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Design activity could be treated as state transition computationally. In stepwise processing, in-between form-states are not easily observed. However, in this research time-based concept is introduced and applied in order to bridge the gap. In architecture, folding is one method of form manipulation and architects also want to search for alternatives by this operation. Besides, folding operation has to be defined and parameterized before time factor is involved as a variable of folding. As a result, time-based transformation provides sequential form states and redirects design activity.
Electromagnetic wave propagation is currently present in the vast majority of situations which occur in veryday life, whether in mobile communications, DTV, satellite tracking, broadcasting, etc. Because of this the study of increasingly complex means of propagation of lectromagnetic waves has become necessary in order to optimize resources and increase the capabilities of the devices as required by the growing demand for such services.
Within the electromagnetic wave propagation different parameters are considered that characterize it under various circumstances and of particular importance are the reflectance and transmittance. There are several methods or the analysis of the reflectance and transmittance such as the method of approximation by boundary condition, the plane wave expansion method (PWE), etc., but this work focuses on the WKB and SPPS methods.
The implementation of the WKB method is relatively simple but is found to be relatively efficient only when working at high frequencies. The SPPS method (Spectral Parameter Powers Series) based on the theory of pseudoanalytic functions, is used to solve this problem through a new representation for solutions of Sturm Liouville equations and has recently proven to be a powerful tool to solve different boundary value and eigenvalue problems. Moreover, it has a very suitable structure for numerical implementation, which in this case took place in the Matlab software for the valuation of both conventional and turning points profiles.
The comparison between the two methods allows us to obtain valuable information about their perfor mance which is useful for determining the validity and propriety of their application for solving problems where these parameters are calculated in real life applications.
The application of a recent method using formal power series is proposed. It is based on a new representation for solutions of Sturm-Liouville equations. This method is used to calculate the transmittance and reflectance coefficients of finite inhomogeneous layers with high accuracy and efficiency. Tailoring the refraction index profile defining the inhomogeneous media it is possible to develop very important applications such as optical filters. A number of profiles were evaluated and then some of them selected in order to perform an improvement of their characteristics via the modification of their profiles.
A UNIFIED APPROACH FOR THE TREATMENT OF SOME HIGHER DIMENSIONAL DIRAC TYPE EQUATIONS ON SPHERES
(2010)
Using Clifford analysis methods, we provide a unified approach to obtain explicit solutions of some partial differential equations combining the n-dimensional Dirac and Euler operators, including generalizations of the classical time-harmonic Maxwell equations. The obtained regular solutions show strong connections between hypergeometric functions and homogeneous polynomials in the kernel of the Dirac operator.
In this paper we study the structure of the solutions to higher dimensional Dirac type equations generalizing the known λ-hyperholomorphic functions, where λ is a complex parameter. The structure of the solutions to the system of partial differential equations (D- λ) f=0 show a close connection with Bessel functions of first kind with complex argument. The more general system of partial differential equations that is considered in this paper combines Dirac and Euler operators and emphasizes the role of the Bessel functions. However, contrary to the simplest case, one gets now Bessel functions of any arbitrary complex order.
This contribution will be freewheeling in the domain of signal, image and surface processing and touch briefly upon some topics that have been close to the heart of people in our research group. A lot of the research of the last 20 years in this domain that has been carried out world wide is dealing with multiresolution. Multiresolution allows to represent a function (in the broadest sense) at different levels of detail. This was not only applied in signals and images but also when solving all kinds of complex numerical problems. Since wavelets came into play in the 1980's, this idea was applied and generalized by many researchers. Therefore we use this as the central idea throughout this text. Wavelets, subdivision and hierarchical bases are the appropriate tools to obtain these multiresolution effects. We shall introduce some of the concepts in a rather informal way and show that the same concepts will work in one, two and three dimensions. The applications in the three cases are however quite different, and thus one wants to achieve very different goals when dealing with signals, images or surfaces. Because completeness in our treatment is impossible, we have chosen to describe two case studies after introducing some concepts in signal processing. These case studies are still the subject of current research. The first one attempts to solve a problem in image processing: how to approximate an edge in an image efficiently by subdivision. The method is based on normal offsets. The second case is the use of Powell-Sabin splines to give a smooth multiresolution representation of a surface. In this context we also illustrate the general method of construction of a spline wavelet basis using a lifting scheme.
