Topological defects in lattice models and affine Temperley-Lieb algebra

This paper is the first in a series where we attempt to define defects in critical lattice models that give rise to conformal field theory topological defects in the continuum limit. We focus mostly on models based on the Temperley-Lieb algebra, with future applications to restricted solid-on-solid (also called anyonic chains) models, as well as non-unitary models like percolation or self-avoiding walks. Our approach is essentially algebraic and focusses on the defects from two points of view: the "crossed channel" where the defect is seen as an operator acting on the Hilbert space of the models, and the "direct channel" where it corresponds to a modification of the basic Hamiltonian with some sort of impurity. Algebraic characterizations and constructions are proposed in both points of view. In the crossed channel, this leads us to new results about the center of the affine Temperley-Lieb algebra; in particular we find there a special subalgebra with non-negative integer structure constants that are interpreted as fusion rules of defects. In the direct channel, meanwhile, this leads to the introduction of fusion products and fusion quotients, with interesting mathematical properties that allow to describe representations content of the lattice model with a defect, and to describe its spectrum.Comment: 41

Signal processing algorithms for enhanced image fusion performance and assessment

The dissertation presents several signal processing algorithms for image fusion in noisy multimodal conditions. It introduces a novel image fusion method which performs well for image sets heavily corrupted by noise. As opposed to current image fusion schemes, the method has no requirements for a priori knowledge of the noise component. The image is decomposed with Chebyshev polynomials (CP) being used as basis functions to perform fusion at feature level. The properties of CP, namely fast convergence and smooth approximation, renders it ideal for heuristic and indiscriminate denoising fusion tasks. Quantitative evaluation using objective fusion assessment methods show favourable performance of the proposed scheme compared to previous efforts on image fusion, notably in heavily corrupted images. The approach is further improved by incorporating the advantages of CP with a state-of-the-art fusion technique named independent component analysis (ICA), for joint-fusion processing based on region saliency. Whilst CP fusion is robust under severe noise conditions, it is prone to eliminating high frequency information of the images involved, thereby limiting image sharpness. Fusion using ICA, on the other hand, performs well in transferring edges and other salient features of the input images into the composite output. The combination of both methods, coupled with several mathematical morphological operations in an algorithm fusion framework, is considered a viable solution. Again, according to the quantitative metrics the results of our proposed approach are very encouraging as far as joint fusion and denoising are concerned. Another focus of this dissertation is on a novel metric for image fusion evaluation that is based on texture. The conservation of background textural details is considered important in many fusion applications as they help define the image depth and structure, which may prove crucial in many surveillance and remote sensing applications. Our work aims to evaluate the performance of image fusion algorithms based on their ability to retain textural details from the fusion process. This is done by utilising the gray-level co-occurrence matrix (GLCM) model to extract second-order statistical features for the derivation of an image textural measure, which is then used to replace the edge-based calculations in an objective-based fusion metric. Performance evaluation on established fusion methods verifies that the proposed metric is viable, especially for multimodal scenarios

A Hybrid Chebyshev-ICA Image Fusion Method Based on Regional Saliency

An image fusion method that performs robustly for image sets heavily corrupted by noise is presented in this paper. The approach combines the advantages of two state-of-the-art fusion techniques, namely Independent Component Analysis (ICA) and Chebyshev Poly-nomial Analysis (CPA) fusion. Fusion using ICA performs well in transferring the salient features of the input images into the composite output, but its performance deteriorates severely under mild to moderate noise conditions. CPA fusion is robust under severe noise conditions, but eliminates the high frequency information of the images involved. We pro-pose to use ICA fusion within high activity image areas, identified by edges and strong textured surfaces and CPA fusion in low activity areas identified by uniform background regions and weak texture. A binary image map is used for selecting the appropriate method, which is constructed by a standard edge detector followed by morphological operators. The results of the proposed approach are very encouraging as far as joint fusion and denoising is concerned. The works presented may prove beneficial for future image fusion tasks in real world applications such as surveillance, where noise is heavily present

