T. Linder, R. Zamir, and K. Zeger, "High-resolution source coding for nondifference distortion measures: Multidimensional companding," IEEE Trans. Inform. Theory, vol. 45, pp. 548--561, Mar. 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....vol. XVII, pp. 70 75, 1997. 13] A. N. Haroutunian, Investigation of achievable interdependence between coding rates and reliability for several classes of sources, Thesis of Kandidat of Science (in Russian) Inst. Inform. Automation Problems of the NAS of RA and of YSU, Yerevan, Nov. 1997. [14] E. A. Haroutunian, A. N. Haroutunian, and A. R. Kazarian, On ratereliabilities distortions function of source with many receivers, in Proc. Joint Session of 6th Prague Symp. Asymptotic Statistics and 13th Prague Conf. Information Theory, Statistical Decision Functions and Random Processes, ....

....of this scheme will be arbitrarily close to the upper bound of Theorem 2. Under the conditions for asymptotic tightness in Theorem 2, the coding rate will asymptotically achieve (D) for small distortions. A rigorous proof of both of these claims can be given using the techniques developed in [14]. APPENDIX A Lemma 1: Let ND denote a zero mean Gaussian random variable with variance D which is independent of (X; Y ) IfEE E[X ] h(X) and EE E[m(X;Y ) are finite, then Y h(X j Y ) A.1) Proof: By [8, Appendix B] if n : 0; 1) is such that EE E[n(X) h(X (ND=n(X) ....

T. Linder, R. Zamir, and K. Zeger, "High-resolution source coding for nondifference distortion measures: Multidimensional companding," IEEE Trans. Inform. Theory, vol. 45, pp. 548--561, Mar. 1999.

.... small then the behavior of d(x; y; x) is determined by the second order term in its Taylor expansion with respect to x around (x; y; x) That is, o(jx 0 xj ) 3) as jx 0 xj 0, where This definition generalizes the notion of locally quadratic input weighted distortion measures [8] to side information dependent distortion measures. Notice, however, that the dependence on the side information is only through the coefficient of the quadratic term; the optimum reconstruction given x and y is still x, independent of the side information y. Locally quadratic input weighted ....

....and for smooth sources RX j Y (D) h(X j Y )0 1 2 log(2eD) 1 EE E[log m(X;Y ) o(1) 4) where o(1) 0 as D 0, and where h(X j Y ) denotes the conditional differential entropy of the source given the side information. This result generalizes a recently derived asymptotic formula [8] for the rate distortion function of a smooth source relative to locally quadratic nondifference distortion measures, to conditional rate distortion functions and to distortion measures which depend on the side information. In contrast, determining the asymptotics of R (D) for distortion ....

[Article contains additional citation context not shown here]

T. Linder and R. Zamir, "High-resolution source coding for nondifference distortion measures: The rate distortion function," IEEE Trans. Inform. Theory, vol. 45, pp. 533--547, Mar. 1999.

.... and 2) the behavior of image coding is not well understood for so called non difference distortion measures where the distortion is not measured by a function of the difference between the source and the decoded images, however non difference distortion measures occur naturally in image coding, [8], 9] In this paper, an alternative criterion for coder selection is proposed as follows. It often happens that the structure of the images cannot be determined exactly due to various reasons (e.g, it is possible that some of the details may not be observable or the observer who makes an attempt ....

Linder, T., Zamir, R., Zeger, K. "High-resolution source coding for non-difference distortion measures: multidimensional companding," IEEE Trans. on Information Theory, Vol. 45, No. 2, pp. 548-561, (1999).

....the statistics at the quantizer output, and in case of mismatch between the entropy coder and the those statistics, it is no longer true that linear companding is best. A natural extension of the scalar compander quantizer is the multi dimensional one (see, e.g. 3] 4] and more recent studies [7], 8] It turns out that for memoryless sources and additive distortion measures, there is no interaction between the different dimensions, and hence optimal multi dimensional companding simply breaks to separate scalar compandings in each dimension. With this background on companding and its ....

T. Linder, R. Zamir, and K. Zeger, "High-resolution source coding for non-difference distortion measures: multidimensional companding," preprint 1997.

