On Y. Nievergelt's Inversion Formula for the Radon Transform

Mathematics Subject Classification 2010: 42C40, 44A12.

Tight frames of k-plane ridgelets and the problem of representing objects that are smooth away from d-dimensional singularities in Rn

For each pair (n, k) with 1 ≤ k < n, we construct a tight frame (ρλ : λ ∈ Λ) for L2 (Rn), which we call a frame of k-plane ridgelets. The intent is to efficiently represent functions that are smooth away from singularities along k-planes in Rn. We also develop tools to help decide whether k-plane ridgelets provide the desired efficient representation. We first construct a wavelet-like tight frame on the X-ray bundle χn,k—the fiber bundle having the Grassman manifold Gn,k of k-planes in Rn for base space, and for fibers the orthocomplements of those planes. This wavelet-like tight frame is the pushout to χn,k, via the smooth local coordinates of Gn,k, of an orthonormal basis of tensor Meyer wavelets on Euclidean space Rk(n−k) × Rn−k. We then use the X-ray isometry [Solmon, D. C. (1976) J. Math. Anal. Appl. 56, 61–83] to map this tight frame isometrically to a tight frame for L2(Rn)—the k-plane ridgelets. This construction makes analysis of a function f ∈ L2(Rn) by k-plane ridgelets identical to the analysis of the k-plane X-ray transform of f by an appropriate wavelet-like system for χn,k. As wavelets are typically effective at representing point singularities, it may be expected that these new systems will be effective at representing objects whose k-plane X-ray transform has a point singularity. Objects with discontinuities across hyperplanes are of this form, for k = n − 1.

Radon-domain detection of the nipple and the pectoral muscle in mammograms

In this paper, methods are presented for automatic detection of the nipple and the pectoral muscle edge in mammograms via image processing in the Radon domain. Radon-domain information was used for the detection of straight-line candidates with high gradient. The longest straight-line candidate was used to identify the pectoral muscle edge. The nipple was detected as the convergence point of breast tissue components, indicated by the largest response in the Radon domain. Percentages of false-positive (FP) and false-negative (FN) areas were determined by comparing the areas of the pectoral muscle regions delimited manually by a radiologist and by the proposed method applied to 540 mediolateral-oblique (MLO) mammographic images. The average FP and FN were 8.99% and 9.13%, respectively. In the detection of the nipple, an average error of 7.4 mm was obtained with reference to the nipple as identified by a radiologist on 1,080 mammographic images (540 MLO and 540 craniocaudal views).

Statistical Inversion Theory in X-ray Tomography

In this thesis the X-ray tomography is discussed from the Bayesian statistical viewpoint. The unknown parameters are assumed random variables and as opposite to traditional methods the solution is obtained as a large sample of the distribution of all possible solutions. As an introduction to tomography an inversion formula for Radon transform is presented on a plane. The vastly used filtered backprojection algorithm is derived. The traditional regularization methods are presented sufficiently to ground the Bayesian approach. The measurements are foton counts at the detector pixels. Thus the assumption of a Poisson distributed measurement error is justified. Often the error is assumed Gaussian, altough the electronic noise caused by the measurement device can change the error structure. The assumption of Gaussian measurement error is discussed. In the thesis the use of different prior distributions in X-ray tomography is discussed. Especially in severely ill-posed problems the use of a suitable prior is the main part of the whole solution process. In the empirical part the presented prior distributions are tested using simulated measurements. The effect of different prior distributions produce are shown in the empirical part of the thesis. The use of prior is shown obligatory in case of severely ill-posed problem.

The vertical structure of oceanic Rossby waves: a comparison of high-resolution model data to theoretical vertical structures

