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Theorem: Let M be an embedded IA hypermatrix, and the corresponding central quantiser for the highest rate ( p ¼ 0) be uniform. The central quantisers are uniform at all quantisation levels ( p, 0 p P) if and only if for any p in each hyperblock Bpj1 ... jM 6¼ [0] (1 jm Jmp, 1 m M) a constant number of consecutive nonzero indices are mapped. Proof: (if) If, for any p, consecutive indices are mapped in any hyperblock Bpj1 ... jM 6¼ [0], then the corresponding central quantiser * is the cells are connected cells of size nnz(Bpj1 ... jM ) C, where C 2 Rþ central description’s constant partition size for level p ¼ 0. By assumption, the number of nonzero elements for all Bpj1 ... jM 6¼ [0] at each p is constant: nnz(Bjp1 ... jM ) ¼ nnz(Bpi1 ... iM ), 8 jm, im, 1 jm, im Jmp for any [Bpj1 ... jM ] 6¼ [0], [Bip1 ... iM ] 6¼ [0] and 0 p P. Hence, the central quantiser at any level p is uniform with the cell size given by p ( p) C ¼ nnz(Bj1 ... jM )C. (only if) Assume by contradiction that there exists Bpj1 ... jM 6¼ [0] for which the indices mapped in Bpj1 ... jM are not consecutive. This implies that the corresponding central quantiser cell is disconnected, which contradicts the assumption that the central quantiser is uniform at any p.
ELECTRONICS LETTERS 20th January 2005
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Generating embedded index assignment (IA): Consider an Mdimensional IA matrix M in which a number of N elements are mapped. Recursively splitting the hypermatrix M along each dimension m, 1 m M for a number of P levels defines M as a block hypermatrix of the form: M ¼ [Bpj1 ... jM ]1jm jmp, 8p, 0 p P (Jmp represents the number of block elements at level p for the dimension m). Such an M is referred to as a generalised embedded IA hypermatrix. Consequently, a set of embedded side quantisers Q0m, Q1m, . . . , QPm, and the corresponding set of embedded central quantisers Q0, Q1, . . . , QP are associated with such an embedded IA. For any p, p < P, the cells of any quantiser Qmp and Q p are embedded in the cells of quantisers QmP, . . . , Qmpþ1 and QP, . . . , Q pþ1, respectively. Notice that a number of Jmp cells are contained in Qmp. As previously indicated in the literature on multiple description scalar quantisation, an entropy-constrained uniform quantiser is verynearly optimal [4]. For embedded quantisation, a notable example where all embedded quantisers can be optimal is the uniform case [5]. The theorem hereunder sets the conditions to be satisfied by the generalised embedded IA to yield embedded uniform central quantisers. Denote by [0] the zero hypermatrix, and define the operator nnz(M) determining the number of nonzero elements contained in a hypermatrix M.
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Introduction: Multiple description coding (MDC) has now become an appropriate approach for efficient communication over error-prone channels, e.g. packet networks and low-power wireless links. Additionally, data transmission over variable-bandwidth channels requires the source adaptation to the available bit-rate and user’s needs; hence, a fine-grain scalability of each description is essential. In this context, to provide an arbitrary number of fine-grain refinable descriptions, we have previously proposed embedded multiple description scalar quantisers (EMDSQs), which provide state-of-the-art results, as demonstrated in [1–3]. The major advantage of EMDSQs is that both the side and central quantisers are embedded, and that the central quantisers are uniform at each quantisation level. In this Letter a condition to design generalised EMDSQs is proposed. All the EMDSQs of [1–3] are seen as instantiations of the proposed framework. Additionally, besides handling the overall redundancy, advanced redundancy tuning between the descriptions at every distinct quantisation level is achieved via a novel redundancy control mechanism.
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In the framework of multiple description coding a condition to design generalised embedded multiple description scalar quantisers yielding embedded uniform central quantisers is proposed. In addition, a control mechanism that enables redundancy tuning between the descriptions at every distinct quantisation level is designed.
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A.I. Gavrilescu, A. Munteanu, J. Cornelis and P. Schelkens
Additionally, for any Bpj1 ... jM 6¼ [0], the size of the central quantiser cell is given by nnz(Bpj1 ... jM ) C. The assumption that the central quantiser is uniform at every level p implies that nnz(Bpj1 ... jM ) ¼ const, for all [Bpj1 ... jM ] 6¼ [0] with 1 jm Jmp, 1 m M. It can be easily verified that the embedded IAs corresponding to the EMDSQs we proposed in [1–3] satisfy the conditions of the above theorem; hence, they represent instantiations of the proposed framework. Consider that any of the hyperblocks Bpj1þ1 ... jM from level p þ 1 dimension m resulting, are split into the same number Lpm along the QM at the lower level p, in a number Lp ¼ m¼1Lpm of hyperblocks hypermatrix of the form: Bpq1 ... qM . That is, Bpj1þ1 ... jM is a block p pþ1 P P ¼ [B ] p p , 1 jm J m . Therefore, if Jm ¼ Lm , then p Bpj1þ1 1q L ... jM m q1 ... qM QP m p P p Qm will contain Jm ¼ i¼p Lm cells. The recursive splitting of a bidimensional M is illustrated in Fig. 1.