.... have proved useful in perceptual coding, the input weighted quadratic distortion measures of the form d(x; x) x Gamma x) t W x (x Gamma x) 21) where W x is a positive definite matrix that depends on the input, cf. 258] 259] 257] 224] 387] 386] 150] 186] 316] 323] [325]. Most of the theory and design techniques considered here extend to such measures, as will be discussed later. We also assume that d(x; x) 0 if and only if x = x, an assumption that involves no genuine loss of generality and allows us to consider a lossless code as a code for which d(x; ....

.... results for Shannon lower bounds to the ratedistortion function have been developed for this family of distortion measures by Linder and Zamir [323] and results for multidimensional companding with lattice codes for similar distortion measures have been developed by Linder, Zamir, and Zeger [325]. H. Rigorous Approaches to High Resolution Theory Over the years, high resolution analyses have been presented in several styles. Informal analyses of distortion, such as those used in this paper to obtain Delta 2 =12 and Bennett s integral (26) generally ignore overload distortion and ....

[Article contains additional citation context not shown here]

T. Linder, R. Zamir, K. Zeger "High resolution source coding for non-difference distortion measures: multidimensional companding, " submitted to IEEE Trans. Inform. Theory.

.... that have proved useful in perceptual coding, the input weighted quadratic distortion measures of the form d(x; x) x Gamma x) t W x (x Gamma x) 21) where W x is a positive definite matrix that depends on the input, cf. 258] 259] 257] 224] 387] 386] 150] 186] 316] [323], 325] Most of the theory and design techniques considered here extend to such measures, as will be discussed later. We also assume that d(x; x) 0 if and only if x = x, an assumption that involves no genuine loss of generality and allows us to consider a lossless code as a code for which ....

....since then det(B(x) 1. Note in particular that the optimal point density for the entropy constrained case is not in general a uniform density. Parallel results for Shannon lower bounds to the ratedistortion function have been developed for this family of distortion measures by Linder and Zamir [323] and results for multidimensional companding with lattice codes for similar distortion measures have been developed by Linder, Zamir, and Zeger [325] H. Rigorous Approaches to High Resolution Theory Over the years, high resolution analyses have been presented in several styles. Informal ....

[Article contains additional citation context not shown here]

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: The rate distortion function, " Proc. 1997 IEEE Int'l Symp. Inform. Theory, Ulm, Germany, p. 187, June 1997. Also submitted to IEEE Trans. Inform. Theory.

....generalized to a class of distortion measures which have proved useful in perceptual coding, the input weighted quadratic distortion measures of the form d(x; x) x Gamma x) t B x (x Gamma x) 18) where B x is a nonnegative definite matrix that depends on the input. See, for example, [276, 275, 115, 141, 227, 234, 233]. Most of the theory and design techniques considered here extend to such measures. We also assume that d(x; x) 0 if and only if x = x, an assumption that involves no genuine loss of generality and allows us to consider a lossless code as a code for which d(x; fi(ff(x) 0 for all inputs ....

.... integral has been extended to this type of distortion and approximations for both fixed rate and variable rate operational distortion rate functions have been developed [141, 227] Parallel results for Shannon lower bounds to the rate distortion function have been developed by Linder and Zamir [234] and similar results for multidimensional companding with lattice codes for similar distortion measures have been developed by Linder, Zamir, and Zeger [233] Using the Bennett high resolution approximations as described for the squared error case, it can be shown for the fixed rate case that ....

[Article contains additional citation context not shown here]

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: The rate distortion function," Proc. 1997 IEEE International Symposium on Information Theory, Ulm, Germany, p. 187, June 1997. Also submitted to IEEE Trans. Information Theory.

....generalized to a class of distortion measures which have proved useful in perceptual coding, the input weighted quadratic distortion measures of the form d(x; x) x Gamma x) t B x (x Gamma x) 18) where B x is a nonnegative definite matrix that depends on the input. See, for example, [276, 275, 115, 141, 227, 234, 233]. Most of the theory and design techniques considered here extend to such measures. We also assume that d(x; x) 0 if and only if x = x, an assumption that involves no genuine loss of generality and allows us to consider a lossless code as a code for which d(x; fi(ff(x) 0 for all inputs ....

