we proposed the improvement of the algorithm used in image coding based on wedgelets - a kind of geometrical wavelets. The proposed algorithm is based on second order wedgelets which are based not only on straight edges but also on fragments of second degree curves in order to ensure more sparse approximation of an image in rate distortion sense. The performed experiments confirmed that the use of second order wedgelets ensures better compression properties in image coding than the use of wedgelets.

Smoothlets—Multiscale Functions for Adaptive Representation of Images

In this paper a special class of functions called smoothlets is presented. They are defined as a generalization of wedgelets and second-order wedgelets. Unlike all known geometrical methods used in adaptive image approximation, smoothlets are continuous functions. They can adapt to location, size, rotation, curvature, and smoothness of edges. The M-term approximation of smoothlets is O(M-3) . In this paper, an image compression scheme based on the smoothlet transform is also presented. From the theoretical considerations and experiments, both described in the paper, it follows that smoothlets can assure better image compression than the other known adaptive geometrical methods, namely, wedgelets and second-order wedgelets.

Moments-Based Fast Wedgelet Transform

In the paper the moments-based fast wedgelet transform has been presented. In order to perform the classical wedgelet transform one searches the whole wedgelets’ dictionary to find the best matching. Whereas in the proposed method the parameters of wedgelet are computed directly from an image basing on moments computation. Such parameters describe wedgelet reflecting the edge present in the image. However, such wedgelet is not necessarily the best one in the meaning of Mean Square Error. So, to overcome that drawback, the method which improves the matching result has also been proposed. It works in the way that the better matching one needs to obtain the longer time it takes. The proposed transform works in linear time with respect to the number of pixels of the full quadtree decomposition of an image. More precisely, for an image of size N×N pixels the time complexity of the proposed wedgelet transform is O(N2log 2N).

Image denoising with second-order wedgelets

In this paper, the new adaptive geometrical multiresolution technique of image denoising based on second-order wedgelets is proposed. The results of denoising are compared with the other well-known multiresolution techniques, which are based on wavelets, curvelets and wedgelets. These methods are seen as representatives of different approaches of image approximation. All these approaches were described in detail and compared in the area of image denoising. From the experiments performed on tested images, it follows that the proposed method gives better results of denoising in comparison with the other described methods, especially for images with a well-defined geometry.

Polynomiography Based on the Nonstandard Newton-Like Root Finding Methods

A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.

Polynomiography via Ishikawa and Mann Iterations

The aim of this paper is to present some modifications of the complex polynomial roots finding visualization process. In this paper Ishikawa and Mann iterations are used instead of the standard Picard iteration. The name polynomiography was introduced by Kalantari for that visualization process and the obtained images are called polynomiographs. Polynomiographs are interesting both from educational and artistic points of view. By the use of different iterations we obtain quite new polynomiographs that look aestheatically pleasing comparing to the ones from standard Picard iteration. As examples we present some polynomiographs for complex polynomial equation z 3 − 1 = 0, permutation and doubly stochastic matrices. We believe that the results of this paper can inspire those who may be interested in created automatically aesthetic patterns. They also can be used to increase functionality of the existing polynomiography software.

Geometrical Multiscale Noise Resistant Method of Edge Detection

In the paper the multiscale geometrical noise resistant method of edge detection based on second order wedgelets has been presented. Unlike the other known methods the proposed one can detect arc edges as segments of second degree curves instead of a set of straight lines. Such curve edges are parameterized by only one additional parameter reflecting the curvature of the edge. That approach allows for more compact representation of edges in an image also better reflecting the image geometry than the other well known methods. Thanks to that the new method can be used as the first step in high performance object recognition techniques. The experiments confirmed high effectiveness of the method in edge detection including noisy images.

Automatic generation of aesthetic patterns with the use of dynamical systems

The aim of this paper is to present some modifications of the orbits generation algorithm of dynamical systems. The well-known Picard iteration is replaced by the more general one - Krasnosielskij iteration. Instead of one dynamical system, a set of them may be used. The orbits produced during the iteration process can be modified with the help of a probabilistic factor. By the use of aesthetic orbits generation of dynamical systems one can obtain unrepeatable collections of nicely looking patterns. Their geometry can be enriched by the use of the three colouring methods. The results of the paper can inspire graphic designers who may be interested in subtle aesthetic patterns created automatically.

Smoothlet Transform: Theory and Applications

Abstract A smoothlet is a generalization of a wedgelet achieved mainly by introducing continuity. In such a form, a smoothlet can adapt to a blurred edge better than a wedgelet. Because the edges present in images are usually more or less blurred, the blur horizon function is considered as the mathematical model of an edge instead of a horizon function used in the case of wedgelets. Here, the smoothlet transform is presented. It is defined in the way that the longer computations are performed, the better the result of an image approximation. The transform is defined basing on the moments-based fast wedgelet transform. Its computational complexity is O(N2log2N) for an image of size N×N pixels. The usefulness of the transform is presented on two image processing tasks: namely, image compression and denoising. In both applications, the smoothlet transform leads to the best results compared to other state-of-the-art methods.

Multiwedgelets in Image Denoising

In this paper the definition of a multiwedgelet is introduced. The multiwedgelet is defined as a vector of wedgelets. In order to use a multiwedgelet in image approximation its visualization and computation methods are also proposed. The application of multiwedgelets in image denoising is presented, as well. As follows from the experiments performed multiwedgelets assure better denoising results than the other known state-of-the-art methods.