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)