Page - 140 - in Joint Austrian Computer Vision and Robotics Workshop 2020

Proof: Assume s is an interior point ofA, which means there is a open setU with s ∈ U ⊂ A. s being a saddle point means there is a neighborhood V ⊂ U so thatV− := V ∩ [f < f(s)] as well as V+ :=V ∩ [f >f(s)] decompose into two or more connected components. Pick a1,a2 from different components of V− as well as b1,b2 from different components ofV+. a1 anda2havetobeconnectedbyamonotonicpath,but this path has to move outside ofV since the points are from different components ofV− and by virtue of being monotonic, the path cannot go throughV+. Analogue forb1 andb2. Again by the Jordan Curve Theorem, these two paths have to cross in some point c, which again yieldsa contradiction. f(c)≤max(f(a1),f(a2))< <f(s)<min(f(b1),f(b2))≤f(c) Thus the assumption that s is a interior point has to be false. Example 3.5. Let Ω =A=R3. Let f be the dis- tance from the unit circle laying in thex-y-plane. f :R3→R : (x,y,z) 7→   ∥∥∥(x− x||x,y||2,y− y||x,y||2,z)∥∥∥2 ‖(x,y)‖2 6= 0 ‖(1,0,z)‖2 ‖(x,y)‖2 = 0 Again, letus lookat thelevelsets toshowA isaslope region. The levelset of f ≡ 0 is the unit circle. For 0 < f < 1 the levelsets are tori. f ≡ 1 marks a transition and the levelset is a torus with its hole closed. Then, for f > 1 the levelsets look like the exterior surface of a self intersecting torus, topolog- ically equivalent to a sphere. All these levelsets are connected. Thus,A is indeedaslope region. Now consider the point (0,0,0). Along thexand y-direction it isa localmaximum,howeveralong the z-direction it isa localminimum. Thus, it is a saddle point. Therefore, theorem3.4doesnotholdinhigher dimensions. 4. Motivating The Border Propagation (BP) Algorithm Now we will work our way to the central in- sights on which the border propagation algorithm (BP) hinges. Let us develop ideas for smooth Figure4. evolutionofdiscretized regionsduring thealgo- rithm (hyper-)surfaces first, and deal with discrete variants in thenext section. Slope regions can be constructed and grown in a straight-forward iterative manner by sweeping through the function values from lowest to highest. This is similar to the intuition employed in Morse theory[4, Section 1.4]. Visualize a smooth, compact 2D surface in 3D space. We want to decompose this surface into slope regions. Initially, our decomposi- tion is empty, i.e. there are no slope regions (thus we don’t have an actual decomposition yet). This is shown inFigure4, Image1. Imagine a water level rising from below the sur- face, up to the point of first contact. Starting at this global minimum, we add a new region, containing only the argmin (i.e. a single point on the 2D image where theminimalvalue is taken). Now, theremightbemanypointswhere theglobal minimum is taken. This will either be due to a con- nected region (plateau) on the surface, which we want to include into the single existing region, or it will be due to individual dales, which all have their lowestpointat thesameheight. In thiscase,wecan’t 140