57 research outputs found

    The Study of Properties of n-D Analytic Signals and Their Spectra in Complex and Hypercomplex Domains

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    In the paper, two various representations of a n-dimensional (n-D) real signal u(x1,x2,
,xn) are investigated. The first one is the n-D complex analytic signal with a single-orthant spectrum defined by Hahn in 1992 as the extension of the 1-D Gabor’s analytic signal. It is compared with two hypercomplex approaches: the known n-D Clifford analytic signal and the Cayley-Dickson analytic signal defined by the Author in 2009. The signal-domain and frequency-domain definitions of these signals are presented and compared in 2-D and 3-D. Some new relations between the spectra in 2-D and 3-D hypercomplex domains are presented. The paper is illustrated with the example of a 2-D separable Cauchy pulse

    Octonion special affine fourier transform: pitt's inequality and the uncertainty principles

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    The special affine Fourier transform (SAFT) is an extended version of the classical Fourier transform and incorporates various signal processing tools which include the Fourier transforms, the fractional Fourier transform, the linear canonical transform, and other related transforms. This paper aims to introduce a novel octonion special affine Fourier transform (O-SAFT) and establish several classes of uncertainty inequalities for the proposed transform. We begin by studying the norm split and energy conservation properties of the proposed (O-SAFT). Afterwards, we generalize several uncertainty relations for the (O-SAFT) which include Pitt's inequality, Heisenberg-Weyl inequality, logarithmic uncertainty inequality, Hausdorff-Young inequality, and local uncertainty inequalities. Finally, we provide an illustrative example and some possible applications of the proposed transform

    Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images

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    In the present work hypercomplex spectral methods of the processing and analysis of images are introduced. The thesis is divided into three main chapters. First the quaternionic Fourier transform (QFT) for 2D signals is presented and its main properties are investigated. The QFT is closely related to the 2D Fourier transform and to the 2D Hartley transform. Similarities and differences of these three transforms are investigated with special emphasis on the symmetry properties. The Clifford Fourier transform is presented as nD generalization of the QFT. Secondly the concept of the phase of a signal is considered. We distinguish the global, the local and the instantaneous phase of a signal. It is shown how these 1D concepts can be extended to 2D using the QFT. In order to extend the concept of global phase we introduce the notion of the quaternionic analytic signal of a real signal. Defining quaternionic Gabor filters leads to the definition of the local quaternionic phase. The relation between signal structure and local signal phase, which is well-known in 1D, is extended to 2D using the quaternionic phase. In the third part two application of the theory are presented. For the image processing tasks of disparity estimation and texture segmentation there exist approaches which are based on the (complex) local phase. These methods are extended to the use of the quaternionic phase. In either case the properties of the complex approaches are preserved while new features are added by using the quaternionic phase

    Connecting spatial and frequency domains for the quaternion Fourier transform

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    The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency domains: the convolution of two quaternion signals does not map to the pointwise product of their qFT images. The recently defined ‘Mustard’ convolution behaves nicely in the frequency domain, but complicates the corresponding spatial domain analysis. The present paper analyses in detail the correspondence between classical convolution and the new Mustard convolution. In particular, an expression is derived that allows one to write classical convolution as a finite linear combination of suitable Mustard convolutions. This result is expected to play a major role in the further development of quaternion image processing, as it yields a formula for the qFT spectrum of the classical convolution

    Hilbert transforms in Clifford analysis

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    The Hilbert transform on the real line has applications in many fields. In particular in one–dimensional signal processing, the Hilbert operator is used to extract global as well as instantaneous characteristics, such as frequency, amplitude and phase, from real signals. The multidimensional approach to the Hilbert transform usually is a tensorial one, considering the so-called Riesz transforms in each of the cartesian variables separately. In this paper we give an overview of generalized Hilbert transforms in Euclidean space, developed within the framework of Clifford analysis. Roughly speaking, this is a function theory of higher dimensional holomorphic functions, which is particularly suited for a treatment of multidimensional phenomena since all dimensions are encompassed at once as an intrinsic feature

    A Quaternionic Version Theory related to Spheroidal Functions

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    In dieser Arbeit wird eine neue Theorie der quaternionischen Funktionen vorgestellt, welche das Problem der Bestapproximation von Familien prolater und oblater sphĂ€roidalen Funktionen im HilbertrĂ€umen behandelt. Die allgemeine Theorie beginnt mit der expliziten Konstruktion von orthogonalen Basen fĂŒr RĂ€ume, definiert auf sphĂ€roidalen Gebieten mit beliebiger ExzentrizitĂ€t, 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 durchgefĂŒhrt. Der Begriff der kontragenen Funktion hĂ€ngt vom Definitionsbereich ab und ist daher keine lokale Eigenschaft, wĂ€hrend die Begriffe der harmonischen und monogenen Funktionen lokal sind. Es werden verschiedene Umwandlungsformeln vorgestellt, die Systeme harmonischer, monogener und kontragener Funktionen auf SphĂ€roiden unterschiedlicher ExzentrizitĂ€t in Beziehung setzen. DarĂŒber hinaus wird die Existenz gemeinsamer nichttrivialer kontragener Funktionen fĂŒr SphĂ€roide jeglicher ExzentrizitĂ€t gezeigt. Der zweite wichtige Beitrag dieser Arbeit betrifft eine quaternionische Raumfrequenztheorie fĂŒr 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. DarĂŒber 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 RĂ€ume kugelförmig sind, kann der Winkelanteil dieser Signale explizit gefunden werden, was zur Lösung von mehreren eindimensionalen radialen Integralgleichungen fĂŒhrt. Wir nutzen die theoretischen Ergebnisse und harmonische Konjugierten um Klassen monogener Funktionen in verschiedenen RĂ€umen zu konstruieren. Zur Charakterisierung der monogenen gewichteten Hardy- und Bergman-RĂ€ume in der Einheitskugel werden zwei konstruktive Algorithmen vorgeschlagen. FĂŒr eine reelle harmonische Funktion, die zu einem gewichteten Hardy- und Bergman-Raum gehört, werden die harmonischen Konjugiert in den gleichen RĂ€umen gefunden. Die BeschrĂ€nktheit der zugrundeliegenden harmonischen Konjugationsoperatoren wird in den angegebenen gewichteten RĂ€umen bewiesen. ZusĂ€tzlich wird ein quaternionisches GegenstĂŒck zum Satz von Bloch fĂŒr 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
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