Multiple Description Lattice Vector Quantization 1 ... - Semantic Scholar
In the Proceedings of the 1999 IEEE Data Compression Conference, DCC'99.
Multiple Description Lattice Vector Quantization Sergio D. Servettoy
Vinay A. Vaishampayanz
N. J. A. Sloanez
Abstract
We consider the problem of designing a lattice-based multiple description vector quantizer for a two-channel diversity system. The design of such a quantizer can be reduced to the problem of assigning pair labels to points of a vector quantizer codebook. A general labeling procedure based on the structure of the lattice is presented, along with detailed results for the hexagonal lattice: algorithms, asymptotic performance, and numerical simulations. Asymptotically, when compared with the lattice Z , the resulting quantizer achieves the standard second-moment gain of the hexagonal lattice for the central distortion, and, surprisingly, achieves the two-dimensional sphere gain for the side distortion.
1 Introduction
1.1 Multiple Description Source Coding
A diversity-based communication system provides multiple channels to send information between a transmitter and a receiver so that even if a channel fails, some data can still be delivered to the receiver. Consider a system with two channels: if identical information is sent over each channel and both work, half the information received has no value. The theory of multiple description source coding studies methods for sending di erent information over the channels, in a way such that if only one channel works, the information received is sucient to guarantee a minimum delity in the reconstruction at the receiver; however, should both channels work, the information from both channels can be combined to yield a higher delity reconstruction. The general problem of designing source codes of this kind was posed by Gersho, Witsenhausen, Wolf, Wyner, Ziv and Ozarow at the September 1979 IEEE Information Theory Workshop; it is a generalization of Shannon's problem of source coding with a delity criterion [10]. Initial progress on this problem was made by El-Gamal and Cover [7] and by Ozarow [9]. In [7], for an arbitrary memoryless source and a bounded distortion Work done at AT&T Labs, between 6/98 and 9/98, while S. Servetto was a summer intern. y Beckman Institute and Dept. of Computer Science, University of Illinois at Urbana-Champaign.
405 N. Matthews St., Urbana IL 61801. e-mail: [email protected]. z AT&T Labs - Research, Shannon Laboratory. 180 Park Av., Florham Park, NJ 07932. e-mail: fvinay,[email protected].
1
measure, the authors nd an achievable rate region of pairs (R1; R2), for a given distortion vector (D0; D1; D2). Ozarow [9] then showed that, if the source is gaussian and the distortion measure is squared error, the achievable rate region of El-Gamal and Cover is, in fact, the exact MD rate/distortion region for the source. El-Gamal and Cover also conjectured, but were not able to prove, that the region of achievable rates they computed is indeed the MD rate/distortion region for other sources and other distortion measures; this conjecture was disproved by Zhang and Berger [14]. To this date, Ozarow's result for the gaussian source and squared error metric is the only one presenting a complete characterization of a MD rate/distortion region.
1.2 Multiple Description Scalar Quantization
The rst practical results on MDs were presented in [11], where a simple procedure is given for designing MD scalar quantizers with remarkably good asymptotic properties. An MD scalar quantizer consists of two main components: a scalar quantizer (in the classical sense), and an index assignment that splits the information about each sample into two complementary and possibly redundant descriptions of the same sample. The problem of designing good index assignments is studied in [11], using Ozarow's characterization of the MD rate/distortion region for the gaussian source as a guiding principle. Under the assumption of equal and high rates for both descriptions, and for a squared error distortion measure, a construction of a large class of index assignments is presented in [11] for which the exponential rate of decay of the MSE is exactly that predicted by Ozarow's result.1 Berger-Wolf and Reingold [2] generalize the construction of index assignments from matrices for two channels to arbitrary k-dimensional tensors for any number of channels. The exact coecient of quantization (i.e., the sub-exponential terms in the error expression) of the MD scalar quantizer was computed in [1], for general source densities and m-th power distortion measures [8]. Perhaps the most interesting case is that of a gaussian density and m = 2. Because of the large number of parameters involved (rates for each of the channels and distortions for all subsets of working/nonworking channels), comparisons between MD systems are complicated. However, it is argued in [1] that in the balanced case (i.e., when the rates on the channels are equal), the product d0d1 of the side and central distortions is a good gure of merit for high rates. For a gaussian source and m = 2, it was shown in [1] that there is a gap of 8.69dB between the distortion product of the MD scalar quantizer and the MD rate/distortion bound; this gap is reduced to 3.06dB when the entropy-constrained MD scalar quantizer [13] is considered.
1.3 Multiple Description Vector Quantization
For the squared error distortion measure, it is known from classical quantization theory [8] that the error of a scalar quantizer with cubic Voronoi regions is 1:53dB higher However, the sub-exponential terms in the MSE are not the same: the performance of the MD scalar quantizer is bounded away from the gaussian MD rate/distortion bound, as one would intuitively expect. 1
2
than the classical rate/distortion bound. The performance loss of the scalar quantizer occurs because the normalized second moment of an N -dimensional hypercube is exactly 1:53dB higher than that of an equivalent hypersphere, in the limit as N ! 1. Although it is not possible in nite dimensions to design quantizers whose Voronoi cells are exactly spheres (spheres are not space- lling polytopes in any dimension 2), the gap can be signi cantly reduced by designing quantizers with more \spherical" cells, i.e., with a smaller normalized second moment than a hypercube. In the MD case, it is a remarkable fact that for a gaussian source, the performance gap of 3:06dB between the distortion product of the two-channel entropy-constrained MD scalar quantizer and the two-channel MD rate/distortion bound is exactly twice as large as the single description gap. It was shown in [12], via a random coding argument, that multiple description vector quantizers can be constructed which achieve gains over scalar quantization (or equivalently, the Z lattice) of 1.53dB simultaneously for the central and side distortion, as the vector dimension goes to in nity. Our main goal in this work is to design MD quantizers with smaller granular central and side distortion than that achieved by the Z lattice. Since we are looking for a systematic and constructive approach, we have chosen to design MD lattice quantizers, more speci cally, quantizers whose codebook is a lattice with good second moment properties and that admits fast encoding algorithms. To accomplish this we must design appropriate index assignments, and the algebraic structure of the lattice is helpful for this purpose.
