Data compression and harmonic analysis

In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing

Espaços de funções com suavidade generalizada e integrabilidade variável

Suavidade generalizada e integrabilidade variável são dois importantes tópicos de investigação na teoria de espaços de funções. Neste trabalho consideramos tanto espaços de Besov com suavidade generalizada como espaços de Lebesgue (e de Sobolev correspondentes) com parâmetro de integração variável. São estudadas no caso geral propriedades de interpolação com parâmetro função de espaços de Besov generalizados, as quais estendem resultados já conhecidos formulados no caso Banach. Além disso, obtemos decomposições em "wavelets" para estes espaços através do uso de técnicas de interpolação apropriadas, que por sua vez são usadas para obter novos resultados de interpolação. Potenciais de Riesz e de Bessel são tratados no âmbito dos espaços de Lebesgue com expoente variável. Em particular, estudamos a inversão do operador potencial de Riesz e apresentamos uma caracterização, tanto para os espaços de potenciais de Riesz como para os espaços de potenciais de Bessel, em termos de convergência de integrais hipersingulares. Também lidamos com desigualdades pontuais no quadro dos espaços de Sobolev com parâmetro de integração variável. Tais desigualdades são usadas para generalizar imersões de Sobolev clássicas ao contexto de expoentes variáveis, no caso em que, para além de condições naturais de regularidade, o expoente toma valores superiores à dimensão do espaço euclidiano. Por outro lado, são dados resultados acerca da limitação de operadores hipersingulares definidos em espaços de Sobolev variáveis em domínios limitados.Generalized smoothness and variable integrability are two important research topics in the theory of function spaces. In this work, we consider both Besov spaces with generalized smoothness and Lebesgue (and corresponding Sobolev) spaces with variable exponent. Interpolation properties with function parameter of generalized Besov spaces are studied in the general case, which extends already known results stated for the Banach case. Moreover, we obtain wavelet decompositions for these spaces by using suitable interpolation techniques, which in turn are used to get new interpolation results. Riesz and Bessel potentials are considered within the framework of the variable exponent Lebesgue spaces. In particular, we study the inversion of the Riesz potential operator and give a characterization both for the Riesz potential spaces and for the Bessel potential spaces, in terms of convergence of hypersingular integrals. We also deal with pointwise inequalities in the context of the variable Sobolev spaces. Such inequalities are used to extend some classical Sobolev embeddings to the variable exponent setting, in the case when, besides natural assumptions, the exponent takes values greater than the dimension of the Euclidean space. Furthermore, boundedness results for hypersingular integral operators on variable Sobolev spaces over bounded domains are given

Poiesis in/between the Transferential Matrix: Insight, Imagination and the Relational Interpretation

The most important question for the Psychoanalytic Process Research is presumably what Mitchell calls the problem of “bootstrapping” the transferential matrix: how do the members of the dyad manage to disengage from being ‘heard’ according to old or unsuitable affective categories? On the grounds of a bi-phasic Conceptual and in-depth Analysis of the Psychoanalytic Complexity literature, I construct a minimal model of the psychoanalytic process as a theoretical context for conducting Process Research. According to the ‘story’ that I have read in the literature four main themes describe the process: a) the gradual emergence of a ‘phenomenological’ language that facilitates the flow of experience, b) the coupling, synchronicity and coordination of analyst and analysand, in ‘phase’ and ‘anti-phase’ at several levels, c) the shifting of the mental states and the thin and delicate slicing and sampling of experience that actualizes the emergence of mental objects and finally, d) Scaling that involves all those ‘mental’ processes that correct for the excesses or the deficiencies that are made evident during the shifting of mental states. Experience is generated as we ‘couple and shift’, and generative tensions appear as we ‘scale’ through this coupling and shifting process. Enactments, role-responsive transferences and countertransferences, testing of the transference and alliance or communication ruptures appear as coupled oscillating patterns that have both a repetitive and a developmental dynamic. Regarding the question of how we should study ‘Coupling, Shifting and Scaling’ I propose the adoption of an Enactivist epistemological framework which perceives the mind not as the workings of a representational machine but as a living process and the expression of an embodied living organism which in a “precarious” state of “needful freedom” (Jonas, in Thompson, 2007) strives to make sense of its environment. On the grounds of this framework I defend the view that we should study Scaling as an expression of the ‘radical dialogicality’ of the human mind that underlies the ‘structuring of experience’. I examine this ‘radical dialogicality’ at the level of inter-hemispheric differences, psychopathology and the enactive structuring of experience and the horizon of affective affordances in the clinical process. Finally, on the grounds of this conceptual analysis and its application to a case-study, I try to defend the view that, adopting relevant “dialogical” and micro-analytic methodological tools, we can achieve an appropriate level of ‘resolution’ so as to study “bootstrapping” at the moment-to-moment shifts in the experiential states or the shifts in attitudes that appear at bifurcation points in the system’s evolution. Through Scaling, the clinical dyad strives for a “maximum grip” of those experiential dimensions that carry the potential to expand the shared reality as a generative field and engage those surfaces of experience that bridge lost connections and separations, by fractalizing the dimensionality of the generative space. A detailed examination of the Scaling processes may bring us closer to a better understanding of the problem of “bootstrapping”

A Search for Home: Navigating Change in \u3ci\u3eBattlestar Galactica\u3c/i\u3e

This dissertation explores the various ways in which the multiple leaders portrayed in the science fiction television series Battlestar Galactica (2003-2009) navigate extreme conditions of continual change. In addition, the dissertation contains a discussion of the larger narrative themes of love, forgiveness, redemption, and embracing the Other as principles effective leaders must cultivate. Through an interpretation of this specific popular media text, a deeper emotional sensitivity to and understanding of leadership, positive and negative, during extreme crises is gained. Furthermore, the series serves as a vehicle through which viewers can reflect on and engage in their own self-awareness about issues surrounding leadership and reconsider personal paradigms based on the depiction presented in the narrative. The choice for using an interpretive hermeneutic method for this dissertation comes from the specific desire to understand the visual text of Battlestar Galactica in relation to leadership studies. The goal is neither to predict behaviors nor to examine an individual case against theory. My intent is to develop our further and deeper understanding of leadership in extremis, while questioning how the visual text may influence our perceptions of leadership theory and practice. The electronic version of this dissertation is at OhioLink ETD Center, www.ohiolink.edu/et