## Generalized multiple description coding with correlating transforms (2001)

Venue: | IEEE Trans. Inform. Theory |

Citations: | 61 - 2 self |

### BibTeX

@ARTICLE{Goyal01generalizedmultiple,

author = {Vivek K Goyal and Jelena Kovačević and Senior Member},

title = {Generalized multiple description coding with correlating transforms},

journal = {IEEE Trans. Inform. Theory},

year = {2001},

volume = {47},

pages = {2199--2224}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract—Multiple description (MD) coding is source coding in which several descriptions of the source are produced such that various reconstruction qualities are obtained from different subsets of the descriptions. Unlike multiresolution or layered source coding, there is no hierarchy of descriptions; thus, MD coding is suitable for packet erasure channels or networks without priority provisions. Generalizing work by Orchard, Wang, Vaishampayan, and Reibman, a transform-based approach is developed for producing descriptions of an-tuple source,. The descriptions are sets of transform coefficients, and the transform coefficients of different descriptions are correlated so that missing coefficients can be estimated. Several transform optimization results are presented for memoryless Gaussian sources, including a complete solution of the aP, aPcase with arbitrary weighting of the descriptions. The technique is effective only when independent components of the source have differing variances. Numerical studies show that this method performs well at low redundancies, as compared to uniform MD scalar quantization. Index Terms—Erasure channels, integer-to-integer transforms, packet networks, robust source coding.

### Citations

8563 | Elements of Information Theory - Cover, Thomas - 1991 |

4676 |
Matrix Analysis
- Horn, Johnson
- 1985
(Show Context)
Citation Context ...ric. Notice the effect of transposing this matrix. Since and are symmetric, the transpose is ; thus, and commute. Diagonalizable matrices commute if and only if they are simultaneously diagonalizable =-=[57]-=-. Furthermore, for a matrix with distinct eigenvalues, the orthogonal transform that diagonalizes and leaves the diagonal in decreasing order is unique up to sign choices. On one hand, is diagonalized... |

645 | Vector Quantization
- Gray
- 1984
(Show Context)
Citation Context ... performance of optimal fixed-rate MDSQ is and the performance of optimal entropy-constrained MDSQ is (10) The factors in parentheses are familiar from high-resolution quantization theory (see, e.g., =-=[41]-=- and the references therein). These features of high-rate MDSQ, however, tell us very little about how MDSQ performs at low rates. Later, in Section III-E, the performance of MD based on correlating t... |

397 |
Sphere packings, lattices, and groups
- Conway, Sloane
- 1998
(Show Context)
Citation Context ...The uniform scalar quantization of such a representation produces nonsquare partition cells. 11 These partition cells have higher normalized second moments than square cells, and are thus undesirable =-=[44]-=-. The insight attributed to Vaishampayan is that a correlating transform can be applied after quantization has been performed in an orthogonal basis representation. This ensures that the partition cel... |

383 |
Internet Protocol, Version 6 (IPv6) Specification, RFC 1883, Xerox PARC and Ipsilon Networks
- Deering, Hinden
- 1995
(Show Context)
Citation Context ...re is nonzero. The emphasis in this paper is on situations in which long block codes cannot be used, which is often the case. For example, consider a network using Internet Protocol, Version 6 (IPv6) =-=[19]-=-. An IPv6 node is required to handle 576-byte packets without fragmentation, and it is recommended that larger packets be accommodated. 3 Accounting for packet head3 Without the “Jumbo Payload” option... |

263 | Wavelet Transform that Map Integers to Integers
- Calderbank, Daubechies, et al.
- 1998
(Show Context)
Citation Context ...ENDIX I PSEUDOLINEAR DISCRETE TRANSFORMS Invertible discrete-domain to discrete-domain transforms are alternatively described as reversible, integer-to-integer, or invertible in finite precision [53]–=-=[55]-=-. These transforms are based on factorizations of matrices which make information flow in a simple, regular way. Inversion can then be achieved by reversing the information flow. For example, one can ... |