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Generalisation of embedded multiple description scalar quantisers
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Fig. 1 Recursive block-matrix decomposition
Consider a block hypermatrix M of the form: M ¼ [Bpj1 ... jM ]1jm jmp. The number of blocks Bpj1 ... jM 6¼ [0] contained in M is determined via the nonzero-blocks operator nzb(M, p). It is noticeable that nzb(M, 0) ¼ nnz(M). Pursuant to this definition and based on the above theorem, for any of pþ1 p the block matrices Bpj1þ1 ... jM ¼ [Bq1 ... qM ]1qm Lpm, 1 jm J m , the number of blocks Bpq1 ... qM 6¼ [0] is constant and given by Np ¼ nzb(Bpj1þ1 ... jM , p). Additionally, at level P, NP ¼ nzb(M, P) represents the number of hyperblocks Bpj1 ... jM 6¼ [0] within M. The total number of indices mappedQ in each hyperblock Bpj1 ... jM at level p is then nnz(Bpj1 ... jM ) ¼ pi¼0 Ni. It can be subsequently shown that the analytical expression of the central EMDSQ for any level p, 0 p P is: jxj ð1Þ Q p ðxÞ ¼ signðxÞ Qp C i¼0 Ni QP The number of cells contained in Q p is i¼p Np. Redundancy control: Denote by Rm the rates and by Dm(Rm) the corresponding side-description distortions. Also, denote by D0 the central distortion. In single-description coding (SDC) one minimises D0 for a given rate R0. The redundancy is the bit-rate sacrificed by MDC compared to SDC in order to achieve the same central D0 distortion: r¼
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Q QP Ni). From (2) we In our case, Rm ¼ log2( Pi¼pLim) and R0 ¼ log2( i¼p obtain the normalised redundancy: PM QP i log L 2 m¼1 i¼p m rp ¼ 1 ð3Þ QP log2 i¼p Ni We can conclude that for any EMDSQ instantiation, the redundancy is directly dependent on the quantisation level and is controlled by changing the ratio Np=Lp (Np Lp). Fig. 2 shows variations of the redundancy against the level p (as expressed by (3)) for two descriptions and practical instantiations of the Lp and Np parameters.
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Acknowledgments: This work was supported by the Federal Office for Scientific, Technical and Cultural Affairs (IAP-V, Mobile Multimedia). P. Schelkens and A. Munteanu have post-doctoral fellowships with the Fund for Scientific Research, Flanders (FWO). Note: All papers by A.I. Gavrilescu et al. may be downloaded from the URL: www.etro.vub.ac.be=Members=gavrilescu.augustin=publications.html # IEE 2005 Electronics Letters online no: 20057118 doi: 10.1049/el:20057118
22 September 2004
Fig. 2 Redundancy r against quantisation level p, 0 p 3 for Lp ¼ 9 and L3 ¼ 4, L2 ¼ 9, L1 ¼ 16, L0 ¼ 25
A.I. Gavrilescu, A. Munteanu, J. Cornelis and P. Schelkens (Department of Electronics and Information Processing, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium)
a Lp ¼ 9 b L3 ¼ 4, L2 ¼ 9, L1 ¼ 16, L0 ¼ 25
E-mail:
[email protected] References
Experimental results: To demonstrate the redundancy-control mechanism we assess the rate-distortion behaviour for several EMDSQ instantiations where we decrease the redundancy between the descriptions after a certain quantisation level. Performance results for the central description EMDSQ instantiations of [3] operating on a memoryless, unit variance Laplacian source are shown in Fig. 3a. Fig. 3b shows the rate-distortion behaviour obtained with a practical multiple description QuadTree (MD-QT) wavelet-based coding system (see [6]), incorporating EMDSQ instantiations. Based on these results, we conclude that it is possible (i) to tune the redundancy at each level, hence to control the trade-off between coding efficiency and resilience to errors, and (ii) to improve the resilience by increasing the redundancy in the important layers of the bit-stream.
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Gavrilescu, A.I., et al.: ‘Embedded multiple description scalar quantisers’, Electron. Lett., 2003, 39, (13), pp. 979–980 Gavrilescu, A.I., et al.: ‘High-redundancy embedded multipledescription scalar quantizers for robust communication over unreliable channels’. Proc. WIAMIS’04, Lisbon, Portugal, 2004, CD-version cr1073.pdf Gavrilescu, A.I., et al.: ‘A new family of embedded multiple description scalar quantizers’. Proc. IEEE ICIP’04, Singapore, 2004 (accepted) Vaishampayan, V.A., and Domaszewicz, J.: ‘Design of entropyconstrained multiple description scalar quantizers’, IEEE Trans. Inf. Theory, 1994, 40, (1), pp. 245–250 Taubman, D., and Marcelin, M.W.: ‘JPEG2000: Image compression fundamentals, standards, and practice’ (Kluwer Academic Publishers, 2002) Gavrilescu, A.I., et al.: ‘Embedded multiple description scalar quantizers and wavelet-based quadtree coding for progressive image transmission over unreliable channels’. Proc. EUSIPCO’04, Vienna, Austria, 2004, Vol. 1, pp. 645–648
Fig. 3 Comparative central rate-distortion performance a For EMDSQ at different overall redundancies applied on a Laplacian source b Lena image coded with MD-QT codec of [6] employing EMDSQ at different overall redundancies
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