Tests of the new Rossby wave theories that have been developed over the past decade to account for discrepancies between theoretical wave speeds and those observed by satellite altimeters have focused primarily on the surface signature of such waves. It appears, however, that the surface signature of the waves acts only as a rather weak constraint, and that information on the vertical structure of the waves is required to better discriminate between competing theories. Due to the lack of 3-D observations, this paper uses high-resolution model data to construct realistic vertical structures of Rossby waves and compares these to structures predicted by theory. The meridional velocity of a section at 24° S in the Atlantic Ocean is pre-processed using the Radon transform to select the dominant westward signal. Normalized profiles are then constructed using three complementary methods based respectively on: (1) averaging vertical profiles of velocity, (2) diagnosing the amplitude of the Radon transform of the westward propagating signal at different depths, and (3) EOF analysis. These profiles are compared to profiles calculated using four different Rossby wave theories: standard linear theory (SLT), SLT plus mean flow, SLT plus topographic effects, and theory including mean flow and topographic effects. Our results support the classical theoretical assumption that westward propagating signals have a well-defined vertical modal structure associated with a phase speed independent of depth, in contrast with the conclusions of a recent study using the same model but for different locations in the North Atlantic. The model structures are in general surface intensified, with a sign reversal at depth in some regions, notably occurring at shallower depths in the East Atlantic. SLT provides a good fit to the model structures in the top 300 m, but grossly overestimates the sign reversal at depth. The addition of mean flow slightly improves the latter issue, but is too surface intensified. SLT plus topography rectifies the overestimation of the sign reversal, but overestimates the amplitude of the structure for much of the layer above the sign reversal. Combining the effects of mean flow and topography provided the best fit for the mean model profiles, although small errors at the surface and mid-depths are carried over from the individual effects of mean flow and topography respectively. Across the section the best fitting theory varies between SLT plus topography and topography with mean flow, with, in general, SLT plus topography performing better in the east where the sign reversal is less pronounced. None of the theories could accurately reproduce the deeper sign reversals in the west. All theories performed badly at the boundaries. The generalization of this method to other latitudes, oceans, models and baroclinic modes would provide greater insight into the variability in the ocean, while better observational data would allow verification of the model findings.

Parâmetro de regularização em problemas inversos: estudo numérico com a transformada de Radon

In general, an inverse problem corresponds to find a value of an element x in a suitable vector space, given a vector y measuring it, in some sense. When we discretize the problem, it usually boils down to solve an equation system f(x) = y, where f : U Rm ! Rn represents the step function in any domain U of the appropriate Rm. As a general rule, we arrive to an ill-posed problem. The resolution of inverse problems has been widely researched along the last decades, because many problems in science and industry consist in determining unknowns that we try to know, by observing its effects under certain indirect measures. Our general subject of this dissertation is the choice of Tykhonov´s regulaziration parameter of a poorly conditioned linear problem, as we are going to discuss on chapter 1 of this dissertation, focusing on the three most popular methods in nowadays literature of the area. Our more specific focus in this dissertation consists in the simulations reported on chapter 2, aiming to compare the performance of the three methods in the recuperation of images measured with the Radon transform, perturbed by the addition of gaussian i.i.d. noise. We choosed a difference operator as regularizer of the problem. The contribution we try to make, in this dissertation, mainly consists on the discussion of numerical simulations we execute, as is exposed in Chapter 2. We understand that the meaning of this dissertation lays much more on the questions which it raises than on saying something definitive about the subject. Partly, for beeing based on numerical experiments with no new mathematical results associated to it, partly for being about numerical experiments made with a single operator. On the other hand, we got some observations which seemed to us interesting on the simulations performed, considered the literature of the area. In special, we highlight observations we resume, at the conclusion of this work, about the different vocations of methods like GCV and L-curve and, also, about the optimal parameters tendency observed in the L-curve method of grouping themselves in a small gap, strongly correlated with the behavior of the generalized singular value decomposition curve of the involved operators, under reasonably broad regularity conditions in the images to be recovered

Topics in geometry, analysis and inverse problems

The thesis consists of three independent parts. Part I: Polynomial amoebas We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1. Part II: Differential equations in the complex plane We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform. Part III: Radon transforms and tomography This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.

Generalized Backscattering and the Lax-Phillips Transform

2000 Mathematics Subject Classification: 35P25, 35R30, 58J50.

Dynamics of the Dow Jones and the NASDAQ stock indexes

The goal of this study is the analysis of the dynamical properties of financial data series from worldwide stock market indices. We analyze the Dow Jones Industrial Average ( ∧ DJI) and the NASDAQ Composite ( ∧ IXIC) indexes at a daily time horizon. The methods and algorithms that have been explored for description of physical phenomena become an effective background, and even inspiration, for very productive methods used in the analysis of economical data. We start by applying the classical concepts of signal analysis, Fourier transform, and methods of fractional calculus. In a second phase we adopt a pseudo phase plane approach.

Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography.

Differential X-ray phase-contrast tomography (DPCT) refers to a class of promising methods for reconstructing the X-ray refractive index distribution of materials that present weak X-ray absorption contrast. The tomographic projection data in DPCT, from which an estimate of the refractive index distribution is reconstructed, correspond to one-dimensional (1D) derivatives of the two-dimensional (2D) Radon transform of the refractive index distribution. There is an important need for the development of iterative image reconstruction methods for DPCT that can yield useful images from few-view projection data, thereby mitigating the long data-acquisition times and large radiation doses associated with use of analytic reconstruction methods. In this work, we analyze the numerical and statistical properties of two classes of discrete imaging models that form the basis for iterative image reconstruction in DPCT. We also investigate the use of one of the models with a modern image reconstruction algorithm for performing few-view image reconstruction of a tissue specimen.

