Spectral Frame Analysis and Learning through Graph Structure

This dissertation investigates the connection between spectral analysis and frame theory. When considering the spectral properties of a frame, we present a few novel results relating to the spectral decomposition. We first show that scalable frames have the property that the inner product of the scaling coefficients and the eigenvectors must equal the inverse eigenvalues. From this, we prove a similar result when an approximate scaling is obtained. We then focus on the optimization problems inherent to the scalable frames by first showing that there is an equivalence between scaling a frame and optimization problems with a non-restrictive objective function. Various objective functions are considered, and an analysis of the solution type is presented. For linear objectives, we can encourage sparse scalings, and with barrier objective functions, we force dense solutions. We further consider frames in high dimensions, and derive various solution techniques. From here, we restrict ourselves to various frame classes, to add more specificity to the results. Using frames generated from distributions allows for the placement of probabilistic bounds on scalability. For discrete distributions (Bernoulli and Rademacher), we bound the probability of encountering an ONB, and for continuous symmetric distributions (Uniform and Gaussian), we show that symmetry is retained in the transformed domain. We also prove several hyperplane-separation results. With the theory developed, we discuss graph applications of the scalability framework. We make a connection with graph conditioning, and show the in-feasibility of the problem in the general case. After a modification, we show that any complete graph can be conditioned. We then present a modification of standard PCA (robust PCA) developed by Cand\`es, and give some background into Electron Energy-Loss Spectroscopy (EELS). We design a novel scheme for the processing of EELS through robust PCA and least-squares regression, and test this scheme on biological samples. Finally, we take the idea of robust PCA and apply the technique of kernel PCA to perform robust manifold learning. We derive the problem and present an algorithm for its solution. There is also discussion of the differences with RPCA that make theoretical guarantees difficult.

LARGE MOTION VISUALIZATION AND ESTIMATION FOR FLAPPING WING SYSTEMS

Studies of fluid-structure interactions associated with flexible structures such as flapping wings require the capture and quantification of large motions of bodies that may be opaque. Motion capture of a free flying insect is considered by using three synchronized high-speed cameras. A solid finite element representation is used as a reference body and successive snapshots in time of the displacement fields are reconstructed via an optimization procedure. An objective function is formulated, and various shape difference definitions are considered. The proposed methodology is first studied for a synthetic case of a flexible cantilever structure undergoing large deformations, and then applied to a Manduca Sexta (hawkmoth) in free flight. The three-dimensional motions of this flapping system are reconstructed from image date collected by using three cameras. The complete deformation geometry of this system is analyzed. Finally, a computational investigation is carried out to understand the flow physics and aerodynamic performance by prescribing the body and wing motions in a fluid-body code. This thesis work contains one of the first set of such motion visualization and deformation analyses carried out for a hawkmoth in free flight. The tools and procedures used in this work are widely applicable to the studies of other flying animals with flexible wings as well as synthetic systems with flexible body elements.

FREQUENCY DOMAIN CHARACTERIZATION OF OPTIC FLOW AND VISION-BASED OCELLAR SENSING FOR ROTATIONAL MOTION

The structure of an animal’s eye is determined by the tasks it must perform. While vertebrates rely on their two eyes for all visual functions, insects have evolved a wide range of specialized visual organs to support behaviors such as prey capture, predator evasion, mate pursuit, flight stabilization, and navigation. Compound eyes and ocelli constitute the vision forming and sensing mechanisms of some flying insects. They provide signals useful for flight stabilization and navigation. In contrast to the well-studied compound eye, the ocelli, seen as the second visual system, sense fast luminance changes and allows for fast visual processing. Using a luminance-based sensor that mimics the insect ocelli and a camera-based motion detection system, a frequency-domain characterization of an ocellar sensor and optic flow (due to rotational motion) are analyzed. Inspired by the insect neurons that make use of signals from both vision sensing mechanisms, advantages, disadvantages and complementary properties of ocellar and optic flow estimates are discussed.