1.4 Main Contributions and Organization of the Paper
The main contribution of this paper is the construction of good vector index assignments for lattices. This is accomplished by labeling lattice points with pairs of points in some similar coarse sublattice. This step can be thought of as a \requantization" of lattice points, but in a way such that knowing the output of both coarse quantizers allows us to retrieve the exact original point (i.e., the ne quantizer); how coarse the sublattice is determines the amount of redundancy in the MD code, in a manner analogous to the number of diagonals in scalar index assignments. This paper is organized as follows. In Section 2, we de ne the structure of multiple description lattice quantizers. In Section 3, we give a formal description of the algorithm and optimization problems involved in the construction of index assignments for any of the lattices considered. Then, in Section 4, we apply these general techniques to the hexagonal lattice, analyze the asymptotic performance, and present results of numerical simulations when the proposed quantizer is applied to a uniform source. Section 5 gives a summary.
2 Structure of Multiple Description Lattice Quantizers A Multiple Description Lattice Vector Quantizer (MDLVQ) is a triplet Q = (; 0; `), where: 3
is a lattice. 0 is a sublattice of that is geometrically similar2 3 to . Each lattice point 2 gets mapped by ` to a pair of sublattice points ;
(0red; 0green) that uniquely identi es ; i.e., ` must be an injection: 1-1! `() 0 0: ` is referred to as the vector index assignment, and the two points in the image under ` of a point are referred to as the red and green descriptions. The amount of redundancy in a lattice quantizer is controlled by N = j=0j, the index of 0 in . Given any sublattice point 0 2 0, we require that ` is such that the total number of distinct lattice points 2 for which 0 is used to describe is exactly N ; i.e., jf : red(`()) = 0gj = jf : green(`()) = 0gj = N; where red(red; green) = red, and similarly for green. In other words, lattice points are labeled with pairs of sublattice points (it is these sublattice points that get actually transmitted over each channel), and each sublattice point is used exactly N times; the larger is N , the higher the uncertainty about the original lattice point when one of the channels fails. A key property of good index assignments ` is that the set of central cells that share a given label must be as localized in space as possible, in order to achieve low distortion in the case of a channel failure. This is analogous to the idea that for a scalar quantizer the spread of a side cell must be minimized [11]. We now give a method for constructing a labeling function `.
3 Multiple Descriptions of Lattice Points The construction we propose is based on the following steps. (We assume the dimension is 2, for simplicity. The technical modi cations needed to extend the algorithm to higher dimensions will be discussed elsewhere.) 1. Given a lattice and a positive integer N , nd a sublattice 0 of index N , geometrically similar to . Choose a subgroup4 W Aut() of orthogonal transformations of determinant +1 (i.e., rotations); denote by o(W ) the order of W . These concepts are illustrated in Fig. 1. In two dimensions we may take W to be the full rotation subgroup of Aut().5
Two lattices and 0 are said to be similar i there is a c 2 R; c > 0 and an orthogonal matrix A 2 Rnn, such that 0 = cA; i.e., if and 0 di er only by a rotation and a change of scale [4, 5]. 3 Similarity is not enough to guarantee correctness of our construction: we also need the sublattice to have the property that no lattice points occur at the boundary of the Voronoi region of any sublattice point. 4 Aut() is the group of automorphisms of , i.e., the set of isometries of the space that x the origin and leave the lattice invariant. 5 In higher dimensions we could take W to be the commutator subgroup of Aut(). 2
4
Figure 1: Lattices and similar sublattices. Examples based on the hexagonal lattice: left, a
similar sublattice of index 7; right, a similar sublattice of index 13. For later reference, pairs of sublattice points within a bounded distance of each other are also indicated. In the case of index 7, there is only one orbit of the group, which for the hexagonal lattice consists of rotations by 60 . For index 13, there are two orbits of the group, which result in sublattice pairs of two di erent lengths, and among the long pairs, three disjoint similar sublattices that are translates of each other. o
2. Partition into equivalence classes according to a relation de ned by:
1 2 , 9 2 W : 1 ? (1 ) = (2 ? (2)); 0
0
where denotes the nearest neighbor map of any point to the sublattice 0. Two points are equivalent i they belong to the same orbit of W , relative to their nearest sublattice neighbors. De ne the set E 0 0 consisting of all pairs of sublattice points 01; 02, such that jj01 ? 02 jj2 C .6 That is, the valid labels are all pairs of sublattice points at a bounded distance from each other. For any edge (01; 02) 2 E , let 00 0 be the sublattice which contains 01 ? 02 as a minimal vector and is geometrically similar to . Then partition E into equivalence classes E= 0, where two pairs are equivalent i they induce the same sublattice 00. Color the pairs in E with two colors, red and green, in a way such that along any straight line, the colors of adjacent pairs (i.e., of pairs that share an endpoint) strictly alternate. De ne a matching between equivalence classes: pair up orbits of the group (classes of ) with distinct edges of E (classes of 0). There are many possible ways of de ning this matching, and they will have signi cant impact on the distortion of the single channel quantizer; later we show how to pick this matching in an optimal way. 0
3.
N
4.
5. 6.
CN = jjcAjj2, where cA is the similarity such that 0 = cA, and 2 is a point of maximal norm within the Voronoi cell of 0 in 0. 6
5
7. Extend the matching de ned in terms of equivalence classes to the rest of the lattice and sublattice pairs, using the group and the sublattice. After doing so, the matching will have been de ned between pairs of sublattice points and pairs of lattice points: resolve this ambiguity using the colors of pairs in E (recall that we want a matching between pairs of sublattice points and individual lattice points). This extended matching is the desired function `. In the rest of this section, we give details about the implementation of each of these steps.