247 |
Design of multiple description scalar quantizers
- Vaishampayan
- 1993
(Show Context)
Citation Context ...alar quantizers to give two descriptions of a scalar source sample, with an additional central decoder that makes use of both descriptions. This approach was expounded and popularized by Vaishampayan =-=[37]-=-, though Reudink introduced very similar methods in earlier unpublished work [38]. The simplest example is to have scalar quantizers with nested thresholds, as shown in Fig. 3(a). Each quantizer outpu... |

238 |
Coding theorems for a discrete source with a fidelity criterion
- Shannon
- 1959
(Show Context)
Citation Context ...bounds to the achievable rate region for MD coding of any continuous-valued memoryless source with squared-error distortion. This is an extension of Shannon’s bounds on rate-distortion functions (see =-=[35]-=-, [60], [36]). C. Practical Codes The bounds of the previous section can be approached when a long sequence of source symbols is coded. In contrast, the focus in this paper is on simple codes for fini... |

193 | Achievable Rates for Multiple Descriptions - Gamal, Cover - 1982 |

190 |
Rate Distortion Theory
- Berger
- 1971
(Show Context)
Citation Context ...e achievable rate region for MD coding of any continuous-valued memoryless source with squared-error distortion. This is an extension of Shannon’s bounds on rate-distortion functions (see [35], [60], =-=[36]-=-). C. Practical Codes The bounds of the previous section can be approached when a long sequence of source symbols is coded. In contrast, the focus in this paper is on simple codes for finite-length so... |

173 | Successive refinement of information
- Equitz, Cover
- 1991
(Show Context)
Citation Context ...; or channel 1 is always present. Successive refinement was first studied by Koshelev [23]–[25]. The conditions for perfect successive refinement—where (1) and (2) hold with equality—are described in =-=[26]-=-. The result follows from the tightness of the achievable region of [2] for the no excess rate sum case [21] (see also [27]). Finally, it should also be noted that there is a substantial literature on... |

137 | Quantized frame expansions with erasures
- Goyal, Kovačević, et al.
(Show Context)
Citation Context ...t. A recent surge of interest in MD coding seems to be due primarily to this application (see [1], [9]–[12]) and yet the present work was among the first to effectively use more than two packets [13]–=-=[17]-=-. Retransmission of lost packets, when feasible, is an effective technique for maintaining constant quality despite losses. The use of a retransmission protocol, such as TCP [18], requires at a minimu... |

136 | A Study of Vector Quantization for Noisy Channels - Farvardin - 1990 |

104 |
Dynamic adaptation in an image trans-coding proxy for mobile WWW browsing
- Han, Bhagwat, et al.
- 1998
(Show Context)
Citation Context ...yless Gaussian Source The achievable rate-distortion region is completely known only for a memoryless Gaussian source with mean-squared error 4 This is not an arbitrary figure. Based on statistics in =-=[20]-=-, the average WWW image is only 3.9 kbytes; an image of this size would not be fragmented into more than eight packets. Of course, this depends on the sampling of the WWW and may change. (MSE) distort... |

88 |
Successive refinement of information: characterization of the achievable rates
- Rimoldi
- 1994
(Show Context)
Citation Context ... successive refinement—where (1) and (2) hold with equality—are described in [26]. The result follows from the tightness of the achievable region of [2] for the no excess rate sum case [21] (see also =-=[27]-=-). Finally, it should also be noted that there is a substantial literature on the error sensitivity of compressed data and more generally on tradeoffs between source and channel coding. The reader is ... |

81 | Implementation of compression with reversible embedded wavelets - Schwartz, Zandi, et al. - 1995 |

80 |
On a Source-Coding Problem with Two Channels and Three Receivers
- Ozarow
- 1980
(Show Context)
Citation Context ...ers (the side decoders) receive information only over their respective channels. The transmission rate over channel is denoted by , ; i.e., signaling over channel uses at most 1 We may infer from [3]–=-=[5]-=- and private communications that the problem originated with Gersho and Witsenhausen. Gersho proposed a scheme (unpublished [6]) that utilized the redundancy in coded speech waveforms to mitigate the ... |

