We present important results from Hilbert space and functional analysis for understanding the subject ofReproducing kernel Hilbert spaces. We then showcase the underlying theory and properties of Reproducingkernel Hilbert Spaces. Finally, we show how the theory of reproducing kernel Hilbert spaces is applicable inboth interpolation and machine learning.

The Hilbert transform is a widely used transform in signal processing. In this thesis we explore its use for three different applications: electrocardiography, the Hilbert-Huang transform and modulation. For electrocardiography, we examine how and why the Hilbert transform can be used for QRS complex detection. Also, what are the advantages and limitations of this method? The Hilbert-Huang transform is a very popular method for spectral analysis for nonlinear and/or nonstationary processes. We examine its connection with the Hilbert transform and show limitations of the method. Lastly, the connection between the Hilbert transform and single-sideband modulation is investigated.

The Fourier transform (spectrum) of a signal is a complex function and is characterized by the magnitude and phase spectra. Phase retrieval is the reconstruction of the phase spectrum from the measurements of the magnitude spectrum. Such problems are encountered in imaging modalities such as X-ray crystallography, frequency-domain optical coherence tomography (FDOCT), quantitative phase microscopy, digital holography, etc., where only the magnitudes of the wavefront are detected by the sensors. The phase retrieval problem is ill-posed in general, since an in nite number of signals can have the same magnitude spectrum. Typical phase retrieval techniques rely on certain prior knowledge about the signal, such as its support or sparsity, to reconstruct the signal. A classical result in phase retrieval is that minimum-phase signals have log-magnitude and phase spectra that satisfy the Hilbert integral equations, thus facilitating exact phase retrieval. In this thesis, we demonstrate that there exist larger classes of signals beyond minimum-phase signals, for which exact phase retrieval is possible. We generalize Hilbert integral equations to 2-D, and also introduce a variant that we call the composite Hilbert transform in the context of 2-D periodic signals. Our first extension pertains to a particular type of parametric modelling of 2-D signals. While 1-D minimum-phase signals have a parametric representation, in terms of poles and zeros, there exists no such 2-D counterpart. We introduce a new class of parametric 2-D signals that possess the exact phase retrieval property, that is, their magnitude spectrum completely characterizes the signal. Starting from the magnitude spectrum, a sequence of non-linear operations lead us to a sum-of-exponentials signal, from which the parameters are computed employing concepts from high-resolution spectral estimation such as the annihilating filter and algebraically coupled matrix-pencil methods. We demonstrate that, for this new class of signals, our method outperforms existing techniques even in the presence of noise. Our second extension is to continuous-domain signals that lie in a principal shift-invariant space spanned by a known basis. Such signals are characterized by the basis combining coefficients. These signals need not be minimum-phase, but certain conditions on the coefficients lead to exact phase retrieval of the continuous-domain signal. In particular, we introduce the concept of causal, delta dominant (CDD) sequences, and show that such signals are characterized by their magnitude spectra. This condition pertains to the time/spatial-domain description of the signal, in contrast to the minimum-phase condition, which is described in the spectral domain. We show that there exist CDD sequences that are not minimum-phase, and vice versa. However, finite-length CDD sequences are always minimum-phase. Our method reconstructs the signal from the magnitude spectrum up to ma-chine precision. We thus have a class of continuous-domain signals that are neither causal nor minimum phase, and yet allow for exact phase retrieval. The shift-invariant structure is applicable to modelling signals encountered in imaging modalities such as FDOCT. We next present an application of 2-D phase retrieval to continuous-domain CDD signals in the context of quantiative phase microscopy. We develop sufficient conditions on the interfering reference wave for exact phase retrieval from magnitude measurements. In particular, we show that when the reference wave is a plane wave with magnitude greater that the intensity of the object wave, and when the carrier frequency is larger than the band-width of the object wave, we can reconstruct the object wave exactly. We demonstrate high-resolution reconstruction of our method on USAF target images. Our final and perhaps the most unifying contribution is in developing Hilbert integral equations for 2-D first-quadrant signals and in introducing the notion of generalized minimum-phase signals for both 1-D and 2-D signals. For 2-D continuous-domain, first-quadrant signals, we establish partial Hilbert transform relations between the real and imaginary parts of the spectrum. In the context of 2-D discrete-domain signals, we show that the partial Hilbert transform does not suffice and introduce the notion of composite Hilbert transform and establish the integral equations. We then introduce four classes of signals (combinations of 1-D/2-D and continuous/discrete-domain) that we call generalized minimum-phase signals, which satisfy corresponding Hilbert integral equations between log-magnitude and phase spectra, hence facilitating exact phase retrieval. This class of generalized minimum-phase signals subsumes the well known class of minimum-phase signals. We further show that, akin to minimum-phase signals, these signals also have stable inverses, which are also generalized minimum-phase signals.