3.1 Matching of Lattice/Sublattice Orbits
The rst problem we need to deal with in designing an algorithm to label lattice points is that of nding a meaningful reduction of the size of the problem: among many other reasons, perhaps the most elementary one is that there are uncountably many maps ` : ! 0 0, but only countably many algorithms! That is what we use the equivalence for. To de ne the index assignment on equivalence classes we rst select a pair of points from each class. These de ne a parallelogram as shown in Fig. 2; the best matching between equivalence classes and pairs of sublattice points is the one that minimizes the sum of squares of the lengths of each side of this parallelogram. 1 0 0 1
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Figure 2: Construction of the matching. For a sublattice of index 13 of the hexagonal lattice there are two possible orbits to be matched with two possible pair of sublattice points of di erent lengths: the left picture shows the two possible ways of matching them. Once a matching is de ned among classes, it has to be extended to the rest of the lattice and sublattice pairs: the right picture illustrates how this extension is computed, with the aid of the sublattice and the group.
The optimal matching is found by reducing this problem to a standard network
ow problem [6]. Once the matching is de ned between a small set of points in the lattice and a small set of points in the sublattice, we use the group to extend it to the rest of the lattice. 6
3.2 Coloring of Sublattice Edges
To resolve the ambiguity left in the extended matching, we color sublattice edges using two colors, red and green. The following conventions are used: The main diagonal of the parallelogram determines the color of the long sides of the parallelogram, and the short sides get the opposite color. If a point 2 n 0 is red-joined to 01 and green-joined to 02, the ambiguity is resolved by setting `() = (01; 02). For points 0 2 0 the construction of the parallelogram does not apply: in this case, ` is de ned by `(0 ) = (0; 0) (i.e., sublattice points are their own label). With these conventions, it is clear that in order to maintain balance among descriptions it is sucient to make sure that colors alternate along any straight line. This is because red edges result in lower side distortion for one of the descriptions, while green edges do the same thing for the other: by making them alternate, equal side distortion is achieved on average. The coloring mechanism is illustrated in Fig. 3.
Figure 3: Coloring of sublattice edges. To implement the coloring, rst we de ne it on a simple graph as shown in this gure, de ned in terms of the minimal vectors of ; then we use the similarity between and 0 to reduce a coloring of the graph of sublattice edges to the problem of coloring a number of disjoint copies of the simpler graph.
3.3 Vector Index Assignment Encoding/Decoding
To encode a given lattice point, we retrieve its corresponding sublattice edge, and nd its color; depending on the color we return the edge as is, or we ip it. To nd the lattice point corresponding to a given sublattice edge, rst we nd the pair of lattice points that make a parallelogram with the given edge; then, depending on the color of the edge, we return one or other of the points. The central decoder uses the inverse index assignment to nd the hexagonal cell to use for decoding. The side decoder, having access only to one of the sublattice points, uses that point for decoding. We should mention that although centroid decoding would result in slightly lower distortion for the side decoders, according to our measurements this produces an improvement of at most 0.23dB, and vanishingly small improvements at high rates. We have therefore decided to use sublattice decoding instead. 7
4 An Example: the Hexagonal Lattice In this section we analyze the performance of the MD quantizer that results from using the construction presented in Section 3 for the hexagonal lattice A2. Although this lattice does not o er signi cant gains in terms of attainable mean squared error7, we decided to start our study of the creation of multiple descriptions of lattices with this one for a number of reasons: This lattice is isomorphic to the ring of Eisenstein integers, and similar sublattices of this lattice are characterized by the property that they are ideals in this ring, with the sublattice index given by the lattice norm of the generator of the ideal [3]. In developing and testing our rst algorithms, being able to \painlessly" obtain good sublattices of various indices, as well as having a simple way of deciding for what indices similar sublattices exist, was crucial in making progress. In two dimensions we can draw pictures! We found these to be an invaluable tool in developing the intuition behind the results we present here. The determinant 1 isometries that leave the hexagonal lattice invariant are rotations by multiples of 60 . This is a cyclic group W of order 6. Part of the implementation of the proposed quantizer involves the solution of an equation over W : but since W is small and cyclic, the trivial search algorithm that scans the entire group by taking powers of a primitive element is feasible. In two dimensions, A2 solves the problem of packing the largest possible number of spheres, of arranging a maximal number of spheres all of the same size such that they all touch another one, of covering space with overlapping spheres of the smallest size, and of quantizing a uniform source. In two dimensions this is the best lattice to use, from almost every point of view. The real \quantum leap" occurs in going from one dimension to higher dimensions; but once we know how to solve the problem in some space of dimension 2, at least under the restricted assumptions we make, solving the problem for other dimensions should be less of a challenge. o
4.1 Asymptotic Performance
We computed the central and side distortions of the proposed hexagonal MD quantizer, in the limit as the rate becomes high, obtaining d1 = 41 2?2 (1? ); d0 = 41 5p 2?2 (1+ ); 36 3 where 0 < a < 1 is a parameter that trades o distortion of the central and side quantizers, and R is the rate; at high rates, both distortions decay exponentially fast, h
R
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h
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The second moment of apVoronoi region in the cubic lattice is 1=12 = 0:08333:::, whereas for the hexagonal lattice it is 5=(36 3) 0:080188; i.e., a gain of 0.1671dB, out of the available 1.53dB. 7
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but at di erent rates. Note that the distortion product d0 d1 is independent of a, and that 24 d0 d1 0:001595. For comparison purposes, for the cubic lattice we have d0 = 481 2?2 (1+ ); d1 = 121 2?2 (1? ); and the resulting distortion product is 24 d0d1 0:001736, 0:3674dB worse. Observe that the gap between the second moments of the hexagonal and cubic lattices is 0:1671dB, so in this case we are gaining slightly more than that per channel. h
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4.2 Numerical Simulations
To verify these results, we implemented the proposed quantizers. We simulated a uniform source over a region of space that is very large compared to the size of a quantizer cell8, we quantized 224 pairs of numbers, dequantized them using both the side and central inverse quantizers, and computed the resulting average distortions. The results of this simulation are shown in Fig. 4. Zoom into large indices
Numerical evaluation of the MSE for the lattice A
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Figure 4: Numerical approximation of the average central and side distortions, for a uniform source. The horizontal axis shows the average p central distortion, scaled by the volume of a fundamental polytope of the lattice A2 ( 3=2), and by the number of central cells that comprise a side cell (N , the sublattice index). The vertical axis shows the average side distortion d1 , with the same scalings. The scalings are performed in order to enable fair com-
parisons between quantizers of di erent dimensions and based on lattices with fundamental polytopes of di erent volumes. For comparison, we show the performance of the MD scalar quantizer on the same source. For large indices, the performance of the hexagonal lattice quantizer is consistently superior to that of the scalar quantizer by 0.3dB, as predicted by our asymptotic analysis.