.... been developed [141, 227] Parallel results for Shannon lower bounds to the rate distortion function have been developed by Linder and Zamir [234] and similar results for multidimensional companding with lattice codes for similar distortion measures have been developed by Linder, Zamir, and Zeger [233]. Using the Bennett high resolution approximations as described for the squared error case, it can be shown for the fixed rate case that D(q) k k 2 C Gamma 2 k k N Gamma 2 k aeZ [f k (x) det(B(x) 1 k ] 1 1 2=k dx oe k 2 k (41) with equality if the point density is ....

[Article contains additional citation context not shown here]

T. Linder, R. Zamir, K. Zeger "High resolution source coding for non-differencedistortion measures: multidimensionalcompanding," submitted to IEEE Trans. Information Theory.

.... results for the same distortion measure by Linder and Zamir [13] on Shannon lower bounds to the rate distortion function (which provide an approximation to the rate distortion function for asymptotically small distortion, corresponding to our asymptotically high rate) and Linder, Zamir, and Zeger [14] on multidimensional companding with lattice codes for similar distortion measures. II Preliminaries Let X be a k dimensional random vector taking sample values x as described by a joint probability density function p(x) where x = x 1 ; x k ) 2 k , k dimensional Euclidean space. ....

T. Linder, R. Zamir, and K. Zeger, "High-resolution Source Coding for Non-difference Distortion Measures: Multidimensional Companding," submitted to IEEE Transactions on Information Theory.

....[10] Hence the notation parallels that of [11] and [10] to facilitate reference. We note that the results generalizing the Bennett integral to input dependent quadratic distortion measures complement and are consistent with recent results for the same distortion measure by Linder and Zamir [13] on Shannon lower bounds to the rate distortion function (which provide an approximation to the rate distortion function for asymptotically small distortion, corresponding to our asymptotically high rate) and Linder, Zamir, and Zeger [14] on multidimensional companding with lattice codes for ....

T. Linder and R. Zamir, "High-resolution Source Coding for Non-difference Distortion Measures: the Rate Distortion Function," submitted to IEEE Transactions on Information Theory.

....this scheme will be arbitrarily close to the upper bound of Theorem 2. Under the conditions for asymptotic tightness in Theorem 2, the coding rate will asymptotically achieve R WZ (D) for small distortions. A rigorous proof of both of these claims can be given using the techniques developed in [14]. Appendix A Lemma 1 Let ND denote a zero mean Gaussian random variable with variance D which is independent of (X; Y ) If E[X 2 ] h(X) and E[m(X; Y ) Gamma1 ] are finite, then 12 lim sup D 0 h i X ND p m(X;Y ) fi fi fi Y j h(XjY ) A.1) Proof. By [8, Appendix B] if n : ....

T. Linder, R. Zamir, and K. Zeger, "High-resolution source coding for non-difference distortion measures: multidimensional companding," IEEE Trans. Inform. Theory, vol. IT-45, pp. 548--561, Mar. 1999.

.... with respect to x around (x; y; x) That is, d(x; y; x) m(x; y) x Gamma x) 2 o(jx Gamma xj 2 ) 3) as jx Gamma xj 0, where m(x; y) 1 2 2 d(x; y; x) x 2 fi fi fi x=x : This definition generalizes the notion of locally quadratic input weighted distortion measures [8] to side information dependent distortion measures. Notice, however, that the dependence on the side information is only through the coefficient of the quadratic term; the optimum reconstruction given x and y is still x, independent of the side information y. Locally quadratic input weighted ....

....for smooth sources, RXjY (D) h(XjY ) Gamma 1 2 log(2eD) 1 2 E Theta log m(X;Y ) o(1) 4) where o(1) 0 as D 0, and where h(XjY ) denotes the conditional differential entropy of the source given the side information. This result generalizes a recently derived asymptotic formula [8] for the rate distortion function of a smooth source relative to locally quadratic non difference distortion measures, to conditional rate distortion functions and to distortion measures which depend on the side information. In contrast, determining the asymptotics of R WZ (D) for distortion ....

[Article contains additional citation context not shown here]

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: the rate distortion function," IEEE Trans. Inform. Theory, vol. IT-45, pp. 533--547, Mar. 1999.

