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Thesis (M.Sc.Eng.) PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / This thesis considers the efficient Hessian computation in inverse problems with specific application to the elastography inverse problem. Inverse problems use measurements of observable parameters to infer information about model parameters, and tend to be ill-posed. They are typically formulated and solved as regularized constrained optimization problems, whose solutions best fit the measured data. Approaching the same inverse problem from a probabilistic Bayesian perspective produces the same optimal point called the maximum a posterior (MAP) estimate of the parameter distribution, but also produces a posterior probability distribution of the parameter estimate, from which a measure of the solution's uncertainty may be obtained. This probability distribution is a very high dimensional function with which it can be difficult to work. For example, in a modest application with N = 104 optimization variables, representing this function with just three values in each direction requires 3^10000 U+2248 10^5000 variables, which far exceeds the number of atoms in the universe. The uncertainty of the MAP estimate describes the shape of the probability distribution and to leading order may be parameterized by the covariance. Directly calculating the Hessian and hence the covariance, requires O(N) solutions of the constraint equations. Given the size of the problems of interest (N = O(10^4 - 10^6)), this is impractical. Instead, an accurate approximation of the Hessian can be assembled using a Krylov basis. The ill-posed nature of inverse problems suggests that its Hessian has low rank and therefore can be approximated with relatively few Krylov vectors. This thesis proposes a method to calculate this Krylov basis in the process of determining the MAP estimate of the parameter distribution. Using the Krylov space based conjugate gradient (CG) method, the MAP estimate is computed. Minor modifications to the algorithm permit storage of the Krylov approximation of the Hessian. As the accuracy of the Hessian approximation is directly related to the Krylov basis, long term orthogonality amongst the basis vectors is maintained via full reorthogonalization. Upon reaching the MAP estimate, the method produces a low rank approximation of the Hessian that can be used to compute the covariance. / 2031-01-01

Tuberculosis (TB), a deadly infectious disease caused by the bacillus Mycobacterium tuberculosis (MTB), is the leading infectious disease killer globally, ranking in the top 10 overall causes of death despite being curable with a timely diagnosis and the correct treatment [3]. As such, eradicating tuberculosis (TB) is one of the targets of the Sustainable Development Goals (SDGs) for global health as approved by the World Health Assembly (WHA) in 2014 [2,3]. This work describes an automated method of screening and determining the severity, or count, of the TB infection in patients via images of fluorescent TB on a sputum smear. Using images from a previously published dataset [9], the algorithm involves a vessel filter which uses the second derivative information in an image by looking at the eigenvalues of the Hessian matrix. Finally, filtering for size and by using background subtraction techniques, each bacillus is effectively isolated in the image. The primary objective was to develop an image processing algorithm in Python that can accurately detect Mycobacteria bacilli in an image for a later deployment in an automated microscope that can improve the timeliness of accurate screenings for acid-fast bacilli (AFB) in a high-volume healthcare setting. Major findings include comparable average and overall object level precision, recall, and F1-score results as compared to the support vector machine (SVM) based algorithm from Chang et al. [9]. Furthermore, this work's algorithm is more accurate on the field level infectiousness accuracy, based on F1-score results, and has a high visual semantic accuracy. / Master of Science / Tuberculosis (TB), a deadly infectious disease caused by the bacillus Mycobacterium tuberculosis (MTB), is the leading infectious disease killer globally, ranking in the top 10 overall causes of death despite being curable with a timely diagnosis and correct treatment [3]. Furthermore, 3.2 billion are part of an at risk population for contracting tuberculosis, yet 90 % of TB related deaths occur in countries across Africa and other Low and Middle Income Countries (LMICs) [2]. This occurrence is, at least in part, due to a lack of the skilled human resources in LMIC laboratories necessary to scan large numbers of patient specimens and properly screen for TB. Sputum smear microscopy of acid-fast bacilli (AFB) is essential in the screening of TB in high-prevalence countries. With the high rates of TB found in LMICs, there is a need to develop affordable, time-efficient alternatives for lab technicians to effectively screen large volumes of patients. This work describes the development of an automated method of screening and determining the severity, or count, of the TB infection in patients via images of fluorescent TB on a sputum smear using images from a previously published dataset [9]. The primary objective of this study was to write a program that can accurately detect tuberculosis in an image for a later deployment in an automated microscope that can improve the timeliness of accurate screening for AFB in a high-volume healthcare setting. Major findings include improved accuracy compared to that of Chang et al.’s machine learning algorithm that was used on this dataset [9].

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Call number: LD2668 .T4 1986 E42 / Master of Science / Entomology

A fundamental result in the theory of elliptic PDEs shows that the hessian of solutions of uniformly elliptic PDEs belong to the Sobolev space ��^2,ε. New results show that for the right choice of c, the optimal hessain integrability exponent ε* is given by ε* = ������ ����(1−������) / ����(1−��), �� ∈ (0,1) Through the techniques of asymptotic analysis, the behavior and properties of this function are better understood to establish improved quantitative estimates for the optimal integrability exponent in the ��^2,ε-regularity theory.