Browsing by Author "Alderson, Troy F."

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    Multiresolution by Repeated Invertible Averaging - With Applications in Digital Earth
    (2019-11) Alderson, Troy F.; Samavati, Faramarz; Stefanakis, Emmanuel; Akleman, Ergun; Prusinkiewicz, Przemysław W.; Alim, Usman R.
    In 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.
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    RIAS: Repeated Invertible Averaging for Surface Multiresolution of Arbitrary Degree
    (2020-02-11) Alderson, Troy F.; Mahdavi-Amiri, Ali; Samavati, Faramarz
    In this paper, we introduce two local surface averaging operators with local inverses and use them to devise a method for surface multi-resolution (subdivision and reverse subdivision) of arbitrary degree. Similar to previous works by Stam, Zorin, and Schröder that achieved forward subdivision only, our averaging operators involve only direct neighbours of a vertex, and can be configured to generalize B-Spline multi-resolution to arbitrary topology surfaces. Our subdivision surfaces are hence able to exhibit Cd continuity at regular vertices (for arbitrary values of d) and appear to exhibit C1 continuity at extraordinary vertices. Smooth reverse and non-uniform subdivisions are additionally supported.