No description available.

In the present dissertation, we characterize the range of the attenuated Radon transform of zero, one, and two tensor fields, supported in strictly convex set. The approach is based on a Hilbert transform associated with A-analytic functions of A. Bukhgeim. We first present new necessary and sufficient conditions for a function to be in the range of the attenuated Radon transform of a sufficiently smooth function supported in the convex set. The approach is based on an explicit Hilbert transform associated with traces of the boundary of A-analytic functions in the sense of A. Bukhgeim. We then uses the range characterization of the Radon transform of functions to characterize the range of the attenuated Radon transform of vector fields as they appear in the medical diagnostic techniques of Doppler tomography. As an application we determine necessary and sufficient conditions for the Doppler and X-ray data to be mistaken for each other. We also characterize the range of real symmetric second order tensor field using the range characterization of the Radon transform of zero tensor field.

No description available.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: An off-line signature verification system attempts to authenticate the identity of an individual by examining his/her handwritten signature, after it has been successfully extracted from, for example, a cheque, a debit or credit card transaction slip, or any other legal document. The questioned signature is typically compared to a model trained from known positive samples, after which the system attempts to label said signature as genuine or fraudulent. Classifier fusion is the process of combining individual classifiers, in order to construct a single classifier that is more accurate, albeit computationally more complex, than its constituent parts. A combined classifier therefore consists of an ensemble of base classifiers that are combined using a specific fusion strategy. In this dissertation a novel off-line signature verification system, using a multi-hypothesis approach and classifier fusion, is proposed. Each base classifier is constructed from a hidden Markov model (HMM) that is trained from features extracted from local regions of the signature (local features), as well as from the signature as a whole (global features). To achieve this, each signature is zoned into a number of overlapping circular retinas, from which said features are extracted by implementing the discrete Radon transform. A global retina, that encompasses the entire signature, is also considered. Since the proposed system attempts to detect high-quality (skilled) forgeries, it is unreasonable to assume that samples of these forgeries will be available for each new writer (client) enrolled into the system. The system is therefore constrained in the sense that only positive training samples, obtained from each writer during enrolment, are available. It is however reasonable to assume that both positive and negative samples are available for a representative subset of so-called guinea-pig writers (for example, bank employees). These signatures constitute a convenient optimisation set that is used to select the most proficient ensemble. A signature, that is claimed to belong to a legitimate client (member of the general public), is therefore rejected or accepted based on the majority vote decision of the base classifiers within the most proficient ensemble. When evaluated on a data set containing high-quality imitations, the inclusion of local features, together with classifier combination, significantly increases system performance. An equal error rate of 8.6% is achieved, which compares favorably to an achieved equal error rate of 12.9% (an improvement of 33.3%) when only global features are considered. Since there is no standard international off-line signature verification data set available, most systems proposed in the literature are evaluated on data sets that differ from the one employed in this dissertation. A direct comparison of results is therefore not possible. However, since the proposed system utilises significantly different features and/or modelling techniques than those employed in the above-mentioned systems, it is very likely that a superior combined system can be obtained by combining the proposed system with any of the aforementioned systems. Furthermore, when evaluated on the same data set, the proposed system is shown to be significantly superior to three other systems recently proposed in the literature. / AFRIKAANSE OPSOMMING: Die doel van ’n statiese handtekening-verifikasiestelsel is om die identiteit van ’n individu te bekragtig deur sy/haar handgeskrewe handtekening te analiseer, nadat dit suksesvol vanaf byvoorbeeld ’n tjek,’n debiet- of kredietkaattransaksiestrokie, of enige ander wettige dokument onttrek is. Die bevraagtekende handtekening word tipies vergelyk met ’n model wat afgerig is met bekende positiewe voorbeelde, waarna die stelsel poog om die handtekening as eg of vervals te klassifiseer. Klassifiseerder-fusie is die proses waardeer individuele klassifiseerders gekombineer word, ten einde ’n enkele klassifiseerder te konstrueer, wat meer akkuraat, maar meer berekeningsintensief as sy samestellende dele is. ’n Gekombineerde klassifiseerder bestaan derhalwe uit ’n ensemble van basis-klassifiseerders, wat gekombineer word met behulp van ’n spesifieke fusie-strategie. In hierdie projek word ’n nuwe statiese handtekening-verifikasiestelsel, wat van ’n multi-hipotese benadering en klassifiseerder-fusie gebruik maak, voorgestel. Elke