Mountain Scholar :: Browsing by Author "Peterson, Chris, committee member"

Browsing by Author "Peterson, Chris, committee member"

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Open Access
Abelian surfaces with real multiplication over finite fields
(Colorado State University. Libraries, 2014) Freese, Hilary, author; Achter, Jeffrey, advisor; Pries, Rachel, committee member; Peterson, Chris, committee member; Tavani, Daniele, committee member
Given a simple abelian surface A/Fq, the endomorphism algebra, End(A) ⊗ Q, contains a unique real quadratic subfield. We explore two different but related questions about when a particular real quadratic subfield K+ is the maximal real subfield of the endomorphism algebra. First, we compute the number of principally polarized abelian surfaces A/Fq such that K+ ⊂ End(A) ⊗ Q. Second, we consider an abelian surface A/Q, and its reduction Ap = A mod p, then ask for which primes p is K+ ⊂ End(A) ⊗ Q. The result from the first question leads to a heuristic for the second question, namely that the number of p < χ for which K+ ⊂ End(A) ⊗ Q grows like √χ/log(c).
Open Access
Algorithms for feature selection and pattern recognition on Grassmann manifolds
(Colorado State University. Libraries, 2015) Chepushtanova, Sofya, author; Kirby, Michael, advisor; Peterson, Chris, committee member; Bates, Dan, committee member; Ben-Hur, Asa, committee member
This dissertation presents three distinct application-driven research projects united by ideas and topics from geometric data analysis, optimization, computational topology, and machine learning. We first consider hyperspectral band selection problem solved by using sparse support vector machines (SSVMs). A supervised embedded approach is proposed using the property of SSVMs to exhibit a model structure that includes a clearly identifiable gap between zero and non-zero feature vector weights that permits important bands to be definitively selected in conjunction with the classification problem. An SSVM is trained using bootstrap aggregating to obtain a sample of SSVM models to reduce variability in the band selection process. This preliminary sample approach for band selection is followed by a secondary band selection which involves retraining the SSVM to further reduce the set of bands retained. We propose and compare three adaptations of the SSVM band selection algorithm for the multiclass problem. We illustrate the performance of these methods on two benchmark hyperspectral data sets. Second, we propose an approach for capturing the signal variability in data using the framework of the Grassmann manifold (Grassmannian). Labeled points from each class are sampled and used to form abstract points on the Grassmannian. The resulting points have representations as orthonormal matrices and as such do not reside in Euclidean space in the usual sense. There are a variety of metrics which allow us to determine distance matrices that can be used to realize the Grassmannian as an embedding in Euclidean space. Multidimensional scaling (MDS) determines a low dimensional Euclidean embedding of the manifold, preserving or approximating the Grassmannian geometry based on the distance measure. We illustrate that we can achieve an isometric embedding of the Grassmann manifold using the chordal metric while this is not the case with other distances. However, non-isometric embeddings generated by using the smallest principal angle pseudometric on the Grassmannian lead to the best classification results: we observe that as the dimension of the Grassmannian grows, the accuracy of the classification grows to 100% in binary classification experiments. To build a classification model, we use SSVMs to perform simultaneous dimension selection. The resulting classifier selects a subset of dimensions of the embedding without loss in classification performance. Lastly, we present an application of persistent homology to the detection of chemical plumes in hyperspectral movies. The pixels of the raw hyperspectral data cubes are mapped to the geometric framework of the Grassmann manifold where they are analyzed, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows the time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmannian. This motivates the search for topological structure, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the manifold. The proposed framework affords the processing of large data sets, such as the hyperspectral movies explored in this investigation, while retaining valuable discriminative information. For a particular choice of a distance metric on the Grassmannian, it is possible to generate topological signals that capture changes in the scene after a chemical release.
