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DEEP LEARNING FOR IMAGE PROCESSING WITH APPLICATIONS TO MEDICAL IMAGING
Title
DEEP LEARNING FOR IMAGE PROCESSING WITH APPLICATIONS TO MEDICAL IMAGING
Creator
Zarshenas, Amin
Date
2019
Description
Deep Learning is a subfield of machine learning concerned with algorithms that learn hierarchical data representations. Deep learning has...
Show moreDeep Learning is a subfield of machine learning concerned with algorithms that learn hierarchical data representations. Deep learning has proven extremely successful in many computer vision tasks including object detection and recognition. In this thesis, we aim to develop and design deep-learning models to better perform image processing and tackle three important problems: natural image denoising, computed tomography (CT) dose reduction, and bone suppression in chest radiography (“chest x-ray”: CXR). As the first contribution of this thesis, we aimed to answer to probably the most critical design questions, under the task of natural image denoising. To this end, we defined a class of deep learning models, called neural network convolution (NNC). We investigated several design modules for designing NNC for image processing. Based on our analysis, we design a deep residual NNC (R-NNC) for this task. One of the important challenges in image denoising regards to a scenario in which the images have varying noise levels. Our analysis showed that training a single R-NNC on images at multiple noise levels results in a network that cannot handle very high noise levels; and sometimes, it blurs the high-frequency information on less noisy areas. To address this problem, we designed and developed two new deep-learning structures, namely, noise-specific NNC (NS-NNC) and a DeepFloat model, for the task of image denoising at varying noise levels. Our models achieved the highest denoising performance comparing to the state-of-the-art techniques.As the second contribution of the thesis, we aimed to tackle the task of CT dose reduction by means of our NNC. Studies have shown that high dose of CT scans can increase the risk of radiation-induced cancer in patients dramatically; therefore, it is very important to reduce the radiation dose as much as possible. For this problem, we introduced a mixture of anatomy-specific (AS) NNC experts. The basic idea is to train multiple NNC models for different anatomic segments with different characteristics, and merge the predictions based on the segmentations. Our phantom and clinical analysis showed that more than 90% dose reduction would be achieved using our AS NNC model.We exploited our findings from image denoising and CT dose reduction, to tackle the challenging task of bone suppression in CXRs. Most lung nodules that are missed by radiologists as well as by computer-aided detection systems overlap with bones in CXRs. Our purpose was to develop an imaging system to virtually separate ribs and clavicles from lung nodules and soft-tissue in CXRs. To achieve this, we developed a mixture of anatomy-specific, orientation-frequency-specific (ASOFS) expert deep NNC model. While our model was able to decompose the CXRs, to achieve an even higher bone suppression performance, we employed our deep R-NNC for the bone suppression application. Our model was able to create bone and soft-tissue images from single CXRs, without requiring specialized equipment or increasing the radiation dose.
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Fast Automatic Bayesian Cubature Using Matching Kernels and Designs
Title
Fast Automatic Bayesian Cubature Using Matching Kernels and Designs
Creator
Rathinavel, Jagadeeswaran
Date
2019
Description
Automatic cubatures approximate multidimensional integrals to user-specified error tolerances. In many real-world integration problems, the...
Show moreAutomatic cubatures approximate multidimensional integrals to user-specified error tolerances. In many real-world integration problems, the analytical solution is either unavailable or difficult to compute. To overcome this, one can use numerical algorithms that approximately estimate the value of the integral. For high dimensional integrals, quasi-Monte Carlo (QMC) methods are very popular. QMC methods are equal-weight quadrature rules where the quadrature points are chosen deterministically, unlike Monte Carlo (MC) methods where the points are chosen randomly.The families of integration lattice nodes and digital nets are the most popular quadrature points used. These methods consider the integrand to be a deterministic function. An alternative approach, called Bayesian cubature, postulates the integrand to be an instance of a Gaussian stochastic process. For high dimensional problems, it is difficult to adaptively change the sampling pattern. But one can automatically determine the sample size, $n$, given a fixed and reasonable sampling pattern. We take this approach using a Bayesian perspective. We assume a Gaussian process parameterized by a constant mean and a covariance function defined by a scale parameter and a function specifying how the integrand values at two different points in the domain are related. These parameters are estimated from integrand values or are given non-informative priors. This leads to a credible interval for the integral. The sample size, $n$, is chosen to make the credible interval for the Bayesian posterior error no greater than the desired error tolerance. However, the process just outlined typically requires vector-matrix operations with a computational cost of $O(n^3)$. Our innovation is to pair low discrepancy nodes with matching kernels, which lowers the computational cost to $O(n \log n)$. We begin the thesis by introducing the Bayesian approach to calculate the posterior cubature error and define our automatic Bayesian cubature. Although much of this material is known, it is used to develop the necessary foundations. Some of the major contributions of this thesis include the following: 1) The fast Bayesian transform is introduced. This generalizes the techniques that speedup Bayesian cubature when the kernel matches low discrepancy nodes. 2) The fast Bayesian transform approach is demonstrated using two methods: a) rank-1 lattice sequences and shift-invariant kernels, and b) Sobol' sequences and Walsh kernels. These two methods are implemented as fast automatic Bayesian cubature algorithms in the Guaranteed Automatic Integration Library (GAIL). 3) We develop additional numerical implementation techniques: a) rewriting the covariance kernel to avoid cancellation error, b) gradient descent for hyperparameter search, and c) non-integer kernel order selection.The thesis concludes by applying our fast automatic Bayesian cubature algorithms to three sample integration problems. We show that our algorithms are faster than the basic Bayesian cubature and that they provide answers within the error tolerance in most cases. The Bayesian cubatures that we develop are guaranteed for integrands belonging to a cone of functions that reside in the middle of the sample space. The concept of a cone of functions is also explained briefly.
