Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves

AbstractA “double” Pythagorean-hodograph (DPH) curve r(t) is characterized by the property that |r′(t)| and |r′(t)×r″(t)| are both polynomials in the curve parameter t. Such curves possess rational Frenet frames and curvature/torsion functions, and encompass all helical polynomial curves as special cases. As noted by Beltran and Monterde, the Hopf map representation of spatial PH curves appears better suited to the analysis of DPH curves than the quaternion form. A categorization of all DPH curve types up to degree 7 is developed using the Hopf map form, together with algorithms for their construction, and a selection of computed examples of (both helical and non-helical) DPH curves is included, to highlight their attractive features. For helical curves, a separate constructive approach proposed by Monterde, based upon the inverse stereographic projection of rational line/circle descriptions in the complex plane, is used to classify all types up to degree 7. Criteria to distinguish between the helical and non-helical DPH curves, in the context of the general construction procedures, are also discussed

On the formulation and uses of SVD-based generalized curvatures

2016 Summer.Includes bibliographical references.In this dissertation we consider the problem of computing generalized curvature values from noisy, discrete data and applications of the provided algorithms. We first establish a connection between the Frenet-Serret Frame, typically defined on an analytical curve, and the vectors from the local Singular Value Decomposition (SVD) of a discretized time-series. Next, we expand upon this connection to relate generalized curvature values, or curvatures, to a scaled ratio of singular values. Initially, the local singular value decomposition is centered on a point of the discretized time-series. This provides for an efficient computation of curvatures when the underlying curve is known. However, when the structure of the curve is not known, for example, when noise is present in the tabulated data, we propose two modifications. The first modification computes the local singular value decomposition on the mean-centered data of a windowed selection of the time-series. We observe that the mean-center version increases the stability of the curvature estimations in the presence of signal noise. The second modification is an adaptive method for selecting the size of the window, or local ball, to use for the singular value decomposition. This allows us to use a large window size when curvatures are small, which reduces the effects of noise thanks to the use of a large number of points in the SVD, and to use a small window size when curvatures are large, thereby best capturing the local curvature. Overall we observe that adapting the window size to the data, enhances the estimates of generalized curvatures. The combination of these two modifications produces a tool for computing generalized curvatures with reasonable precision and accuracy. Finally, we compare our algorithm, with and without modifications, to existing numerical curvature techniques on different types of data such as that from the Microsoft Kinect 2 sensor. To address the topic of action segmentation and recognition, a popular topic within the field of computer vision, we created a new dataset from this sensor showcasing a pose space skeletonized representation of individuals performing continuous human actions as defined by the MSRC-12 challenge. When this data is optimally projected onto a low-dimensional space, we observed each human motion lies on a distinguished line, plane, hyperplane, etc. During transitions between motions, either the dimension of the optimal subspace significantly, or the trajectory of the curve through pose space nearly reverses. We use our methods of computing generalized curvature values to identify these locations, categorized as either high curvatures or changing curvatures. The geometric characterization of the time-series allows us to segment individual,or geometrically distinct, motions. Finally, using these segments, we construct a methodology for selecting motions to conjoin for the task of action classification

Path Planning For Persistent Surveillance Applications Using Fixed-Wing Unmanned Aerial Vehicles

This thesis addresses coordinated path planning for fixed-wing Unmanned Aerial Vehicles (UAVs) engaged in persistent surveillance missions. While uniquely suited to this mission, fixed wing vehicles have maneuver constraints that can limit their performance in this role. Current technology vehicles are capable of long duration flight with a minimal acoustic footprint while carrying an array of cameras and sensors. Both military tactical and civilian safety applications can benefit from this technology. We make three main contributions: C1 A sequential path planner that generates a C2 flight plan to persistently acquire a covering set of data over a user designated area of interest. The planner features the following innovations: • A path length abstraction that embeds kino-dynamic motion constraints to estimate feasible path length • A Traveling Salesman-type planner to generate a covering set route based on the path length abstraction • A smooth path generator that provides C2 routes that satisfy user specified curvature constraints C2 A set of algorithms to coordinate multiple UAVs, including mission commencement from arbitrary locations to the start of a coordinated mission and de-confliction of paths to avoid collisions with other vehicles and fixed obstacles iv C3 A numerically robust toolbox of spline-based algorithms tailored for vehicle routing validated through flight test experiments on multiple platforms. A variety of tests and platforms are discussed. The algorithms presented are based on a technical approach with approximately equal emphasis on analysis, computation, dynamic simulation, and flight test experimentation. Our planner (C1) directly takes into account vehicle maneuverability and agility constraints that could otherwise render simple solutions infeasible. This is especially important when surveillance objectives elevate the importance of optimized paths. Researchers have devel oped a diverse range of solutions for persistent surveillance applications but few directly address dynamic maneuver constraints. The key feature of C1 is a two stage sequential solution that discretizes the problem so that graph search techniques can be combined with parametric polynomial curve generation. A method to abstract the kino-dynamics of the aerial platforms is then presented so that a graph search solution can be adapted for this application. An A* Traveling Salesman Problem (TSP) algorithm is developed to search the discretized space using the abstract distance metric to acquire more data or avoid obstacles. Results of the graph search are then transcribed into smooth paths based on vehicle maneuver constraints. A complete solution for a single vehicle periodic tour of the area is developed using the results of the graph search algorithm. To execute the mission, we present a simultaneous arrival algorithm (C2) to coordinate execution by multiple vehicles to satisfy data refresh requirements and to ensure there are no collisions at any of the path intersections. We present a toolbox of spline-based algorithms (C3) to streamline the development of C2 continuous paths with numerical stability. These tools are applied to an aerial persistent surveillance application to illustrate their utility. Comparisons with other parametric poly nomial approaches are highlighted to underscore the benefits of the B-spline framework. Performance limits with respect to feasibility constraints are documented

A New Shape Similarity Framework for brain fibers classification

Diffusion Magnetic Resonance Imaging (dMRI) techniques provide a non-invasive way to explore organization and integrity of the white matter structures in human brain. dMRI quantifies in each voxel, the diffusion process of water molecules which are mechanically constrained in their motion by the axons of the neurons. This technique can be used in surgical planning and in the study of anatomical connectivity, brain changes and mental disorders. From dMRI data, white matter fiber tracts can be reconstructed using a class of technique called tractography. The dataset derived by tractography is composed by a large number of streamlines, which are sequences of points in 3D space. To simplify the visualization and analysis of white matter fiber tracts obtained from tracking algorithms, it is often necessary to group them into larger clusters or bundles. This step is called clustering. In order to perform clustering, a mathematical definition of fiber similarity (or more commonly a fiber distance) must be specified. On the basis of this metric, pairwise fiber distance can be computed and used as input for a clustering algorithm. The most common metrics used for distance measure are able to capture only the local relationship between streamlines but not the global structure of the fiber. The global structure refers to the variability of the shape. Together, local and global information, can define a better metric of similarity. We have extracted the global information using a mathematical representation based on the study of the tract with Frénet equations. In particular, we have defined some intrinsic parameters of the fibers that led to a classification of the tracts based on global geometrical characteristics. Using these parameters, a new distance metric for fiber similarity has been developed. For the evaluation of the goodness of the new metric, indices were used for a qualitative study of the results