Open Access
Algorithms in numerical algebraic geometry and applications
(Colorado State University. Libraries, 2015) Hanson, Eric M., author; Bates, Daniel J., advisor; Peterson, Chris, committee member; Cavalieri, Renzo, committee member; Maciejewski, Anthony, committee member
The topics in this dissertation, while independent, are unified under the field of numerical algebraic geometry. With ties to some of the oldest areas in mathematics, numerical algebraic geometry is relatively young as a field of study in its own right. The field is concerned with the numerical approximation of the solution sets of systems of polynomial equations and the manipulation of these sets. Given a polynomial system ƒ : CN → Cn, the methods of numerical algebraic geometry produce numerical approximations of the isolated solutions of ƒ(z) = 0, as well as points on any positive-dimensional components of the solution set, V(ƒ). In a short time, the work done in numerical algebraic geometry has significantly pushed the boundary of what is computable. This dissertation aims to further this work by contributing new algorithms to the field and using cutting edge techniques of the field to expand the scope of problems that can be addressed using numerical methods. We begin with an introduction to numerical algebraic geometry and subsequent chapters address independent topics: perturbed homotopies, exceptional sets and fiber products, and a numerical approach to finding unit distance embeddings of finite simple graphs. One of the most recent advances in numerical algebraic geometry is regeneration, an equation-by-equation homotopy method that is often more efficient than other approaches. However, the basic form of regeneration will not necessarily find all isolated singular solutions of a polynomial system without the additional cost of using deflation. In the second chapter, we present an alternative to deflation in the form of perturbed homotopies for solving polynomial systems. In particular, we propose first solving a perturbed version of the polynomial system, followed by a parameter homotopy to remove the perturbation. The aim of this chapter is two-fold. First, such perturbed homotopies are sometimes more efficient than regular homotopies, though they can also be less efficient. Second, a useful consequence is that the application of this perturbation to regeneration will yield all isolated solutions, including all singular isolated solutions. The third chapter considers families of polynomial systems which depend on parameters. There is a typical dimension for the variety defined by a system in the family; however, this dimension may jump for parameters in algebraic subsets of the parameter space. Sommese and Wampler exploited fiber products to give a numerical method for identifying these special parameter values. In this chapter, we propose a refined numerical approach to fiber products, which uses recent advancements in numerical algebraic geometry, such as regeneration extension. We show that this method is sometimes more efficient then known techniques. This gain in efficiency is due to the fact that regeneration extension allows the construction of the fiber product to be restricted to specified irreducible components. This work is motivated by applications in Kinematics - the study of mechanisms. As such we use an algebraic model of a two link arm to illustrate the algorithms developed in this chapter. The topic of the last chapter is the identification of unit distance embeddings of finite simple graphs. Given a graph G(V,E), a unit distance embedding is a map ɸ from the vertex set V into a metric space M such that if {vi,vj} is an element of E then the distance between ɸ (vi) and ɸ (vj) in M is one. Given G, we cast the question of the existence of a unit distance embedding in Rn as the question of the existence of a real solution to a system of polynomial equations. As a consequence, we are able to develop theoretic algorithms for determining the existence of a unit distance embedding and for determining the smallest dimension of Rn for which a unit distance embedding of G exists (that is, we determine the minimal embedding dimension of G). We put these algorithms into practice using the methods of numerical algebraic geometry. In particular, we consider unit distance embeddings of the Heawood Graph. This is the smallest example of a point-line incidence graph of a finite projective plan. In 1972, Chvátal conjectured that point-line incidence graphs of finite projective planes do not have unit-distance embeddings into R². In other words, Chvátal conjectured that the minimal embedding dimension of any point-line incidence graph of a finite projective plane is at least 3. We disprove this conjecture, adding hundreds of counterexamples to the 11 known counterexamples found by Gerbracht.
Open Access
Automated detection of circulating cells using low level features
(Colorado State University. Libraries, 2013) Emerson, Tegan Halley, author; Kirby, Michael, advisor; Peterson, Chris, committee member; Nyborg, Jennifer, committee member
This thesis addresses the problem of detection of high definition circulating tumor cells using data driven feature selection. We propose techniques in pattern analysis and computer vision to achieve this goal. Specifically, we determine a set of low level features which can structurally differentiate between different cell types of interest to contribute to the treatment and monitoring of patients. We have implemented three image representation techniques on a curated data set. The curated data set consists of digitized images of 1000 single cells: 500 of which are high definition circulating tumor cells or other cells of high interest, and 500 of which are white blood cells. None of the three image representation techniques have been previously applied to this data set. One image representation is a novel contribution and is based on the characterization of a cell in terms of its concentric Fourier rings. The Fourier Ring Descriptors (FRDs) exploit the size variations and morphological differences between events of high and low interest while being rotationally invariant. Using the low level descriptors, FRDs, as a representation with a linear support vector machine decision tree classifier we have been able to average 99.34% accuracy on the curated data set and 99.53% on non-curated data. FRDs exhibit robustness to rotation and segmentation error. We discuss the applications of the results to clinical use in context of data provided by The Kuhn Laboratory at The Scripps Research Institute.
