Samavati, FaramarzAlderson, Troy F.2019-11-272019-11-272019-11Alderson, T. F. (2019). Multiresolution by Repeated Invertible Averaging - With Applications in Digital Earth (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.http://hdl.handle.net/1880/111254In this thesis, we present a general-purpose, arbitrary-degree framework for the multiscale representation of various types of graphics objects. These include curves in Euclidean and non-Euclidean spaces (particularly the surfaces of spheres and ellipsoids) as well as polygonal mesh surfaces. The core framework, which operates on curves, is based on simple yet fundamental modifications to the Lane-Riesenfeld algorithm and its generalizations. The algorithm’s averaging step is replaced with invertible variants, defining a repeated invertible averaging approach that supports a class of subdivision and reverse subdivision methods (including those that produce B-Spline curves). These averaging steps and their inverses are defined in terms of sequences of two-point interpolations between neighbouring vertices, which can be easily generalized to several different spaces and manifolds. In addition to developing this core framework, we explore different applications and generalizations of the approach. In particular, we concentrate on applications to Digital Earth, where spherical and ellipsoidal curves can be used to represent geospatial vectors (e.g. nation boundaries, road networks). We use multiscale representations of geospatial vector data to develop fast algorithms for offsetting queries and inside/outside tests. A fast offsetting algorithm for rasterized vectors in a DGGS is also presented. Generalizations of the approach include a modification that allows our framework to produce multiscale NURBS curves on the sphere and ellipsoid. This is accomplished by incorporating vertex weights into the interpolation parameters of individual operations, preserving the framework's generalizability. We additionally present a generalization of our framework to the multiscale representation of polygonal meshes. Similarly to the curve case, our framework for surfaces is defined in terms of local mesh operations that involve only direct vertex neighbours. Smooth reverse and non-uniform surface subdivisions are additionally supported.engUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.MultiresolutionSubdivisionReverse SubdivisionDigital EarthComputer Sciencedoctoral thesis10.11575/PRISM/37274