79 |
Multiple description image coding for noisy channels by pairing transform coefficients
- Wang, Orchard, et al.
- 1997
(Show Context)
Citation Context ...term MD correlating transform (MDCT) is used for the present technique. 2) Pairwise Correlating Transforms: A considerably different approach to MD coding was introduced by Wang, Orchard, and Reibman =-=[10]-=-. Instead of using MDSQ to produce two indexes that describe the same quantity, the MD character is achieved with a linear transform that introduces correlation between a pair of random variables; qua... |

68 |
New Results in Binary Multiple Descriptions
- Zhang, Berger
- 1987
(Show Context)
Citation Context ...ound is achieved by using truncated Reed–Solomon codes. A similar result holds in the more general setting of [7]. The situation with three channels and seven decoders was studied by Zhang and Berger =-=[8]-=-. The MD codes constructed in this paper apply to the generalized MD problem. Unfortunately, no tight achievability bounds are known for generalized MD coding. 2 Since the distortion metrics may diffe... |

66 | Tradeoff between source and channel coding on a Gaussian channel
- Hochwald
- 1998
(Show Context)
Citation Context ...ould also be noted that there is a substantial literature on the error sensitivity of compressed data and more generally on tradeoffs between source and channel coding. The reader is referred to [28]–=-=[32]-=- for a sampling of the results. B. Theoretical Bounds for a Memoryless Gaussian Source The achievable rate-distortion region is completely known only for a memoryless Gaussian source with mean-squared... |

65 | Multiple description transform coding of images
- Goyal, Kovacevic, et al.
(Show Context)
Citation Context ...r MD coding; their image coding results appear in [10]. (No image coding results appear in their later papers [1], [11].) Image coding results using the generalized framework presented here appear in =-=[15]-=-. These experiments indicate the expected behavior: at the expense of a slight increase in bit rate, image representations can be made more robust to packet losses. Conventional channel codes adding s... |

60 |
Redundancy rate distortion analysis of multiple description coding using pairwise correlating transforms
- Orchard, Wang, et al.
- 1997
(Show Context)
Citation Context ...source coding is introduced. This is a generalization to more arbitrary transforms of a technique for two channels (or packets, or descriptions) introduced by Orchard, Wang, Vaishampayan, and Reibman =-=[1]-=-. The organization of the paper is as follows. First, MD coding and its connection to packet communication are described in the remainder of this section. Then, in Section II, a few theoretical and pr... |

53 |
Linear Algebra and its applications, 3 rd ed
- Strang
- 1988
(Show Context)
Citation Context ...ersing the calculations in (58) The existence of such a factorization follows from the fact that any nonsingular matrix can be reduced to an identity matrix by multiplication with elementary matrices =-=[56]-=-. Since our original matrix has determinant , it can be represented as the product of the following three types of elementary matrices: • , to subtract a multiple of row from row ; • , to exchange row... |

48 | Source coding for multiple description - Wolf, Wyner, et al. - 1980 |

47 |
Optimal multiple description transform coding of Gaussian vectors
- Goyal, Kova˘cević
- 1998
(Show Context)
Citation Context ...levant. A recent surge of interest in MD coding seems to be due primarily to this application (see [1], [9]–[12]) and yet the present work was among the first to effectively use more than two packets =-=[13]-=-–[17]. Retransmission of lost packets, when feasible, is an effective technique for maintaining constant quality despite losses. The use of a retransmission protocol, such as TCP [18], requires at a m... |

47 | The rate-distortion region for multiple descriptions without excess rate
- Ahlswede
- 1985
(Show Context)
Citation Context ...te region later published in [2]. Ozarow’s contribution was to show the tightness of the El Gamal–Cover region for memoryless Gaussian sources and squared-error distortion [5]. Subsequently, Ahlswede =-=[21]-=- showed that the El Gamal–Cover region is tight in the “no excess rate sum” case (where there is equality in (1)), and Zhang and Berger [8] showed that this region is not tight when there is excess ra... |

45 | Multiple description transform coding: robustness to erasures using tight frame expansions
- Goyal, Kovačević, et al.
- 1998
(Show Context)
Citation Context ...Vaishampayan use the term MD transform coding (MDTC) to refer to MDSQ applied to coefficients from an orthogonal transform. The same term could be used to describe the techniques developed here or in =-=[14]-=-, [16], [17]; however, to avoid confusion the term MD correlating transform (MDCT) is used for the present technique. 2) Pairwise Correlating Transforms: A considerably different approach to MD coding... |