Open Access
Bridgeland stability of line bundles on smooth projective surfaces
(Colorado State University. Libraries, 2014) Miles, Eric W., author; Cavalieri, Renzo, advisor; Achter, Jeff, committee member; Peterson, Chris, committee member; Prasad, Ashok, committee member; Pries, Rachel, committee member
Bridgeland Stability Conditions can be thought of as tools for creating and varying moduli spaces parameterizing objects in the derived category of a variety X. Line bundles on the variety are fundamental objects in its derived category, and we characterize the Bridgeland stability of line bundles on certain surfaces. Evidence is provided for an analogous characterization in the general case. We find stability conditions for P1 × P1 which can be seen as giving the stability of representations of quivers, and we deduce projective structure on the Bridgeland moduli spaces in this situation. Finally, we prove a number of results on objects and a construction related to the quivers mentioned above.
Open Access
Classifying simplicial dissections of convex polyhedra with symmetry
(Colorado State University. Libraries, 2021) Mukthineni, Tarun, author; Betten, Anton, advisor; Peterson, Chris, committee member; Rajopadhye, Sanjay, committee member
A convex polyhedron is the convex hull of a finite set ofpoints in R3. A triangulation of a convex polyhedron is a decomposition into a finite number of 3-simplices such that any two intersect in a common face or are disjoint. A simplicial dissection is a decomposition into a finite number of 3-simplices such that no two share an interior point. We present an algorithm to classify the simplicial dissections of a regular polyhedron under the symmetry group of the polyhedron.
Open Access
Computer vision approach to classification of circulating tumor cells
(Colorado State University. Libraries, 2013) Hopkins, David, author; Kirby, Michael, advisor; Peterson, Chris, committee member; Givens, Geof, committee member
Current research into the detection and characterization of circulating tumor cells (CTCs) in the bloodstream can be used to assess the threat to a potential cancer victim. We have determined specific goals to further the understanding of these cells. 1) Full automation of an algorithm to overcome the current methods challenges of being labor-intensive and time-consuming, 2) Detection of single CTC cells amongst several million white blood cells given digital imagery of a panel of blood, and 3) Objective classification of white blood cells, CTCs, and potential sub-types. We demonstrate in this paper the developed theory, code and implementation necessary for addressing these goals using mathematics and computer vision techniques. These include: 1) Formation of a completely data-driven methodology, and 2) Use of Bag of Features computer vision technique coupled with custom-built pixel-centric feature descriptors, 3) Use of clustering techniques such as K-means and Hierarchical clustering as a robust classification method to glean insights into cell characteristics. To objectively determine the adequacy of our approach, we test our algorithm against three benchmarks: sensitivity/specificity in classification, nontrivial event detection, and rotational invariance. The algorithm performed well with the first two, and we provide possible modifications to improve performance on the third. The results of the sensitivity and specificity benchmark are important. The unfiltered data we used to test our algorithm were images of blood panels containing 44,914 WBCs and 39 CTCs. The algorithm classified 67.5 percent of CTCs into an outlier cluster containing only 300 cells. A simple modification brought the classification rate up to 80 percent of total CTCs. This modification brings the cluster count to only 400 cells. This is a significant reduction in cells a pathologist would sort through as it is only .9 percent of the total data.
