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1 On the Potential Optimality of the Weaire-Phelan Partition avin Kashyap David. euho Department of Electrical Engineering and Computer Science University of Michigan, Ann Arbor, MI USA Abstract Until recently, the solution to the Kelvin problem of nding a partition of R into equal-volume cells with the least surface area was believed to be tessellation by the truncated octahedron. In 994, D. Weaire and R. Phelan described a partition that outperformed the truncated octahedron partition in this respect. This raises the question of whether the Weaire-Phelan (WP) partition can outperform the truncated octahedron partition in terms of normalized moment of inertia (MI), thus providing a counterexample to Gersho's conjecture that the truncated octahedron partition has the least MI among all partitions of R. In this paper, we show that the eective MI of the WP partition is larger than that of the truncated octahedron partition. We also show that if the WP partition is used as the partition of a -dimensional vector quantizer, then the resulting quantization error is white. We then show that the eective MI of the WP partition cannot be reduced by passing it through an invertible linear transformation. Another contribution of this paper is a proof of the fact that the quantization error corresponding to an optimal periodic partition is white, which generalizes a result of Zamir and Feder. Introduction Consider a source uniformly distributed over a large ball B in R k, centered at the origin. Suppose that this source is quantized using a k-dimensional vector quantizer (VQ) specied by a partition S = fs ; S 2 ; : : : ; S g of B, and corresponding codebook C = fc ; c 2 ; : : : ; c g of points in R k. et V i denote the volume of the cell S i, so that vol(b) = P i= V i, and let M(S i ; c i ) = R S i jjx? c i jj 2 dx denote the moment of inertia (MI) of the cell S i about the point c i. The mean squared error (MSE) or distortion, per dimension, of this quantizer is then given by X Z P i= M(S i ; c i ) P i= V i D = jjx? c i jj 2 k S i= i vol(b) dx = k P = i= M(S i ; c i ) vol(b) 2=k vol(b) 2=k k P i= +2=k = m(s; C) () V i The quantity m(s; C) is called the eective normalized moment of inertia of the partition S about the points in C. This quantity is invariant to any scaling, translation or rotation of the partition and the points. This denition generalizes the usual denition of normalized moment of inertia (MI) of a cell S about a point c in R k : m(s; c) = k M(S; c)=(vol(s))+2=k. This work was supported by SF Grant CCR

2 One special case of interest is when we have a periodic partition of R k, i.e., a partition composed of translates of a fundamental unit consisting of cells S ; S 2 ; : : : ; S, and a codebook C consisting of the centroids of the cells forming the partition. By suitably scaling the partition so as to be able to (barely) t points of the codebook into the ball B, we obtain a partition S of B. It is clear from () that the eective MI of this partition, in the limit as!, is given by m(s; C) = k P i= M(S i ; c i ) +2=k (2) P i= vol(s i ) where c i is the centroid of S i. We shall take this to be the denition of the eective MI of a periodic partition about the centroids of its cells. It is well-known (cf. [], p. 50) that for a xed codebook C, the MSE in () is minimized by choosing the partition S to be the Voronoi partition with respect to C, which we shall denote by Vor(C). As is also well-known, for asymptotically large, the least distortion of any k-dimensional VQ applied to a source that is uniformly distributed over B has the form m k (vol(b)=)2=k, where m k is Zador's constant [2]. In our formulation, m k = lim inf! fcr k :jcj=g m(vor(c); C) () It should be noted that since m(vor(c); C) is invariant to any scaling or translation of C, m k is independent of the actual choice of the ball B. So far, the value of m k is known only for k = (m = =2), and for k = 2 (m 2 = 5=(6 p )) []. It has been conjectured [4] that m = 9=(92 p 2), which is the eective MI of the Voronoi partition of the BCC lattice, 4Z [ 4Z + (2; 2; 2), about the centroids of the cells, which are the lattice points themselves. Due to the lattice structure, this partition is a periodic partition with its fundamental unit being a truncated octahedron. Until recently, tessellation by the truncated octahedron was also believed to be the solution to the Kelvin problem of nding a partition of R into cells of equal volume with the least surface area (normalized so as to be invariant to scaling) [5]. In 994, Weaire and Phelan [6] described a partition of R that was better in this respect than tessellation by the truncated octahedron. Given the similarities between the two problems presented here (the sphere minimizes both surface area and MI among all -dimensional bodies of a given volume), it is natural to ask whether the Weaire-Phelan partition can outperform the truncated octahedron in terms of MI. In this paper, we calculate the eective MI of the Weaire-Phelan partition and show that this is not better than the MI of the truncated octahedron. In addition, we show that if the cells comprising the Weaire-Phelan partition are taken to be the cells of a quantizer whose codebook consists of the centroids of these cells, then the resulting quantization error is white (cf. [7]). Consequently, a generalization of the arguments in [7] shows that the eective MI of the Weaire- Phelan partition cannot be improved by passing the partition through a linear transformation. In the process, we also show that the quantization error corresponding to an optimal periodic partition is white, which generalizes a result of Zamir and Feder ([7], Theorem ) for lattice partitions. 2

