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Optical coherence elastography allows the characterization of the mechanical properties of tissues, and can be performed through estimating local displacement maps from subsequent acquisitions of a sample under different loads. This displacement estimation is limited by noise in the images, which can be high in dynamic systems due to the inability to perform long exposures or B-scan averaging. In this work, we propose a framework for simultaneously enhancing both the image quality and displacement map for elastography, by motion compensated denoising with the block-matching and 4D filtering (BM4D) method, followed by a re-estimation of displacement. We adopt the interferometric synthetic aperture microscopy (ISAM) method to enhance the lateral resolution away from the focal plane, and use sub-pixel cross correlation block matching for non-uniform deformation estimation. We validate this approach on data from a commercial spectral domain optical coherence tomography system, whereby we observe an enhancement of both image and displacement accuracy of up to 33% over a standard approach., Comment: 7 pages, 6 figures, presented at SPIE BiOS 2019

We present a computational method for full-range interferometric synthetic aperture microscopy (ISAM) under dispersion encoding. With this, one can effectively double the depth range of optical coherence tomography (OCT), whilst dramatically enhancing the spatial resolution away from the focal plane. To this end, we propose a model-based iterative reconstruction (MBIR) method, where ISAM is directly considered in an optimization approach, and we make the discovery that sparsity promoting regularization effectively recovers the full-range signal. Within this work, we adopt an optimal nonuniform discrete fast Fourier transform (NUFFT) implementation of ISAM, which is both fast and numerically stable throughout iterations. We validate our method with several complex samples, scanned with a commercial SD-OCT system with no hardware modification. With this, we both demonstrate full-range ISAM imaging, and significantly outperform combinations of existing methods., Comment: 17 pages, 7 figures. The images have been compressed for arxiv - please follow DOI for full resolution

Scatter can account for large errors in cone-beam CT (CBCT) due to its wide field of view, and its complicated nature makes its compensation difficult. Iterative polyenergetic reconstruction algorithms offer the potential to provide quantitative imaging in CT, but they are usually incompatible with scatter contaminated measurements. In this work, we introduce a polyenergetic convolutional scatter model that is directly fused into the reconstruction process, and exploits information readily available at each iteration for a fraction of additional computational cost. We evaluate this method with numerical and real CBCT measurements, and show significantly enhanced electron density estimation and artifact mitigation over pre-calculated fast adaptive scatter kernel superposition (fASKS). We demonstrate our approach has two levels of benefit: reducing the bias introduced by estimating scatter prior to reconstruction; and adapting to the spectral and spatial properties of the specimen., Comment: 28 pages, 12 figures

Quantifying material mass and electron density from computed tomography (CT) reconstructions can be highly valuable in certain medical practices, such as radiation therapy planning. However, uniquely parameterising the X-ray attenuation in terms of mass or electron density is an ill-posed problem when a single polyenergetic source is used with a spectrally indiscriminate detector. Existing approaches to single source polyenergetic modelling often impose consistency with a physical model, such as water--bone or photoelectric--Compton decompositions, which will either require detailed prior segmentation or restrictive energy dependencies, and may require further calibration to the quantity of interest. In this work, we introduce a data centric approach to fitting the attenuation with piecewise-linear functions directly to mass or electron density, and present a segmentation-free statistical reconstruction algorithm for exploiting it, with the same order of complexity as other iterative methods. We show how this allows both higher accuracy in attenuation modelling, and demonstrate its superior quantitative imaging, with numerical chest and metal implant data, and validate it with real cone-beam CT measurements., Comment: 23 pages, 12 figures

Due to its wide field of view, cone-beam computed tomography (CBCT) is plagued by large amounts of scatter, where attenuated photons hit the detector, and corrupt the linear models used for reconstruction. Given that one can generate a good estimate of scatter however, then image accuracy can be retained. In the context of adaptive radiotherapy, one usually has a low-scatter planning CT image of the same patient at an earlier time. Correcting for scatter in the subsequent CBCT scan can either be self consistent with the new measurements or exploit the prior image, and there are several recent methods that report high accuracy with the latter. In this study, we will look at the accuracy of various scatter estimation methods, how they can be effectively incorporated into a statistical reconstruction algorithm, along with introducing a method for matching off-line Monte-Carlo (MC) prior estimates to the new measurements. Conclusions we draw from testing on a neck cancer patient are: statistical reconstruction that incorporates the scatter estimate significantly outperforms analytic and iterative methods with pre-correction; and although the most accurate scatter estimates can be made from the MC on planning image, they only offer a slight advantage over the measurement based scatter kernel superposition (SKS) in reconstruction error., Comment: 13 pages, submitted to Medical Image Understanding and Analysis (MIUA) 2017 conference