Open Access
Computing syzygies of homogeneous polynomials using linear algebra
(Colorado State University. Libraries, 2014) Hodges, Tim, author; Bates, Dan, advisor; Peterson, Chris, committee member; Böhm, A. P. Willem, committee member
Given a ideal generated by polynomials ƒ1,...,ƒn in polynomial ring of m variables a syzygy is an n-tuple α1,.., αn, & αi in our polynomial ring of m variables such that our n-tuple holds the orthogonal property on the generators above. Syzygies can be computed by Buchberger's algorithm for computing Gröbner Bases. However, Gröbner bases have been computationally impractical as the number of variables and number of polynomials increase. The aim of this thesis is to describe a way to compute syzygies without the need for Grobner bases but still retrieve some of the same information as Gröbner bases. The approach is to use the monomial structure of the polynomials in our generating set to build syzygies using Nullspace computations.
Open Access
Consistent hidden Markov models
(Colorado State University. Libraries, 2014) Narayana Rao Gari, Pradyumna Kumar, author; Draper, Bruce A., advisor; Beveridge, Ross, committee member; Peterson, Chris, committee member
Activity recognition in Computer Vision involves recognizing the appearance of an object of interest along with its action, and its relation to the scene or other important objects. There exist many methods that give this information about an object. However, these methods are noisy and are independent of each other. So, the mutual information between the labels is lost. For example, an object might be predicted to be a tree, whereas its action might be predicted as walk. But, trees can't walk. However, the compositional structure of the events is reflected by the compositional structure of natural language. The object of interest is the predicate, usually a noun, the action is the verb, and its relation to the scene may be a preposition or adverb. The lost mutual information that says that trees can't walk is present in natural language. The contribution of this thesis is a method of visual information fusion based on exploiting the mutual information from Natural language databases. Although Hidden Markov Models (HMM) are the traditional way to smooth noisy stream of data by integrating information across time, they can't account for the lost mutual information. This thesis proposes an extension to HMM (Consistent HMM) that can integrate visual information to the lost mutual information by exploiting the knowledge from language databases. Consistent HMM performs better than other state of the art HMMs on synthetic data generated to simulate the real world behavior. Although the performance gain of integrating the knowledge from language databases both during training phase and run-time is better, when considered individually, the performance gain is more when the knowledge is integrated during run-time than training.
Open Access
Demonstrating that dataset domains are largely linearly separable in the feature space of common CNNs
(Colorado State University. Libraries, 2020) Dragan, Matthew R., author; Beveridge, J. Ross, advisor; Ortega, Francisco, committee member; Peterson, Chris, committee member
Deep convolutional neural networks (DCNNs) have achieved state of the art performance on a variety of tasks. These high-performing networks require large and diverse training datasets to facilitate generalization when extracting high-level features from low-level data. However, even with the availability of these diverse datasets, DCNNs are not prepared to handle all the data that could be thrown at them. One major challenges DCNNs face is the notion of forced choice. For example, a network trained for image classification is configured to choose from a predefined set of labels with the expectation that any new input image will contain an instance of one of the known objects. Given this expectation it is generally assumed that the network is trained for a particular domain, where domain is defined by the set of known object classes as well as more implicit assumptions that go along with any data collection. For example, some implicit characteristics of the ImageNet dataset domain are that most images are taken outdoors and the object of interest is roughly in the center of the frame. Thus the domain of the network is defined by the training data that is chosen. Which leads to the following key questions: Does a network know the domain it was trained for? and Can a network easily distinguish between in-domain and out-of-domain images? In this thesis it will be shown that for several widely used public datasets and commonly used neural networks, the answer to both questions is yes. The presence of a simple method of differentiating between in-domain and out-of-domain cases has significant implications for work on domain adaptation, transfer learning, and model generalization.
