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Analysis of the Truncated Hilbert Transform Arising in Limited Data Computerized Tomography

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Dissertation Department of Mathematics Analysis of the truncated Hilbert transform arising in limited data computerized tomography Author: Thesis advisors: Reema Al-Aifari Prof. Ingrid Daubechies Prof. Michel Defrise May 4, 2014 ii Contents I Introduction 1 1 Preliminaries 9 1.1 Functional analysis . 10 1.1.1 Bounded operators . 10 1.1.2 Unbounded operators . 16 1.2 Linear inverse problems . 19 1.2.1 Regularization of ill-posed problems . 21 1.2.2 Miller's theory and stability estimates . 26 1.3 Sturm{Liouville problems . 27 1.3.1 Regular Sturm{Liouville problems . 28 1.3.2 Two-point singular Sturm{Liouville problems . 30 1.3.3 Singular Sturm{Liouville problems on two intervals . 33 1.3.4 Local behavior of solutions . 35 1.3.5 Global behavior of solutions . 37 1.4 2D Computerized Tomography . 41 1.4.1 Reconstruction from limited data . 45 1.4.2 The Hilbert transform . 48 2 Spectral analysis 51 2.1 Introduction . 52 2.2 Introducing a differential operator . 56 2.2.1 Self-adjoint realizations . 58 2.3 The spectrum of LS ......................... 60 2.4 Singular value decomposition of HT ................ 65 2.5 Accumulation points of the singular values . 69 2.6 Numerical illustration . 73 3 Asymptotic Analysis of the SVD 77 3.1 Introduction . 78 3.2 Two accumulation points in σ(LS)................. 78 3.3 A procedure for finding the asymptotics . 81 3.3.1 Outline of the construction of vn for λn + ...... 82 3.3.2 Validity of the approach . .! . .1 . 82 iii iv CONTENTS 3.4 Asymptotic analysis of the SVD for σn 0............ 84 3.4.1 The WKB approximation and its! region of validity . 85 3.4.2 The Bessel solutions and their region of validity . 86 3.4.3 Overlap region of validities . 87 3.4.4 Derivation of the asymptotics . 87 3.4.5 Asymptotic behavior of the singular values accumulating at zero . 92 3.5 Asymptotic analysis for the case of σn 1 ............ 95 3.6 Comparison of numerics and asymptotics! . 97 4 Lower bounds and stability estimates 101 4.1 Introduction . 102 4.2 Statement of results . 104 4.2.1 Functions of bounded variation . 104 4.2.2 Weakly differentiable functions . 105 4.2.3 A quantitative result for functions with bounded variation 106 4.2.4 Stable reconstruction . 106 4.2.5 An improved estimate for the overlap case . 107 4.3 The Proofs . 109 5 Numerical illustration 117 5.1 Regularized reconstruction in one dimension . 118 5.2 ROI reconstruction in 2D . 130 A Appendices 139 A.1 Normalization of v on (a1; a3).................... 139 A.2 Proof of Equation 4.7 . 144 A.3 Proof of Relation (4.8) . 147 A.3.1 Evaluation of Ia ....................... 148 A.3.2 Evaluation of Ib ....................... 149 A.4 Proof of Relation (4.9) . 150 Introduction 1 2 INTRODUCTION It was the discovery of X-rays by Wilhelm R¨ontgen in 1895 that laid the foundation of Computerized Tomography (CT). He was the first to produce a radiographic image. It showed his wife's hand (see Fig. 1). This finding came at a time when the only way for doctors to observe the inside of the patient's body was through surgery. Consequently, R¨ontgen's discovery was of great impact and only a year later X-ray radiography appeared in hospitals. Over the next decades, X-ray imaging saw major improvements to image quality and safety of the method. However, what would turn out to have the biggest impact on clinical imaging, only appeared in the early 1970s: the emergence of computers. Until then, X-ray images were taken from one fixed position, resulting in a two- dimensional (2D) projection of the three-dimensional (3D) body. The idea of CT was essentially to obtain an actual image of a 2D cross-section or "slice" of the human body instead of a mere projection. One-dimensional (1D) X-ray projec- tions obtained from different angles were combined and processed by a computer to form a 2D cross-sectional image of the human body. The first commercial CT scanner was developed by Hounsfield in 1971 and the first scan took about 5 minutes for a single cross-sectional scan. The scanning and computation time of course reduced drastically over the time, as the performance of computers improved. Nowadays, scanners can perform several hundreds of cross-sectional scans simultaneously in just a fraction of a second. This performance allows to stack the 2D cross-sections together to obtain a 3D reconstruction. Not only is the development of new devices constantly bringing improvement in terms of speed, but also in terms of accuracy and reduction of radiation dose. Depend- ing on the application, either 2D or 3D reconstructions might be of