A General Geometric Fourier Transform

The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature. We show which constraints are additionally necessary to obtain certain features like linearity or a shift theorem. As a result, we provide guidelines for the target-oriented design of yet unconsidered transforms that fulfill requirements in a specific application context. Furthermore, the standard theorems do not need to be shown in a slightly different form every time a new geometric Fourier transform is developed since they are proved here once and for all.Comment: First presented in Proc. of The 9th Int. Conf. on Clifford Algebras and their Applications, (2011

A General Geometric Fourier Transform Convolution Theorem

The large variety of Fourier transforms in geometric algebras inspired the straight forward definition of ``A General Geometric Fourier Transform`` in Bujack et al., Proc. of ICCA9, covering most versions in the literature. We showed which constraints are additionally necessary to obtain certain features like linearity, a scaling, or a shift theorem. In this paper we extend the former results by a convolution theorem

Hardy spaces of solutions of generalized Riesz and Moisil-Teodorescu systems

Hardy spaces of solutions of generalized Riesz and generalized Moisil-Teodorescu systems in half space Rm+1,+ , and of their non-tangential L2-boundary values in Rm are characterized

Convolution Theorems for Clifford Fourier Transform and Properties

The non-commutativity of the Clifford multiplication gives different aspects from the classical Fourier analysis.We establish main properties of convolution theorems for the Clifford Fourier transform. Some properties of these generalized convolutionsare extensions of the corresponding convolution theorems of the classical Fourier transform.DOI : http://dx.doi.org/10.22342/jims.20.2.143.125-14

Conformal model of Euclidean space R^n and Möbius-transformations of the model

Avaruus on matematiikan perustavimpia käsitteitä. Yksinkertaisimmillaan avaruus on joukko, jonka alkioilla ei ole muuta ominaisuutta kuin kuuluminen joukkoon. Joukon ja sen alkioiden hedelmällinen tutkiminen vaatii kuitenkin lisärakenteiden muodostamista joukkoon. Tämä vaatimus korostuu erityisesti, kun avaruutta käytetään mallintamaan fysikaalista todellisuutta. Tässä työssä mallinnamme yhtä matemaattista struktuuria toisella matemaattisella rakenteella. Malliesimerkin tästä tarjoaa tason R^2 vektoreiden mallintaminen kompleksilukujen avulla. Tällöin kompleksialgebra laajentaa vektoreiden laskennallista käsittelyä. Tässä työssä tutkimme euklidisen avaruuden R^n konformista mallia, jossa pohjana oleva perusavaruus mallinnetaan (n+2)-ulotteisessa avaruudessa R^n+2. Konformisessa mallissa sisätulolla on mielekäs geometrinen tulkinta ja erilaisille geometrisille konstruktioille saadaan esitykset duaalivektoreiden avulla. Laskennallisena työkaluna käytämme Cliffordin geometrista algebraa. Työn toisena osuutena on tutkia kompleksianalyysistä tuttuja Möbius-kuvauksia konformisessa mallissa. Tällöin kaikki Möbius-kuvaukset esitetään Cliffordin algebran peilauksina ja rotaatioina. Nämä ovat algebran ortogonaalikuvauksia, jotka voidaan esittää hyvin tehokkaasti. Konforminen malli siis homogenisoi Möbius-kuvausten esityksiä. Perinteisesti Möbius-kuvaukset voidaan ajatella geometrisesti peilauksina erilaisten geometristen kontruktioiden suhteen. Esimerkiksi inversio vastaa peilausta ympyrän suhteen. Koska konformisessa mallissa geometriset konstruktiot voidaan esittää duaalisten vektoreiden avulla, Cliffordin algebra ja konforminen malli yhdessä tuovat Möbius-kuvausten esityksiin mukaan niiden taustalla olevat geometriset ideat

Contributions to automated realtime underwater navigation

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2012This dissertation presents three separate–but related–contributions to the art of underwater navigation. These methods may be used in postprocessing with a human in the loop, but the overarching goal is to enhance vehicle autonomy, so the emphasis is on automated approaches that can be used in realtime. The three research threads are: i) in situ navigation sensor alignment, ii) dead reckoning through the water column, and iii) model-driven delayed measurement fusion. Contributions to each of these areas have been demonstrated in simulation, with laboratory data, or in the field–some have been demonstrated in all three arenas. The solution to the in situ navigation sensor alignment problem is an asymptotically stable adaptive identifier formulated using rotors in Geometric Algebra. This identifier is applied to precisely estimate the unknown alignment between a gyrocompass and Doppler velocity log, with the goal of improving realtime dead reckoning navigation. Laboratory and field results show the identifier performs comparably to previously reported methods using rotation matrices, providing an alignment estimate that reduces the position residuals between dead reckoning and an external acoustic positioning system. The Geometric Algebra formulation also encourages a straightforward interpretation of the identifier as a proportional feedback regulator on the observable output error. Future applications of the identifier may include alignment between inertial, visual, and acoustic sensors. The ability to link the Global Positioning System at the surface to precision dead reckoning near the seafloor might enable new kinds of missions for autonomous underwater vehicles. This research introduces a method for dead reckoning through the water column using water current profile data collected by an onboard acoustic Doppler current profiler. Overlapping relative current profiles provide information to simultaneously estimate the vehicle velocity and local ocean current–the vehicle velocity is then integrated to estimate position. The method is applied to field data using online bin average, weighted least squares, and recursive least squares implementations. This demonstrates an autonomous navigation link between the surface and the seafloor without any dependence on a ship or external acoustic tracking systems. Finally, in many state estimation applications, delayed measurements present an interesting challenge. Underwater navigation is a particularly compelling case because of the relatively long delays inherent in all available position measurements. This research develops a flexible, model-driven approach to delayed measurement fusion in realtime Kalman filters. Using a priori estimates of delayed measurements as augmented states minimizes the computational cost of the delay treatment. Managing the augmented states with time-varying conditional process and measurement models ensures the approach works within the proven Kalman filter framework–without altering the filter structure or requiring any ad-hoc adjustments. The end result is a mathematically principled treatment of the delay that leads to more consistent estimates with lower error and uncertainty. Field results from dead reckoning aided by acoustic positioning systems demonstrate the applicability of this approach to real-world problems in underwater navigation.I have been financially supported by: the National Defense Science and Engineering Graduate (NDSEG) Fellowship administered by the American Society for Engineering Education, the Edwin A. Link Foundation Ocean Engineering and Instrumentation Fellowship, and WHOI Academic Programs office