Page - 154 - in Joint Austrian Computer Vision and Robotics Workshop 2020

half-space median can be applied meaningfully to data for which no physically meaningful Euclidean structure inRn can be assumed (e.g., if different di- mensionsof thedataspacerefer to incommensurable physicalquantities). 3.PDELimitofHalf-SpaceMedianFiltering Ourmaintheoretical result is thefollowingpropo- sition which was already stated as a conjecture in [15]. It specifies the space- and time-continuous image evolution which is approximated by iterated space-continuous half-space median filtering in the limitcasewhentheradiusoftheslidingwindowgoes to zero, thereby generalising the result from [5] for the univariate median filter and results from [14] for other multivariate median filters. In particular, the approximatedPDEis identicalwith theoneapproxi- mated by theOjamedianfilter, see [14,15]. Proposition1 Letu : R2 ⊃ Ω→ R2, (x,y) 7→ (u,v) be a smooth bivariate image over a compact domainΩ. At any regular locationx= (x,y)∈Ω, i.e., for which the Jacobian Du(x) is of rank 2, one step of space-continuous half-space median filtering withadisc-shapedwindowof radius%approximates a time step of an explicit time discretisation of the PDE ut= 2∆u+A(uyy−uxx)+Buxy (1) with time step size %2/24, where the coefficient ma- tricesA≡A(Du),B≡B(Du)aregivenby A= 1 uxvy−uyvx ( uxvy+uyvx −2uxuy 2vxvy −uxvy−uyvx ) , (2) B= 2 uxvy−uyvx ( uxvx−uyvy −u2x+u2y v2x−v2y −uxvx+uyvy ) . (3) The proof of this result relies on the following lemma. Lemma2 LetubeasinProposition1,andletx0= 0∈Ω be a regular point for whichu(x0) =0, and Du(x0) is the 2×2 unit matrix. Then one step of space-continuous half-space median filtering with a disc-shaped window of radius%approximates atx0 a time step of an explicit time discretisation of the PDEsystem ut=uxx+3uyy−2vxy , (4) vt= 3vxx+vyy−2uxy (5) with timestepsize%2/24. Note that the lemma states the approximation re- sult of the proposition for a specific geometric con- figuration where the gradients of the componentsu, v ofu are locally aligned with the x, y coordinate axes and of unit magnitude. This special geometric situationalsohelps inunderstanding theeffectof the PDE of the proposition. A more detailed discussion isfoundin[14,Sect.3.1.3]fromwhichweshortlyre- call themainfacts. First, theright-handsidecontains termswhichplayasimilar roleas themeancurvature motionapproximatedby theunivariatemedianfilter: in the lemma, uyy and vxx represent separate mean curvature motion contributions for theuandv chan- nel. Second, therearecouplingterms–inthelemma: vxy in theequation foru, anduxy in theequation for v – that promote a joint evolution of the channels. Third, there is an isotropic diffusion term ∆uwhich has no counterpart in the univariate case. Remember that alsoFigure3showsaslight edge-blurringeffect ofmultivariatemedianfiltering. Proof of Lemma 2. By Taylor expansion of u around0weobtainwithin the%-discD% around0 u . =x+ax2+by2+cxy , (6) v . =y+dx2+ey2+fxy . (7) where .= denotes equality up toO(%3) terms. The inverse functioncanbewrittenas x . =u−au2−bv2−cuv , (8) y . =v−du2−ev2−fuv . (9) Coarse estimates yield that the median of the values u(x,y) for(x,y) inD%differs from0byO(%2). Let thereforeamediancandidatepoint in the(u,v)plane begivenasµ= (λ%2,µ%2)Twithλ,µ=O(1) (i.e., bounded for %→ 0). To determine the half-space depth ofµ, we consider straight lines throughµ in the (u,v)plane. Aparametric representationof such a lineL=L(ϕ) is u(t) =λ%2+ tp , v(t) =µ%2+ tq (10) wherep= cosϕ, q= sinϕwith the angleϕdenot- ing the direction of the line, and t is a real parameter whichalsodeterminesanorientationofL. We are interested in the total weightw(ϕ) of the densityofvaluesuwithin thehalf-planeon the right side ofL(ϕ). The half-space depth ofµ is propor- tional to theminimumofw(ϕ) forϕ∈ [0,2pi]. 154