Affine-invariant querying of spatial data using a triangle-based logic
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Affine-invariant querying of spatial data using a triangle-based logic Sofie Haesevoets1 · Bart Kuijpers2 · Peter Z. Revesz3 Received: 26 March 2019 / Revised: 21 February 2020 / Accepted: 27 March 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract If the same land area is photographed from two different angles, then a computer can recognize the two images to show the same land area by proving that the images are affineinvariant. In practice, images are abstracted as a type of spatial data that is a finite set of triangles. Hence we propose a simple affine-generic query language with variables over finite sets of triangles. The proposed query language is intuitive and has the same expressive power as the affine-generic fragment of first-order logic over the reals on triangle databases. Keywords Affine-invariant queries · Spatial query languages · Weak perspective assumption
1 Introduction The need for affine-invariance arises naturally from many spatial applications where different images such as various aerial photographs of some land area have to be recognized to belong together. For example, after the recent floods in the U.S. Midwest an aerial survey had to be made to assess the damages. In bringing together the “before” and the “after” images, a computer program needs to solve two distinct problems. First, the change of the landscape due to the flood. Second, even where there is no flood, due to different viewing
Bart Kuijpers
[email protected] Sofie Haesevoets [email protected] Peter Z. Revesz [email protected] 1
Hexagon Geospatial, Arenberg Research-Park, Leuven, Belgium
2
Hasselt University, Data Science Institute, Databases and Theoretical Computer Science Research Group, Agoralaan, Gebouw D, 3590 Diepenbeek, Belgium
3
Department of Computer Science and Engineering, University of Nebraska-Lincoln, Avery Hall 358, Lincoln, NE 68588-0115, USA
Geoinformatica
angles, the “before” and the “after” images are not perfect copies but only affine-invariant images of each other. Hence affine-invariant similarity measures are needed between pairs of images [19, 21, 24]. After matching the “before” and the “after” images, the finer details of the flood can be examined by a spatial database query. In this paper, we propose a practical approach to combine the two steps and use affine-invariant spatial database queries. The main idea of our affine-invariant spatial database queries, which we call affine triangle logic, is to use sets of triangles as a basic data representation. This can be considered an abstract spatial data type [28]. Geographic information systems also rely on triangles as the fundamental basis of representation, particularly in Triangulated Irregular Networks [25, 36]. In computer graphics and traffic network simulations data is approximated by triangular meshes (e.g., [4, 6, 7, 43]), and in many spatial interpolation algorithms those triangular meshes serve as the basis for the interpolation [3, 26]. If in all these areas the data is rep
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