Open Access
Discrete-time topological dynamics, complex Hadamard matrices, and oblique-incidence ion bombardment
(Colorado State University. Libraries, 2014) Motta, Francis Charles, author; Shipman, Patrick D., advisor; Dangelmayr, Gerhard, committee member; Peterson, Chris, committee member; Bradley, Mark, committee member
The topics covered in this dissertation are not unified under a single mathematical discipline. However, the questions posed and the partial solutions to problems of interest were heavily influenced by ideas from dynamical systems, mathematical experimentation, and simulation. Thus, the chapters in this document are unified by a common flavor which bridges several mathematical and scientific disciplines. The first chapter introduces a new notion of orbit density applicable to discrete-time dynamical systems on a topological phase space, called the linear limit density of an orbit. For a fixed discrete-time dynamical system, Φ(χ) : M → M defined on a bounded metric space, we introduce a function E : {γχ : χ ∈ Mg} → R∪{∞} on the orbits of Φ, γχ ≐ {Φt(χ) : t ∈ N}, and interpret E(γχ) as a measure of the orbit's approach to density; the so-called linear limit density (LLD) of an orbit. We first study the family of dynamical systems Rθ : [0; 1) → [0; 1)(θ ∈ (0; 1)) defined by Rθ(χ) = (χ + θ) mod 1. Utilizing a formula derived from the Three-Distance theorem, we compute the exact value of E({RtΦ(χ) : t ∈ N}, χ ∈ [0; 1)), where Φ = √5 – 1) /2. We further compute E({Rtθ(χ) : t ∈ N}; χ ∈ [0, 1)) for a class of irrational rotation angles θ = [j, j,…] with period-1 continued fraction expansions and discuss how this measure distinguishes the topologically transitive behavior of different choices of θ. We then expand our focus to a much broader class of orientation-preserving homeomorphisms of the circle and extend a result of R. Graham and J.H. van Lint about optimal irrational rotations. Finally, we consider the LLD of orbits of the Bernoulli shift map acting on sequences defined over a finite alphabet and prove bounds for a class of sequences built by recursive extension of de Bruijn sequences. To compute approximations of E(γχ) for orbits of the Bernoulli shift map, we develop an efficient algorithm which determines a point in the set of all words of a fixed length over a finite alphabet whose distance to a distinguished subset is maximal. Chapter two represents a departure from a dynamical systems problem by instead exploring the structure of the space of complex Hadamard matrices and mutually unbiased bases (MUBs) of complex Hilbert space. Although the problem is not intrinsically dynamical, our mechanisms for experimentation and exploration include an algorithm which can be viewed as a discrete-time dynamical system as well as a gradient system of ordinary differential equations (ODEs) whose fixed points are dephased complex Hadamards. We use our discrete system to produce numerical evidence which supports existing conjectures regarding complex Hadamards and mutually unbiased bases, including that the maximal size of a set of 6 x 6 MUBs is four. By applying center-manifold theory to our gradient system, we introduce a novel method to analyze the structure of Hadamards near a fixed matrix. In addition to formalizing this technique, we apply it to prove that a particular 9 x 9 Hadamard does not belong to a continuous family of inequivalent matrices, despite having a positive defect. This is the first known example of this type. The third chapter explores the phenomenon of pattern formation in dynamical systems by considering a model of off-normal incidence ion bombardment (OIIB) of a binary material. We extend the Bradley-Shipman theory of normal-incidence ion bombardment of a binary material by analyzing a system of partial differential equations that models the off-normal incidence ion bombardment of a binary material by coupling surface topography and composition. In this chapter we perform linear and non-linear analysis of the equations modeling the interaction between surface height and composition and derive a system of ODEs which govern the time-evolution of the unstable modes, allowing us to identify parameter ranges which lead to patterns of interest. In particular, we demonstrate that an unusual "dots-on-ripples" topography can emerge for nonzero angles of ion incidence θ. In such a pattern, nanodots arranged in a hexagonal array sit atop a ripple topography. We find that if dots-on-ripples are supplanted by surface ripples as θ or the ion energy are varied, the transition is continuous.