basis-klassifiseerder word vanuit ’n verskuilde Markov-model (HMM) gekonstrueer, wat afgerig word met kenmerke wat vanuit lokale gebiede in die handtekening (lokale kenmerke), sowel as vanuit die handtekening in geheel (globale kenmerke), onttrek is. Ten einde dit te bewerkstellig, word elke handtekening in ’n aantal oorvleulende sirkulêre retinas gesoneer, waaruit kenmerke onttrek word deur die diskrete Radon-transform te implementeer. ’n Globale retina, wat die hele handtekening in beslag neem, word ook beskou. Aangesien die voorgestelde stelsel poog om hoë-kwaliteit vervalsings op te spoor, is dit onredelik om te verwag dat voorbeelde van hierdie handtekeninge beskikbaar sal wees vir elke nuwe skrywer (kliënt) wat vir die stelsel registreer. Die stelsel is derhalwe beperk in die sin dat slegs positiewe afrigvoorbeelde, wat bekom is van elke skrywer tydens registrasie, beskikbaar is. Dit is egter redelik om aan te neem dat beide positiewe en negatiewe voorbeelde beskikbaar sal wees vir ’n verteenwoordigende subversameling van sogenaamde proefkonynskrywers, byvoorbeeld bankpersoneel. Hierdie handtekeninge verteenwoordig ’n gereieflike optimeringstel, wat gebruik kan word om die mees bekwame ensemble te selekteer. ’n Handtekening, wat na bewering aan ’n wettige kliënt (lid van die algemene publiek) behoort, word dus verwerp of aanvaar op grond van die meerderheidstem-besluit van die basis-klassifiseerders in die mees bekwame ensemble. Wanneer die voorgestelde stelsel op ’n datastel, wat hoë-kwaliteit vervalsings bevat, ge-evalueer word, verhoog die insluiting van lokale kenmerke en klassifiseerder-fusie die prestasie van die stelsel beduidend. ’n Gelyke foutkoers van 8.6% word behaal, wat gunstig vergelyk met ’n gelyke foutkoers van 12.9% (’n verbetering van 33.3%) wanneer slegs globale kenmerke gebruik word. Aangesien daar geen standard internasionale statiese handtekening-verifikasiestelsel bestaan nie, word die meeste stelsels, wat in die literatuur voorgestel word, op ander datastelle ge-evalueer as die datastel wat in dié projek gebruik word. ’n Direkte vergelyking van resultate is dus nie moontlik nie. Desnieteenstaande, aangesien die voorgestelde stelsel beduidend ander kenmerke en/of modeleringstegnieke as dié wat in bogenoemde stelsels ingespan word gebruik, is dit hoogs waarskynlik dat ’n superieure gekombineerde stelsel verkry kan word deur die voorgestelde stelsel met enige van bogenoemde stelsels te kombineer. Voorts word aangetoon dat, wanneer op dieselfde datastel geevalueerword, die voorgestelde stelstel beduidend beter vaar as drie ander stelsels wat onlangs in die literatuur voorgestel is.

Depuis l'avènement des premiers appareils imageurs par rayonnement ionisant initié par les prix Nobel Godfrey Newbold Hounsfield et Allan MacLeod Cormack en 1979, le besoin en de nouvelles techniques d'imagerie non invasives n'a cessé de croître. Ces techniques s'appuient sur les propriétés de pénétration dans la matière des rayonnements X et gamma pour détecter une structure cachée sans avoir à détruire le milieu exposé. Elles sont employées dans de nombreux domaines allant de l'imagerie médicale au contrôle non destructif en passant par le contrôle environnemental. Cependant les techniques utilisées jusqu'à maintenant subissent de fortes dégradations dans la qualité des mesures et des images reconstruites. Généralement approchées par un bruit, ces dégradations exigent d'être compensées ou corrigées par des dispositifs de collimation et de filtrage souvent coûteux. Ces dégradations sont principalement dues aux phénomènes de diffusion qui peuvent constituer jusqu'à 80 % du rayonnement émis en imagerie biomédicale. Dès les années 80 un nouveau concept a vu le jour pourcontourner cette difficulté : la tomographie Compton. Cette nouvelle approche propose de mesurer le rayonnement dit diffusé en se plaçant dans des gammes d'énergie (140−511 keV) où l'effet Compton est le phénomène de diffusion prépondérant. L'exploitation de tels dispositifs d'imagerie nécessite une compréhension profonde des interactions rayonnement/matière afin de proposer un modèle, cohérent avec les données mesurées, indispensable à la reconstruction d'images. Dans les systèmes d'imagerie conventionnels (qui mesurent le rayonnement primaire), la transformée de Radon définie sur les lignes droites est apparue comme le modèle naturel. Mais en tomographie Compton, l'information mesurée est liée à l'énergie de diffusion et ainsi à l'angle de diffusion.Ainsi la géométrie circulaire induite par le phénomène de diffusion rend la transformée de Radon classique inadaptée. Dans ce contexte, il devient nécessaire de proposer des transformées de type Radon sur des variétés géométriques plus larges.L'étude de la transformée de Radon sur de nouvelles diversités de courbes devient alors nécessaire pour répondre aux besoins d'outils analytiques de nouvelles techniques d'imagerie. Cormack, lui-même, fut le premier à étendre les propriétés de la transformée de Radon classique à une famille de courbes du plan. Par la suite plusieurs travaux ont été menés dans le but d'étudier la transformée de Radon définie sur différentes variétés de cercles, des sphères, des lignes brisées pour ne citer qu'eux. En 1994 S.J. Norton proposa la première modalité de tomography Compton modélisable par une transformée de Radon sur lesarcs de cercle, la CART1. En 2010 Nguyen et Truong établirent l'inversion de la transformée de Radon sur les arcs de cercle, CART2, permettant de modéliser la formation d'image dans une nouvelle modalité de tomographie Compton. La géométrie des supports d'intégration impliqués dans