interest. Reconstructions in 3D can be either obtained by combining a fine discretization of independent 2D cross-sections, or by scanning and reconstructing the full 3D density. The latter has emerged in the 1980s and is often referred to as fully or truly 3D reconstruction. Both 2D and 3D imaging are based on the same principle of tomographic reconstruction, which is widely used for but not limited to medical applications. A 2D or 3D object is illuminated by a penetrating beam (usually X-rays) from multiple directions, and a detector records the attenuation of the beam after it has traversed the body. These measurements are processed by a computer to determine the corresponding attenuation coefficients, with which the full 2D or 3D density is then reconstructed. The terms full data or complete data are used to describe the case when the beams are sufficiently wide to fully embrace the object and when the beams from a sufficiently dense set of directions around the object can be used. Under these conditions, the problem of tomographic reconstruction and its solution are well understood [Nat01]. Roughly speaking, the data collected from a CT scanner can be modeled as the Radon transform of the underlying density, i.e. the collection of line integrals of the density along straight lines. Different methods have been proposed for CT imaging, the most famous of which is the Filtered Back-Projection or FBP. Inversion of the Radon transform is an ill-posed process and thus, reconstruction methods such as the FBP or iterative algorithms are concerned with providing a stable solution to the problem. Although these methods yield accurate results when full data is 3 Figure 1: First X-ray image taken by Wilhelm R¨ontgen of his wife Anna Bertha's hand, 1895. available, they are in general less useful when the data are more limited. There, the image reconstruction problem becomes much more challenging. Cases of limited data can occur for example when only a reduced range of directions can be used or only a part of the object can be illuminated. On the other hand, limited data cases are interesting to study because they might allow to reduce the radiation dose when only a subregion of the 2D or 3D object is of actual interest. The question is then whether a reconstruction of this subregion can be obtained by using only beams that penetrate the subregion instead of the full 2D or 3D object. Reconstruction from limited data requires the identification of specific sub- sets of line integrals that allow for an exact and stable reconstruction of some specified subregion. One class of such configurations that have already been identified, relies on the reduction of the 2D and 3D reconstruction problem to a family of 1D problems. The Radon transform can be related to the 1D Hilbert transform along certain lines by differentiation and back-projection of the Radon transform data (Differentiated Back-Projection or DBP). Inversion of the Hilbert transform along a family of lines covering a subregion of the object (region of interest or ROI) then allows for the reconstruction within the ROI. This method goes back to a result by Gelfand & Graev [GG91]. Its ap- plication to tomography was formulated by Finch [Fin02] and was later made 4 INTRODUCTION explicit for 2D in [NCP04, YYWW07, ZPS05] and for 3D in [PNC05, YZYW05, ZLNC04, ZP04]. To reconstruct from data obtained by the DBP method, it is necessary to solve a family of 1D problems which consist of inverting the Hilbert transform data on a finite segment of the line. If the Hilbert transform Hf of a 1D function f was given on all of R, then the inversion would be trivial, since H−1 = H. In case f is compactly supported, it can be reconstructed even if − Hf is not known on all of R. Due to an explicit inversion formula that can be found e.g. in Tricomi [Tri85], f can be found from measuring Hf on only an interval that covers the support of f. However, a limited field of view might result in configurations in which the Hilbert transform is known on only a seg- ment that does not completely cover the object support. Then, the question is whether stable inversion is still possible. This leads to the formulation of the main topic of the work presented here: In 1D, given the compact support [a; b] of a function f and supposing its Hilbert transform is measured on a segment [c; d] that does not contain the support of f, we seek to analyze the inverse problem of reconstruction from these trun- cated Hilbert transform data. In particular, we will discuss uniqueness of the solution, the dependence on measurement errors in the Hilbert transform data and possible stability estimates under a-priori assumptions on the solution f.
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