## Asymmetric multiple description lattice vector quantizers (2002)

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Venue: | IEEE Trans. Inf. Theory |

Citations: | 24 - 2 self |

### BibTeX

@ARTICLE{Diggavi02asymmetricmultiple,

author = {Suhas N. Diggavi and N. J. A. Sloane and Vinay A. Vaishampayan},

title = {Asymmetric multiple description lattice vector quantizers},

journal = {IEEE Trans. Inf. Theory},

year = {2002},

pages = {174}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract—We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, along with a general design procedure. The high-rate asymptotic performance of the quantizer is also studied. We evaluate the rate-distortion performance of the quantizer and compare it to known information-theoretic bounds. The high-rate asymptotic analysis is compared to the performance of the quantizer. Index Terms—Cubic lattice, high-rate quantization, lattice quantization, multiple descriptions, quantization, source coding, successive refinement, vector quantization. I.

### Citations

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Citation Context ...(20) 3We usually omit the word “discrete” when referring to this set.4. Matching the edges to the lattice points in the Voronoi set is now a straightforward and easily solved assignment problem (cf. =-=[18]-=-). To formulate this assignment problem we compute the cost given by (12) for each lattice point and each equivalence class of edges modulo Λs (taking the minimum over the edge class). We use only one... |

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Citation Context ...cated. For example, it is necessary to distinguish between left gcd’s and right gcd’s. Both are well-defined in H1 and also in H0 as long as at least one of the quaternions involved has odd norm [8], =-=[15]-=-. We plan to discuss this theory and its applications to the study of sublattices of Z 4 and D4 elsewhere. In the present paper we will restrict our attention to a narrow class of sublattices, which h... |

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Citation Context ...all sets of four vectors corresponding to the Coxeter diagram of Fig. 4. Given a sublattice Λ ′, we compute the equations defining an octahedral face of the Voronoi cell from (31) and (33). Then AMPL =-=[12]-=- and CPLEX [7] were used to verify that in every case there was a point of D4 on the face. The preceding discussion has shown that the lattices Z, Z 2 , Z 4k for k ≥ 1 and D4 have a plentiful supply o... |

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Citation Context ...5 9,1 5,5 1 5,1 2,9 6 5 10,5 9 11 12,6 12 10 10 9 −6 −6 −4 −2 0 2 4 6 Figure 3: Labels generated by the algorithm. are the numbers 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20,... (22) (Sequence A1481 of =-=[24]-=-). If Λ = A2 = J , the planar hexagonal lattice, multiplication by the Eisenstein integer ξ = a+bω ∈ J gives a similar sublattice Λ ′ = ξΛ of index N = a 2 +ab+b 2 . A number N is of the form a 2 +ab+... |

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Citation Context ...5. Given a sublattice , we compute the equations defining an octahedral face of the Voronoi cell from (31) and184 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 1, JANUARY 2002 (33). ThenAMPL =-=[12]-=- and CPLEX [7] were used to verify that in every case there was a point of on the face. The preceding discussion has shown that the lattices , , for and have a plentiful supply of clean, geometrically... |

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Citation Context ...ic aspects of the multiple description problem see [9], [10], [22], [29], [30]. The problem of designing quantizers for the multiple description problem has been considered in [11], [17], [19], [25], =-=[26]-=-, [28]. The work presented in [28] considered only the balanced/symmetric lattice quantizer design. Unlike the work in [11], we do not use a training approach; instead, we use the geometry of the unde... |

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Citation Context ...without necessarily waiting for the more important descriptions to arrive (as in successive refinement). For previous work on the information-theoretic aspects of the multiple description problem see =-=[9]-=-, [10], [22], [29], [30]. The problem of designing quantizers for the multiple description problem has been considered in [11], [17], [19], [25], [26], [28]. The work presented in [28] considered only... |

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Citation Context ...ut necessarily waiting for the more important descriptions to arrive (as in successive refinement). For previous work on the information-theoretic aspects of the multiple description problem see [9], =-=[10]-=-, [22], [29], [30]. The problem of designing quantizers for the multiple description problem has been considered in [11], [17], [19], [25], [26], [28]. The work presented in [28] considered only the b... |

114 |
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Citation Context ...nd relatively prime, then the sublattice D4ξ is clean, where ξ is given by (34). Suppose M is a product of primes congruent to 1 (mod 4). From the classical theory of quadratic forms (see for example =-=[6]-=-), we know that M = p2 + q2 with p even, q odd and gcd(p, q) = 1. We now simply set α = p + q and β = |p − q|.It remains to discuss the case M = 7. For this we can multiply on the left or on the righ... |

112 |
An Introduction to the Theory of Groups
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Citation Context ...= Λ1 ∪ Λ2, as shown in Fig. 6. The join is the lattice generated by the vectors of both Λ1 and Λ2 (and in general is not simply their union). From the second isomorphism theorem of group theory (e.g. =-=[23]-=-) the indices and determinants of these lattices are related by [Λ∪ : Λ1] = [Λ2 : Λ∩], [Λ∪ : Λ2] = [Λ1 : Λ∩] , (40) det Λ1 det Λ2 = det Λ∪ det Λ∩ . (41) There are now in general many ways to find a “p... |