....treatment of the same lower bound is given in [8] and a new lower bound on the variable rate (i.e. entropy coded) performance is developed using optimal point densities. It is also pointed out in [8] that some important perceptual distortion measures in image coding are locally quadratic. In [12], an asymptotically tight expression for the rate distortion function is derived for locally quadratic distortion measures. As will be shown in this paper, the expression given in [12] plays the same important role in high resolution quantization for these distortion measures as does the Shannon ....

....pointed out in [8] that some important perceptual distortion measures in image coding are locally quadratic. In [12] an asymptotically tight expression for the rate distortion function is derived for locally quadratic distortion measures. As will be shown in this paper, the expression given in [12] plays the same important role in high resolution quantization for these distortion measures as does the Shannon lower bound in quantizing for squared error loss. To develop the basics of a high resolution quantization theory for locally quadratic distortion measures, we investigate variable rate ....

[Article contains additional citation context not shown here]

T. Linder and R. Zamir, "High-resolution source coding for nondifference distortion measures: The rate distortion function," this issue, pp. 000--000.

....this scheme will be arbitrarily close to the upper bound of Theorem 2. Under the conditions for asymptotic tightness in Theorem 2, the coding rate will asymptotically achieve R WZ (D) for small distortions. A rigorous proof of both of these claims can be given using the techniques developed in [14]. Appendix A Lemma 1 Let ND denote a zero mean Gaussian random variable with variance D which is independent of (X; Y ) If E[X 2 ] h(X) and E[m(X; Y ) Gamma1 ] are finite, then lim sup D 0 h i X ND p m(X;Y ) fi fi fi Y j h(XjY ) A.1) Proof. By [8, Appendix B] if n : R ....

T. Linder, R. Zamir, and K. Zeger, "High-resolution source coding for non-difference distortion measures: multidimensional companding," IEEE Trans. Inform. Theory, vol. IT-45, pp. 548--561, Mar. 1999.

.... with respect to x around (x; y; x) That is, d(x; y; x) m(x; y) x Gamma x) 2 o(jx Gamma xj 2 ) 3) as jx Gamma xj 0, where m(x; y) 1 2 2 d(x; y; x) x 2 fi fi fi x=x : This definition generalizes the notion of locally quadratic input weighted distortion measures [8] to side information dependent distortion measures. Notice, however, that the dependence on the side information is only through the coefficient of the quadratic term; the optimum reconstruction given x and y is still x, independent of the side information y. Locally quadratic input weighted ....

....for smooth sources, RXjY (D) h(XjY ) Gamma 1 2 log(2eD) 1 2 E Theta log m(X;Y ) o(1) 4) where o(1) 0 as D 0, and where h(XjY ) denotes the conditional differential entropy of the source given the side information. This result generalizes a recently derived asymptotic formula [8] for the rate distortion function of a smooth source relative to locally quadratic non difference distortion measures, to conditional rate distortion functions and to distortion measures which depend on the side information. In contrast, determining the asymptotics of R WZ (D) for distortion ....

[Article contains additional citation context not shown here]

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: the rate distortion function," IEEE Trans. Inform. Theory, vol. IT-45, pp. 533--547, Mar. 1999.

....The proof of Theorem 6 is given in two parts. First we show in Lemma 3 that the right hand side of (42) is an asymptotic lower bound on RX n(D) Then a matching upper bound is proved in Lemma 4. Our method of proof is based partially on [26] but with the help of techniques developed in [25] and [28], we have managed to give simpler proofs of more general results. 27 Lemma 3 Assume X n is of the mixture form (40) and conditions (a) and (b) hold. Let = 1 n P N j=1 ff j c j . Then we have lim inf D 0 RX n(D) 2 log(2eD= 1 n H(V ) 1 n N X j=1 ff j H( b X (j) ....

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: the rate distortion function." IEEE Trans. Inform. Theory under revision, 1998.

....this scheme will be arbitrarily close to the upper bound of Theorem 2. Under the conditions for asymptotic tightness in Theorem 2, the coding rate will asymptotically achieve R WZ (D) for small distortions. A rigorous proof of both of these claims can be given using the techniques developed in [14]. Appendix A Lemma 1 Let ND denote a zero mean Gaussian random variable with variance D which is independent of (X; Y ) If E[X 2 ] h(X) and E[m(X; Y ) 1 ] are finite, then lim sup D 0 h X ND p m(X;Y ) Y h(XjY ) A.1) Proof. By [8, Appendix B] if n : R [0; 1) is ....