Open Access
EEG subspace analysis and classification using principal angles for brain-computer interfaces
(Colorado State University. Libraries, 2015) Ashari, Rehab Bahaaddin, author; Anderson, Charles W., advisor; Ben-Hur, Asa, committee member; Draper, Bruce, committee member; Peterson, Chris, committee member
Brain-Computer Interfaces (BCIs) help paralyzed people who have lost some or all of their ability to communicate and control the outside environment from loss of voluntary muscle control. Most BCIs are based on the classification of multichannel electroencephalography (EEG) signals recorded from users as they respond to external stimuli or perform various mental activities. The classification process is fraught with difficulties caused by electrical noise, signal artifacts, and nonstationarity. One approach to reducing the effects of similar difficulties in other domains is the use of principal angles between subspaces, which has been applied mostly to video sequences. This dissertation studies and examines different ideas using principal angles and subspaces concepts. It introduces a novel mathematical approach for comparing sets of EEG signals for use in new BCI technology. The success of the presented results show that principal angles are also a useful approach to the classification of EEG signals that are recorded during a BCI typing application. In this application, the appearance of a subject's desired letter is detected by identifying a P300-wave within a one-second window of EEG following the flash of a letter. Smoothing the signals before using them is the only preprocessing step that was implemented in this study. The smoothing process based on minimizing the second derivative in time is implemented to increase the classification accuracy instead of using the bandpass filter that relies on assumptions on the frequency content of EEG. This study examines four different ways of removing outliers that are based on the principal angles and shows that the outlier removal methods did not help in the presented situations. One of the concepts that this dissertation focused on is the effect of the number of trials on the classification accuracies. The achievement of the good classification results by using a small number of trials starting from two trials only, should make this approach more appropriate for online BCI applications. In order to understand and test how EEG signals are different from one subject to another, different users are tested in this dissertation, some with motor impairments. Furthermore, the concept of transferring information between subjects is examined by training the approach on one subject and testing it on the other subject using the training subject's EEG subspaces to classify the testing subject's trials.
Open Access
Element rearrangement for action classification on product manifolds
(Colorado State University. Libraries, 2013) Kadappan, Karthik, author; Beveridge, J. Ross, advisor; Maciejewski, Anthony A., committee member; Peterson, Chris, committee member; Rajopadhye, Sanjay, committee member
Conventional tensor-based classification algorithms unfold tensors into matrices using the standard mode-k unfoldings and perform classification using established machine learning algorithms. These methods assume that the standard mode-k unfolded matrices are the best 2-dimensional representations of N-dimensional structures. In this thesis, we ask the question: "Is there a better way to unfold a tensor?" To address this question, we design a method to create unfoldings of a tensor by rearranging elements in the original tensor and then applying the standard mode-k unfoldings. The rearrangement of elements in a tensor is formulated as a combinatorial optimization problem and tabu search is adapted in this work to solve it. We study this element rearrangement problem in the context of tensor-based action classification on product manifolds. We assess the proposed methods using a publicly available video data set, namely Cambridge-Gesture data set. We design several neighborhood structures and search strategies for tabu search and analyze their performance. Results reveal that the proposed element rearrangement algorithm developed in this thesis can be employed as a preprocessing step to increase classification accuracy in the context of action classification on product manifolds method.
Open Access
Evaluating cluster quality for visual data
(Colorado State University. Libraries, 2013) Wigness, Maggie, author; Draper, Bruce, advisor; Beveridge, Ross, committee member; Howe, Adele, committee member; Peterson, Chris, committee member
Digital video cameras have made it easy to collect large amounts of unlabeled data that can be used to learn to recognize objects and actions. Collecting ground-truth labels for this data, however, is a much more time consuming task that requires human intervention. One approach to train on this data, while keeping the human workload to a minimum, is to cluster the unlabeled samples, evaluate the quality of the clusters, and then ask a human annotator to label only the clusters believed to be dominated by a single object/action class. This thesis addresses the task of evaluating the quality of unlabeled image clusters. We compare four cluster quality measures (and a baseline method) using real-world and synthetic data sets. Three of these measures can be found in the existing data mining literature: Dunn Index, Davies-Bouldin Index and Silhouette Width. We introduce a novel cluster quality measure as the fourth measure, derived from recent advances in approximate nearest neighbor algorithms from the computer vision literature, called Proximity Forest Connectivity (PFC). Experiments on real-world data show that no cluster quality measure performs "best" on all data sets; however, our novel PFC measure is always competitive and results in more top performances than any of the other measures. Results from synthetic data experiments show that while the data mining measures are susceptible to over-clustering typically required of visual data, PFC is much more robust. Further synthetic data experiments modeling features of visual data show that Davies-Bouldin is most robust to large amounts of class-specific noise. However, Davies-Bouldin, Silhouette and PFC all perform well in the presence of data with small amounts of class-specific noise, whereas Dunn struggles to perform better than random.