103 |
Source Coding Theory
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Citation Context ...d we have assumed that is the centroid of its Voronoi cell. This is true for the uniform density. For nonuniform densities, there is an error term which goes to zero with the size of the Voronoi cell =-=[14]-=-. The first term in (7) is the two-channel distortion and the second term is the excess distortion which is incurred when only description is available. Note that, for a given , only the excess distor... |

84 |
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Citation Context ...essarily waiting for the more important descriptions to arrive (as in successive refinement). For previous work on the information-theoretic aspects of the multiple description problem see [9], [10], =-=[22]-=-, [29], [30]. The problem of designing quantizers for the multiple description problem has been considered in [11], [17], [19], [25], [26], [28]. The work presented in [28] considered only the balance... |

68 |
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Citation Context ...ting for the more important descriptions to arrive (as in successive refinement). For previous work on the information-theoretic aspects of the multiple description problem see [9], [10], [22], [29], =-=[30]-=-. The problem of designing quantizers for the multiple description problem has been considered in [11], [17], [19], [25], [26], [28]. The work presented in [28] considered only the balanced/symmetric ... |

64 |
An introduction to the theory of numbers, 5th ed
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Citation Context ... more complicated. For example, it is necessary to distinguish between left ’s and right ’s. Both are well-defined in and also in as long as at least one of the quaternions involved has odd norm [8], =-=[15]-=-. We plan to discuss this theory and its applications to the study of sublattices of and elsewhere. In the present paper, we will restrict our attention to a narrow class of sublattices, which, howeve... |

60 |
Redundancy rate distortion analysis of multiple description coding using pairwise correlating transforms
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(Show Context)
Citation Context ...blem. Other approaches to multiple description coding based on overcomplete expansions are presented in [1], [2], [13] and methods based on optimizing transforms and predictors are presented in [16], =-=[21]-=-, [27]. The paper is organized as follows. The source coding problem is formulated in Section II, the design method is described in Section III, properties of the lattices and sublattices needed for t... |

49 |
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Citation Context ...ly waiting for the more important descriptions to arrive (as in successive refinement). For previous work on the information-theoretic aspects of the multiple description problem see [9], [10], [22], =-=[29]-=-, [30]. The problem of designing quantizers for the multiple description problem has been considered in [11], [17], [19], [25], [26], [28]. The work presented in [28] considered only the balanced/symm... |

46 | Graceful degradation over packet erasure channels through forward error correction
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Citation Context ...tion-theoretic aspects of the multiple description problem see [9], [10], [22], [29], [30]. The problem of designing quantizers for the multiple description problem has been considered in [11], [17], =-=[19]-=-, [25], [26], [28]. The work presented in [28] considered only the balanced/symmetric lattice quantizer design. Unlike the work in [11], we do not use a training approach; instead, we use the geometry... |

39 | Quantized frame expansions as sourcechannel codes for erasure channels
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Citation Context ...proach; instead, we use the geometry of the underlying lattice to solve a labeling problem. Other approaches to multiple description coding based on overcomplete expansions are presented in [1], [2], =-=[13]-=- and methods based on optimizing transforms and predictors are presented in [16], [21], [27]. The paper is organized as follows. The source coding problem is formulated in Section II, the design metho... |

37 |
Primes of the Form x + ny
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Citation Context ...dd, positive, and relatively prime, then the sublattice is clean, where is given by (34). Suppose is a product of primes congruent to . From the classical theory of quadratic forms (see, for example, =-=[6]-=-), we know that with even, odd, and . We now simply set and . It remains to discuss the case . For this we can multiply on the left or on the right by either of the quaternions or and denote its rows ... |

31 |
Algebras and Their Arithmetics
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Citation Context ...ry is more complicated. For example, it is necessary to distinguish between left ’s and right ’s. Both are well-defined in and also in as long as at least one of the quaternions involved has odd norm =-=[8]-=-, [15]. We plan to discuss this theory and its applications to the study of sublattices of and elsewhere. In the present paper, we will restrict our attention to a narrow class of sublattices, which, ... |

29 | On the existence of similar sublattices
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- 1999
(Show Context)
Citation Context ...een the lattice and and thus the index represents the scaling between the respective volumes of the fundamental regions of the lattices. We will also usually assume that the sublattices and are clean =-=[3]-=-, that is, no point of lies on the boundary of the Voronoi cells of or . Our algorithm still applies if this condition is not satisfied, but the book-keeping becomes more complicated. Finally, we requ... |

29 |
Generalized multiple description vector quantization
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Citation Context ...ears, multiple description coders have received considerable attention, driven by the interest in packet voice and video communications (see the bibliography). Most of the work (with the exception of =-=[11]-=-) has centered around the successively refinable case and the balanced/symmetric case, which are in a sense two extremes of the distortion profile. Successive refinement coders find application in net... |