44 | Self-synchronizing huffman codes - Ferguson, Rabinowitz - 1984 |

42 | Linear Algebra and Its Applications, Third Edition - Strang - 1988 |

39 | Quantized frame expansions as sourcechannel codes for erasure channels
- Goyal, Kovacevic, et al.
- 1999
(Show Context)
Citation Context ...mpayan use the term MD transform coding (MDTC) to refer to MDSQ applied to coefficients from an orthogonal transform. The same term could be used to describe the techniques developed here or in [14], =-=[16]-=-, [17]; however, to avoid confusion the term MD correlating transform (MDCT) is used for the present technique. 2) Pairwise Correlating Transforms: A considerably different approach to MD coding was i... |

29 | Gaussian codes and Shannon bounds for multiple descriptions
- Zamir
- 1999
(Show Context)
Citation Context ...ch more impact if dedicated to reducing the side distortion than if dedicated to reducing the central distortion. The rate-distortion region is not completely known for any non-Gaussian source. Zamir =-=[34]-=- has found inner and outer bounds to the achievable rate region for MD coding of any continuous-valued memoryless source with squared-error distortion. This is an extension of Shannon’s bounds on rate... |

27 |
Design of entropyconstrained multiple-description scalar quantizers
- Vaishampayan, Domaszewicz
- 1994
(Show Context)
Citation Context ..., the central distortion is smaller—at the cost of higher side distortions—than for an MDSQ as in Fig. 3(a). The optimal design for fixed-rate MDSQ in [37] was extended to entropy-constrained MDSQ in =-=[39]-=-. One of the most satisfying aspects of the theory of MDSQ is that at high rates, all the decay exponent tradeoffs discussed in Section II-B can be obtained. Furthermore, the factor by which the disto... |

27 | Optimal pairwise correlating transforms for multiple description coding - Wang, Orchard, et al. - 1998 |

25 |
DPCM system design for diversity systems with applications to packetized speech
- Ingle, Vaishampayan
- 1995
(Show Context)
Citation Context ...ain the analogy with the classical MD problem and because the source coding in that case is irrelevant. A recent surge of interest in MD coding seems to be due primarily to this application (see [1], =-=[9]-=-–[12]) and yet the present work was among the first to effectively use more than two packets [13]–[17]. Retransmission of lost packets, when feasible, is an effective technique for maintaining constan... |

25 | Error recovery for variables length codes - Maxted, Robinson - 1985 |

25 | Sphere Packing, Lattices and Groups, volume 290 of Grundlehren der math - Conway, Sloane - 1988 |

24 | Asymptotic analysis of multiple description quantizers
- Vaishampayan, Batllo
- 1998
(Show Context)
Citation Context ...gh rates, all the decay exponent tradeoffs discussed in Section II-B can be obtained. Furthermore, the factor by which the distortion product exceeds the bounds of Theorem 1 is approximately constant =-=[40]-=-. For and , the bound for large rates is approximately 6 There is no “central encoder,” but and effectively implement a quantizer with cells given by the intersections of the cells of the individual q... |

24 | Asymptotic performance of multiple description transform codes
- Batllo, Vaishampayan
- 1997
(Show Context)
Citation Context ...in providing useful operating points for low base rates and redundancies. Applications of MDSQ are described in [9], [42], [12]. Joint optimization of an orthogonal transform and MDSQ is discussed in =-=[43]-=-. For the purposes of this paper, it is sufficient to note that an identity transform is optimal for a source vector with independent components. Note that Batllo and Vaishampayan use the term MD tran... |

24 | Transform coding with backward adaptive updates
- Goyal, Zhuang, et al.
- 2000
(Show Context)
Citation Context ... justified by high-resolution analysis. As long as the rate allocated to the lower variance component is at least about 1 bit/sample, an optimal rate allocation only negligibly reduces the distortion =-=[45]-=-. Fig. 6. Correlating transform structure for MD coding of a two-tuple source. and then computing the transform with intermediate roundings (15) The properties of are described in Appendix I. Though n... |

