Physical Sciences and Mathematics Commons^™

Advances In Graph-Cut Optimization: Multi-Surface Models, Label Costs, And Hierarchical Costs, Andrew T. Delong

Electronic Thesis and Dissertation Repository

Computer vision is full of problems that are elegantly expressed in terms of mathematical optimization, or energy minimization. This is particularly true of "low-level" inference problems such as cleaning up noisy signals, clustering and classifying data, or estimating 3D points from images. Energies let us state each problem as a clear, precise objective function. Minimizing the correct energy would, hypothetically, yield a good solution to the corresponding problem. Unfortunately, even for low-level problems we are confronted by energies that are computationally hard—often NP-hard—to minimize. As a consequence, a rather large portion of computer vision research is dedicated to proposing …

Arrangements Of Submanifolds And The Tangent Bundle Complement, Priyavrat Deshpande

Electronic Thesis and Dissertation Repository

Drawing parallels with the theory of hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold $X$ we consider a finite collection $\A$ of locally flat codimension $1$ submanifolds that intersect like hyperplanes. To such an arrangement we associate two posets: the \emph{poset of faces} (or strata) $\FA$ and the \emph{poset of intersections} $L(\A)$. We also associate two topological spaces to $\A$. First, the complement of the union of submanifolds in $X$ which we call the \emph{set of chambers} and denote by $\Ch$. Second, the complement of union of tangent bundles of these …

Characteristic Polynomial Of Arrangements And Multiarrangements, Mehdi Garrousian

Electronic Thesis and Dissertation Repository

This thesis is on algebraic and algebraic geometry aspects of complex hyperplane arrangements and multiarrangements. We start by examining the basic properties of the logarithmic modules of all orders such as their freeness, the cdga structure, the local properties and close the first chapter with a multiarrangement version of a theorem due to M. Mustata and H. Schenck.

In the next chapter, we obtain long exact sequences of the logarithmic modules of an arrangement and its deletion-restriction under the tame conditions. We observe how the tame conditions transfer between an arrangement and its deletion-restriction.

In chapter 3, we use some …