Open Access
Expected distances on homogeneous manifolds and notes on pattern formation
(Colorado State University. Libraries, 2023) Balch, Brenden, author; Shipman, Patrick, advisor; Bradley, Mark, committee member; Shonkwiler, Clay, committee member; Peterson, Chris, committee member; Chen, Hua, committee member
Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In Chapter 1 of this dissertation, we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data. Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In Chapter 2, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the kth moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces. Motivated by previous results showing that the addition of a linear dispersive term to the two-dimensional Kuramoto-Sivashinsky equation has a dramatic effect on the pattern formation, we study the Swift-Hohenberg equation with an added linear dispersive term, the dispersive Swift-Hohenberg equation (DSHE) in Chapter 3. The DSHE produces stripe patterns with spatially extended defects that we call seams. A seam is defined to be a dislocation that is smeared out along a line segment that is obliquely oriented relative to an axis of reflectional symmetry. In contrast to the dispersive Kuramoto-Sivashinsky equation, the DSHE has a narrow band of unstable wavelengths close to an instability threshold. This allows for analytical progress to be made. We show that the amplitude equation for the DSHE close to threshold is a special case of the anisotropic complex Ginzburg-Landau equation (ACGLE) and that seams in the DSHE correspond to spiral waves in the ACGLE. Seam defects and the corresponding spiral waves tend to organize themselves into chains, and we obtain formulas for the velocity of the spiral wave cores and for the spacing between them. In the limit of strong dispersion, a perturbative analysis yields a relationship between the amplitude and wavelength of a stripe pattern and its propagation velocity. Numerical integrations of the ACGLE and the DSHE confirm these analytical results. Chapter 4 explores the measurement and characterization of order in non-equilibrium pattern forming systems. The study focuses on the use of topological measures of order, via persistent homology and the Wasserstein metric. We investigate the quantification of order with respect to ideal lattice patterns and demonstrate the effectiveness of the introduced measures of order in analyzing imperfect three-dimensional patterns and their time evolution. The paper provides valuable insights into the complex pattern formation and contributes to the understanding of order in three dimensions.
Open Access
FDOA-based passive source localization: a geometric perspective
(Colorado State University. Libraries, 2018) Cameron, Karleigh J., author; Bates, Dan, advisor; Cheney, Margaret, committee member; Peterson, Chris, committee member; Fosdick, Bailey, committee member
We consider the problem of passively locating the source of a radio-frequency signal using observations by several sensors. Received signals can be compared to obtain time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements. The geometric relationship satisfied by these measurements allow us to make inferences about the emitter's location. In this research, we choose to focus on the FDOA-based source localization problem. This problem has been less widely studied and is more difficult than solving for an emitter's location using TDOA measurements. When the FDOA-based source localization problem is formulated as a system of polynomials, the source's position is contained in the corresponding algebraic variety. This provides motivation for the use of methods from algebraic geometry, specifically numerical algebraic geometry (NAG), to solve for the emitter's location and gain insight into this system's interesting structure.
Open Access
Generalizations of persistent homology
(Colorado State University. Libraries, 2021) McCleary, Alexander J., author; Patel, Amit, advisor; Adams, Henry, committee member; Ben Hur, Asa, committee member; Peterson, Chris, committee member
Persistent homology typically starts with a filtered chain complex and produces an invariant called the persistence diagram. This invariant summarizes where holes are born and die in the filtration. In the traditional setting the filtered chain complex is a chain complex of vector spaces filtered over a totally ordered set. There are two natural directions to generalize the persistence diagram: we can consider filtrations of more general chain complexes and filtrations over more general partially ordered sets. In this dissertation we develop both of these generalizations by defining persistence diagrams for chain complexes in an essentially small abelian category filtered over any finite lattice.
