Page - 155 - in Joint Austrian Computer Vision and Robotics Workshop 2020

ThelineL ismappedtosomecurveC inthe(x,y) plane by the inverse functionu 7→ x from (8), (9). Then,w(ϕ) is proportional to the area of the part of D% that lies on the right side ofC. We will there- fore study in the following the part ofCwithinD%. For sufficiently small%, this is a curve segment cor- respondingtoaparameter interval [t−,t+] for t,with t±=O(%). We calculate a parametric representation ofC by inserting (10) into (8), (9) toobtain x(t) . =λ%2+ tp−at2p2−bt2q2−ct2pq , (11) y(t) . =µ%2+ tq−dt2p2−et2q2−ft2pq . (12) By easy estimates, one has t± .=±%+ r±%2 with r± = O(1). Thus, the intersection points x+ = (x(t+),y(t+))T,x− = (x(t−),y(t−))T ofCwith the boundaryofD% aregivenby x± .=±% ( p q ) +%2η± , (13) η±= ( λ+r±p−ap2−bq2−cpq µ+r±q−dp2−eq2−fpq ) . (14) Asx± are to lie on the boundary ofD%, we have for their Euclidean norms |x±| that |x±|2 = %2. By p2+q2= 1, this implies 〈(p,q)T,η±〉=O(%) and thusr±= r+O(%)andη±=η+O(%)with r=ap3+(c+d)p2q+(b+f)pq2+eq3 , (15) η=        λ+ap4 +(c+d)p3q+(b+f)p2q2+epq3 −(λp+µq)p−ap2−bq2−cpq µ+ap3q +(c+d)p2q2+(b+f)pq3+eq4 −(λp+µq)q−dp2−eq2−fpq        . (16) From(14) it isevident that the intersectionpointsx± differ from the intersection points±%2(p,q)T of the diameter δϕ ofD% in directionϕwith the boundary ofD% justbyanoffset%2η+O(%3). Thecomponent of thisoffsetperpendicular toδϕ is 〈%2η,(−q,p)T〉=%2(µp−λq−dp3 +(a−f)p2q+(c−e)pq2+bq3) . (17) Up to higher order termsO(%3), the entire curveC is approximated by a parabola over the diameter δϕ with heighth(t) = %2(µp−λq)+ t2(−dp3+(a− f)p2q+(c−e)pq2+bq3) for t∈ [t−,t+]. Thearea on the right ofC (i.e., belowC) differs from that of thehalf-discbelowthediameterδϕby ∆(ϕ) = ∫ t+ t− h(t)+O(%3)dt = 2%3(µp−λq)+ 43%3 (−dp3+(a−f)p2q +(c−e)pq2+bq3)+O(%4) . (18) Thehalf-spacedepthofµ isproportional to themin- imumofpi%2/2+∆(ϕ) forϕ∈ [0,2pi]. The sought half-space median is therefore given by those λ, µ for which the minimum of ∆(ϕ) is largest. It can be proven that the minimum of ∆(ϕ) differs only by higher-order terms w.r.t. % from that of ∆˜(ϕ) = ( 3µ− 34d+ 14(c−e) ) cosϕ + (−3λ+ 14(a−f)+ 34b)sinϕ + (−14d− 14(c−e))cos(3ϕ) + ( 1 4(a−f)− 14b ) sin(3ϕ) (19) where we have inserted p = cosϕ, q = sinϕ, and addition theorems. This function is the super- position of a shifted2pi-periodic sine function (com- bining the cosϕ, sinϕ contributions) and a shifted 2pi/3-periodicsinefunction(combiningthecos(3ϕ), sin(3ϕ)contributions). Moreover, ∆˜ is an odd func- tion, such that is maximum and minimum are of equal magnitude and opposite sign. Since only the 2pi-periodic part of ∆˜ depends onλ,µ, it is easy to see that the amplitude of ∆˜ is minimised (and thus the minimum is maximised) if and only ifλ, µ are chosen such that the 2pi-periodic contribution van- ishes. Again, the neglection of higher order terms in∆(ϕ)above entails only a higher-order error inλ, µ. Therefore, the sought median is determined up to higherorder termsby λ= a12+ b 4− f12 , µ= d4+ e12− c12 (20) fromwhich theclaimof the lemmafollowsbyvirtue of a = uxx/2, b = uyy/2, c = uxy, d = vxx/2, e=vyy/2,f=vxy. Proof of Proposition 1. The transfer of the lemma to the general geometric situation of the proposition is analogous to [14, Sect. 3.1.2]. It relies on the ob- servation that for any regular pointx ∈ Ω, trans- forming the valuesu in its neighbourhood via the affine transform uˆ= (Du(x))−1u leads to a trans- formedfunction uˆwithDuˆ= diag(1,1)as required by the lemma. Due to the affine equivariance of the 155