Optimal Consensus set for nD Fixed Width Annulus Fitting

International audienceThis paper presents a method for fitting a nD fixed width spherical shell to a given set of nD points in an image in the presence of noise by maximizing the number of inliers, namely the consensus set. We present an algorithm, that provides the optimal solution(s) within a time complexity O(N n+1 log N) for dimension n, N being the number of points. Our algorithm guarantees optimal solution(s) and has lower complexity than previous known methods

A perceptual learning model to discover the hierarchical latent structure of image collections

Biology has been an unparalleled source of inspiration for the work of researchers in several scientific and engineering fields including computer vision. The starting point of this thesis is the neurophysiological properties of the human early visual system, in particular, the cortical mechanism that mediates learning by exploiting information about stimuli repetition. Repetition has long been considered a fundamental correlate of skill acquisition andmemory formation in biological aswell as computational learning models. However, recent studies have shown that biological neural networks have differentways of exploiting repetition in forming memory maps. The thesis focuses on a perceptual learning mechanism called repetition suppression, which exploits the temporal distribution of neural activations to drive an efficient neural allocation for a set of stimuli. This explores the neurophysiological hypothesis that repetition suppression serves as an unsupervised perceptual learning mechanism that can drive efficient memory formation by reducing the overall size of stimuli representation while strengthening the responses of the most selective neurons. This interpretation of repetition is different from its traditional role in computational learning models mainly to induce convergence and reach training stability, without using this information to provide focus for the neural representations of the data. The first part of the thesis introduces a novel computational model with repetition suppression, which forms an unsupervised competitive systemtermed CoRe, for Competitive Repetition-suppression learning. The model is applied to generalproblems in the fields of computational intelligence and machine learning. Particular emphasis is placed on validating the model as an effective tool for the unsupervised exploration of bio-medical data. In particular, it is shown that the repetition suppression mechanism efficiently addresses the issues of automatically estimating the number of clusters within the data, as well as filtering noise and irrelevant input components in highly dimensional data, e.g. gene expression levels from DNA Microarrays. The CoRe model produces relevance estimates for the each covariate which is useful, for instance, to discover the best discriminating bio-markers. The description of the model includes a theoretical analysis using Huber’s robust statistics to show that the model is robust to outliers and noise in the data. The convergence properties of themodel also studied. It is shown that, besides its biological underpinning, the CoRe model has useful properties in terms of asymptotic behavior. By exploiting a kernel-based formulation for the CoRe learning error, a theoretically sound motivation is provided for the model’s ability to avoid local minima of its loss function. To do this a necessary and sufficient condition for global error minimization in vector quantization is generalized by extending it to distance metrics in generic Hilbert spaces. This leads to the derivation of a family of kernel-based algorithms that address the local minima issue of unsupervised vector quantization in a principled way. The experimental results show that the algorithm can achieve a consistent performance gain compared with state-of-the-art learning vector quantizers, while retaining a lower computational complexity (linear with respect to the dataset size). Bridging the gap between the low level representation of the visual content and the underlying high-level semantics is a major research issue of current interest. The second part of the thesis focuses on this problem by introducing a hierarchical and multi-resolution approach to visual content understanding. On a spatial level, CoRe learning is used to pool together the local visual patches by organizing them into perceptually meaningful intermediate structures. On the semantical level, it provides an extension of the probabilistic Latent Semantic Analysis (pLSA) model that allows discovery and organization of the visual topics into a hierarchy of aspects. The proposed hierarchical pLSA model is shown to effectively address the unsupervised discovery of relevant visual classes from pictorial collections, at the same time learning to segment the image regions containing the discovered classes. Furthermore, by drawing on a recent pLSA-based image annotation system, the hierarchical pLSA model is extended to process and representmulti-modal collections comprising textual and visual data. The results of the experimental evaluation show that the proposed model learns to attach textual labels (available only at the level of the whole image) to the discovered image regions, while increasing the precision/ recall performance with respect to flat, pLSA annotation model

Quantum Entanglement in Time

In this doctoral thesis we provide one of the first theoretical expositions on a quantum effect known as entanglement in time. It can be viewed as an interdependence of quantum systems across time, which is stronger than could ever exist between classical systems. We explore this temporal effect within the study of quantum information and its foundations as well as through relativistic quantum information. An original contribution of this thesis is the design of one of the first applications of entanglement in time.Comment: 271 pages, PhD Thesis (Victoria University of Wellington

