Seite - 137 - in Joint Austrian Computer Vision and Robotics Workshop 2020

BorderPropagation: ANovelApproachToDetermineSlopeRegion Decompositions FlorianBogner TUWien e1225415@student.tuwien.ac.at Co-FirstAuthor AlexanderPalmrich TUWien apalmrich@gmail.com Co-FirstAuthor WalterG.Kropatsch TUWien krw@prip.tuwien.ac.at SupervisoryAuthor Abstract. Slope regions are a useful tool in pattern recognition. We review theory about slope regions andproveatheoremlinkingmonotonicpathsandthe connectedness of levelsets. Unexpected behavior of slope regions in higher dimensions is illustrated by two examples. We introduce the border propagation (BP) algorithm, which decomposes ad-dimensional array (d∈N) of scalar values into slope regions. It isnovelas it allowsmore than2-dimensionaldata. Figure1.gray-scale toheight-mapconversion 1. Introduction In thissectionwedevelopan intuitiveunderstand- ingof the termsloperegion [3]anditsgeneralization to higher dimensions. The concise definition of the terms already employed here is reserved for the next section. Consider an image, either gray-scale or in color. If it is a color image, it can be decomposed into its colorchannels (red-green-blue),whichcan individu- ally be read as gray-scale images. We consider pixel intensity of one such gray-scale image as the height of a landscape, yielding a 2D surface in 3D space. Thesurfacewillhavepeaks inareaswhere the image isbright, andwill havedales indarkareas. Nowouraimistopartitionthesurfaceintoregions (i.e. subsets) in a particular way: We require each region to consist only of a single slope, by which we mean that we can ascend (or descend) from any given point of the region, to any other given point of the region, along a path that runs entirely within the region. Such a decomposition is not unique, but we canat least trytogetapartitionascoarseaspossible, meaning that we merge slope regions if the resulting subset is still a slope region, and we iterate this un- til no further change occurs. There might be many different coarsest slopedecompositions. The criterion we used to describe slopes, any two points being connected by either an ascending or a descendingpath, caneasilybeused inhigherdimen- sions. Think of a computed tomography scan, which will yield gray-scale data, but not just on a 2D im- age,but ratherona3Dvolume. Wewant topartition the 3D volume, such that any two points in a region canbeconnectedviaaneitherascendingordescend- ingpathwithin the region. Recall thatascendingand descending refers to the intensityvalueof the tomog- raphyscanaswemove in thevolume. Forpiecewise linear functions on a volume, decompositions were introduced in [1]. By abstracting from image and tomography to a real function f : Ω→Rdefined on some subset of Rn (thinkof itas thepixel intensityfunction),andby rigorously defining a coarsest slope decomposition, we can lift the concept to arbitrary dimensions in a mathematicallyconcise fashion. 2.DefiningSlopeRegions In this and the following chapters we will con- sider a topological space (Ω,T ) with a continuous function f : Ω→ R. In practice or for ease of imagination, (Ω,T ) will typically be a rectangle or cuboid subset of R2 or R3 equipped with the eu- 137