Image processing has been much inspired by the human vision, in particular with regard to early vision. The latter refers to the earliest stage of visual processing responsible for the measurement of local structures such as points, lines, edges and textures in order to facilitate subsequent interpretation of these structures in higher stages (known as high level vision) of the human visual system. This low level visual computation is carried out by cells of the primary visual cortex. The receptive field profiles of these cells can be interpreted as the impulse responses of the cells, which are then considered as filters. According to the Gaussian derivative theory, the receptive field profiles of the human visual system can be approximated quite well by derivatives of Gaussians. Two mathematical models suggested for these receptive field profiles are on the one hand the Gabor model and on the other hand the Hermite model which is based on analysis filters of the Hermite transform. The Hermite filters are derivatives of Gaussians, while Gabor filters, which are defined as harmonic modulations of Gaussians, provide a good approximation to these derivatives. It is important to note that, even if the Gabor model is more widely used than the Hermite model, the latter offers some advantages like being an orthogonal basis and having better match to experimental physiological data. In our earlier research both filter models, Gabor and Hermite, have been developed in the framework of Clifford analysis. Clifford analysis offers a direct, elegant and powerful generalization to higher dimension of the theory of holomorphic functions in the complex plane. In this paper we expose the construction of the Hermite and Gabor filters, both in the classical and in the Clifford analysis framework. We also generalize the concept of complex Gaussian derivative filters to the Clifford analysis setting. Moreover, we present further properties of the Clifford-Gabor filters, such as their relationship with other types of Gabor filters and their localization in the spatial and in the frequency domain formalized by the uncertainty principle.
Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centered around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions, called Hermitean (or h-) monogenic functions, of two Hermitean Dirac operators which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Clifford-Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Previously, a Hermitean Clifford-Cauchy integral formula has been established by means of a matrix approach. This formula reduces to the traditional Martinelli-Bochner formula for holomorphic functions of several complex variables when taking functions with values in an appropriate part of complex spinor space. This means that the theory of Hermitean monogenic functions should encompass also other results of several variable complex analysis as special cases. At present we will elaborate further on the obtained results and refine them, considering fundamental solutions, Borel-Pompeiu representations and the Teoderescu inversion, each of them being developed at different levels, including the global level, handling vector variables, vector differential operators and the Clifford geometric product as well as the blade level were variables and differential operators act by means of the dot and wedge products. A rich world of results reveals itself, indeed including well-known formulae from the theory of several complex variables.
The one-dimensional continuous wavelet transform is a successful tool for signal and image analysis, with applications in physics and engineering. Clifford analysis offers an appropriate framework for taking wavelets to higher dimension. In the usual orthogonal case Clifford analysis focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator ∂, defined in terms of an orthogonal basis for the quadratic space Rm underlying the construction of the Clifford algebra R0,m. An intrinsic feature of this function theory is that it encompasses all dimensions at once, as opposed to a tensorial approach with products of one-dimensional phenomena. This has allowed for a very specific construction of higher dimensional wavelets and the development of the corresponding theory, based on generalizations of classical orthogonal polynomials on the real line, such as the radial Clifford-Hermite polynomials introduced by Sommen. In this paper, we pass to the Hermitian Clifford setting, i.e. we let the same set of generators produce the complex Clifford algebra C2n (with even dimension), which we equip with a Hermitian conjugation and a Hermitian inner product. Hermitian Clifford analysis then focusses on the null solutions of two mutually conjugate Hermitian Dirac operators which are invariant under the action of the unitary group. In this setting we construct new Clifford-Hermite polynomials, starting in a natural way from a Rodrigues formula which now involves both Dirac operators mentioned. Due to the specific features of the Hermitian setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. These polynomials are used to introduce a new continuous wavelet transform, after thorough investigation of all necessary properties of the involved polynomials, the mother wavelet and the associated family of wavelet kernels.
In earlier research, generalized multidimensional Hilbert transforms have been constructed in m-dimensional Euclidean space, in the framework of Clifford analysis. Clifford analysis, centred around the notion of monogenic functions, may be regarded as a direct and elegant generalization to higher dimension of the theory of the holomorphic functions in the complex plane. The considered Hilbert transforms, usually obtained as a part of the boundary value of an associated Cauchy transform in m+1 dimensions, might be characterized as isotropic, since the metric in the underlying space is the standard Euclidean one. In this paper we adopt the idea of a so-called anisotropic Clifford setting, which leads to the introduction of a metric dependent m-dimensional Hilbert transform, showing, at least formally, the same properties as the isotropic one. The Hilbert transform being an important tool in signal analysis, this metric dependent setting has the advantage of allowing the adjustment of the co-ordinate system to possible preferential directions in the signals to be analyzed. A striking result to be mentioned is that the associated anisotropic (m+1)-dimensional Cauchy transform is no longer uniquely determined, but may stem from a diversity of (m+1)-dimensional "mother" metrics.