24 |
Capacity definitions and coding strategies for general channels with receiver side information
- Effros, Goldsmith
- 1998
(Show Context)
Citation Context ... For this reason, “description,” “channel,” and “packet” have been used interchangeably. However, this is not the only communication environment in which MD coding may be useful. Effros and Goldsmith =-=[52]-=- have studied the various notions of capacity for general time-varying channels. One of their results is that more information can be reliably received than can be reliably transmitted. With some thou... |

23 | Communication Networks: A First Course - Walrand - 1991 |

21 |
Multiple Description Source Coding with No Excess Marginal Rate
- Zhang, Berger
- 1995
(Show Context)
Citation Context ...region is not tight when there is excess rate. The complementary situation, where (2) and (3) hold with equality, is called the “no excess marginal rate” case and was also studied by Zhang and Berger =-=[22]-=-. MD coding includes as a special case the better known successive refinement or multiresolution coding. The successive refinement problem can also be described by Fig. 1, but the interest is only in ... |

18 |
On source networks with minimal breakdown degradation,” Bell Syst
- Witsenhausen
- 1980
(Show Context)
Citation Context ...ecoders (the side decoders) receive information only over their respective channels. The transmission rate over channel is denoted by , ; i.e., signaling over channel uses at most 1 We may infer from =-=[3]-=-–[5] and private communications that the problem originated with Gersho and Witsenhausen. Gersho proposed a scheme (unpublished [6]) that utilized the redundancy in coded speech waveforms to mitigate ... |

18 |
Low-delay communication for Rayleigh fading channels: an application of the multiple description quantizer
- Yang, Vaishampayan
- 1995
(Show Context)
Citation Context ...perating points. (a) Central distortion h aP . (b) Central distortion h aP . UMDSQ in providing useful operating points for low base rates and redundancies. Applications of MDSQ are described in [9], =-=[42]-=-, [12]. Joint optimization of an orthogonal transform and MDSQ is discussed in [43]. For the purposes of this paper, it is sufficient to note that an identity transform is optimal for a source vector ... |

18 | Beyond traditional transform coding - Goyal - 1998 |

15 | Multiple-description wavelet based image coding
- Servetto, Ramchandran, et al.
- 1998
(Show Context)
Citation Context ...the analogy with the classical MD problem and because the source coding in that case is irrelevant. A recent surge of interest in MD coding seems to be due primarily to this application (see [1], [9]–=-=[12]-=-) and yet the present work was among the first to effectively use more than two packets [13]–[17]. Retransmission of lost packets, when feasible, is an effective technique for maintaining constant qua... |

13 | To compress or not to compress - Buch, Burkert, et al. - 1996 |

13 |
Single and Multiple Description Transform Coding with Bases and Frames
- Goyal
- 2001
(Show Context)
Citation Context ... increase in the rate of decay of . The second way to interpret the tradeoff between central and side distortion begins with rearranging (5), under the assumption of equal side distortion , to obtain =-=[33]-=- Now separate the coding rate into a base rate and a redundancy . The base rate corresponds to the quality of the joint description and the redundancy is the additional rate added to make the side dis... |

13 | Multiple description perceptual audio coding with correlating transforms
- Arean, Kovacevic, et al.
- 2000
(Show Context)
Citation Context ...tual Audio Coder (PAC) [50]. Experiments confirm that the new MD audio coder is robust and gracefully degrades as the number of lost packets increases. The implementation and results are described in =-=[51]-=-. Since the results cannot be adequately conveyed numerically or visually, the reader is invited to hear them. Audio files are provided on-line in aiff, wave, and next formats at http: //cm.bell-labs.... |

12 |
The perceptual audio coder
- Sinha, Johnston, et al.
- 1998
(Show Context)
Citation Context ...ith one or two components lost. Removing restrictions further by allowing a general R 2 R transform allows a further decrease in two-erasure distortion. coder: Bell Labs’ Perceptual Audio Coder (PAC) =-=[50]-=-. Experiments confirm that the new MD audio coder is robust and gracefully degrades as the number of lost packets increases. The implementation and results are described in [51]. Since the results can... |