Figure : The Weaire-Phelan equal-volume partition 2 The Weaire-Phelan Partition Weaire and Phelan's partition of R into equal-volume cells (shown in Figure ) is, in fact, one of a class of partitions

3 Figure : The Weaire-Phelan equal-volume partition 2 The Weaire-Phelan Partition Weaire and Phelan's partition of R into equal-volume cells (shown in Figure ) is, in fact, one of a class of partitions based on the Voronoi partition of the non-lattice packing consisting of the points of the BCC lattice and half of its Voronoi corners (a detailed description can be found in [8]). In other words, the packing consists of the cubic lattice 4Z and its translates by the points (2; 2; 2) and cyc(; 0; 2) (i.e., the points (; 0; 2) and their cyclic shifts). Each member of the Weaire-Phelan (WP) class of partitions is parametrized by a real number 0 < < =2, and is a periodic partition composed of translates by 4Z of a fundamental unit comprising two pentagonal dodecahedra, which are centered at the BCC lattice points (0,0,0) and (2,2,2), and six 4-hedra, which are centered at the non-bcc points cyc(2,,0) and cyc(2,,0). The dodecahedron, C 2;0, at the origin has vertices (2=; 2=; 2=) and cyc(0; =2; ). The dodecahedron at (2,2,2) is congruent to C 2;0, i.e., it can be transformed to C 2;0 by a rigid motion that consists of a translation followed by an orthogonal linear transformation. On the other hand, each 4-hedron in the fundamental unit is congruent to a 4-hedron, C 4;0, centered at the origin, with vertices (?2=; (2?2=); 2=), (2=?; 2=; (2?2=)), (?; (2?=2); 0), (? ; 0; (2? =2)), (; (2? ); =2), (?; =2; (2? )), (?; ; 0) and (; 0; ). The WP partition with parameter is a true Voronoi partition of the packing for = 5=4; for other values of, this is a \weighted" Voronoi partition, with BCC points and non-bcc points A point to be noted is that C 4;0 is not a cell belonging to the Weaire-Phelan partition.

4 given dierent weights. A weighted Voronoi partition simply means that each face of a cell passes through a weighted average of the two relevant packing points, while still being perpendicular to the line joining those points. Moreover, the centroid of each cell in any WP parition is the packing point that lies within the cell. We now describe the cells of the partition in more detail. The 2 faces of the dodecahedron C 2;0 are all congruent pentagons, each with area p 5 2 =2, and each face lies at a distance of 2= p 5 from the origin. The volume of the dodecahedron is thus 2 (= p 5 2 =2 2= p 5) = 4. The faces of the 4-hedron C 4;0 are of three types two congruent hexagonal faces each with area 4? 2, each lying at a distance of from the origin; four pentagonal faces that are congruent to the faces of C 2;0, each lying at a distance of (5? 2)= p 5 from the origin; eight other congruent pentagonal faces each with area p 6(? 2 )= and lying at a distance p 6=2 from the origin. From this data, the volume of the 4-hedron is calculated to be 4(8? )=. It is easy to verify that the partition consists of equal-volume cells when = p 2. To calculate the eective MI of a WP partition about the centroids of its cells, let M 2 denote the (unnormalized) moment of inertia of a dodecahedral cell of the partition about its centroid, and let M 4 denote the same for a 4-hedral cell. Similarly, let V 2 and V 4 be the volumes of the respective cells. Then, (2) shows that the eective MI of the WP partition is given by M M 4 2 V V 5= (4) 4 2 In order to determine M 2 and M 4, we use a well-known formula ([], Chapter 2, Theorem ) for determining the moment of inertia of a polyhedron about a point in its interior. This formula, when applied to the dodecahedron C 2;0, yields M 2 = 7 5 =0, and when applied to the 4-hedron C 4;0, yields M 4 = (200? ? 7 5 )=90. The details of these calculations can be found in [9]. Putting the expressions for M 2, M 4, V 2 and V 4 into (4), we nd that the eective MI of the WP partition with parameter is given by m W P () = 96 (4? 5 + 0) (5) The value of this expression is a minimum at = 5=4, which is to be expected since this corresponds to the Voronoi WP partition. The eective MI of the Voronoi partition is m W P (5=4) = 645=892 = 0:07875 : : :, which is greater than the MI of the truncated octahedron (9=(92 p 2) = 0:07854 : : :). For the equal-volume WP partition, we have m W P ( p 2) = 2?= = 0: : : :. Remarkably, this is precisely the MI of the rhombic dodecahedron, which is the Voronoi cell for the face-centered cubic (FCC) lattice []. Minimizing MI via inear Transforms Although the Voronoi WP partition is not better than the truncated octahedron in terms of MI, we can use it as the starting point of a search for a partition with lower eective MI. In this section, we focus our search on partitions that are the images of WP partitions under invertible linear transformations. 4