27 | Multipledescription vector quantization with lattice codebooks: Design and analysis
- Vaishampayan, Sloane, et al.
- 2001
(Show Context)
Citation Context ...ects of the multiple description problem see [9], [10], [22], [29], [30]. The problem of designing quantizers for the multiple description problem has been considered in [11], [17], [19], [25], [26], =-=[28]-=-. The work presented in [28] considered only the balanced/symmetric lattice quantizer design. Unlike the work in [11], we do not use a training approach; instead, we use the geometry of the underlying... |

25 |
BDPCM system design for diversity systems with applications to packetized speech
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Citation Context ...ng problem. Other approaches to multiple description coding based on overcomplete expansions are presented in [1], [2], [13] and methods based on optimizing transforms and predictors are presented in =-=[16]-=-, [21], [27]. The paper is organized as follows. The source coding problem is formulated in Section II, the design method is described in Section III, properties of the lattices and sublattices needed... |

24 | Multiple description vector quantization with lattice codebooks: design and analysis
- Vaishampayan, Sloane, et al.
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(Show Context)
Citation Context ... work on the information theoretic aspects of the multiple description problem see [9, 10, 22, 29, 30]. The problem of designing quantizers for the multiple description problem has been considered in =-=[11, 17, 19, 25, 26, 28]-=-. The work presented here extends that in [28], which considered only the balanced/symmetric case. Unlike the work in [11], we do not use a training approach; instead we use the geometry of the underl... |

21 | The On-line Encyclopedia of Integer Sequences”. The OEIS Foundation Inc. http://oeis.org - Sloane |

19 | The cell structures of certain lattices
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Citation Context ...e will show that under certain conditions, and are clean sublattices. We only give the proof for , the other case being completely analogous. The Voronoi cell for is a -cell, with 24 octahedral faces =-=[4]-=-, [5]. A typical face (they are all equivalent) is that lying in the hyperplane (31) having center and six vertices (27) The four rows of correspond to the nodes of the Coxeter diagram for shown in Fi... |

16 |
An Introduction to the Theory of Groups, 4th ed
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Citation Context ... their join , as shown in Fig. 7. The join is the lattice generated by the vectors of both and (and, in general, is not simply their union). From the Second Isomorphism Theorem of group theory (e.g., =-=[23]-=-) the indexes and determinants of these lattices are related by (40) (41) There are now in general many ways to find a “product” sublattice with (42) Let be one of , ,or , and let , be geometrically s... |

13 |
BMultiple description trellis coded quantization
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Citation Context ...nformation-theoretic aspects of the multiple description problem see [9], [10], [22], [29], [30]. The problem of designing quantizers for the multiple description problem has been considered in [11], =-=[17]-=-, [19], [25], [26], [28]. The work presented in [28] considered only the balanced/symmetric lattice quantizer design. Unlike the work in [11], we do not use a training approach; instead, we use the ge... |

12 |
Vector quantizer design for diversity systems
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Citation Context ...heoretic aspects of the multiple description problem see [9], [10], [22], [29], [30]. The problem of designing quantizers for the multiple description problem has been considered in [11], [17], [19], =-=[25]-=-, [26], [28]. The work presented in [28] considered only the balanced/symmetric lattice quantizer design. Unlike the work in [11], we do not use a training approach; instead, we use the geometry of th... |

7 |
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Citation Context ...Other approaches to multiple description coding based on overcomplete expansions are presented in [1], [2], [13] and methods based on optimizing transforms and predictors are presented in [16], [21], =-=[27]-=-. The paper is organized as follows. The source coding problem is formulated in Section II, the design method is described in Section III, properties of the lattices and sublattices needed for the con... |

6 |
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Citation Context ...ng approach; instead, we use the geometry of the underlying lattice to solve a labeling problem. Other approaches to multiple description coding based on overcomplete expansions are presented in [1], =-=[2]-=-, [13] and methods based on optimizing transforms and predictors are presented in [16], [21], [27]. The paper is organized as follows. The source coding problem is formulated in Section II, the design... |

5 |
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Citation Context ...raining approach; instead, we use the geometry of the underlying lattice to solve a labeling problem. Other approaches to multiple description coding based on overcomplete expansions are presented in =-=[1]-=-, [2], [13] and methods based on optimizing transforms and predictors are presented in [16], [21], [27]. The paper is organized as follows. The source coding problem is formulated in Section II, the d... |

1 |
A Catalogue of Lattices. [Online]. Available: www.research.att.com/~njas/lattices
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Citation Context ...es and [5]. The analysis could be extended to treat other lattices such as , , the 12-dimensional Coxeter–Todd lattice, the 16-dimensional Barnes–Wall lattice or the 24-dimensional Leech lattice [5], =-=[20]-=-, but we shall not discuss these here. A. The Construction of Similar Sublattices We begin with the observation that multiplication of points in the square lattice (regarded as points in the complex p... |