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Pacific Graphics 2016 Volume 35 (2016), Number 7 E. Grinspun, B. Bickel, and Y. Dobashi (Guest Editors) SupplementaryMaterial 1041 In this supplementary material, we provide a proof for Eq. 7 in equal to U (C) up to scale, where µ guarantees the unity determinant c our manuscript. Note this proof is the discrete analogy of Sec. 3 constraint. Thus: in [RA15]. ⋆ −1 1 −1 (∆ ) =|Uc(C)|d Uc (C). ✷ Lemma:ForafixedclusterC,the minimization of our defined color homogeneity, which is References ¯ ⊤ −1 ¯ Ecolor(C) = min ∑(I(x)−c) ∆ (I(x)−c). [PP12] PETERSEN K. B., PEDERSEN M. S.: The matrix cookbook, 2012. ¯ ⊤ c,∆=∆ ,|∆|=1 x∈C 1 ¯⋆ [RA15] RICHTER R., ALEXA M.: Mahalanobis centroidal Voronoi tes- can be obtained by the optimal observation point c and the optimal sellations. Computers & Graphics 46, 0 (2015), 48–54. 1 ⋆ −1 variation matrix(∆ ) in Eq. 7 of our manuscript: ¯⋆ c = ∑ I(x)/|C|, x∈C ⋆ −1 1 −1 (∆ ) =|U (C)|d U (C). c c Proof: Note we have assumed ∆ is symmetry and positive def- inite. For this constrained optimization problem, we consider the Lagrangian: ¯ ⊤ −1 ¯ −1 L= (I(x)−c) ∆ (I(x)−c)+µ( ∆ −1). ∑ x∈C ¯⋆ Theoptimal observation point c can be obtained by: ∂L −1 ⋆ ⋆ ¯ 0= |¯ ¯ =∆ (I(x)−c ). ¯ c=c ∑ ∂c x∈C Thusoptimalobservationpointisthemeanvalueoftheobservations regardless of the variation matrix: ¯⋆ c = ∑ I(x)/|C|. x∈C ⋆ −1 The optimal variation matrix(∆ ) can be obtained by the same approach. From [PP12], we note that: ¯ ⊤ −1 ¯ ∂ ∑ (I(x)−c) ∆ (I(x)−c) x∈C =U(C), −1 c ∂∆ −1 ∂ ∆ −1 ⊤ = ∆ ∆ . ∂∆−1 ⋆ −1 Let us write the condition for the optimal variation matrix(∆ ) : ∂L ⋆ −1 ⋆ ⊤ 0= | −1 ⋆ −1 = U (C)+µ (∆ ) (∆ ) . (1) −1 (∆) =(∆ ) c ∂(∆) FromEq.1,wecanclearlyseethattheoptimalvariationmatrix∆⋆ is submitted to Pacific Graphics (2016)
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