Open Access
Hierarchical cluster guided labeling: efficient label collection for visual classification
(Colorado State University. Libraries, 2015) Wigness, Maggie, author; Draper, Bruce, advisor; Beveridge, Ross, committee member; Howe, Adele, committee member; Peterson, Chris, committee member
Visual classification is a core component in many visually intelligent systems. For example, recognition of objects and terrains provides perception during path planning and navigation tasks performed by autonomous agents. Supervised visual classifiers are typically trained with large sets of images to yield high classification performance. Although the collection of raw training data is easy, the required human effort to assign labels to this data is time consuming. This is particularly problematic in real-world applications with limited labeling time and resources. Techniques have emerged that are designed to help alleviate the labeling workload but suffer from several shortcomings. First, they do not generalize well to domains with limited a priori knowledge. Second, efficiency is achieved at the cost of collecting significant label noise which inhibits classifier learning or requires additional effort to remove. Finally, they introduce high latency between labeling queries, restricting real-world feasibility. This thesis addresses these shortcomings with unsupervised learning that exploits the hierarchical nature of feature patterns and semantic labels in visual data. Our hierarchical cluster guided labeling (HCGL) framework introduces a novel evaluation of hierarchical groupings to identify the most interesting changes in feature patterns. These changes help localize group selection in the hierarchy to discover and label a spectrum of visual semantics found in the data. We show that employing majority group-based labeling after selection allows HCGL to balance efficiency and label accuracy, yielding higher performing classifiers than other techniques with respect to labeling effort. Finally, we demonstrate the real-world feasibility of our labeling framework by quickly training high performing visual classifiers that aid in successful mobile robot path planning and navigation.
Open Access
Hyperovals, Laguerre planes and hemisystems - an approach via symmetry
(Colorado State University. Libraries, 2013) Bayens, Luke, author; Penttila, Tim, advisor; Achter, Jeff, committee member; Bohm, Willem, committee member; Peterson, Chris, committee member
In 1872, Felix Klein proposed the idea that geometry was best thought of as the study of invariants of a group of transformations. This had a profound effect on the study of geometry, eventually elevating symmetry to a central role. This thesis embodies the spirit of Klein's Erlangen program in the modern context of finite geometries -- we employ knowledge about finite classical groups to solve long-standing problems in the area. We first look at hyperovals in finite Desarguesian projective planes. In the last 25 years a number of infinite families have been constructed. The area has seen a lot of activity, motivated by links with flocks, generalized quadrangles, and Laguerre planes, amongst others. An important element in the study of hyperovals and their related objects has been the determination of their groups -- indeed often the only way of distinguishing them has been via such a calculation. We compute the automorphism group of the family of ovals constructed by Cherowitzo in 1998, and also obtain general results about groups acting on hyperovals, including a classification of hyperovals with large automorphism groups. We then turn our attention to finite Laguerre planes. We characterize the Miquelian Laguerre planes as those admitting a group containing a non-trivial elation and acting transitively on flags, with an additional hypothesis -- a quasiprimitive action on circles for planes of odd order, and insolubility of the group for planes of even order. We also prove a correspondence between translation ovoids of translation generalized quadrangles arising from a pseudo-oval O and translation flocks of the elation Laguerre plane arising from the dual pseudo-oval O*. The last topic we consider is the existence of hemisystems in finite hermitian spaces. Hemisystems were introduced by Segre in 1965 -- he constructed a hemisystem of H(3,32) and rasied the question of their existence in other spaces. Much of the interest in hemisystems is due to their connection to other combinatorial structures, such as strongly regular graphs, partial quadrangles, and association schemes. In 2005, Cossidente and Penttila constructed a family of hemisystems in H(3,q2), q odd, and in 2009, the same authors constructed a family of hemisystem in H(5,q2), q odd. We develop a new approach that generalizes the previous constructions of hemisystems to H(2r - 1,q2), r > 2, q odd.
Open Access
Intersections of ψ classes on Hassett spaces of rational curves
(Colorado State University. Libraries, 2018) Sharma, Nand, author; Cavalieri, Renzo, advisor; Peterson, Chris, committee member; Achter, Jeff, committee member; Prasad, Ashok, committee member
Hassett spaces are moduli spaces of weighted stable pointed curves. In this work, we consider such spaces of curves of genus 0 with weights all 1/q , q being a positive integer greater than or equal to 2. These spaces are interesting as they have different universal families and different intersection theory when compared with classical moduli spaces of pointed stable rational curves. We develop closed formulas for intersections of ψ-classes on such spaces. In our main result, we encode the formula for top intersections in a generating function obtained by applying an exponential differential operator to the Witten-potential.