The key element to observe here is how, for high indices, the performance of the MD hexagonal quantizer is consistently better than that of the MD scalar quantizer, by approximately 0:3dB, as predicted by theory. The minimal vectors of the lattice are taken of length 1, and the uniform distribution is taken over a square of side 10000. 8
9
C code implementing the MD hexagonal quantizer, a number of optimal matchings, and a full color (and much more detailed) version of this paper can be found at any of our webpages: http://www.ifp.uiuc.edu/~servetto/research/more/md/, http://www.research.att.com/~vinay/, http://www.research.att.com/~njas/.
5 Conclusions In this paper we presented an overview of work in progress on the construction of multiple description lattice vector quantizers. We presented a construction of a function to label lattice points that holds for a general class of lattices, and we presented speci c results for the two-dimensional hexagonal lattice; for this lattice, we showed performance gains over the multiple description scalar quantizer (based on the cubic lattice) at least as large as those predicted by single description quantization theory (i.e., in excess of 0.1671dB/channel). The MDLVQ de ned in this work is based on the lattice generated by the root system A2. We plan to repeat our construction for other lattices. Good candidates for this purpose are the lattice E8 and the Leech lattice: the densest known sphere packings in 8 and 24 dimensions. The gains of these lattices over the cubic lattice are 0.65dB and 1.03dB respectively, signi cantly more than the 0.1671dB gained by the hexagonal lattice. Our current research e orts are focused on working out the details of our construction for the lattice E8.
References
[1] J.-C. Batllo and V. A. Vaishampayan. Asymptotic Analysis of Multiple Description Quantizers. IEEE Trans. Inform. Theory, 44(1):278{283, 1998.
[2] T. Y. Berger-Wolf and E. M. Reingold. Optimal Index Assignment for Multichannel Communication. In Proc. SIAM Symp. Discrete Algorithms, Baltimore, MD, 1999. Extended abstract available from http://emr.cs.uiuc.edu/~reingold/algorithms.shtml. [3] M. Bernstein, N. J. A. Sloane, and P. E. Wright. On Sublattices of the Hexagonal Lattice. Discrete Math., 170:29{39, 1997. [4] J. H. Conway, E. M. Rains, and N. J. A. Sloane. On the Existence of Similar Sublattices. Canad. J. Math., 1999. To appear. [5] J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups. Springer Verlag, 3rd edition, 1998. [6] T. Cormen, C. Leiserson, and R. Rivest. Introduction to Algorithms. MIT Press, 1990. [7] A. A. El Gamal and T. Cover. Achievable Rates for Multiple Descriptions. IEEE Trans. Inform. Theory, IT-28(6):851{857, 1982. [8] A. Gersho and R. M. Gray. Vector Quantization and Signal Compression. Kluwer Academic Publishers, 1992. [9] L. H. Ozarow. On a Source Coding Problem with Two Channels and Three Receivers. Bell System Technical Journal, 59(10):1909{ 1921, 1980. [10] C. E. Shannon. Coding Theorems for a Discrete Source with a Fidelity Criterion. IRE Nat. Conv. Rec., 4:142{163, 1959. [11] V. A. Vaishampayan. Design of Multiple Description Scalar Quantizers. IEEE Trans. Inform. Theory, 39(3):821{834, 1993. [12] V. A. Vaishampayan, J.-C. Batllo, and A. R. Calderbank. On Reducing Granular Distortion in Multiple Description Quantization. In Proceedings of the IEEE International Symposium on Information Theory, Cambridge, MA, 1998. [13] V. A. Vaishampayan and J. Domaszewicz. Design of Entropy-Constrained Multiple Description Scalar Quantizers. IEEE Trans. Inform. Theory, 40(1):245{250, 1994. [14] Z. Zhang and T. Berger. New Results in Binary Multiple Descriptions. IEEE Trans. Inform. Theory, IT-33:502{521, 1987.
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Multiple Description Lattice Vector Quantization Sergio D. Servettoy
Vinay A. Vaishampayanz
N. J. A. Sloanez
Abstract
We consider the problem of designing a lattice-based multiple description vector quantizer for a two-channel diversity system. The design of such a quantizer can be reduced to the problem of assigning pair labels to points of a vector quantizer codebook. A general labeling procedure based on the structure of the lattice is presented, along with detailed results for the hexagonal lattice: algorithms, asymptotic performance, and numerical simulations. Asymptotically, when compared with the lattice Z , the resulting quantizer achieves the standard second-moment gain of the hexagonal lattice for the central distortion, and, surprisingly, achieves the two-dimensional sphere gain for the side distortion.