Made available in DSpace on 2014-07-29T16:02:19Z (GMT). No. of bitstreams: 1 Dissertacao Bruno Trindade Reis.pdf: 1549283 bytes, checksum: 850cae1de80dba723aabf95e990ddd6a (MD5) Previous issue date: 2011-08-31 / We begin with a definition of the algebras Qn, who originated the study of algebra associated to directed graphs. Then, we define key concepts such as Hilbert series, graded and filtered algebras. Among the quadratic algebras, we introduce the Koszul algebras. The Hilbert series is a useful tool to study the Koszulity of a quadratic algebra. The homological interpretation of the coefficients of the Hilbert series of algebras associated with direct graphs allowed us to give conditions Koszulity these algebras in terms of the homological properties of the graph. We use this interpretation to construct algebras with Hilbert series prescribed. / Começamos definindo as álgebras Qn, que originaram o estudo das álgebras associadas a grafos orientados em níveis. Em seguida, definimos conceitos importantes, tais como séries de Hilbert , álgebras graduadas e álgebras filtradas. Entre as álgebras quadráticas, introduzimos as álgebras de Koszul. As séries de Hilbert são instrumentos úteis para estudar a Koszulidade de álgebras quadráticas. A interpretação homológica dos coeficientes da série de Hilbert de álgebras associadas a grafos em níveis nos permite dar condições de Koszulidade dessas álgebras em termos das propriedades homológicas do grafo. Usamos essa interpretação para construir álgebras com séries de Hilbert préestabelecidas.

Le rehaussement de la parole est un domaine du traitement du signal qui prend de plus en plus d'ampleur. En effet, dans un monde où la télécommunication connaît un véritable essor, les technologies se doivent d'être de plus en plus performantes afin de satisfaire au plus grand nombre. Les applications qui nécessitent un rehaussement de la parole sont très nombreuses, la plus évidente étant sans doute celle de la téléphonie mobile, où de nombreux bruits environnants peuvent gêner la qualité et l'intelligibilité du signal de parole transmis. Il existe à ce jour de nombreuses techniques de rehaussement de la parole. Celles-ci peuvent d'ores et déjà se décliner en deux catégories distinctes. En effet, certaines techniques utilisent plusieurs microphones et sont qualifiées de multi-capteur, tandis que d'autres techniques n'en utilisent qu'un seul et sont alors qualifiées de mono-capteur.Le présent sujet de recherche se situe dans la catégorie des techniques mono-capteurs qui utilisent principalement les propriétés statistiques de la parole et du bruit afin de réduire au mieux le signal de bruit. La Décomposition Modale Empirique, ou EMD, est une méthode de transformée de signaux qui est apparue récemment et qui suscite de plus en plus l'intérêt des chercheurs en rehaussement de la parole. L'EMD s'avère être une méthode de décomposition de signal très efficace car, contrairement aux transformées plus classiques, l'EMD est une transformée non linéaire et non stationnaire. Ses propriétés statistiques, en réponse au bruit blanc gaussien, ont permis de conclure sur le comportement de cette approche similaire à un banc de filtres quasi-dyadique. Les méthodes existantes de rehaussement de la parole basée sur la modélisation par EMD s'appuient toutes sur ce comportement dans leur démarche de réduction de bruit, et leur efficacité n'est validée que dans le cas de signaux de parole corrompus par du bruit blanc gaussien. Cependant, un algorithme de réduction de bruit n'est intéressant que s'il est efficace sur des bruits environnants de tous les jours. Ces travaux de recherche visent ainsi à déterminer les caractéristiques de l'EMD face à des signaux de parole corrompus par des bruits"réels", avant de comparer ces caractéristiques à ceux issues de signaux de parole corrompus par du bruit blanc gaussien. Les conclusions de cette étude sont finalement mises en pratiques dans le développement d'un système de réduction de bruit qui vise à séparer au mieux le bruit du signal de parole, et ce quel que soit le type de bruit rencontré.

International Telemetering Conference Proceedings / October 26-29, 1998 / Town & Country Resort Hotel and Convention Center, San Diego, California / This paper analyzes the performance of a modified correlation, or delay-locked loop (DLL). These devices typically cross-correlate the received signal with a differentiated version of the originally transmitted signal. This paper describes some interesting properties the loop assumes when the differentiator is replaced by a Hilbert transform. The loop will still track the timing offset of the code, but it will also be able to acquire the signal when the initial offset is greater than one chip time. The new loop may also be superior to traditional DLL in low SNR environments, since it is much less likely to lose lock. Since the new loop is highly non-linear, it is studied through the use of computer simulations.

