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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.
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.
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.
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.
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.
This paper examines the interactions between issues such as ideology, neoliberalism, institution-building and sustainable development. The central focus of the paper is to demonstrate that the neoliberalism hasn’t succeeded just because it is an economic strategy which better serves the interests of the capitalist class (though it must be stressed that this fact has obviously contributed) but mainly because it has been promoted as an attractive economic strategy by respected, well organized and transnationalized institutions all around the globe, and understand the reasons behind this institutional support. After reviewing the interrelationships between the Bretton-Woods institutions and their relationship with the concepts of neoliberalism and sustainable development, the paper concludes claiming that the so-called neoliberal ideology operates behind the discourses of growth, progress and sustainability and that it is in charge of softening domination by diffusing legitimating ideas and granting concessions to subordinate forces, thus implying significations and values that transcend the possible manipulation of the world as an object.
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.
The paper gives the results of scientific research, which, being based on probabilistic and statistical modeling, identifies the relationship of certain socio-economic factors and the number of people killed in road accidents in the Russian Federation regions. It notes the identity of processes in various fields, in which there is loss of life. Scientific methods and techniques were used in the process of data processing and study findings: systematic approach, methods of system analysis (algorithmization, mathematical programming) and mathematical statistics. The scientific novelty lies in the formulation, formalization and solving problems related to the analysis of regional road traffic accidents, its modeling taking into account the factors of socio-economic impact.
MICROPLANE MODEL WITH INITIAL AND DAMAGE-INDUCED ANISOTROPY APPLIED TO TEXTILE-REINFORCED CONCRETE
(2010)
The presented material model reproduces the anisotropic characteristics of textile reinforced concrete in a smeared manner. This includes both the initial anisotropy introduced by the textile reinforcement, as well as the anisotropic damage evolution reflecting fine patterns of crack bridges. The model is based on the microplane approach. The direction-dependent representation of the material structure into oriented microplanes provides a flexible way to introduce the initial anisotropy. The microplanes oriented in a yarn direction are associated with modified damage laws that reflect the tension-stiffening effect due to the multiple cracking of the matrix along the yarn.
In this paper we consider the time independent Klein-Gordon equation on some conformally flat 3-tori with given boundary data. We set up an explicit formula for the fundamental solution. We show that we can represent any solution to the homogeneous Klein-Gordon equation on the torus as finite sum over generalized 3-fold periodic elliptic functions that are in the kernel of the Klein-Gordon operator. Furthermore we prove Cauchy and Green type integral formulas and set up a Teodorescu and Cauchy transform for the toroidal Klein-Gordon operator. These in turn are used to set up explicit formulas for the solution to the inhomogeneous version of the Klein-Gordon equation on the 3-torus.
ON THE NAVIER-STOKES EQUATION WITH FREE CONVECTION IN STRIP DOMAINS AND 3D TRIANGULAR CHANNELS
(2006)
The Navier-Stokes equations and related ones can be treated very elegantly with the quaternionic operator calculus developed in a series of works by K. Guerlebeck, W. Sproeossig and others. This study will be extended in this paper. In order to apply the quaternionic operator calculus to solve these types of boundary value problems fully explicitly, one basically needs to evaluate two types of integral operators: the Teodorescu operator and the quaternionic Bergman projector. While the integral kernel of the Teodorescu transform is universal for all domains, the kernel function of the Bergman projector, called the Bergman kernel, depends on the geometry of the domain. With special variants of quaternionic holomorphic multiperiodic functions we obtain explicit formulas for three dimensional parallel plate channels, rectangular block domains and regular triangular channels. The explicit knowledge of the integral kernels makes it then possible to evaluate the operator equations in order to determine the solutions of the boundary value problem explicitly.
