Fractional fourier transforms of hypercomplex signals

An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given

The Clifford-Fourier integral kernel in even dimensional Euclidean space

AbstractRecently, we devised a promising new multi-dimensional integral transform within the Clifford analysis setting, the so-called Fourier–Bessel transform. In the specific case of dimension two, it coincides with the Clifford–Fourier transform introduced earlier as an operator exponential. Moreover, the L2-basis elements, consisting of generalized Clifford–Hermite functions, appear to be simultaneous eigenfunctions of both integral transforms. In the even dimensional case, this allows us to express the Clifford–Fourier transform in terms of the Fourier–Bessel transform, leading to a closed form of the Clifford–Fourier integral kernel

Convolution products for hypercomplex Fourier transforms

Hypercomplex Fourier transforms are increasingly used in signal processing for the analysis of higher-dimensional signals such as color images. A main stumbling block for further applications, in particular concerning filter design in the Fourier domain, is the lack of a proper convolution theorem. The present paper develops and studies two conceptually new ways to define convolution products for such transforms. As a by-product, convolution theorems are obtained that will enable the development and fast implementation of new filters for quaternionic signals and systems, as well as for their higher dimensional counterparts.Comment: 18 pages, two columns, accepted in J. Math. Imaging Visio

Cauchy-Kowalevski extensions and monogenic plane waves in Clifford analysis

This paper deals with axially and biaxial monogenic functions that are derived using two fundamental methods of Clifford analysis, namely the Cauchy-Kowalevski extension and monogenic plane waves

Translation and convolution in Clifford analysis

In this paper we study an integral transform in Clifford analysis with a general kernel expressed as an infinite series in terms of Bessel functions and Gegenbauer polynomials. After giving some examples, we construct the inverse of this general transform on the Schwartz space. Moreover, we define a generalized translation operator and four types of generalized convolution associated to the general kernel

Setting control of completely recyclable concrete with slag and aluminate cements

A completely recyclable concrete (CRC) is designed to have a chemical composition equivalent to the one of general raw materials for cement production. By doing so, this CRC can be used at the end of its service life in cement manufacturing without the need for ingredient adjustments. In one of the designed CRC compositions, blast-furnace slag cement (BFSC) was combined with calcium aluminate cement (CAC), which resulted in fast setting. In an attempt to control this fast setting, different retarders and/or the combination of lime and calcium sulfate were added to the system. The workability (slump and flow), setting time (ultrasonic transmission measurements and Vicat), strength development (compressive strength tests), and hydration behavior (isothermal calorimetry) were studied. It was found that the combined addition of lime and calcium sulfate results in a workable mixture that becomes even more workable if a retarder is also added to the system

Multi-dimensional continuous wavelet transforms and generalized fourier transforms in Clifford analysis

This work covers three mathematical analysis domains: Continuous Wavelet Transform, Fourier Transform and Clifford Analysis, and consists of three parts in which these domains interact. The one-dimensional continuous wavelet transform (CWT) is a successful tool for signal and image analysis, with applications in mathematics, physics and engineering. Higher dimensional CWTs typically originate as tensor products of one-dimensional phenomena. Clifford analysis offers a natural generalization to higher dimension of the theory of holomorphic functions in the complex plane. The generalized holomorphic functions, known as monogenic functions, are null-solutions of the so-called Dirac operator, a first order rotationally invariant differential operator factorizing the Laplacian in higher dimensions. This factorization of the Laplace operator establishes a special relationship between monogenic functions and harmonic functions of several variables, in that the properties of monogenic functions constitute a refinement of those of harmonic functions. An intrinsic feature of Clifford analysis is that it encompasses all dimensions at once, as opposed to the usual tensorial approaches. This true multi-dimensional nature allows for a very specific construction of higher dimensional wavelets and the development of the corresponding CWT-theory, based on generalizations to higher dimension of classical orthogonal polynomials on the real line. In Part I this wavelet construction procedure is presented within the usual, orthogonal Clifford analysis framework, while in Part III we generalize it to the metric dependent setting of Clifford analysis. The latter gives rise to so-called anisotropic Clifford-wavelets which are adaptable to preferential, not necessarily orthogonal, directions in the signals or textures to be analysed. The central topic of Part II is the development of a new multi-dimensional Fourier transform in the framework of Clifford analysis, the so-called Clifford-Fourier transform. It arises as a theoretical construct quite naturally in the spirit of the above mentioned refinement of harmonic functions by monogenic ones

THE FOURIER-BESSEL TRANSFORM

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

Life cycle assessment of completely recyclable concrete

Since the construction sector uses 50% of the Earth. s raw materials and produces 50% of its waste, the development of more durable and sustainable building materials is crucial. Today, Construction and Demolition Waste (CDW) is mainly used in low level applications, namely as unbound material for foundations, e.g., in road construction. Mineral demolition waste can be recycled as crushed aggregates for concrete, but these reduce the compressive strength and affect the workability due to higher values of water absorption. To advance the use of concrete rubble, Completely Recyclable Concrete (CRC) is designed for reincarnation within the cement production, following the Cradle-to-Cradle (C2C) principle. By the design, CRC becomes a resource for cement production because the chemical composition of CRC will be similar to that of cement raw materials. If CRC is used on a regular basis, a closed concrete-cement-concrete material cycle will arise, which is completely different from the current life cycle of traditional concrete. Within the research towards this CRC it is important to quantify the benefit for the environment and Life Cycle Assessment (LCA) needs to be performed, of which the results are presented in a this paper. It was observed that CRC could significantly reduce the global warming potential of concrete