Projective Methods of Image Recognition

We propose a method for image recognition on the base of projections. Radon transform gives an opportunity to map image into space of its projections. Projection properties allow constructing informative features on the base of moments that can be successfully used for invariant recognition. Offered approach gives about 91-97% of correct recognition.

Resolução em tomografia de tempo de trânsito poço-a-poço: a dependência da iluminação

The key aspect limiting resolution in crosswell traveltime tomography is illumination, a well known result but not as well exemplified. Resolution in the 2D case is revisited using a simple geometric approach based on the angular aperture distribution and the Radon Transform properties. Analitically it is shown that if an interface has dips contained in the angular aperture limits in all points, it is correctly imaged in the tomogram. By inversion of synthetic data this result is confirmed and it is also evidenced that isolated artifacts might be present when the dip is near the illumination limit. In the inverse sense, however, if an interface is interpretable from a tomogram, even an aproximately horizontal interface, there is no guarantee that it corresponds to a true interface. Similarly, if a body is present in the interwell region it is diffusely imaged in the tomogram, but its interfaces - particularly vertical edges - can not be resolved and additional artifacts might be present. Again, in the inverse sense, there is no guarantee that an isolated anomaly corresponds to a true anomalous body because this anomaly can also be an artifact. Jointly, these results state the dilemma of ill-posed inverse problems: absence of guarantee of correspondence to the true distribution. The limitations due to illumination may not be solved by the use of mathematical constraints. It is shown that crosswell tomograms derived by the use of sparsity constraints, using both Discrete Cosine Transform and Daubechies bases, basically reproduces the same features seen in tomograms obtained with the classic smoothness constraint. Interpretation must be done always taking in consideration the a priori information and the particular limitations due to illumination. An example of interpreting a real data survey in this context is also presented.

Resolução em tomografia de tempo de trânsito poço-a-poço: a dependência da iluminação

The key aspect limiting resolution in crosswell traveltime tomography is illumination, a well known result but not as well exemplified. Resolution in the 2D case is revisited using a simple geometric approach based on the angular aperture distribution and the Radon Transform properties. Analitically it is shown that if an interface has dips contained in the angular aperture limits in all points, it is correctly imaged in the tomogram. By inversion of synthetic data this result is confirmed and it is also evidenced that isolated artifacts might be present when the dip is near the illumination limit. In the inverse sense, however, if an interface is interpretable from a tomogram, even an aproximately horizontal interface, there is no guarantee that it corresponds to a true interface. Similarly, if a body is present in the interwell region it is diffusely imaged in the tomogram, but its interfaces - particularly vertical edges - can not be resolved and additional artifacts might be present. Again, in the inverse sense, there is no guarantee that an isolated anomaly corresponds to a true anomalous body because this anomaly can also be an artifact. Jointly, these results state the dilemma of ill-posed inverse problems: absence of guarantee of correspondence to the true distribution. The limitations due to illumination may not be solved by the use of mathematical constraints. It is shown that crosswell tomograms derived by the use of sparsity constraints, using both Discrete Cosine Transform and Daubechies bases, basically reproduces the same features seen in tomograms obtained with the classic smoothness constraint. Interpretation must be done always taking in consideration the a priori information and the particular limitations due to illumination. An example of interpreting a real data survey in this context is also presented.

Plane detection from monocular image sequences

AIRES, Kelson R. T. ; ARAÚJO, Hélder J. ; MEDEIROS, Adelardo A. D. . Plane Detection from Monocular Image Sequences. In: VISUALIZATION, IMAGING AND IMAGE PROCESSING, 2008, Palma de Mallorca, Spain. Proceedings..., Palma de Mallorca: VIIP, 2008

Fourier transform infrared spectroscopy reveals a rigid α-helical assembly for the tetrameric Streptomyces lividans K+ channel

The structure of the tetrameric K+ channel from Streptomyces lividans in a lipid bilayer environment was studied by polarized attenuated total reflection Fourier transform infrared spectroscopy. The channel displays approximately 43% α-helical and 25% β-sheet content. In addition, H/D exchange experiments show that only 43% of the backbone amide protons are exchangeable with solvent. On average, the α-helices are tilted 33° normal to the membrane surface. The results are discussed in relationship to the lactose permease of Escherichia coli, a membrane transport protein.