The Moment Problem in Hypercomplex Systems

This paper is devoted to give the necessary and sufficient conditions guarantees that the product of two generalized moment functions defined in a hypercomplex system L1(Q,m) is also generalized moment function in L1(Q,m) . Also, we prove that a bounded continuous function  in a hypercomplex system L1(Q,m) is conditionally positive definite if and only if is generalized moment functions defined in L1(Q,m) . Moreover, we will give the integral representations of a generalized moment functions defined in L1 (Q, m)

Connected Attribute Filtering Based on Contour Smoothness

A new attribute measuring the contour smoothness of 2-D objects is presented in the context of morphological attribute filtering. The attribute is based on the ratio of the circularity and non-compactness, and has a maximum of 1 for a perfect circle. It decreases as the object boundary becomes irregular. Computation on hierarchical image representation structures relies on five auxiliary data members and is rapid. Contour smoothness is a suitable descriptor for detecting and discriminating man-made structures from other image features. An example is demonstrated on a very-high-resolution satellite image using connected pattern spectra and the switchboard platform

The HR-Calculus: Enabling Information Processing with Quaternion Algebra

From their inception, quaternions and their division algebra have proven to be advantageous in modelling rotation/orientation in three-dimensional spaces and have seen use from the initial formulation of electromagnetic filed theory through to forming the basis of quantum filed theory. Despite their impressive versatility in modelling real-world phenomena, adaptive information processing techniques specifically designed for quaternion-valued signals have only recently come to the attention of the machine learning, signal processing, and control communities. The most important development in this direction is introduction of the HR-calculus, which provides the required mathematical foundation for deriving adaptive information processing techniques directly in the quaternion domain. In this article, the foundations of the HR-calculus are revised and the required tools for deriving adaptive learning techniques suitable for dealing with quaternion-valued signals, such as the gradient operator, chain and product derivative rules, and Taylor series expansion are presented. This serves to establish the most important applications of adaptive information processing in the quaternion domain for both single-node and multi-node formulations. The article is supported by Supplementary Material, which will be referred to as SM

Compressed sensing: Reconstruction of non-uniformly sampled multidimensional NMR data

© 2018 Wiley Periodicals, Inc. Nuclear magnetic resonance (NMR) spectroscopy is widely used across the physical, chemical, and biological sciences. A core component of NMR studies is multidimensional experiments, which enable correlation of properties from one or more NMR-active nuclei. In high-resolution biomolecular NMR, common nuclei are1H,15N, and13C, and triple resonance experiments using these three nuclei form the backbone of NMR structural studies. In other fields, a range of other nuclei may be used. Multidimensional NMR experiments provide unparalleled information content, but this comes at the price of long experiment times required to achieve the necessary resolution and sensitivity. Non-uniform sampling (NUS) techniques to reduce the required data sampling have existed for many decades. Recently, such techniques have received heightened interest due to the development of compressed sensing (CS) methods for reconstructing spectra from such NUS datasets. When applied jointly, these methods provide a powerful approach to dramatically improve the resolution of spectra per time unit and under suitable conditions can also lead to signal-to-noise ratio improvements. In this review, we explore the basis of NUS approaches, the fundamental features of NUS reconstruction using CS and applications based on CS approaches including the benefits of expanding the repertoire of biomolecular NMR experiments into higher dimensions. We discuss some of the recent algorithms and software packages and provide practical tips for recording and processing NUS data by CS

