Two-Dimensional Source Coding by Means of Subblock Enumeration Takahiro Ota Hiroyoshi Morita Dept. of Computer & Systems Engineering Graduate School of Informatics and Engineering Nagano Prefectural Institute of Technology The University of Electro-Communications 813-8, Shimonogo, Ueda, Nagano, 386-1211, JAPAN 1-5-1, Chofugaoka, Chofu, Tokyo, 182-8585, JAPAN Email: ota@pit-nagano.ac.jp Email: morita@uec.ac.jp 7 1 0 Abstract—A technique of lossless compression via substring witha lineof2Dsourceasasymbolofanextendedalphabet. 2 enumeration(CSE)attainscompressionratiosaswellaspopular AttheinitialstepoftheCSEencodingprocess,theCSEneeds n lossless compressors for one-dimensional (1D) sources. The CSE to outputfrequenciesof all symbolsof the extendedalphabet. a utilizesa probabilisticmodel builtfrom thecircularstringof an To reduce the computational time, we propose a new CSE J input source for encoding the source. The CSE is applicable to two-dimensional (2D) sources such as images by dealing with a for a 2D source which utilizes the flat torus of an input 2D 4 lineof pixels of 2D source as a symbol of an extended alphabet. source as a probabilistic model instead of the circular string 2 At the initial step of the CSE encoding process, we need to of the source. In the initial step, the total number of output outputthenumberofoccurrencesofallsymbolsoftheextended ] blocksisconstantsincethenewCSEworksinblock-by-block. T alphabet,sothatthetimecomplexityincreaseexponentiallywhen Moreover, we evaluate the limit of the average codeword I the size of source becomes large. To reducethe time complexity, . we propose a new CSE which can encode a 2D source in block- length of the proposed algorithm for general sources. s c by-block instead of line-by-line. The proposed CSE utilizes the [ flat torus of an input 2D source as a probabilistic model for II. BASIC NOTATIONS AND DEFINITIONS encoding the source instead of the circular string of the source. A. Alphabet and Block 1 Moreover, we analyze the limit of the average codeword length v of the proposed CSE for general sources. Let X be a finite source alphabet {0,1,...,J−1} and let 3 |X| be a cardinality of X, that is |X| = J. Let X[m,n] be 3 I. INTRODUCTION the set of all m×n finite blocks p = (p ) 7 (i,j) 1≤i≤m,1≤j≤n In 2010, Dube´ and Beaudoin proposed an efficient off- over X, where p ∈ X is the element of p at (i,j)- 6 (i,j) 0 line data compression algorithm for a binary source known coordinate. Furthermore, let X[∗,∗] be ∪m,n≥0X[m,n], where . as Compression via Substring Enumeration (CSE) [1]. In [2], X[m,n] includes the empty block λ[m,n] when at least one 1 YokooproposedauniversalCSEalgorithmforabinarysource of m and n is 0. For convenience, X[m,0] and X[0,n] are 0 7 and various versions of the CSE for a binary source have defined as {λ[m,0]} and {λ[0,n]}, respectively.For p∈X[∗,∗], 1 been proposed so far [3]–[5]. It is reported that performance let |p| and |p| be the length of row (the height) and the r c : of the CSE [4] is as well as that of an efficient off-line lengthofcolumn(thewidth),respectively.Forexample,when v i datacompressionalgorithmusingtheBurrows-Wheelertrans- X ={0,1},Fig.1illustratesp∈X[3,3] where|p|r=|p|c=3. X formation (BWT) [6]. In [7], it is proved that an encoder, r which is a deterministic finite automaton, of the CSE and 1 0 0 a ! ! ! an encoder without sinks of the antidictionary coding [8] are 1 1 0 ! ! ! ! ! isomorphic for a binary source. Moreover, an antidictionary 0 0 1 ! ! ! ! ! ! codingproposedin[9]providedthefirstCSEforq-ary(q>2) alphabetsources as a byproduct.Iwata and Arimura proposed Fig.1.A3×3 Fig.2.πc(p),σc(p),πr(p),andσr(p)ofpinFig.1. the modified algorithm and evaluated the maximum redun- blockp. dancy rate of the CSE for the kth order Markovsources [10]. Forencodinganinputsource,theCSEutilizesaprobabilis- B. Subblock, Concatenation, and Dictionary tic model built from the circular string which is obtained by Forp∈X[m,n], a subblock p(i+k−1,j+l−1)∈X[k,l] isdefinedas concatenatingthefirstsymboltothelastsymbolofthesource. (i,j) A probabilistic model of the circular string is also useful for λ[0,l] (k≤0 and l≥0), the BWT and antidictionary coding [7], [9], and in [11], it  λ[k,0] (k≥0 and l≤0), is shown that an antidictionary built from the circular string iasciudse(DfuNlAfo)r. Hgeonwoemveer,cofomrpaari2sDonssouucrhceassudcheoxasyriabnonimucalgeeic, p((ii+,jk)−1,j+l−1):=  p(i...,j) ·.