de nouvelles modalitésde tomographie Compton les conduirent à démontrer l'invertibilité de la transformée de Radon définie sur une famille de courbes de type Cormack, appelée C_alpha. Ils illustrèrent la procédure d'inversion dans le cadre d'une nouvelle transformée, la CART3 modélisant une nouvelle modalité de tomographie Compton.En nous basant sur les travaux de Cormack et de Truong et Nguyen, nous proposons d'établir plusieurs propriétés de la transformée de Radon définie sur la famille C_alpha et plus particulièrement sur C1. Nous avons ainsi démontré deux formules d'inversion qui reconstruisent l'image d'origine via sa décompositionharmonique circulaire et celle de sa transformée et qui s'apparentent à celles établies par Truong and Nguyen. Nous avons enfin établi la bien connue rétroprojection filtrée ainsi que la décomposition en valeurs singulières dans le cas alpha = 1. L'ensemble des résultats établis dans le cadre de cette étude apporte des réponses concrètes a / Since the advent of the first ionizing radiation imaging devices initiated by Godfrey Newbold Hounsfield and Allan MacLeod Cormack, Nobel Prizes in 1979, the requirement for new non-invasive imaging techniques has grown. These techniques rely upon the properties of penetration in the matter of X and gamma radiation for detecting a hidden structure without destroying the illuminated environment. They are used in many fields ranging from medical imaging to non-destructive testing through. However, the techniques used so far suffer severe degradation in the quality of measurement and reconstructed images. Usually approximated by a noise, these degradations require to be compensated or corrected by collimating devices and often expensive filtering. These degradation is mainly due to scattering phenomena which may constitute up to 80% of the emitted radiation in biological tissue. In the 80's a new concept has emerged to circumvent this difficulty : the Compton scattering tomography (CST).This new approach proposes to measure the scattered radiation considering energy ranges ( 140-511 keV) where the Compton effect is the phenomenon of leading broadcast. The use of such imaging devices requires a deep understanding of the interactions between radiation and matter to propose a modeling, consistent with the measured data, which is essential to image reconstruction. In conventional imaging systems (which measure the primary radiation) the Radon transformdefined on the straight lines emerged as the natural modeling. But in Compton scattering tomography, the measured information is related to the scattering energy and thus the scattering angle. Thus the circular geometry induced by scattering phenomenon makes the classical Radon transform inadequate.In this context, it becomes necessary to provide such Radon transforms on broader geometric manifolds.The study of the Radon transform on new manifolds of curves becomes necessary to provide theoretical needs for new imaging techniques. Cormack, himself, was the first to extend the properties of the conventional Radon transform of a family of curves of the plane. Thereafter several studies have been done in order to study the Radon transform defined on different varieties of circles, spheres, broken lines ... . In 1994 S.J. Norton proposed the first modality in Compton scattering tomography modeled by a Radon transform on circular arcs, the CART1 here. In 2010, Nguyen and Truong established the inversion formula of a Radon transform on circular arcs, CART2, to model the image formation in a new modality in Compton scattering tomography. The geometry involved in the integration support of new modalities in Compton scattering tomography lead them to demonstrate the invertibility of the Radon transform defined on a family of Cormack-type curves, called C_alpha. They illustrated the inversion procedure in the case of a new transform, the CART3, modeling a new modeling of Compton scattering tomography. Based on the work of Cormack and Truong and Nguyen, we propose to establish several properties of the Radon transform on the family C_alpha especially on C1. We have thus demonstrated two inversion formulae that reconstruct the original image via its circular harmonic decomposition and itscorresponding transform. These formulae are similar to those established by Truong and Nguyen. We finally established the well-known filtered back projection and singular value decomposition in the case alpha = 1. All results established in this study provide practical problems of image reconstruction associated with these new transforms. In particular we were able to establish new inversion methods for transforms CART1,2,3 as well as numerical approaches necessary for the implementation of these transforms. All these results enable to solve problems of image formation and reconstruction related to three Compton scattering tomography modalities.In addition we propose to improve models and algorithms es

In this paper, a new millimeter wave 3D imaging radar is proposed. The user just needs to move the radar along a circular track, and high resolution 3D imaging can be generated. The proposed radar uses the movement of itself to synthesize a large aperture in both the azimuth and elevation directions. It can utilize inverse Radon transform to resolve 3D imaging. To improve the sensing result, the compressed sensing approach is further investigated. The simulation and experimental result further illustrated the design. Because a single transceiver circuit is needed, a light, affordable and high resolution 3D mmWave imaging radar is illustrated in the paper.

