284 research outputs found
The Set Structure of Precision: Coherent Probabilities on Pre-Dynkin-Systems
In literature on imprecise probability little attention is paid to the fact
that imprecise probabilities are precise on some events. We call these sets
system of precision. We show that, under mild assumptions, the system of
precision of a lower and upper probability form a so-called
(pre-)Dynkin-system. Interestingly, there are several settings, ranging from
machine learning on partial data over frequential probability theory to quantum
probability theory and decision making under uncertainty, in which a priori the
probabilities are only desired to be precise on a specific underlying set
system. At the core of all of these settings lies the observation that precise
beliefs, probabilities or frequencies on two events do not necessarily imply
this precision to hold for the intersection of those events. Here,
(pre-)Dynkin-systems have been adopted as systems of precision, too. We show
that, under extendability conditions, those pre-Dynkin-systems equipped with
probabilities can be embedded into algebras of sets. Surprisingly, the
extendability conditions elaborated in a strand of work in quantum physics are
equivalent to coherence in the sense of Walley (1991, Statistical reasoning
with imprecise probabilities, p. 84). Thus, literature on probabilities on
pre-Dynkin-systems gets linked to the literature on imprecise probability.
Finally, we spell out a lattice duality which rigorously relates the system of
precision to credal sets of probabilities. In particular, we provide a hitherto
undescribed, parametrized family of coherent imprecise probabilities
2D growth processes: SLE and Loewner chains
This review provides an introduction to two dimensional growth processes.
Although it covers a variety processes such as diffusion limited aggregation,
it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner
evolutions (SLE) which are Markov processes describing interfaces in 2D
critical systems. It starts with an informal discussion, using numerical
simulations, of various examples of 2D growth processes and their connections
with statistical mechanics. SLE is then introduced and Schramm's argument
mapping conformally invariant interfaces to SLE is explained. A substantial
part of the review is devoted to reveal the deep connections between
statistical mechanics and processes, and more specifically to the present
context, between 2D critical systems and SLE. Some of the SLE remarkable
properties are explained, as well as the tools for computing with SLE. This
review has been written with the aim of filling the gap between the
mathematical and the physical literatures on the subject.Comment: A review on Stochastic Loewner evolutions for Physics Reports, 172
pages, low quality figures, better quality figures upon request to the
authors, comments welcom
Doctor of Philosophy
dissertationImage segmentation entails the partitioning of an image domain, usually two or three dimensions, so that each partition or segment has some meaning that is relevant to the application at hand. Accurate image segmentation is a crucial challenge in many disciplines, including medicine, computer vision, and geology. In some applications, heterogeneous pixel intensities; noisy, ill-defined, or diffusive boundaries; and irregular shapes with high variability can make it challenging to meet accuracy requirements. Various segmentation approaches tackle such challenges by casting the segmentation problem as an energy-minimization problem, and solving it using efficient optimization algorithms. These approaches are broadly classified as either region-based or edge (surface)-based depending on the features on which they operate. The focus of this dissertation is on the development of a surface-based energy model, the design of efficient formulations of optimization frameworks to incorporate such energy, and the solution of the energy-minimization problem using graph cuts. This dissertation utilizes a set of four papers whose motivation is the efficient extraction of the left atrium wall from the late gadolinium enhancement magnetic resonance imaging (LGE-MRI) image volume. This dissertation utilizes these energy formulations for other applications, including contact lens segmentation in the optical coherence tomography (OCT) data and the extraction of geologic features in seismic data. Chapters 2 through 5 (papers 1 through 4) explore building a surface-based image segmentation model by progressively adding components to improve its accuracy and robustness. The first paper defines a parametric search space and its discrete formulation in the form of a multilayer three-dimensional mesh model within which the segmentation takes place. It includes a generative intensity model, and we optimize using a graph formulation of the surface net problem. The second paper proposes a Bayesian framework with a Markov random field (MRF) prior that gives rise to another class of surface nets, which provides better segmentation with smooth boundaries. The third paper presents a maximum a posteriori (MAP)-based surface estimation framework that relies on a generative image model by incorporating global shape priors, in addition to the MRF, within the Bayesian formulation. Thus, the resulting surface not only depends on the learned model of shapes,but also accommodates the test data irregularities through smooth deviations from these priors. Further, the paper proposes a new shape parameter estimation scheme, in closed form, for segmentation as a part of the optimization process. Finally, the fourth paper (under review at the time of this document) presents an extensive analysis of the MAP framework and presents improved mesh generation and generative intensity models. It also performs a thorough analysis of the segmentation results that demonstrates the effectiveness of the proposed method qualitatively, quantitatively, and clinically. Chapter 6, consisting of unpublished work, demonstrates the application of an MRF-based Bayesian framework to segment coupled surfaces of contact lenses in optical coherence tomography images. This chapter also shows an application related to the extraction of geological structures in seismic volumes. Due to the large sizes of seismic volume datasets, we also present fast, approximate surface-based energy minimization strategies that achieve better speed-ups and memory consumption
Representation and estimation of stochastic populations
This work is concerned with the representation and the estimation of populations
composed of an uncertain and varying number of individuals which can randomly
evolve in time. The existing solutions that address this type of problems make the
assumption that all or none of the individuals are distinguishable. In other words,
the focus is either on specific individuals or on the population as a whole. Theses
approaches have complimentary advantages and drawbacks and the main objective
in this work is to introduce a suitable representation for partially-indistinguishable
populations. In order to fulfil this objective, a sufficiently versatile way of quantifying
different types of uncertainties has to be studied. It is demonstrated that this can
be achieved within a measure-theoretic Bayesian paradigm. The proposed representation
of stochastic populations is then used for the introduction of various filtering
algorithms from the most general to the most specific. The modelling possibilities
and the accuracy of one of these filters are then demonstrated in different situations
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