A Quaternionic Version Theory related to Spheroidal Functions

In dieser Arbeit wird eine neue Theorie der quaternionischen Funktionen vorgestellt, welche das Problem der Bestapproximation von Familien prolater und oblater sphäroidalen Funktionen im Hilberträumen behandelt. Die allgemeine Theorie beginnt mit der expliziten Konstruktion von orthogonalen Basen für Räume, definiert auf sphäroidalen Gebieten mit beliebiger Exzentrizität, deren Elemente harmonische, monogene und kontragene Funktionen sind und durch die Form der Gebiete parametrisiert werden. Eine detaillierte Studie dieser grundlegenden Elemente wird in dieser Arbeit durchgeführt. Der Begriff der kontragenen Funktion hängt vom Definitionsbereich ab und ist daher keine lokale Eigenschaft, während die Begriffe der harmonischen und monogenen Funktionen lokal sind. Es werden verschiedene Umwandlungsformeln vorgestellt, die Systeme harmonischer, monogener und kontragener Funktionen auf Sphäroiden unterschiedlicher Exzentrizität in Beziehung setzen. Darüber hinaus wird die Existenz gemeinsamer nichttrivialer kontragener Funktionen für Sphäroide jeglicher Exzentrizität gezeigt. Der zweite wichtige Beitrag dieser Arbeit betrifft eine quaternionische Raumfrequenztheorie für bandbegrenzte quaternionische Funktionen. Es wird eine neue Art von quaternionischen Signalen vorgeschlagen, deren Energiekonzentration im Raum und in den Frequenzbereichen unter der quaternionischen Fourier-Transformation maximal ist. Darüber hinaus werden diese Signale im Kontext der Spektralkonzentration als Eigenfunktionen eines kompakten und selbstadjungierteren quaternionischen Integraloperators untersucht und die grundlegenden Eigenschaften ihrer zugehörigen Eigenwerte werden detailliert beschrieben. Wenn die Konzentrationsgebiete beider Räume kugelförmig sind, kann der Winkelanteil dieser Signale explizit gefunden werden, was zur Lösung von mehreren eindimensionalen radialen Integralgleichungen führt. Wir nutzen die theoretischen Ergebnisse und harmonische Konjugierten um Klassen monogener Funktionen in verschiedenen Räumen zu konstruieren. Zur Charakterisierung der monogenen gewichteten Hardy- und Bergman-Räume in der Einheitskugel werden zwei konstruktive Algorithmen vorgeschlagen. Für eine reelle harmonische Funktion, die zu einem gewichteten Hardy- und Bergman-Raum gehört, werden die harmonischen Konjugiert in den gleichen Räumen gefunden. Die Beschränktheit der zugrundeliegenden harmonischen Konjugationsoperatoren wird in den angegebenen gewichteten Räumen bewiesen. Zusätzlich wird ein quaternionisches Gegenstück zum Satz von Bloch für monogene Funktionen bewiesen.This work presents a novel Quaternionic Function Theory associated with the best approximation problem in the setting of Hilbert spaces concerning families of prolate and oblate spheroidal functions. The general theory begins with the explicit construction of orthogonal bases for the spaces of harmonic, monogenic, and contragenic functions defined in spheroidal domains of arbitrary eccentricity, whose elements are parametrized by the shape of the corresponding spheroids. A detailed study regarding the elements that constitute these bases is carried out in this thesis. The notion of a contragenic function depends on the domain, and, therefore, it is not a local property in contrast to the concepts of harmonic and monogenic functions. Various conversion formulas that relate systems of harmonic, monogenic, and contragenic functions associated with spheroids of differing eccentricity are presented. Furthermore, the existence of standard nontrivial contragenic functions is shown for spheroids of any eccentricity. The second significant contribution presented in this work pertains to a quaternionic space-frequency theory for band-limited quaternionic functions. A new class of quaternionic signals is proposed, whose energy concentration in the space and the frequency domains are maximal under the quaternion Fourier transform. These signals are studied in the context of spatial-frequency concentration as eigenfunctions of a compact and self-adjoint quaternion integral operator. The fundamental properties of their associated eigenvalues are described in detail. When the concentration domains are spherical in both spaces, the angular part of these signals can be found explicitly, leading to a set of one-dimensional radial integral equations. The theoretical framework described in this work is applied to the construction of classes of monogenic functions in different spaces via harmonic conjugates. Two constructive algorithms are proposed to characterize the monogenic weighted Hardy and Bergman spaces in the Euclidean unit ball. For a real-valued harmonic function belonging to a Hardy and a weighted Bergman space, the harmonic conjugates in the same spaces are found. The boundedness of the underlying harmonic conjugation operators is proven in the given weighted spaces. Additionally, a quaternionic counterpart of Bloch’s Theorem is established for monogenic functions