No description available.

This thesis is a collection of the three projects I have worked on at Purdue. The first is a paper on thermoacoustic tomography involving circular integrating detectors that was published in Inverse Problems and Imaging. Results from this paper include demonstrating that the measurement operators involved are Fourier integral operators, as well as proving microlocal uniqueness in certain cases, and also stability. The second paper, submitted to the Journal of Inverse and Ill-Posed Problems, is much more of an application of sampling theory in to the specific case of thermoacoustic tomography. Results from this paper include demonstrating resolution limits imposed by sampling rates, and showing that aliasing artifacts appear in predictable locations in an image when the measurement operator is under sampled in either the time variable or space variables. We also show an application of a basic anti aliasing scheme based on averaging of data. The last project moves slightly away from microlocal analysis and considers the uniqueness in medical imaging of the restricted Radon transform in even dimensions. This is the classical interior problem, and we show a characterization of the range of the Radon transform, and from this are able to obtain a characterization of the kernel of the restricted Radon transform. We include figures throughout to illustrate results.

<p style="text-align: justify;">In this doctoral dissertation several integral transforms are discussed.The first one is the Short time Fourier transform (STFT). We present continuity theorems for the STFT and its adjoint on the test function space <em>K</em>₁(ℝⁿ) and the topological tensor product <em>K</em>₁(ℝⁿ) &otimes; <em>U</em>(<strong>ℂ</strong>ⁿ), where <em>U</em>(<strong>ℂ</strong>ⁿ) is the space of entirerapidly decreasing functions in any horizontal band of&nbsp;<strong>ℂ</strong>ⁿ. We then use such continuity results to develop a framework for the STFT on K&#39;₁(ℝⁿ). Also, we devote one section to the characterization of <em>K</em>&rsquo;₁(ℝⁿ) and related spaces via modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform.</p><p style="text-align: justify;">Part of the thesis is dedicated to the ridgelet and the Radon transform. We define and study the ridgelet transform of (Lizorkin) distributions and we show that the ridgelet transform and the ridgelet synthesis operator can be extended as continuous mappings <em>R</em>_{<em>&psi;&nbsp;</em>}: <em>S</em>&rsquo;₀(ℝⁿ) &rarr; <em>S</em>&rsquo;(<strong>Y</strong>ⁿ⁺¹) and <em>R^t</em>_{<span style="vertical-align: sub;">&psi;</span>}: <em>S</em>&rsquo;(<strong>Y</strong>ⁿ⁺¹) &rarr; <em>S</em>&rsquo;₀(ℝⁿ). We then use our results to develop a distributional framework for the ridgelet transform that is, we treat the ridgelet transform on <em>S</em>&rsquo;₀(ℝⁿ) via a duality approach. Then, the continuity theorems for the ridgelet transform are applied to discuss the continuity of the Radon transform on these spaces and their duals. Finally, we deal with some Abelian and Tauberian theorems relating the quasiasymptotic behavior of distributions with the quasiasymptotics of the its Radon and ridgelet transform.</p><p style="text-align: justify;">The last chapter is dedicated to the MRA of M-exponential distributions. We study the convergence of multiresolution expansions in various test function and distribution spaces and we discuss the pointwise convergence of multiresolution expansions to the distributional point values of a distribution. We also provide a characterization of the quasiasymptotic behavior in terms of multiresolution expansions and give an MRA sufficient condition for the existence of &alpha;-density points of positive measures.