Page - 138 - in Joint Austrian Computer Vision and Robotics Workshop 2020

clidean topology and f will describe a continuous image or3D-scan. Definition2.1. Apath isacontinuousfunctionfrom the real interval [a,b] (witha<b) into a topological spaceΩ. Definition 2.2. Two points x 6= y in a topological spaceΩarecalledpath-connected ifandonlyif there exists a path γ : [a,b]→ Ω with γ(a) = x and γ(b) =y. Definition 2.3. The set of all points which are path- connected to a pointx∈Ω is the connected compo- nentofx: [x] :={y∈Ω|x ispath-connected toy} Any subset of Ω which can be written in above way (forasuitablechoiceofx) iscalledaconnectedcom- ponent. Definition2.4. Apathγ : [a,b]→Ω iscalledmono- tonic if andonly if thewholepath isascendingor the wholepathisdescending,meaningthefirstorsecond formula belowhas tohold, respectively: ∀s,t∈ [a,b] :s<t⇒f(γ(s))≤f(γ(t)) ∀s,t∈ [a,b] :s<t⇒f(γ(s))≥f(γ(t)) Definition 2.5. LetR ⊂ Ω. R is called slope re- gionormonotonicallyconnected if andonly if forall x,y∈R there exists a monotonic pathγ : [a,b]→ Rwithγ(a) =xandγ(b) =y. Definition 2.6. A family of sets{Ai⊂Ω | i∈ I} is called a sloperegiondecomposition if andonly if: • Ai is a slope region for all i∈ I • ∀i,j∈ I : i 6= j⇒Ai∩Aj=∅. • ⋃i∈IAi= Ω Definition 2.7. Consider two slope region de- compositions A = {Ai⊂Ω | i∈ I} and B = {Bj⊂Ω | j∈J}. We callA coarser thanB, al- ternativelyBfiner thanA, in SymbolsA B if and only if ∀j∈J ∃i∈ I :Bj⊂Ai. Theorem2.8. isapartialorder, i.e. fulfills reflex- ivity, antisymmetryand transitivity. Proof: Straight forward. Antisymmetry follows fromthedecompositionproperty. Definition 2.9. A slope region decompositionA is calledmaximallycoarseorsimplycoarse ifandonly if there is no other coarser slope region decomposi- tion. We can apply Zorn’s lemma [6] to the partial or- der , which yields the existence of maximal ele- ments. For this we need to show that chains have upperbounds. Theorem2.10. Foranyascendingchainof slopere- gion decompositions (Ai)i∈I, that is t≥ s⇒At As, there is a slope region decompositionA∞ satis- fying∀i∈ I :A∞ Ai. Proof: Weconsider theequivalencerelation”con- nected inAi” for twopointsx,y∈Ω: x∼i y :⇔∃A∈Ai :x∈A∧y∈A Theequivalence relation is a subsetofΩ2, andAt As implies∼t⊃∼s. This suggests the use of∼∞:=⋃ i∈I ∼i to get an upper bound. Indeed the equiv- alence classes of∼∞ yield a partitionA∞ of Ω, which is coarser than anyAi. But do they form a slope region decomposition? Yes: For any two fixed pointsx,y to be∼∞-connected, they need to be∼i- connected for some i ∈ I. So there is a mono- tonic path linkingx and y inA= [x]∼i ⊂ [x]∼∞, by which they are monotonically connected inA∞. ThereforA∞ is a slope regiondecomposition. HenceeverysetΩhasacoarsedecomposition. Theorem 2.11. LetA⊂Ω be a path-connected set. A is a slope region if and only if all levelsets off in Aarepath-connected, i.e. ∀c∈R :f−1({c})∩A ispath-connected. Proof: ”⇒”viacontraposition: Supposethereexistsac∈RwithL :=f−1(c)∩A not path-connected. We decomposeL in its compo- nents and pick x and y from different components. Since f(x) = f(y) = c a monotonic path between xandywouldhave to lie completely inL.However, since x and y are from different components, they cannotbeconnectedbyapathinLandthereforecan- not be connected with a monotonic path. Therefore, A isnot a slope region. ”⇐”via ironingout anarbitrarypath: Givenx,y∈Awe have to find a monotonic path γ. Without loss of generality supposef(x)≥ f(y). SinceA is path-connected, there exists an (not nec- essarily monotonic) path γ0 : [a,b]→A fromx to 138