A Fuzzy Approach for Topological Data Analysis

Geometry and topology are becoming more powerful and dominant in data analysis because of their outstanding characteristics. It has emerged recently as a promising research area, known as Topological Data Analysis (TDA), for modern computer science. In recent years, the Mapper algorithm, an outstanding TDA representative, is increasingly completed with a stabilized theoretical foundation and practical applications and diverse, intuitive, user-friendly implementations. From a theoretical perspective, the Mapper algorithm is still a fuzzy clustering algorithm, with a visualization capability to extract the shape summary of data. However, its outcomes are still very sensitive to the parameter choice, including resolution and function. Therefore, there is a need to reduce the dependence on its parameters significantly. This idea is exciting and can be solved thanks to the outstanding characteristics of fuzzy clustering. The Mapper clustering ability is getting more potent by the support from well-known techniques. Therefore, this combination is expected to usefully and powerfully solve some problems encountered in many fields. The main research goal of this thesis is to approach TDA by fuzzy theory to create the interrelationships between them in terms of clustering. Explicitly speaking, the Mapper algorithm represents TDA, and the Fuzzy $C$-Means (FCM) algorithm represents fuzzy theory. They are combined to promote their advantages and overcome their disadvantages. On the one hand, the FCM algorithm helps the Mapper algorithm simplify the choice of parameters to obtain the most informative presentation and is even more efficient in data clustering. On the other hand, the FCM algorithm is equipped with the outstanding features of the Mapper algorithm in simplifying and visualizing data with qualitative analysis. This thesis focuses on conquering and achieving the following aims: (1) Summarizing the theoretical foundations and practical applications of the Mapper algorithm in the flow of literature with improved versions and various implementations. (2) Optimizing the cover choice of the Mapper algorithm in the direction of dividing the filter range automatically into irregular intervals with a random overlapping percentage by using the FCM algorithm. (3) Constructing a novel method for mining data that can exhibit the same clustering ability as the FCM algorithm and reveal some meaningful relationships by visualizing the global shape of data supplied by the Mapper algorithm.Geometrie a topologie se stávají silnějšími a dominantnějšími v analýze dat díky svým vynikajícím vlastnostem. Nedávno se objevila jako slibná výzkumná oblast, známá jako topologická analýza dat (TDA), pro moderní informatiku. V posledních letech je algoritmus Mapper, vynikající představitel TDA, stále více doplněn o stabilizovaný teoretický základ a praktické aplikace a rozmanité, intuitivní a uživatelsky přívětivé implementace. Z teoretického hlediska je algoritmus Mapper stále fuzzy shlukovací algoritmus se schopností vizualizace extrahovat souhrn tvaru dat. Jeho výsledky jsou však stále velmi citlivé na volbu parametrů, včetně rozlišení a funkce. Proto je potřeba výrazně snížit závislost na jeho parametrech. Tato myšlenka je vzrušující a lze ji vyřešit díky vynikajícím vlastnostem fuzzy shlukování. Schopnost shlukování Mapperu je stále silnější díky podpoře známých technik. Proto se očekává, že tato kombinace užitečně a účinně vyřeší některé problémy, se kterými se setkáváme v mnoha oblastech. Hlavním výzkumným cílem této práce je přiblížit TDA pomocí fuzzy teorie a vytvořit mezi nimi vzájemné vztahy z hlediska shlukování. Explicitně řečeno, algoritmus Mapper představuje TDA a algoritmus Fuzzy $C$-Means (FCM) představuje fuzzy teorii. Jsou kombinovány, aby podpořily své výhody a překonaly své nevýhody. Na jedné straně algoritmus FCM pomáhá algoritmu Mapper zjednodušit výběr parametrů pro získání nejinformativnější prezentace a je ještě efektivnější při shlukování dat. Na druhé straně je algoritmus FCM vybaven vynikajícími vlastnostmi algoritmu Mapper pro zjednodušení a vizualizaci dat pomocí kvalitativní analýzy. Tato práce se zaměřuje na dobývání a dosažení následujících cílů: (1) Shrnutí teoretických základů a praktických aplikací Mapperova algoritmu v toku literatury s vylepšenými verzemi a různými implementacemi. (2) Optimalizace volby pokrytí algoritmu Mapper ve směru automatického rozdělení rozsahu filtru do nepravidelných intervalů s náhodně se překrývajícím procentem pomocí algoritmu FCM. (3) Vytvoření nové metody pro těžbu dat, která může vykazovat stejnou schopnost shlukování jako algoritmus FCM a odhalit některé smysluplné vztahy vizualizací globálního tvaru dat poskytovaných algoritmem Mapper.460 - Katedra informatikyvyhově