5 et S be any periodic partition of R k, composed of translates of a fundamental unit consisting of cells S ; S 2 ; : : : ; S, at least one of which is non-degenerate (i.e., has positive volume). We take the codebook C to be the set of centroids of the cells forming the partition. Given T 2 G k (R), the group of real, invertible, k k matrices, let T S = ft (S) : S 2 Sg and T C = ft c : c 2 Cg. Since T S is a periodic partition of R k consisting of translates of the fundamental unit made up of the cells T (S ); T (S 2 ); : : : ; T (S ), and T C is the set of centroids of the cells in T S, (2) shows that m(t S; T C) = k P i= M(T (S i ); T c i ) +2=k (6) P i= jdet(t )jvol(s i ) Here, we have used the fact that vol(t (S i )) = jdet(t )jvol(s i ). Our goal now is to nd a T 2 G k (R) for which m(t S; T C) is a minimum. To do this, we need an alternative formulation of the denition of the eective MI of a periodic partition, which we provide next. Consider a random vector X uniformly distributed over the fundamental translational unit, F = [ i= S i, of the partition S. et Q : F! C be the vector quantizer dened by Q(x) = c i, if x 2 S i, for i 2 f; 2; : : : ; g. et E = X? Q(X) be the quantization error, and let R E = E[EE t ] be the corresponding autocorrelation matrix. As explained in [9], since at least one of the S i 's has positive volume, R E is non-singular (and hence positive denite). oting that the trace, tr(r E ), of the matrix R E is E[jjEjj 2 ] = P Pi= i= M(S i ; c i )= vol(s i ), we have from (2), m(s; C) = k tr(r E) X i= vol(s i )!?2=k (7) ow, suppose X 0 is a random vector uniformly distributed over T (F ) = [ i= T (S i). et Q 0 : T (F )! T C be the quantizer dened by Q 0 (x) = T Q(T? x), and let E 0 = X 0? Q 0 (X 0 ) be the corresponding quantization error. ote that X 0 and T X are identically distributed and hence, E 0 has the same probability distribution as T X? Q 0 (T X) = T X? T Q(X) = T E. Therefore, the autocorrelation matrix of E 0 is given by T R E T t. It now follows from (7) that m(t S; T C) = k tr(t R ET t ) det(t R E T t ) =k det(r E) =k using the fact that det(t R E T t ) = jdet(t )j 2 det(r E ). X i= vol(s i )!?2=k (8) Following the lines of the proof of Theorem in [7] (which applies to partitions generated by lattices, rather than periodic partitions in general), we can prove the following result, which shows that T minimizes m(t S; T C) if and only if the corresponding quantization error E 0 = T E is white, i.e., T R E T t = I where I is the k k identity matrix and is some positive real number. Theorem For any T 2 G k (R), m(t S; T C) det(r E ) =k with equality if and only if the quantization error E 0 = T E is white. m (S; C) with equality if and only if E is white. 5 X i= vol(s i )!?2=k = m (S; C) (9) In particular, m(s; C)

