Generalized iterated wreath products of cyclic groups and rooted trees correspondence

Consider the generalized iterated wreath product $\mathbb{Z}_{r_1}\wr \mathbb{Z}_{r_2}\wr \ldots \wr \mathbb{Z}_{r_k}$ where $r_i \in \mathbb{N}$. We prove that the irreducible representations for this class of groups are indexed by a certain type of rooted trees. This provides a Bratteli diagram for the generalized iterated wreath product, a simple recursion formula for the number of irreducible representations, and a strategy to calculate the dimension of each irreducible representation. We calculate explicitly fast Fourier transforms (FFT) for this class of groups, giving literature's fastest FFT upper bound estimate.Comment: 15 pages, to appear in Advances in the Mathematical Science

Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence

Consider the generalized iterated wreath product $S_{r_1}\wr \ldots \wr S_{r_k}$ of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection between the equivalence classes of ordinary irreducible representations of the generalized iterated wreath product and orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. Finally, we give rough upper bound estimates for fast Fourier transforms.Comment: 18 pages, to appear in Advances in the Mathematical Sciences. arXiv admin note: text overlap with arXiv:1409.060

Fast multiresolution contour completion

We consider the problem of improving contour detection by filling gaps between collinear contour pieces. A fast algorithm is proposed which takes into account local edge orientation and local curvature. Each edge point is replaced by a curved elongated patch, whose orientation and curvature match the local edge orientation and edge. The proposed contour completion algorithm is integrated in a multiresolution framework for contour detection. Experimental results show the superiority of the proposed method to other well-established approaches

Edge and corner preserving smoothing for artistic imaging

What visually distinguishes a painting from a photograph is often the absence of texture and the sharp edges: in many paintings, edges are sharper than in photographic images while textured areas contain less detail. Such artistic effects can be achieved by filters that smooth textured areas while preserving, or enhancing, edges and corners. However, not all edge preserving smoothers are suitable for artistic imaging. This study presents a generalization of the well know Kuwahara filter aimed at obtaining an artistic effect. Theoretical limitations of the Kuwahara filter are discussed and solved by the new nonlinear operator proposed here. Experimental results show that the proposed operator produces painting-like output images and is robust to corruption of the input image such as blurring. Comparison with existing techniques shows situations where traditional edge preserving smoothers that are commonly used for artistic imaging fail while our approach produces good results

A multiscale approach to contour detection by texture suppression - art. no. 60640D

In this paper we propose a multiscale biologically motivated technique for contour detection by texture suppression. Standard edge detectors react to all the local luminance changes, irrespective whether they are due to the contours of the objects represented in the scene, rather than to natural texture like grass, foliage, water, etc. Moreover, edges due to texture are often stronger than edges due to true contours. This implies that further processing is needed to discriminate true contours from texture edges. In this contribution we exploit the fact that, in a multiresolution analysis, at coarser scales, only the edges due to object contours are present while texture edges disappear. This is used in combination with surround inhibition, a biologically motivated technique for texture suppression, in order to build a contour detector which is insensitive to texture. The experimental results show that our approach is also robust to additive noise

Inverting Monotonic Nonlinearities by Entropy Maximization

This paper proposes a new method for blind inversion of a monotonic nonlinear map applied to a sum of random variables. Such kinds of mixtures of random variables are found in source separation and Wiener system inversion problems, for example. The importance of our proposed method is based on the fact that it permits to decouple the estimation of the nonlinear part (nonlinear compensation) from the estimation of the linear one (source separation matrix or deconvolution filter), which can be solved by applying any convenient linear algorithm. Our new nonlinear compensation algorithm, the MaxEnt algorithm, generalizes the idea of Gaussianization of the observation by maximizing its entropy instead. We developed two versions of our algorithm based either in a polynomial or a neural network parameterization of the nonlinear function. We provide a sufficient condition on the nonlinear function and the probability distribution that gives a guarantee for the MaxEnt method to succeed compensating the distortion. Through an extensive set of simulations, MaxEnt is compared with existing algorithms for blind approximation of nonlinear maps. Experiments show that MaxEnt is able to successfully compensate monotonic distortions outperforming other methods in terms of the obtained Signal to Noise Ratio in many important cases, for example when the number of variables in a mixture is small. Besides its ability for compensating nonlinearities, MaxEnt is very robust, i.e. showing small variability in the results. 1