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Algorithms for Discrete Data in Statistics and Operations Research
Title
Algorithms for Discrete Data in Statistics and Operations Research
Creator
Schwartz, William K.
Date
2021
Description
This thesis develops mathematical background for the design of algorithms for discrete-data problems, two in statistics and one in operations...
Show moreThis thesis develops mathematical background for the design of algorithms for discrete-data problems, two in statistics and one in operations research. Chapter 1 gives some background on what chapters 2 to 4 have in common. It also defines some basic terminology that the other chapters use.Chapter 2 offers a general approach to modeling longitudinal network data, including exponential random graph models (ERGMs), that vary according to certain discrete-time Markov chains (The abstract of chapter 2 borrows heavily from the abstract of Schwartz et al., 2021). It connects conditional and Markovian exponential families, permutation- uniform Markov chains, various (temporal) ERGMs, and statistical considerations such as dyadic independence and exchangeability. Markovian exponential families are explored in depth to prove that they and only they have exponential family finite sample distributions with the same parameter as that of the transition probabilities. Many new statistical and algebraic properties of permutation-uniform Markov chains are derived. We introduce exponential random ?-multigraph models, motivated by our result on replacing ? observations of a permutation-uniform Markov chain of graphs with a single observation of a corresponding multigraph. Our approach simplifies analysis of some network and autoregressive models from the literature. Removing models’ temporal dependence but not interpretability permitted us to offer closed-form expressions for maximum likelihood estimators that previously did not have closed-form expression available. Chapter 3 designs novel, exact, conditional tests of statistical goodness-of-fit for mixed membership stochastic block models (MMSBMs) of networks, both directed and undirected. The tests employ a ?²-like statistic from which we define p-values for the general null hypothesis that the observed network’s distribution is in the MMSBM as well as for the simple null hypothesis that the distribution is in the MMSBM with specified parameters. For both tests the alternative hypothesis is that the distribution is unconstrained, and they both assume we have observed the block assignments. As exact tests that avoid asymptotic arguments, they are suitable for both small and large networks. Further we provide and analyze a Monte Carlo algorithm to compute the p-value for the simple null hypothesis. In addition to our rigorous results, simulations demonstrate the validity of the test and the convergence of the algorithm. As a conditional test, it requires the algorithm sample the fiber of a sufficient statistic. In contrast to the Markov chain Monte Carlo samplers common in the literature, our algorithm is an exact simulation, so it is faster, more accurate, and easier to implement. Computing the p-value for the general null hypothesis remains an open problem because it depends on an intractable optimization problem. We discuss the two schools of thought evident in the literature on how to deal with such problems, and we recommend a future research program to bridge the gap those two schools. Chapter 4 investigates an auctioneer’s revenue maximization problem in combinatorial auctions. In combinatorial auctions bidders express demand for discrete packages of multiple units of multiple, indivisible goods. The auctioneer’s NP-complete winner determination problem (WDP) is to fit these packages together within the available supply to maximize the bids’ sum. To shorten the path practitioners traverse from from legalese auction rules to computer code, we offer a new wdp formalism to reflect how government auctioneers sell billions of dollars of radio-spectrum licenses in combinatorial auctions today. It models common tie-breaking rules by maximizing a sum of bid vectors lexicographically. After a novel pre-solving technique based on package bids’ marginal values, we develop an algorithm for the WDP. In developing the algorithm’s branch-and-bound part adapted to lexicographic maximization, we discover a partial explanation of why classical WDP has been successful in using the linear programming relaxation: it equals the Lagrangian dual. We adapt the relaxation to lexicographic maximization. The algorithm’s dynamic-programming part retrieves already computed partial solutions from a novel data structure suited specifically to our WDP formalism. Finally we show that the data structure can “warm start” a popular algorithm for solving for opportunity-cost prices.
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