6 ote that the quantization error E 0 = T E can always be whitened by taking T = R?=2 E, the? (unique) real, symmetric, positive denite square root of R E. Therefore, it follows immediately from Theorem that a necessary condition for a periodic partition to be optimal, in the sense of having the least eective MI among all periodic partitions, is that the corresponding quantization error should be white. Theorem also provides an alternative way of looking at (7). Observe that (7) may be rewritten as m(s; C) = (S; C)m (S; C) (0) where (S; C) = k tr(r E)=det(R E ) =k is a term that measures how far the MI of the partition S is from the MI, m (S; C), of the \whitened" version of the partition. We now show that, if we take S to be any WP partition, then the corresponding quantization error E is in fact white. This, by the above theorem, would imply that m(s; C) = m (S; C), and so the eective MI of a WP partition cannot be improved by any invertible linear transformation. et S be the WP partition with parameter. et S and S 2 be the dodecahedra centered at (0,0,0) and (2,2,2) respectively, let S i, i = ; 4; 5, be the 4-hedra centered at cyc(2,,0) and let S i, i = 6; 7; 8, be the 4-hedra centered at cyc(2,,0). Since the random vector X = (X; Y; Z) t is uniformly distributed over the fundamental translational unit of S, each element of the correlation matrix R E is of the form 6V 4 + 2V 2 8X Z i= S i (x? c i; ) p (y? c i;2 ) q (z? c i; ) r dx () for some p; q; r 2 f0; ; 2g such that p + q + r = 2, where c i = (c i; ; c i;2 ; c i; ) is the centroid of S i. Using the symmetries of the packing that generates the WP partition, and some simple changes of variable, we can show that [9] if none of p, q and r is 2, then the expression in () reduces to 0; otherwise the expression evaluates to (6M 4 + 2M 2 )=(6V 4 + 2V 2 ). Hence, the correlation matrix R E is given by R E = 6M 4 + 2M 2 I (2) 6V 4 + 2V 2 where I is the identity matrix, thus showing that E is white. In fact, the technique described above to evaluate the expression in () in eect shows that if, for i = ; 2; : : : ; 8, we dene a random vector X i uniformly distributed over S i, then the corresponding \error" vector E i = X i?c i is white for i = ; 2, and uncorrelated for i = ; : : : ; 8. In other words, the corresponding autocorrelation matrices R Ei are all diagonal, with the diagonal entries being all equal for i = ; 2, but not for i = ; : : : ; 8. The triples formed by the diagonal elements of the autocorrelation matrices for E, E 4 and E 5 are all cyclic permutations of one another, i.e., if R E = diag( ; 2 ; ), then R E4 = diag( 2 ; ; ) and R E5 = diag( ; ; 2 ). Hence, the sum of these three matrices is a scalar multiple of the identity matrix. The same is the case for the autocorrelation matrices corresponding to E 6, E 7 and E 8. Thus, the whiteness of the quantization error over the fundamental unit of a WP partition may be viewed as a consequence of the whiteness over each dodecahedron, and over each of the two sets of three 4-hedra, namely, fs ; S 4 ; S 5 g and fs 6 ; S 7 ; S 8 g. 6

7 Acknowledgment The authors wish to express their gratitude to Thomas Hales for sparking their interest in the Weaire-Phelan partition, and to Dennis Hui for pointing out that it is instructive to reformulate (7) as (0). References [] A. Gersho and R.M. Gray, Vector Quantization and Signal Compression, Boston, MA: Kluwer, 992. [2] P.. Zador, \Development and evaluation of procedures for quantizing multivariate distributions," Ph.D. dissertation, Stanford Univ., 96. [] J.H. Conway and.j.a. Sloane, Sphere Packings, attices and Groups, rd ed., ew York, Y: Springer-Verlag, 998. [4] A. Gersho, \Asymptotically Optimal Block Quantization," IEEE Trans. Inform. Theory, vol. IT-25, no. 4, pp. 7{80, July 979. [5] Denis Weaire (ed.), The Kelvin Problem, ondon, England: Taylor & Francis, 996. [6] D. Weaire and R. Phelan, \A Counter-Example to Kelvin's Conjecture on Minimal Surfaces," Phil. Mag. ett., vol. 69, no. 2, pp. 07{0, 994 (reproduced in [5], pp. 47{5). [7] R. Zamir and M. Feder, \On attice Quantization oise," IEEE Trans. Inform. Theory, vol. 42, no. 4, pp. 52{59, July 996. [8] R. Kusner and J.M. Sullivan, \Comparing the Weaire-Phelan Equal-Volume Foam to Kelvin's Foam," in [5], pp. 7{80. [9]. Kashyap and D.. euho, \On Quantization with the Weaire-Phelan Partition," submitted to IEEE Trans. Inform. Theory. Available via anonymous FTP at ftp.eecs.umich.edu/ people/neuhoff/weaire phelan.submit.ps. 7

I. Introduction Consider a source uniformly distributed over a large ball B in R k, centered at the origin. Suppose that this source is quantized usin

I. Introduction Consider a source uniformly distributed over a large ball B in R k, centered at the origin. Suppose that this source is quantized usin Accepted for publication in IEEE Trans. Inform. Thy., Mar. 2, 200 On Quantization with the Weaire-Phelan Partition * Navin Kashyap and David L. Neuho Department of Electrical Engineering and Computer Science