In classical complex function theory the geometric mapping property of conformality is closely linked with complex differentiability. In contrast to the planar case, in higher dimensions the set of conformal mappings is only the set of Möbius transformations. Unfortunately, the theory of generalized holomorphic functions (by historical reasons they are called monogenic functions) developed on the basis of Clifford algebras does not cover the set of Möbius transformations in higher dimensions, since Möbius transformations are not monogenic. But on the other side, monogenic functions are hypercomplex differentiable functions and the question arises if from this point of view they can still play a special role for other types of 3D-mappings, for instance, for quasi-conformal ones. On the occasion of the 16th IKM 3D-mapping methods based on the application of Bergman's reproducing kernel approach (BKM) have been discussed. Almost all authors working before that with BKM in the Clifford setting were only concerned with the general algebraic and functional analytic background which allows the explicit determination of the kernel in special situations. The main goal of the abovementioned contribution was the numerical experiment by using a Maple software specially developed for that purpose. Since BKM is only one of a great variety of concrete numerical methods developed for mapping problems, our goal is to present a complete different from BKM approach to 3D-mappings. In fact, it is an extension of ideas of L. V. Kantorovich to the 3-dimensional case by using reduced quaternions and some suitable series of powers of a small parameter. Whereas until now in the Clifford case of BKM the recovering of the mapping function itself and its relation to the monogenic kernel function is still an open problem, this approach avoids such difficulties and leads to an approximation by monogenic polynomials depending on that small parameter.
Wolken
(2006)
In vielen öffentlichen Gebäuden besteht ein hohes wirtschaftliches Einsparpotenzial bei den relevanten Energieträgern Wärme und Strom. Projekte zur energetischen Optimierungen refinanzieren sich häufig nach wenigen Jahren. Die notwenigen Investitionsmittel stehen jedoch nur begrenzt zur Verfügung. Zielgerichtete Analysen und Potenzialschätzungen sind erforderlich, um eine Priorisierung optionaler Maßnahmen zu erreichen. Die Studie zeigt anhand eines öffentlichen Portfolios notwendige Untersuchungsschritte auf. Die Einzelpotenziale werden über geeignete Benchmarks ermittelt. Auf Portfolioebene werden u. a. spezifische Potenzial-Matrizen genutzt. Die kennzahlenbasierte Priorisierung von Maßnahmen ist umso wichtiger, je stärker das Potenzial auf wenige Objekte konzentriert ist.
In the past, several types of Fourier transforms in Clifford analysis have been studied. In this paper, first an overview of these different transforms is given. Next, a new equation in a Clifford algebra is proposed, the solutions of which will act as kernels of a new class of generalized Fourier transforms. Two solutions of this equation are studied in more detail, namely a vector-valued solution and a bivector-valued solution, as well as the associated integral transforms.
THE FOURIER-BESSEL TRANSFORM
(2010)
In this paper we devise a new multi-dimensional integral transform within the Clifford analysis setting, the so-called Fourier-Bessel transform. It appears that in the two-dimensional case, it coincides with the Clifford-Fourier and cylindrical Fourier transforms introduced earlier. We show that this new integral transform satisfies operational formulae which are similar to those of the classical tensorial Fourier transform. Moreover the L2-basis elements consisting of generalized Clifford-Hermite functions appear to be eigenfunctions of the Fourier-Bessel transform.
Designing a structure follows a pattern of creating a structural design concept, executing a finite element analysis and developing a design model. A project was undertaken to create computer support for executing these tasks within a collaborative environment. This study focuses on developing a software architecture that integrates the various structural design aspects into a seamless functional collaboratory that satisfies engineering practice requirements. The collaboratory is to support both homogeneous collaboration i.e. between users operating on the same model and heterogeneous collaboration i.e. between users operating on different model types. Collaboration can take place synchronously or asynchronously, and the information exchange is done either at the granularity of objects or at the granularity of models. The objective is to determine from practicing engineers which configurations they regard as best and what features are essential for working in a collaborative environment. Based on the suggestions of these engineers a specification of a collaboration configuration that satisfies engineering practice requirements will be developed.
This paper describes the application of interval calculus to calculation of plate deflection, taking in account inevitable and acceptable tolerance of input data (input parameters). The simply supported reinforced concrete plate was taken as an example. The plate was loaded by uniformly distributed loads. Several parameters that influence the plate deflection are given as certain closed intervals. Accordingly, the results are obtained as intervals so it was possible to follow the direct influence of a change of one or more input parameters on output (in our example, deflection) values by using one model and one computing procedure. The described procedure could be applied to any FEM calculation in order to keep calculation tolerances, ISO-tolerances, and production tolerances in close limits (admissible limits). The Wolfram Mathematica has been used as tool for interval calculation.