This thesis is concerned with the study of the geometry and derived categories associated to the moduli problems of local systems and Higgs bundles in positive characteristic. As a cornerstone of our investigation, we establish a local system analogue of the BNR correspondence for Higgs bundles. This result (Proposition 4.3.1) relates flat connections to certain modules of an Azumaya algebra on the family of spectral curves. We prove properness over the semistable locus of the Hitchin map for local systems introduced by Laszlo–Pauly (Theorem 4.4.1). Moreover, we show that with respect to this Hitchin map, the moduli stack of local systems is étale locally equivalent to the moduli stack of Higgs bundles (Theorem 4.6.3) (with or without stability conditions). Subsequently, we study two-dimensional examples of moduli spaces of parabolic Higgs bundles and local systems (Theorem 5.2.1), given by equivariant Hilbert schemes of cotangent bundles of elliptic curves. Furthermore, the Hilbert schemes of points of these surfaces are equivalent to moduli spaces of parabolic Higgs bundles, respectively local systems (Theorem 5.3.1). The proof for local systems in positive characteristic relies on the properness results for the Hitchin fibration established earlier. The Autoduality Conjecture of Donagi–Pantev follows from Bridgeland–King–Reid’s McKay equivalence in these examples. The last chapter of this thesis is concerned with the con- struction of derived equivalences, resembling a Geometric Langlands Correspondence in positive characteristic, generalizing work of Bezrukavnikov–Braverman. Away from finitely many primes, we show that over the locus of integral spectral curves, the derived category of coherent sheaves on the stack of local systems is equivalent to a derived category of coherent D-modules on the stack of vector bundles. We conclude by establishing the Hecke eigenproperty of Arinkin’s autoduality and thereby of the Geometric Langlands equivalence in positive characteristic.

Management of data with a time dimension increases the overhead of storage and query processing in large database applications especially with the join operation, which is a commonly used and expensive relational operator whose processing is dependent on the size of the input relations. An index-based approach has been shown to improve the processing of a join operation, which in turn, improves the performance of querying historical data. Temporal data consist of tuples associated with a time interval value having a valid life span of different lengths. With join processing on temporal data, since tuples with longer life spans tend to overlap a greater number of joining tuples, they are likely to be accessed more often. The efficient performance of a temporal join depending on index-clustered data is the main theme studied and researched in this work. The presence of intervals having an extended data range in temporal data makes the join evaluation harder because temporal data are intrinsically multidimensional. Some temporal join processing methods create duplicates of tuples with long life spans to achieve clustering of similar data, which improves the performance on tuples that tend to be accessed more frequently. The proposed Hilbert-Temporal Join (Hilbert-TJ) join algorithm overcomes the need of data duplication by mapping temporal data into Hilbert curve space that is inherently clustered, thus allowing for fast retrieval and storage. A balanced B+ tree index structure was implemented to manage and query the data. The query method identifies data pages containing matching tuples that intersect a multidimensional region. Given that data pages consist of contiguously mapped points on the curve, the query process successively traverses along the curve to determine the next page that intersects the query region by iteratively partitioning the data space. The proposed Adaptive Replacement Cache-Temporal Data (ARC-TD) buffer replacement policy is built upon the Adaptive Replacement Cache (ARC) policy by favoring the cache retention of data pages in proportion to the average life span of the tuples in the buffer. By giving preference to tuples having long life spans, a higher cache hit ratio was evident. The caching priority is also balanced between recently and frequently accessed data. An evaluation and comparison study of the proposed Hilbert-TJ algorithm determined the relative performance with respect to a nested-loop join, a sort-merge join, and a partition-based join algorithm that use a multiversion B+ tree (MVBT) index. The metrics are based on a comparison between the processing time (disk I/O time plus CPU time), cache hit ratio, and index storage size needed to perform the temporal join. The study was conducted with comparisons in terms of the Least Recently Used (LRU), Least Frequently Used (LFU), ARC, and the new ARC-TD buffer replacement policy. Under the given conditions, the expected outcome was that by reducing data redundancy and considering the longevity of frequently accessed temporal data, better performance was achieved. Additionally, the Hilbert-TJ algorithm offers support to both valid-time and transaction-time data.

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A Cuntz-Pimsner algebra is a quotient of a generalized Toeplitz algebra. It is completely determined by a C*-correspondence, which consists of a right Hilbert A- module, E, and a *-homomorphism from the C*-algebra A into L(E), the adjointable operators on E. Some familiar examples of C*-algebras which can be recognized as Cuntz-Pimsner algebras include the Cuntz algebras, Cuntz-Krieger algebras, and crossed products of a C*-algebra by an action of the integers by automorphisms. In this dissertation, we construct a Cuntz-Pimsner Algebra associated to a dynam- ical system of a substitution tiling, which provides an alternate construction to the groupoid approach found in [3], and has the advantage of yielding a method for com- puting the K-Theory. / Graduate