·.·. p(i,j+...l−1)  computational time of the CSE is exponential with respect to  p ··· p  linelengthsincetheCSEworksinline-by-line.TheCSEdeals   (i+(kk−>1,j0) and l>0()i+k−1,j+l−1)  where 1≤i≤m, 1≤j≤n, k≤m−i+1, and l≤n−j+1. D. Classifications of Flat Tori and Core Hereinafter, without notice, we assume that the height and For p and k (0≤k ≤m), and l (0≤l≤n), width of p are respectively given by m (≥ 2) and n (≥ 2). In particular, (m−1)×n subblocks p(m−1,n) and p(m,n) are T(p,k,l):={q ∈X[m,n] s.t. N(w|q)=N(w|p), (1,1) (2,1) denotedbyπr(p)andσr(p),respectively.Moreover,m×(n− ∀w ∈X[k,l],q is primitive.} (6) 1) subblocks p(m,n−1) and p(m,n) are denoted by π (p) and (1,1) (1,2) c For example, [p] = T(p,m,n). For 0 ≤ k < n and fixed σ (p),respectively.Forexample,forpinFig.1,Fig.2shows c 0≤l≤n, T(p,k,l) is monotone decreasing with k, that is π (p), σ (p), π (p), and σ (p) from the left-hand side. c c r r T(p,k+1,l)⊂T(p,k,l). Similarly, for fixed 0≤k′≤n and For p, the dictionary of p is defined as the set of all the 0≤l′X,X[m(<,1nm,2]),nta>k,e.s.a.,vXal(t)h}e∞m=m,∞1,n×=1n lmim,n→su∞p−K1wL∈XX[KE,L](cid:20)N(wm|Xn[m,n])(cid:21)log2E(cid:20)N(wm|Xn[m,n])(cid:21) 1 Cartesian productX[m,n] of X [14]. The probabilitydistribu- =limsup− P (w)log P (w) KL X[m,n] 2 X[m,n] tion of a random variable X[m,n] is denoted by PX[m,n]. For m,n→∞ w∈XX[K,L] X, the sup-entropy rate of X is defined as H(X[K,L]) =limsup =Hˆ(X). 1 KL Hˆ(X):= limsup H(X[m,n]). (22) m,n→∞ mn m→∞,n→∞ (ProofofTheorem1): Asforℓ (p),fromtheassumption, is satisfied. Therefore, limsup E ℓ(X[m,n]) ≥Hˆ(X). 0 m,n→∞ mn since m≤n, ℓ (p)≤2(log n+2log log n+7)+⌈log mn⌉ h i 0 2 2 2 2 where (log n+2log log n+7) and ⌈log mn⌉ are costs of FromRemark1.7.3[14],if Xisa stationarysource,Hˆ(X) 2 2 2 2 Elias integer code for n and ǫ(rank(p)), respectively. As for can be expressed by H(X)(:= lim H(X[m,n])), that m,n→∞ mn ℓ1(p), the cost of N(bi) in (P-i) is ⌈log2mn⌉ bits from (18), is the entropy rate of X. Therefore, if X is a stationary sothatℓ1(p)≤(|X|−1)⌈log2mn⌉.Asforℓ2(p),sinceI(bi)≤ source,theaveragecodewordlengthoftheproposedalgorithm mn and I′(bi) ≤ mn, costs of I(bi) and I′(bi) are at most convergesto H(X) as m and n go to infinity. log mn bits. Moreover, since m≤n and K ≤L, 2 VI. CONCLUSION K L ℓ (p)≤ |X|whlog mn≤L2|X|L2log mn For reducing computational time, we proposed a new CSE 2 2 2 hX=1wX=1 fora2Dsourcewhichutilizestheflattorusofthesourcewhile the conventionalCSE utilizes the circular string of the source ≤2(log log n)(log n)(log n). |X| |X| |X| 2 asa probabilisticmodel.Thetotalnumberofoutputblocksof Therefore, thenewCSEispolynomialwhilethatoftheconventionalCSE is exponential with respect to the source size. The new CSE lim (ℓ (p)+ℓ (p)+ℓ (p))/mn=0. (23) m,n→∞ 0 1 2 encodes the source in block-by-block while the conventional CSE does in line-by-line. Moreover, we prove that an upper As for ℓ (p), from (20), cost of N(b ) is 3 i bound on the average codeword length of the proposed CSE −log (|T(B(p),p,i)|/|T(B(p),p,i−1)|) bits. 2 convergesto the sup-entropy rate for a general source as size Cost of the next encoded N(b ) such that N(b ) of the input source goes to infinity. Furthermore, if a general j i has been encoded immediately before N(b ) is source is a stationary source, then the length convergesto the j −log (|T(B(p),p,j)|/|T(B(p),p,j−1)|). From Lemma 3, entropy rate of the source as the size goes to infinity. 2 |T(B(p),p,j−1)|=|T(B(p),p,i)|. Therefore,N(b ) can be j REFERENCES written by −log (|T(B(p),p,j)|/|T(B(p),p,i)|), Hence, 2 thedenominator|T(B(p),p,i)|forp isequaltotheprevious [1] D. Dube´ and V. Beaudoin, “Lossless data compression via substring j enumeration,” in Proc. of the Data Compression Conference 2010, numerator |T(B(p),p,i)| for b , so that they are canceled. i pp.229–238, Mar.2010. Moreover, since |T(B(p),p,|B(p)|)|=|[p]|=mn, [2] H. Yokoo, “Asymptotic optimal lossless compression via the cse technique,” in Proc. of the Data Compression, Communications and Pro- ℓ3(p)=log2|T(B(p),p,S−1)|−log2mn. (24) cessing2011,pp.11–18,June2011. 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Two-Dimensional Source Coding by Means of Subblock Enumeration

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