T. Linder, R. Zamir, and K. Zeger, "High-resolution source coding for non-difference distortion measures: multidimensional companding," IEEE Trans. Inform. Theory, vol. IT-45, pp. 548--561, Mar. 1999.

.... Taylor expansion with respect to x around (x; y; x) That is, d(x; y; x) m(x; y) x x) 2 o(jx xj 2 ) 3) as jx xj 0, where m(x; y) 1 2 2 d(x; y; x) x 2 x=x : This definition generalizes the notion of locally quadratic input weighted distortion measures [8] to side information dependent distortion measures. Notice, however, that the dependence on the side information is only through the coefficient of the quadratic term; the optimum reconstruction given x and y is still x, independent of the side information y. Locally quadratic input weighted ....

....and for smooth sources, RXjY (D) h(XjY ) 1 2 log(2 eD) 1 2 E log m(X;Y ) o(1) 4) where o(1) 0 as D 0, and where h(XjY ) denotes the conditional differential entropy of the source given the side information. This result generalizes a recently derived asymptotic formula [8] for the rate distortion function of a smooth source relative to locally quadratic non difference distortion measures, to conditional rate distortion functions and to distortion measures which depend on the side information. In contrast, determining the asymptotics of R WZ (D) for distortion ....

[Article contains additional citation context not shown here]

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: the rate distortion function," IEEE Trans. Inform. Theory, vol. IT-45, pp. 533--547, Mar. 1999.

....treatment of the same lower bound is given in [8] and a new lower bound on the variable rate (i.e. entropy coded) performance is developed using optimal point densities. It is also pointed out in [8] that some important perceptual distortion measures in image coding are locally quadratic. In [12] an asymptotically tight expression for the rate distortion function is derived for locally quadratic distortion measures. As will be shown in this paper, the expression given in [12] plays the same important role in high resolution quantization for these distortion measures as does the Shannon ....

....pointed out in [8] that some important perceptual distortion measures in image coding are locally quadratic. In [12] an asymptotically tight expression for the rate distortion function is derived for locally quadratic distortion measures. As will be shown in this paper, the expression given in [12] plays the same important role in high resolution quantization for these distortion measures as does the Shannon lower bound in quantizing for squared error loss. To develop the basics of a high resolution quantization theory for locally quadratic distortion measures, we investigate variable rate ....

[Article contains additional citation context not shown here]

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: the rate distortion function." IEEE Trans. Inform. Theory under revision, 1998.

.... matrix g 0 satisfies g 0 (t) W (r Gamma1 (t) 13) where W t (x)W (x) M(x) that is, g 0 (t) t g 0 (t) M(r Gamma1 (t) For example, under the regularity condition given in Section 2, such a function g always exists for n = 1 (scalar case) For the general case see [15]. Then, substituting g(r(X) as the source in (7) and using the identity h(g(Z) h(Z) E log j det g 0 (Z)j, we have as D D min R(X; d; D) R (g[r(X) MSE;D Gamma D min ) 14) Corollaries (12) and (14) are actually implied by the following stronger statement. At high resolution, ....

....distortion D (with respect to the distortion measure d) and thus by (22) its rate distortion performance is within 1 2 log(2 e=12) bit of the rate distortion function of X. A detailed treatment of high resolution variable rate companding for non difference distortion measures is given in [15], where it is shown that g above is the optimal companding function for a uniform scalar quantizer, and where the analysis above is made rigorous and is extended to sources with memory, to lattice quantizers, and to vector distortion measures. 2 Main Result 2.1 Statement of the Main Result Let X ....

[Article contains additional citation context not shown here]

T. Linder, R. Zamir, and K. Zeger, "High-resolution source coding for non-difference distortion measures: multidimensional companding." submitted to IEEE Trans. Inform. Theory, 1997.