1 Introduction
1.1 Multiple Description Source Coding
A diversity-based communication system provides multiple channels to send information between a transmitter and a receiver so that even if a channel fails, some data can still be delivered to the receiver. Consider a system with two channels: if identical information is sent over each channel and both work, half the information received has no value. The theory of multiple description source coding studies methods for sending di erent information over the channels, in a way such that if only one channel works, the information received is sucient to guarantee a minimum delity in the reconstruction at the receiver; however, should both channels work, the information from both channels can be combined to yield a higher delity reconstruction. The general problem of designing source codes of this kind was posed by Gersho, Witsenhausen, Wolf, Wyner, Ziv and Ozarow at the September 1979 IEEE Information Theory Workshop; it is a generalization of Shannon's problem of source coding with a delity criterion [10]. Initial progress on this problem was made by El-Gamal and Cover [7] and by Ozarow [9]. In [7], for an arbitrary memoryless source and a bounded distortion Work done at AT&T Labs, between 6/98 and 9/98, while S. Servetto was a summer intern. y Beckman Institute and Dept. of Computer Science, University of Illinois at Urbana-Champaign.
405 N. Matthews St., Urbana IL 61801. e-mail: [email protected]. z AT&T Labs - Research, Shannon Laboratory. 180 Park Av., Florham Park, NJ 07932. e-mail: fvinay,[email protected].
1
measure, the authors nd an achievable rate region of pairs (R1; R2), for a given distortion vector (D0; D1; D2). Ozarow [9] then showed that, if the source is gaussian and the distortion measure is squared error, the achievable rate region of El-Gamal and Cover is, in fact, the exact MD rate/distortion region for the source. El-Gamal and Cover also conjectured, but were not able to prove, that the region of achievable rates they computed is indeed the MD rate/distortion region for other sources and other distortion measures; this conjecture was disproved by Zhang and Berger [14]. To this date, Ozarow's result for the gaussian source and squared error metric is the only one presenting a complete characterization of a MD rate/distortion region.
1.2 Multiple Description Scalar Quantization
The rst practical results on MDs were presented in [11], where a simple procedure is given for designing MD scalar quantizers with remarkably good asymptotic properties. An MD scalar quantizer consists of two main components: a scalar quantizer (in the classical sense), and an index assignment that splits the information about each sample into two complementary and possibly redundant descriptions of the same sample. The problem of designing good index assignments is studied in [11], using Ozarow's characterization of the MD rate/distortion region for the gaussian source as a guiding principle. Under the assumption of equal and high rates for both descriptions, and for a squared error distortion measure, a construction of a large class of index assignments is presented in [11] for which the exponential rate of decay of the MSE is exactly that predicted by Ozarow's result.1 Berger-Wolf and Reingold [2] generalize the construction of index assignments from matrices for two channels to arbitrary k-dimensional tensors for any number of channels. The exact coecient of quantization (i.e., the sub-exponential terms in the error expression) of the MD scalar quantizer was computed in [1], for general source densities and m-th power distortion measures [8]. Perhaps the most interesting case is that of a gaussian density and m = 2. Because of the large number of parameters involved (rates for each of the channels and distortions for all subsets of working/nonworking channels), comparisons between MD systems are complicated. However, it is argued in [1] that in the balanced case (i.e., when the rates on the channels are equal), the product d0d1 of the side and central distortions is a good gure of merit for high rates. For a gaussian source and m = 2, it was shown in [1] that there is a gap of 8.69dB between the distortion product of the MD scalar quantizer and the MD rate/distortion bound; this gap is reduced to 3.06dB when the entropy-constrained MD scalar quantizer [13] is considered.
1.3 Multiple Description Vector Quantization
For the squared error distortion measure, it is known from classical quantization theory [8] that the error of a scalar quantizer with cubic Voronoi regions is 1:53dB higher However, the sub-exponential terms in the MSE are not the same: the performance of the MD scalar quantizer is bounded away from the gaussian MD rate/distortion bound, as one would intuitively expect. 1
2
than the classical rate/distortion bound. The performance loss of the scalar quantizer occurs because the normalized second moment of an N -dimensional hypercube is exactly 1:53dB higher than that of an equivalent hypersphere, in the limit as N ! 1. Although it is not possible in nite dimensions to design quantizers whose Voronoi cells are exactly spheres (spheres are not space- lling polytopes in any dimension 2), the gap can be signi cantly reduced by designing quantizers with more \spherical" cells, i.e., with a smaller normalized second moment than a hypercube. In the MD case, it is a remarkable fact that for a gaussian source, the performance gap of 3:06dB between the distortion product of the two-channel entropy-constrained MD scalar quantizer and the two-channel MD rate/distortion bound is exactly twice as large as the single description gap. It was shown in [12], via a random coding argument, that multiple description vector quantizers can be constructed which achieve gains over scalar quantization (or equivalently, the Z lattice) of 1.53dB simultaneously for the central and side distortion, as the vector dimension goes to in nity. Our main goal in this work is to design MD quantizers with smaller granular central and side distortion than that achieved by the Z lattice. Since we are looking for a systematic and constructive approach, we have chosen to design MD lattice quantizers, more speci cally, quantizers whose codebook is a lattice with good second moment properties and that admits fast encoding algorithms. To accomplish this we must design appropriate index assignments, and the algebraic structure of the lattice is helpful for this purpose.
1.4 Main Contributions and Organization of the Paper
The main contribution of this paper is the construction of good vector index assignments for lattices. This is accomplished by labeling lattice points with pairs of points in some similar coarse sublattice. This step can be thought of as a \requantization" of lattice points, but in a way such that knowing the output of both coarse quantizers allows us to retrieve the exact original point (i.e., the ne quantizer); how coarse the sublattice is determines the amount of redundancy in the MD code, in a manner analogous to the number of diagonals in scalar index assignments. This paper is organized as follows. In Section 2, we de ne the structure of multiple description lattice quantizers. In Section 3, we give a formal description of the algorithm and optimization problems involved in the construction of index assignments for any of the lattices considered. Then, in Section 4, we apply these general techniques to the hexagonal lattice, analyze the asymptotic performance, and present results of numerical simulations when the proposed quantizer is applied to a uniform source. Section 5 gives a summary.