Segmentation and quantification of spinal cord gray matter–white matter structures in magnetic resonance images

This thesis focuses on finding ways to differentiate the gray matter (GM) and white matter (WM) in magnetic resonance (MR) images of the human spinal cord (SC). The aim of this project is to quantify tissue loss in these compartments to study their implications on the progression of multiple sclerosis (MS). To this end, we propose segmentation algorithms that we evaluated on MR images of healthy volunteers. Segmentation of GM and WM in MR images can be done manually by human experts, but manual segmentation is tedious and prone to intra- and inter-rater variability. Therefore, a deterministic automation of this task is necessary. On axial 2D images acquired with a recently proposed MR sequence, called AMIRA, we experiment with various automatic segmentation algorithms. We first use variational model-based segmentation approaches combined with appearance models and later directly apply supervised deep learning to train segmentation networks. Evaluation of the proposed methods shows accurate and precise results, which are on par with manual segmentations. We test the developed deep learning approach on images of conventional MR sequences in the context of a GM segmentation challenge, resulting in superior performance compared to the other competing methods. To further assess the quality of the AMIRA sequence, we apply an already published GM segmentation algorithm to our data, yielding higher accuracy than the same algorithm achieves on images of conventional MR sequences. On a different topic, but related to segmentation, we develop a high-order slice interpolation method to address the large slice distances of images acquired with the AMIRA protocol at different vertebral levels, enabling us to resample our data to intermediate slice positions. From the methodical point of view, this work provides an introduction to computer vision, a mathematically focused perspective on variational segmentation approaches and supervised deep learning, as well as a brief overview of the underlying project's anatomical and medical background

On the information theory of clustering, registration, and blockchains

Progress in data science depends on the collection and storage of large volumes of reliable data, efficient and consistent inference based on this data, and trusting such computations made by untrusted peers. Information theory provides the means to analyze statistical inference algorithms, inspires the design of statistically consistent learning algorithms, and informs the design of large-scale systems for information storage and sharing. In this thesis, we focus on the problems of reliability, universality, integrity, trust, and provenance in data storage, distributed computing, and information processing algorithms and develop technical solutions and mathematical insights using information-theoretic tools. In unsupervised information processing we consider the problems of data clustering and image registration. In particular, we evaluate the performance of the max mutual information method for image registration by studying its error exponent and prove its universal asymptotic optimality. We further extend this to design the max multiinformation method for universal multi-image registration and prove its universal asymptotic optimality. We then evaluate the non-asymptotic performance of image registration to understand the effects of the properties of the image transformations and the channel noise on the algorithms. In data clustering we study the problem of independence clustering of sources using multivariate information functionals. In particular, we define consistent image clustering algorithms using the cluster information, and define a new multivariate information functional called illum information that inspires other independence clustering methods. We also consider the problem of clustering objects based on labels provided by temporary and long-term workers in a crowdsourcing platform. Here we define budget-optimal universal clustering algorithms using distributional identicality and temporal dependence in the responses of workers. For the problem of reliable data storage, we consider the use of blockchain systems, and design secure distributed storage codes to reduce the cost of cold storage of blockchain ledgers. Additionally, we use dynamic zone allocation strategies to enhance the integrity and confidentiality of these systems, and frame optimization problems for designing codes applicable for cloud storage and data insurance. Finally, for the problem of establishing trust in computations over untrusting peer-to-peer networks, we develop a large-scale blockchain system by defining the validation protocols and compression scheme to facilitate an efficient audit of computations that can be shared in a trusted manner across peers over the immutable blockchain ledger. We evaluate the system over some simple synthetic computational experiments and highlights its capacity in identifying anomalous computations and enhancing computational integrity

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group