Page - 142 - in Joint Austrian Computer Vision and Robotics Workshop 2020

Algorithm 1BorderPropagation Enumerate all values of f and collect points into levelsets. foreach levelset inbottomto toporderdo Add points to regions if they are in the border of a region if an added point is in the border of different region then Find the union of the borders of the in- volved regions Find theconnectedcomponents thereof Assign these to the regions in an arbitrary way end if ifanaddedpointsplits theborderof theregion in two then Reduce the border the region to one com- ponent foreachother componentdo Create a new region containing the component asborder endfor end if for leftover points that cannot be added to any regionsdo Create a new region containing only that point endfor endfor Additional featuresweconsider: • Providing a tolerance parameter, which gov- erns how steep a continuous function might get, before an iso-surface is deemed discon- nectedinthediscretedata. Thiswouldallowfor a trade-off between continuous connectedness and discrete connectedness. Modeling continu- ous connectedness creates fewer slope regions and yields pleasing results on smooth data, but the resulting regions are not monotonically connected (in the discrete sense of connected) in general. Discrete connectedness guaran- tees monotonic connectedness, but it necessar- ily creates significantly more and smaller slope regions. On smooth data the latter tends to pro- duce toofineof adecomposition. • Using established data structures that model smooth level sets from discrete data. There might be performance gains in employing such adata structure. 7.Conclusion In this paper we have shown that slope regions of continuous functions in high dimensions (n ≥ 3) do not have the same critical point properties well-established in 2D. Hence previous graph-based methods of building slope region decompositions by merging regions according to their border extrema will fail in high dimensions. Instead we developed a new, levelset-based method of growing regions, whichyieldssloperegiondecompositionsondiscrete dataofarbitrarydimension. Acknowledgements The BP algorithm as well as this paper are the re- sult of a pattern recognition course held at the Vi- ennaUniversityOfTechnologyfromOct2019tillJan 2020. ProfessorW. KROPATSCH introducedustothe concept of slope regions and posed the challenge to compute them in high dimensions (> 2). We wish to thank him as well as DARSHAN BATAVIA and the anonymous reviewers for theirvaluable input. References [1] H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pas- cucci. Morse-smale complexes for piecewise linear 3-manifolds. InProceedingsof thenineteenthannual symposium on Computational geometry, pages 361– 370,2003. [2] A. Hatcher. Algebraic Topology. Cambridge Univer- sityPress, 2002. [3] W. G. Kropatsch, R. M. Casablanca, D. Batavia, and R. Gonzalez-Diaz. Computing and reducing slope complexes. In International Workshop on Compu- tational Topology in Image Context, pages 12–25. Springer, 2019. [4] Y. Matsumoto. An introduction to Morse theory. Iwanami series in modern mathematics. American Math.Soc.,Providence,RI,2002. [5] F. Riesz. Sur un The´oreme de Maximum de Mm. Hardy et Littlewood. Journal of the London Math- ematicalSociety, 1(1):10–13, 1932. [6] M. Zorn. A remark on method in transfinite alge- bra. Bulletin of the American Mathematical Society, 41(10):667–670,1935. 142