2 Structure of Multiple Description Lattice Quantizers A Multiple Description Lattice Vector Quantizer (MDLVQ) is a triplet Q = (; 0; `), where: 3
is a lattice. 0 is a sublattice of that is geometrically similar2 3 to . Each lattice point 2 gets mapped by ` to a pair of sublattice points ;
(0red; 0green) that uniquely identi es ; i.e., ` must be an injection: 1-1! `() 0 0: ` is referred to as the vector index assignment, and the two points in the image under ` of a point are referred to as the red and green descriptions. The amount of redundancy in a lattice quantizer is controlled by N = j=0j, the index of 0 in . Given any sublattice point 0 2 0, we require that ` is such that the total number of distinct lattice points 2 for which 0 is used to describe is exactly N ; i.e., jf : red(`()) = 0gj = jf : green(`()) = 0gj = N; where red(red; green) = red, and similarly for green. In other words, lattice points are labeled with pairs of sublattice points (it is these sublattice points that get actually transmitted over each channel), and each sublattice point is used exactly N times; the larger is N , the higher the uncertainty about the original lattice point when one of the channels fails. A key property of good index assignments ` is that the set of central cells that share a given label must be as localized in space as possible, in order to achieve low distortion in the case of a channel failure. This is analogous to the idea that for a scalar quantizer the spread of a side cell must be minimized [11]. We now give a method for constructing a labeling function `.
3 Multiple Descriptions of Lattice Points The construction we propose is based on the following steps. (We assume the dimension is 2, for simplicity. The technical modi cations needed to extend the algorithm to higher dimensions will be discussed elsewhere.) 1. Given a lattice and a positive integer N , nd a sublattice 0 of index N , geometrically similar to . Choose a subgroup4 W Aut() of orthogonal transformations of determinant +1 (i.e., rotations); denote by o(W ) the order of W . These concepts are illustrated in Fig. 1. In two dimensions we may take W to be the full rotation subgroup of Aut().5
Two lattices and 0 are said to be similar i there is a c 2 R; c > 0 and an orthogonal matrix A 2 Rnn, such that 0 = cA; i.e., if and 0 di er only by a rotation and a change of scale [4, 5]. 3 Similarity is not enough to guarantee correctness of our construction: we also need the sublattice to have the property that no lattice points occur at the boundary of the Voronoi region of any sublattice point. 4 Aut() is the group of automorphisms of , i.e., the set of isometries of the space that x the origin and leave the lattice invariant. 5 In higher dimensions we could take W to be the commutator subgroup of Aut(). 2
4
Figure 1: Lattices and similar sublattices. Examples based on the hexagonal lattice: left, a
similar sublattice of index 7; right, a similar sublattice of index 13. For later reference, pairs of sublattice points within a bounded distance of each other are also indicated. In the case of index 7, there is only one orbit of the group, which for the hexagonal lattice consists of rotations by 60 . For index 13, there are two orbits of the group, which result in sublattice pairs of two di erent lengths, and among the long pairs, three disjoint similar sublattices that are translates of each other. o
2. Partition into equivalence classes according to a relation de ned by:
1 2 , 9 2 W : 1 ? (1 ) = (2 ? (2)); 0
0
where denotes the nearest neighbor map of any point to the sublattice 0. Two points are equivalent i they belong to the same orbit of W , relative to their nearest sublattice neighbors. De ne the set E 0 0 consisting of all pairs of sublattice points 01; 02, such that jj01 ? 02 jj2 C .6 That is, the valid labels are all pairs of sublattice points at a bounded distance from each other. For any edge (01; 02) 2 E , let 00 0 be the sublattice which contains 01 ? 02 as a minimal vector and is geometrically similar to . Then partition E into equivalence classes E= 0, where two pairs are equivalent i they induce the same sublattice 00. Color the pairs in E with two colors, red and green, in a way such that along any straight line, the colors of adjacent pairs (i.e., of pairs that share an endpoint) strictly alternate. De ne a matching between equivalence classes: pair up orbits of the group (classes of ) with distinct edges of E (classes of 0). There are many possible ways of de ning this matching, and they will have signi cant impact on the distortion of the single channel quantizer; later we show how to pick this matching in an optimal way. 0
3.
N
4.
5. 6.
CN = jjcAjj2, where cA is the similarity such that 0 = cA, and 2 is a point of maximal norm within the Voronoi cell of 0 in 0. 6
5
7. Extend the matching de ned in terms of equivalence classes to the rest of the lattice and sublattice pairs, using the group and the sublattice. After doing so, the matching will have been de ned between pairs of sublattice points and pairs of lattice points: resolve this ambiguity using the colors of pairs in E (recall that we want a matching between pairs of sublattice points and individual lattice points). This extended matching is the desired function `. In the rest of this section, we give details about the implementation of each of these steps.
