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### lecture4

Course: EE 477, Fall 2009
School: Stanford
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Set Slide IV: Lossy Source Coding Lossy Source Code Rate vs. Distortion for Memoryless Sources Rate vs. Distortion for Sources with Memory Universal Lossy Source Coding Shannon Lower Bound EE477, Autumn 04-05, Lec. IV 1 Lossy Source Code Assume X , Y are, respectively, the finite source and reconstruction alphabet. Assume a given single-letter distortion measure : X Y [0, ) which induces a block...

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Set Slide IV: Lossy Source Coding Lossy Source Code Rate vs. Distortion for Memoryless Sources Rate vs. Distortion for Sources with Memory Universal Lossy Source Coding Shannon Lower Bound EE477, Autumn 04-05, Lec. IV 1 Lossy Source Code Assume X , Y are, respectively, the finite source and reconstruction alphabet. Assume a given single-letter distortion measure : X Y [0, ) which induces a block distortion: 1 (xn, y n) = (xi, yi). n i=1 n EE477, Autumn 04-05, Lec. IV 2 Lossy Source Code (cont.) Definition 1. A fixed-rate n-block code is a pair (Cn, n) where Cn Y n is the code-book and n : X n Cn. The rate of the block-code is given by 1 n n n n n log |Cn |. Y = Y (X ) = n (X ) will denote the reconstruction sequence when the n-block code is applied to the source sequence X n. The expected distortion is E(X n, Y n). Definition 2. A variable-rate code for n-blocks is a triple (Cn, n, ln) with Cn and n as in Definition 1. The code operates by mapping a source n-tuple X n into Cn via n, and then encoding the corresponding member of the code-book (denoted Y n in Definition 1) using a uniquely decodable binary code. Letting ln(X n) denote the associated length function, the rate of the 1 code is defined by E n ln(X n). The expected distortion is E(X n, Y n). EE477, Autumn 04-05, Lec. IV 3 Rate Distortion Function The Rate Distortion function associated with the random variable X is R(D) = E(X,Y )D min I(X; Y ) EE477, Autumn 04-05, Lec. IV 4 Rate Distortion Theorem Theorem 1. Let X = {Xi} be an i.i.d. source Xi X. [Direct:] For any > 0 and n sufficiently large there exists an n-block code with rate R(D) + and distortion D + . [Converse:] Any code-book Cn with rate R(D) has distortion D. EE477, Autumn 04-05, Lec. IV 5 Strongly Typical Sequences and Rate Distortion For xn X n let N (a|xn) = |{1 i n : xi = a}|. The strongly typical set TX is the set of all sequences xn X n satisfying 1. 1 N (a|xn) n - PX (a) |X | for all a X with PX (a) > 0 and 2. N (a|xn) = 0 for all a X with PX (a) = 0. TX,Y is defined analogously. Lemma 1. Let {Yi} be i.i.d. Y . For xn TX 2-n(I(X;Y )+ where 1 1) Pr (xn, Y n) TX,Y 2-n(I(X;Y )- 1) , 0 as 0 and n . EE477, Autumn 04-05, Lec. IV 6 Recalling Achievability Idea in Rate Distortion Theorem Let (X, Y ) achieve the rate distortion function. Generate codewords in code-book independently with components i.i.d. Y By Lemma 1, for typical source realizations xn, probability that xn be jointly typical with, say, the first codeword is 2-nI(X;Y ). So, if we generate 2n(I(X;Y )+ ) codewords at least one of them will be jointly typical with the source with high probability Distortion between this codeword and source will be E(X, Y ) Recall, e.g., [Berger71], [Gallager68], [CT91, Sec. 13.6]. EE477, Autumn 04-05, Lec. IV 7 Remarkably Simple Converse Fix a code-book with rate R and distortion D. R 1 H(Y n) n 1 I(X n; Y n) n 1 I(Xi; Yi) n i=1 n n R(E(Xi, Yi)) i=1 1 R n n E(Xi, Yi) i=1 R (D) . EE477, Autumn 04-05, Lec. IV 8 Rate Distortion Function for Sources with Memory The Rate Distortion function associated with X n is 1 I(X n; Y n). R(X , D) = minn E(X n ,Y )D n n The Rate Distortion function associated with the stationary source X is R(X, D) = lim R(X n, D) = inf R(X n, D). n n The existence of the limit, as well as the second equality, will be established in the HW exercise. You will also show that R(X, D) = inf{I(X; Y) : E(X0, Y0) D}, the infimum taken over jointly stationary processes consistent with the 1 distribution of X and I(X; Y) = lim n I(X n; Y n) denotes the mutual information rate. EE477, Autumn 04-05, Lec. IV 9 Rate Distortion Theorem for Stationary Ergodic Sources Theorem 2. Let X be stationary ergodic. [Direct:] For any > 0 and n sufficiently large there exists a code-book for n-blocks with rate R(X, D) + and distortion D + . [Converse:] Any code-book Cn with rate R(X, D) has distortion D. EE477, Autumn 04-05, Lec. IV 10 Achievability Idea Achievability of R(X1, D): Same as i.i.d. case. Achievability of R(X n, D): Similar to achievability of R(X1, D) by considering super alphabet X n Only problem is: "super process" may not be ergodic. But can be separated into, at most, n "ergodic modes". Letting Pn(y n|xn) achieve R(X n, D) and represent i the mode 1 n n n nR(X , D) = I(X ; Y ) Ii(X n; Y n) n i=1 n n i=1 R(X n, Di|i), (notation Ii(X n; Y n) and R(X n, Di|i) to be explained in class) where n 1 Di = E[(X n, Y n)|i] so D = n i=1 Di. EE477, Autumn 04-05, Lec. IV 11 Thus, for large L can find codes of block length nL attaining distortion essentially Di and rate R(X n, Di|i) on the i-th mode. Average distortion n n 1 1 is then D = n i=1 Di and average rate is n i=1 R(X n, Di|i) R(X n, D). Details can be found in [Berger71], [Gallager68]. EE477, Autumn 04-05, Lec. IV 12 Universal Lossy Source Coding David L. Neuhoff, Paul C. Shields; Simplistic Universal Coding, IEEE Trans. Info. Theory, vol. IT-44, pp. 778 - 781, March 1998. [Ziv72] J. Ziv. Coding of sources with unknown statistics - part II: Distortion relative to a fidelity criterion. IEEE Trans. Inform. Theory, 18:389394, May 1972. [YK96] E. H. Yang and J. Kieffer. Simple universal lossy data compression schemes derived from the Lempel-Ziv algorithm. IEEE Trans. Inform. Theory, 42:239245, 1996. We follow [YK96] EE477, Autumn 04-05, Lec. IV 13 Universal Lossy Source Coding (cont.) A Simple Fixed-Rate Universal Lossy Coding Scheme: Let the code-book be given by ^ Bn = {y n X n : lLZ (y n) nR}. The encoding is done by giving the index of y n Bn that minimizes (xn, y n). Note universality of code-book w.r.t. source distribution and also w.r.t. distortion measure ! (1) EE477, Autumn 04-05, Lec. IV 14 Universal Lossy Source Coding (cont.) A Simple Fixed-Distortion Universal Lossy Coding Scheme: ^ Code-book: List the elements of X n in nondecreasing order of LZ codeword length. Encoding: Encode source n-block into the LZ codeword of the first word in code-book for which the resulting distortion is D. EE477, Autumn 04-05, Lec. IV 15 Universality Result Theorem 3. [YK96] For X stationary and ergodic E(X n, Bn) D(X, R) as n . EE477, Autumn 04-05, Lec. IV 16 Proof of Theorem 3 Our proof will use the following lemma: ^ Lemma 2. Let m be a positive integer and : X m {1, 2, . . .} be a length function. Then there exists a sequence of positive real numbers { n} ^ tending to zero such that for all n and y n X n lLZ (y n) min i+m-1 (yi ) ij(mod m),1in-m+1 1jm + nn. Proof of Theorem 3: See white-board. EE477, Autumn 04-05, Lec. IV 17 The Shannon Lower Bound The Shannon lower bound we now derive will be key in our treatment of the compression-based approach to denoising. Let A be an alphabet where addition and subtraction of elements are well-defined. Let be a difference distortion measure (x, y) = (y - x). The m...

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