Adaptive chebyshev fusion of vegetation imagery based on SVM classifier

A novel adaptive image fusion method by using Chebyshev polynomial analysis (CPA), for applications in vegetation satellite imagery, is introduced in this paper. Fusion is a technique that enables the merging of two satellite cameras: panchromatic and multi-spectral, to produce higher quality satellite images to address agricurtural and vegetation issues such as soiling, floods and crop harvesting. Recent studies show Chebyshev polynomials to be effective in image fusion mainly in medium to high noise conditions, as per real-life satellite conditions. However, its application was limited to heuristics. In this research, we have proposed a way to adaptively select the optimal CPA parameters according to user specifications. Support vector machines (SVM) is used as a classifying tool to estimate the noise parameters, from which the appropriate CPA degree is utilised to perform image fusion according to a look-up table. Performance evaluation affirms the approach’s ability in reducing the computational complexity to perform fusion. Overall, adaptive CPA fusion is able to optimize an image fusion system’s resources and processing time. It therefore may be suitably incorporated onto real hardware for use on vegetation satellite imagery

Polynomial fusion rings of W-extended logarithmic minimal models

The countably infinite number of Virasoro representations of the logarithmic minimal model LM(p,p') can be reorganized into a finite number of W-representations with respect to the extended Virasoro algebra symmetry W. Using a lattice implementation of fusion, we recently determined the fusion algebra of these representations and found that it closes, albeit without an identity for p>1. Here, we provide a fusion-matrix realization of this fusion algebra and identify a fusion ring isomorphic to it. We also consider various extensions of it and quotients thereof, and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings. One particular extension is reminiscent of the fundamental fusion algebra of LM(p,p') and offers a natural way of introducing the missing identity for p>1. Working out explicit fusion matrices is facilitated by a further enlargement based on a pair of mutual Moore-Penrose inverses intertwining between the W-fundamental and enlarged fusion algebras.Comment: 48 page

Around multivariate Schmidt-Spitzer theorem

Given an arbitrary complex-valued infinite matrix A and a positive integer n we introduce a naturally associated polynomial basis B_A of C[x0...xn]. We discuss some properties of the locus of common zeros of all polynomials in B_A having a given degree m; the latter locus can be interpreted as the spectrum of the m*(m+n)-submatrix of A formed by its m first rows and m+n first columns. We initiate the study of the asymptotics of these spectra when m goes to infinity in the case when A is a banded Toeplitz matrix. In particular, we present and partially prove a conjectural multivariate analog of the well-known Schmidt-Spitzer theorem which describes the spectral asymptotics for the sequence of principal minors of an arbitrary banded Toeplitz matrix. Finally, we discuss relations between polynomial bases B_A and multivariate orthogonal polynomials

Zero modes' fusion ring and braid group representations for the extended chiral su(2) WZNW model

The zero modes' Fock space for the extended chiral $su(2)$ WZNW model gives room to a realization of the Grothendieck fusion ring of representations of the restricted $U_q sl(2)$ quantum universal enveloping algebra (QUEA) at an even ($2h$-th) root of unity, and of its extension by the Lusztig operators. It is shown that expressing the Drinfeld images of canonical characters in terms of Chebyshev polynomials of the Casimir invariant $C$ allows a streamlined derivation of the characteristic equation of $C$ from the defining relations of the restricted QUEA. The properties of the fusion ring of the Lusztig's extension of the QUEA in the zero modes' Fock space are related to the braiding properties of correlation functions of primary fields of the extended $su(2)_{h-2}$ current algebra model.Comment: 36 pages, 1 figure; version 3 - improvements in Sec. 2 and 3: definitions of the double, as well as R- (and M-)matrix changed to fit the zero modes' one

The tensor structure on the representation category of the Wp\mathcal{W}_p triplet algebra

We study the braided monoidal structure that the fusion product induces on the abelian category $\mathcal{W}_p$-mod, the category of representations of the triplet $W$-algebra $\mathcal{W}_p$. The $\mathcal{W}_p$-algebras are a family of vertex operator algebras that form the simplest known examples of symmetry algebras of logarithmic conformal field theories. We formalise the methods for computing fusion products, developed by Nahm, Gaberdiel and Kausch, that are widely used in the physics literature and illustrate a systematic approach to calculating fusion products in non-semi-simple representation categories. We apply these methods to the braided monoidal structure of $\mathcal{W}_p$-mod, previously constructed by Huang, Lepowsky and Zhang, to prove that this braided monoidal structure is rigid. The rigidity of $\mathcal{W}_p$-mod allows us to prove explicit formulae for the fusion product on the set of all simple and all projective $\mathcal{W}_p$-modules, which were first conjectured by Fuchs, Hwang, Semikhatov and Tipunin; and Gaberdiel and Runkel.Comment: 58 pages; edit: added references and revisions according to referee reports. Version to appear on J. Phys.