3.1 Matching of Lattice/Sublattice Orbits
The rst problem we need to deal with in designing an algorithm to label lattice points is that of nding a meaningful reduction of the size of the problem: among many other reasons, perhaps the most elementary one is that there are uncountably many maps ` : ! 0 0, but only countably many algorithms! That is what we use the equivalence for. To de ne the index assignment on equivalence classes we rst select a pair of points from each class. These de ne a parallelogram as shown in Fig. 2; the best matching between equivalence classes and pairs of sublattice points is the one that minimizes the sum of squares of the lengths of each side of this parallelogram. 1 0 0 1
1 0 0 1
1 0 0 1
1 0 0 1 1 0 1 0
1 0 1 0 1 0 1 0
1 0 1 0 1 0 1 0
1 0 1 0
1 0 1 0 1 0 1 0
1 0 0 1 1 0 0 1
1 0 1 0 1 0 0 1
1 0 0 1
1 0 0 1 1 0 0 1
1 0 0 1 1 0 0 1
1 0 1 0
1 0 1 0
1 0 0 1
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1 0 1 0 1 0 0 1
1 0 0 1 1 0 0 1
1 0 1 0 1 0 1 0
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1 0 0 1
1 0 0 1 1 0 0 1
1 0 1 0 1 0 1 0
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1 0 0 1 1 0 1 0
1 0 1 0
1 0 0 1
1 0 0 1 1 0 1 0
1 0 1 0
1 0 1 0
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1 0 0 1 1 0 1 0
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1 0 0 1 1 0 1 0
1 0 0 1 1 0 0 1
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1 0 1 0 1 0 0 1
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1 0 0 1
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1 0 1 0 1 0 0 1
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1 0 0 1 1 0 0 1
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1 0 1 0 1 0 1 0 1 0 1 0
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1 0 1 0 1 0 1 0
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1 0 0 1 1 0 0 1
1 0 0 1 1 0 0 1
1 0 1 0 1 0 0 1
1 0 1 0 1 0 0 1
Figure 2: Construction of the matching. For a sublattice of index 13 of the hexagonal lattice there are two possible orbits to be matched with two possible pair of sublattice points of di erent lengths: the left picture shows the two possible ways of matching them. Once a matching is de ned among classes, it has to be extended to the rest of the lattice and sublattice pairs: the right picture illustrates how this extension is computed, with the aid of the sublattice and the group.
The optimal matching is found by reducing this problem to a standard network
ow problem [6]. Once the matching is de ned between a small set of points in the lattice and a small set of points in the sublattice, we use the group to extend it to the rest of the lattice. 6
3.2 Coloring of Sublattice Edges
To resolve the ambiguity left in the extended matching, we color sublattice edges using two colors, red and green. The following conventions are used: The main diagonal of the parallelogram determines the color of the long sides of the parallelogram, and the short sides get the opposite color. If a point 2 n 0 is red-joined to 01 and green-joined to 02, the ambiguity is resolved by setting `() = (01; 02). For points 0 2 0 the construction of the parallelogram does not apply: in this case, ` is de ned by `(0 ) = (0; 0) (i.e., sublattice points are their own label). With these conventions, it is clear that in order to maintain balance among descriptions it is sucient to make sure that colors alternate along any straight line. This is because red edges result in lower side distortion for one of the descriptions, while green edges do the same thing for the other: by making them alternate, equal side distortion is achieved on average. The coloring mechanism is illustrated in Fig. 3.
Figure 3: Coloring of sublattice edges. To implement the coloring, rst we de ne it on a simple graph as shown in this gure, de ned in terms of the minimal vectors of ; then we use the similarity between and 0 to reduce a coloring of the graph of sublattice edges to the problem of coloring a number of disjoint copies of the simpler graph.
3.3 Vector Index Assignment Encoding/Decoding
To encode a given lattice point, we retrieve its corresponding sublattice edge, and nd its color; depending on the color we return the edge as is, or we ip it. To nd the lattice point corresponding to a given sublattice edge, rst we nd the pair of lattice points that make a parallelogram with the given edge; then, depending on the color of the edge, we return one or other of the points. The central decoder uses the inverse index assignment to nd the hexagonal cell to use for decoding. The side decoder, having access only to one of the sublattice points, uses that point for decoding. We should mention that although centroid decoding would result in slightly lower distortion for the side decoders, according to our measurements this produces an improvement of at most 0.23dB, and vanishingly small improvements at high rates. We have therefore decided to use sublattice decoding instead. 7
4 An Example: the Hexagonal Lattice In this section we analyze the performance of the MD quantizer that results from using the construction presented in Section 3 for the hexagonal lattice A2. Although this lattice does not o er signi cant gains in terms of attainable mean squared error7, we decided to start our study of the creation of multiple descriptions of lattices with this one for a number of reasons: This lattice is isomorphic to the ring of Eisenstein integers, and similar sublattices of this lattice are characterized by the property that they are ideals in this ring, with the sublattice index given by the lattice norm of the generator of the ideal [3]. In developing and testing our rst algorithms, being able to \painlessly" obtain good sublattices of various indices, as well as having a simple way of deciding for what indices similar sublattices exist, was crucial in making progress. In two dimensions we can draw pictures! We found these to be an invaluable tool in developing the intuition behind the results we present here. The determinant 1 isometries that leave the hexagonal lattice invariant are rotations by multiples of 60 . This is a cyclic group W of order 6. Part of the implementation of the proposed quantizer involves the solution of an equation over W : but since W is small and cyclic, the trivial search algorithm that scans the entire group by taking powers of a primitive element is feasible. In two dimensions, A2 solves the problem of packing the largest possible number of spheres, of arranging a maximal number of spheres all of the same size such that they all touch another one, of covering space with overlapping spheres of the smallest size, and of quantizing a uniform source. In two dimensions this is the best lattice to use, from almost every point of view. The real \quantum leap" occurs in going from one dimension to higher dimensions; but once we know how to solve the problem in some space of dimension 2, at least under the restricted assumptions we make, solving the problem for other dimensions should be less of a challenge. o