</p> / <p>U ovoj doktorskoj disertaciji razmotreno je nekoliko integralnih transformacija. Prva je short time Fourier transform (STFT). Date su i dokazane teoreme o neprekidnosti STFT i njena sinteza na prostoru test funkcije <em>K</em>₁(ℝⁿ) i na prostoru <em>K</em>₁(ℝⁿ) &otimes; <em>U</em>(ℂⁿ), gde je&nbsp;<em>U</em>(ℂⁿ) prostor od celih brzo opadajućih funkcija u proizvoljnom horizontalnom opsegu na ℂⁿ. Onda, ovi rezultati neprekidnosti su iskori&scaron;teni za razvijanje teorije STFT na prostoru <em>K</em>&rsquo;₁(ℝⁿ). Jedno poglavlje je posvećeno karakterizaciji&nbsp;<em>K</em>&rsquo;₁(ℝⁿ) sa srodnih modulaciskih prostora. Dokazani su i različiti Tauberovi rezultata za STFT. Deo teze je posvećen na ridglet i Radon transformacije. Ridgelet transformacija je definisana na (Lizorkin) distribucije i pokazano je da ridgelet transformacija i njen operator sinteze mogu da se pro&scaron;ire kako neprekidna preslikava <em>R</em>_&psi; : <em>S</em>&rsquo;₀(ℝⁿ) &rarr; <em>S</em>&rsquo;(<strong>Y</strong>ⁿ⁺¹) and <em>R</em>^t_&Psi;: <em>S</em>&rsquo;(<strong>Y</strong>ⁿ⁺¹) &rarr; <em>S</em>&rsquo;₀(ℝⁿ).&nbsp;Ridgelet transformacija na <em>S</em>&rsquo;₀(ℝⁿ) je data preko dualnog pristupa. Na&scaron;e teoreme neprekidnosti ridgelet transformacije su primenjene u dokazivanju neprekidnosti Radonove transformacije na Lizorkin test prostorima i njihovim dualima. Na kraju, dajemo Abelovih i Tauberovih teorema koji daju veze izmedju kvaziasimptotike distribucija i kvaziasimptotike rigdelet i Radonovog transfomaciju.<br />Zadnje poglavje je posveceno multirezolucijskog analizu M - eksponencijalnih distrubucije. Proucavamo konvergenciju multirezolucijkog razvoja u razlicitih prostori test funkcije i distribucije i razmotrena je tackasta konvergencija multirezolucijkog razvoju u tacku u distributivnog smislu. Obezbedjena je i karakterizacija kvaziasimptotike u pogled multirezolucijskog razvoju i dat dovoljni uslov za postojanje &alpha;-tacka gustine za pozitivne mere.</p>

La caractérisation tridimensionnelle de matériaux anciens plats est restée une activité non évidente à accomplir par des méthodes classiques de tomographie à rayons X en raison de leur morphologie anisotrope et de leur géométrie aplatie.Pour surmonter les limites de ces méthodologies, une modalité d'imagerie basée sur le rayonnement diffusé Compton est étudiée dans ce travail. La tomographie classique aux rayons X traite les données de diffusion Compton comme du bruit ajouté au processus de formation d'image, tandis que dans la tomographie du rayonnement diffusé, les conditions sont définies de sorte que la diffusion inélastique devienne le phénomène dominant dans la formation d'image. Dans ces conditions, les rotations relatives entre l'échantillon et la configuration d'imagerie ne sont plus nécessaires. Mathématiquement, ce problème est résolu par la transformée de Radon conique. Le problème direct où la sortie du système est l'image spectrale obtenue à partir d'un objet d'entrée est modélisé. Dans le problème inverse une estimation de la distribution tridimensionnelle de la densité électronique de l'objet d'entrée à partir de l'image spectrale est proposée. La faisabilité de cette méthodologie est supportée par des simulations numériques. / Three-dimensional characterization of flat ancient material objects has remained a challenging activity to accomplish by conventional X-ray tomography methods due to their anisotropic morphology and flattened geometry.To overcome the limitations of such methodologies, an imaging modality based on Compton scattering is studied in this work. Classical X-ray tomography treats Compton scattering data as noise in the image formation process, while in Compton scattering tomography the conditions are set such that Compton data become the principal image contrasting agent. Under these conditions, we are able to avoid relative rotations between the sample and the imaging setup. Mathematically this problem is addressed by means of the conical Radon transform. A model of the direct problem is presented where the output of the system is the spectral image obtained from an input object. The inverse problem is addressed to estimate the 3D distribution of the electronic density of the input object from the spectral image. The feasibility of this methodology is supported by numerical simulations.