....treatment of the same lower bound is given in [8] and a new lower bound on the variable rate (i.e. entropy coded) performance is developed using optimal point densities. It is also pointed out in [8] that some important perceptual distortion measures in image coding are locally quadratic. In [12] an asymptotically tight expression for the rate distortion function is derived for locally quadratic distortion measures. As will be shown in this paper, the expression given in [12] plays the same important role in high resolution quantization for these distortion measures as does the Shannon ....

....pointed out in [8] that some important perceptual distortion measures in image coding are locally quadratic. In [12] an asymptotically tight expression for the rate distortion function is derived for locally quadratic distortion measures. As will be shown in this paper, the expression given in [12] plays the same important role in high resolution quantization for these distortion measures as does the Shannon lower bound in quantizing for squared error loss. To develop the basics of a high resolution quantization theory for locally quadratic distortion measures, we investigate variable rate ....

[Article contains additional citation context not shown here]

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: the rate distortion function." submitted to IEEE Trans. Inform. Theory, 1997.

....The proof of Theorem 1 is given in two parts. First we show in Lemma 1 that the right hand side of (13) is an asymptotic lower bound on RX n(D) Then a matching upper bound is proved in Lemma 2. Our method of proof is based partially on [6] but with the help of techniques developed in [4] and [10], we have managed to give simpler proofs of more general results. Lemma 1 Assume X n is of the mixture form (11) and conditions (a) and (b) hold. Let = 1 n P N j=1 ff j c j . Then we have lim inf D 0 RX n(D) 2 log(2 eD= 1 n H(V ) 1 n N X j=1 ff j H( b X (j) 1 ....

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: the rate distortion function." IEEE Trans. Inform. Theory under revision, 1998.

....The proof of Theorem 1 is given in two parts. First we show in Lemma 1 that the right hand side of (13) is an asymptotic lower bound on RX n(D) Then a matching upper bound is proved in Lemma 2. Our method of proof is based partially on [6] but with the help of techniques developed in [4] and [9], we have managed to give simpler proofs of more general results. Lemma 1 Assume X n is of the mixture form (11) and conditions (a) and (b) hold. Let = 1 n P N j=1 ff j c j . Then we have lim inf D 0 RX n(D) 2 log(2 eD= 1 n H(V ) 1 n N X j=1 ff j H( b X (j) 1 ....

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: the rate distortion function." IEEE Trans. Inform. Theory under revision, 1998.

....on Y ) The performance of this scheme will be arbitrarily close to the upper bound of Theorem 2. For the special case of independent X and Y , it will asymptotically achieve R WZ (D) for small distortions. A rigorous proof of both of these claims can be given using the techniques developed in [12]. Appendix A Lemma 1 Let ND denote a zero mean Gaussian random variable with variance D which is independent of (X; Y ) If E[X 2 ] and h(X) are finite, and if E[m(X; Y ) Gamma1 ] 1, then lim sup D 0 h i X ND p m(X;Y ) fi fi fi Y j h(XjY ) A.1) Proof. By [7, Appendix B] if ....

T. Linder, R. Zamir, and K. Zeger, "High-resolution source coding for non-difference distortion measures: multidimensional companding." submitted to IEEE Trans. Inform. Theory, 1997.

.... with respect to x around (x; y; x) That is, d(x; y; x) m(x; y) x Gamma x) 2 o(jx Gamma xj 2 ) 3) as jx Gamma xj 0, where m(x; y) 1 2 2 d(x; y; x) x 2 fi fi fi x=x : This definition generalizes the notion of locally quadratic input weighted distortion measures [7] to side information dependent distortion measures. Locally quadratic input weighted distortion measures are of particular interest because some important perceptual distortion measures for speech and image coding fall into this category [8, 9] Theorem 1 in the next section states that for such ....

....can prove that Z can be restricted to be a real random variable without changing the defining infimum. where o(1) 0 as D 0, and where h(XjY ) denotes the conditional differential entropy of the source given the side information. This result generalizes a recently derived asymptotic formula [7] for the rate distortion function of a smooth source relative to locally quadratic non difference distortion measures, to conditional rate distortion functions and to distortion measures which depend on the side information. In contrast, determining the asymptotics of R WZ (D) for distortion ....

[Article contains additional citation context not shown here]

T. Linder and R. Zamir, "High-resolution source coding for non-difference distortion measures: the rate distortion function." IEEE Trans. Inform. Theory under revision, 1998.

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