4.1 Asymptotic Performance
We computed the central and side distortions of the proposed hexagonal MD quantizer, in the limit as the rate becomes high, obtaining d1 = 41 2?2 (1? ); d0 = 41 5p 2?2 (1+ ); 36 3 where 0 < a < 1 is a parameter that trades o distortion of the central and side quantizers, and R is the rate; at high rates, both distortions decay exponentially fast, h
R
a
h
R
a
The second moment of apVoronoi region in the cubic lattice is 1=12 = 0:08333:::, whereas for the hexagonal lattice it is 5=(36 3) 0:080188; i.e., a gain of 0.1671dB, out of the available 1.53dB. 7
8
but at di erent rates. Note that the distortion product d0 d1 is independent of a, and that 24 d0 d1 0:001595. For comparison purposes, for the cubic lattice we have d0 = 481 2?2 (1+ ); d1 = 121 2?2 (1? ); and the resulting distortion product is 24 d0d1 0:001736, 0:3674dB worse. Observe that the gap between the second moments of the hexagonal and cubic lattices is 0:1671dB, so in this case we are gaining slightly more than that per channel. h
R
h
h
h
c
R
a
c
R
c
R
a
c
4.2 Numerical Simulations
To verify these results, we implemented the proposed quantizers. We simulated a uniform source over a region of space that is very large compared to the size of a quantizer cell8, we quantized 224 pairs of numbers, dequantized them using both the side and central inverse quantizers, and computed the resulting average distortions. The results of this simulation are shown in Fig. 4. Zoom into large indices
Numerical evaluation of the MSE for the lattice A
2
2.6
N=127 N=121
2.5
N=127 N=121
2.4 2.2
N=103
N=103
2 2
N=91 N=81 N=79 N=73 N=67
d1
N=81 1.6
N=79 N=73
N=61
1.4
N=49 (both)
1
N=91
1.8
d1
1.5
N=67 N=61
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N=43 N=37
N=49 (both)
1
N=31
N=43
N=25
0.5
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N=19
N=37 N=31
N=13
0.6
N=7 N=9
0
0
0.002
0.004
0.006 d0
0.008
0.01
1
1.5
2 d0
0.012
2.5
3 −3
x 10
Figure 4: Numerical approximation of the average central and side distortions, for a uniform source. The horizontal axis shows the average p central distortion, scaled by the volume of a fundamental polytope of the lattice A2 ( 3=2), and by the number of central cells that comprise a side cell (N , the sublattice index). The vertical axis shows the average side distortion d1 , with the same scalings. The scalings are performed in order to enable fair com-
parisons between quantizers of di erent dimensions and based on lattices with fundamental polytopes of di erent volumes. For comparison, we show the performance of the MD scalar quantizer on the same source. For large indices, the performance of the hexagonal lattice quantizer is consistently superior to that of the scalar quantizer by 0.3dB, as predicted by our asymptotic analysis.
The key element to observe here is how, for high indices, the performance of the MD hexagonal quantizer is consistently better than that of the MD scalar quantizer, by approximately 0:3dB, as predicted by theory. The minimal vectors of the lattice are taken of length 1, and the uniform distribution is taken over a square of side 10000. 8
9
C code implementing the MD hexagonal quantizer, a number of optimal matchings, and a full color (and much more detailed) version of this paper can be found at any of our webpages: http://www.ifp.uiuc.edu/~servetto/research/more/md/, http://www.research.att.com/~vinay/, http://www.research.att.com/~njas/.
5 Conclusions In this paper we presented an overview of work in progress on the construction of multiple description lattice vector quantizers. We presented a construction of a function to label lattice points that holds for a general class of lattices, and we presented speci c results for the two-dimensional hexagonal lattice; for this lattice, we showed performance gains over the multiple description scalar quantizer (based on the cubic lattice) at least as large as those predicted by single description quantization theory (i.e., in excess of 0.1671dB/channel). The MDLVQ de ned in this work is based on the lattice generated by the root system A2. We plan to repeat our construction for other lattices. Good candidates for this purpose are the lattice E8 and the Leech lattice: the densest known sphere packings in 8 and 24 dimensions. The gains of these lattices over the cubic lattice are 0.65dB and 1.03dB respectively, signi cantly more than the 0.1671dB gained by the hexagonal lattice. Our current research e orts are focused on working out the details of our construction for the lattice E8.
References
[1] J.-C. Batllo and V. A. Vaishampayan. Asymptotic Analysis of Multiple Description Quantizers. IEEE Trans. Inform. Theory, 44(1):278{283, 1998.
[2] T. Y. Berger-Wolf and E. M. Reingold. Optimal Index Assignment for Multichannel Communication. In Proc. SIAM Symp. Discrete Algorithms, Baltimore, MD, 1999. Extended abstract available from http://emr.cs.uiuc.edu/~reingold/algorithms.shtml. [3] M. Bernstein, N. J. A. Sloane, and P. E. Wright. On Sublattices of the Hexagonal Lattice. Discrete Math., 170:29{39, 1997. [4] J. H. Conway, E. M. Rains, and N. J. A. Sloane. On the Existence of Similar Sublattices. Canad. J. Math., 1999. To appear. [5] J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups. Springer Verlag, 3rd edition, 1998. [6] T. Cormen, C. Leiserson, and R. Rivest. Introduction to Algorithms. MIT Press, 1990. [7] A. A. El Gamal and T. Cover. Achievable Rates for Multiple Descriptions. IEEE Trans. Inform. Theory, IT-28(6):851{857, 1982. [8] A. Gersho and R. M. Gray. Vector Quantization and Signal Compression. Kluwer Academic Publishers, 1992. [9] L. H. Ozarow. On a Source Coding Problem with Two Channels and Three Receivers. Bell System Technical Journal, 59(10):1909{ 1921, 1980. [10] C. E. Shannon. Coding Theorems for a Discrete Source with a Fidelity Criterion. IRE Nat. Conv. Rec., 4:142{163, 1959. [11] V. A. Vaishampayan. Design of Multiple Description Scalar Quantizers. IEEE Trans. Inform. Theory, 39(3):821{834, 1993. [12] V. A. Vaishampayan, J.-C. Batllo, and A. R. Calderbank. On Reducing Granular Distortion in Multiple Description Quantization. In Proceedings of the IEEE International Symposium on Information Theory, Cambridge, MA, 1998. [13] V. A. Vaishampayan and J. Domaszewicz. Design of Entropy-Constrained Multiple Description Scalar Quantizers. IEEE Trans. Inform. Theory, 40(1):245{250, 1994. [14] Z. Zhang and T. Berger. New Results in Binary Multiple Descriptions. IEEE Trans. Inform. Theory, IT-33:502{521, 1987.
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