Conscious experience is awash with underlying relationships. Moreover, for various brain regions such as the visual cortex, the system is biased toward some states. Representing this bias using a probability distribution shows that the system can define expected quantities. The mathematical theory in the present paper links these facts by using expected float entropy (efe), which is a measure of the expected amount of information needed, to specify the state of the system, beyond what is already known about the system from relationships that appear as parameters. Under the requirement that the relationship parameters minimise efe, the brain defines relationships. It is proposed that when a brain state is interpreted in the context of these relationships the brain state acquires meaning in the form of the relational content of the associated experience. For a given set, the theory represents relationships using weighted relations which assign continuous weights, from 0 to 1, to the elements of the Cartesian product of that set. The relationship parameters include weighted relations on the nodes of the system and on their set of states. Examples obtained using Monte-Carlo methods (where relationship parameters are chosen uniformly at random) suggest that efe distributions with long left tails are most important., Comment: 33 pages (double spacing), 11 figures, 15 Tables

The theoretical base for consciousness, in particular an explanation of how consciousness is defined by the brain, has long been sought by science. We propose a partial theory of consciousness as relations defined by typical data. The theory is based on the idea that a brain state on its own is almost meaningless but in the context of the typical brain states, defined by the brain's structure, a particular brain state is highly structured by relations. The proposed theory can be applied and tested both theoretically and experimentally. Precisely how typical data determines relations is fully established using discrete mathematics., Comment: 16 pages, 8 figures, First submitted for publication March 2012

UNIFORM algebras have been extensively investigated because of their importance in the theory of uniform approximation and as examples of complex Banach algebras. An interesting question is whether analogous algebras exist when a complete valued field other than the complex numbers is used as the underlying field of the algebra. In the Archimedean setting, this generalisation is given by the theory of real function algebras introduced by S. H. Kulkarni and B. V. Limaye in the 1980s. This thesis establishes a broader theory accommodating any complete valued field as the underlying field by involving Galois automorphisms and using non-Archimedean analysis. The approach taken keeps close to the original definitions from the Archimedean setting. Basic function algebras are defined and generalise real function algebras to all complete valued fields. Several examples are provided. Each basic function algebra is shown to have a lattice of basic extensions related to the field structure. In the non-Archimedean setting it is shown that certain basic function algebras have residue algebras that are also basic function algebras. A representation theorem is established. Commutative unital Banach F-algebras with square preserving norm and finite basic dimension are shown to be isometrically F-isomorphic to some subalgebra of a Basic function algebra. The theory of non-commutative real function algebras was established by K. Jarosz in 2008. The possibility of their generalisation to the non-Archimedean setting is established in this thesis. In the context of complex uniform algebras, a new proof is given using transfinite induction of the Feinstein-Heath Swiss cheese "Classicalisation" theorem., Comment: PhD Thesis 2012, 113 pages, The main results were presented at The Sixth Conference on Function Spaces at SIUE USA 2010

We survey commutative and non-commutative analogs of uniform algebras in the Archimedean settings and also offer some non-Archimedean examples. Constraints on the development of non-complex uniform algebras are also discussed., Comment: 10 pages, accepted to appear in the proceedings of the sixth conference on function spaces, AMS contemporary mathematics book series

A theory of allocation maps has been developed by J. F. Feinstein and M. J. Heath in order to prove a theorem, using Zorn's lemma, concerning the compact plane sets known as Swiss cheese sets. These sets are important since, as domains, they provide a good source of examples in the theory of uniform algebras and rational approximation. In this paper we take a more direct approach when proving their theorem by using transfinite induction and cardinality. An explicit reference to a theory of allocation maps is no longer required. Instead we find that the repeated application of a single operation developed from the final step of the proof by Feinstein and Heath is enough., Comment: 10 pages, 1 figure, Accepted for publication by the Proceedings of the American Mathematical Society, volume 138, year 2010, number 12, pages 4423-4432

During the Arizona Winter School 2008 (held in Tucson, AZ) we worked on the following problems: a) (Expanding a remark by S. Lang). Define $E_0 = \overline{\mathbb{Q}}$ Inductively, for $n \geq 1$, define $E_n$ as the algebraic closure of the field generated over $E_{n-1}$ by the numbers $\exp(x)=e^x$, where $x$ ranges over $E_{n-1}$. Let $E$ be the union of $E_n$, $n \geq 0$. Show that Schanuel's Conjecture implies that the numbers $\pi, \log \pi, \log \log \pi, \log \log \log \pi, \ldots $ are algebraically independent over $E$. b) Try to get a (conjectural) generalization involving the field $L$ defined as follows. Define $L_0 = \overline{\mathbb{Q}}$. Inductively, for $n \geq 1$, define $L_n$ as the algebraic closure of the field generated over $L_{n-1}$ by the numbers $y$, where $y$ ranges over the set of complex numbers such that $e^y\in L_{n-1}$. Let $L$ be the union of $L_n$, $n \geq 0$. We were able to prove that Schanuel's Conjecture implies $E$ and $L$ are linearly disjoint over $\overline{\mathbb{Q}}$., Comment: 8 pages summarizing the results obtained in this project during the AWS08 http://swc.math.arizona.edu/aws/08/08WaldschmidtOutline.pdf