Lossless Compression of Map Contours by Context Tree Modeling of Chain Codes

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1 Lossless Compresson of Map Contours by Context Tree Modelng of Chan Codes Alexander Akmo, Alexander Kolesnko, and Pas Fränt Department of Computer Scence, Unersty of Joensuu, P.O. Box 111, Joensuu, Fnland {akmo, koles, Abstract. We consder lossless compresson of dgtal contours n map mages. The problem s attacked by the use of context-based statstcal modelng and entropy codng of chan codes. We propose to generate an optmal context tree by frst constructng a complete tree up to a predefned depth, and then create the optmal tree by prunng out nodes that do not prode mproement n compresson. Experments show that the proposed method ges lower bt rates than the exstng methods for the set of test mages. 1 Introducton Dgtal maps are usually stored as ector graphcs n a database for retreng the data usng spatal locaton as the search key. The sual outlook of maps representng the same regon ares dependng on the type of the map (topographc or road map), and on the desred scale (local or regonal map). Vector representaton s conenent for zoomng as the maps can be dsplayed n any resoluton defned by the user. The maps can be conerted to raster mages for data transmsson, dstrbuton a nternet, or because of ncompatblty of the ector representatons of dfferent systems. In order to ncrease the effcency of raster map compresson, we consder the arant, when some object, nstead of beng rasterzed, wll be descrbed by chan codes and compressed separately from rest of map data. Ths can lead to a more effcent representaton of the map and, consequently, to mproe of compresson. Chan codng s a common approach for representng dfferent rasterzed shapes such as lne-drawngs, planar cures and contours. We consder thn dgtal cures of one pxel wdth, extracted from the ector data before rasterzaton. The preous works consder dfferent schemes of encodng and chan code representaton [3], [10], [11]. For example, the method n [12] uses second order context model based on 8 drectonal chan codes. Further deelopment of the context-based compresson of chan codes was presented n [4]. The authors hae mproed the performance of the chan codes encodng by ncreasng the sze of fnte context models. The problem of encodng of chan codes by predcton by partal matchng (PPM) algorthm [2] has been consdered n [1]. In prncple, context based compresson can be mproed by usng a larger number of neghborng symbols n the context. But the ncrease of the context sze leads to the H. Kalanen et al. (Eds.): SCIA 2005, LCS 3540, pp , Sprnger-Verlag Berln Hedelberg 2005

2 Lossless Compresson of Map Contours by Context Tree Modelng of Chan Codes 313 problem of context dluton, n whch the statstcs are dstrbuted oer too many contexts, and thus, affects the accuracy of the probablty estmates. Context tree prodes a more flexble approach for modelng the contexts so that a larger number of neghbor pxels can be taken nto account wthout the context dluton problem [13]. The context tree algorthm was orgnally ntroduced n [16], and analyzed n [10]. Practcal solutons for the context tree based compresson algorthms for grey-scale and b-leel mages hae been descrbed at [19] and [13] respectely. In ths paper, we use the context tree approach for encodng the chan codes. We prode algorthm for optmal context tree constructon. We compare the compresson performance of the rasterzed map contours when encoded by JBIG [9], and by the optmal context tree chan codes, representng the same contours. The results show that the proposed method prodes 25% lower bt rate than JBIG, and s 40% faster because only the contour pxels need to be processed. The oerall scheme of the proposed compresson method s as follows: Step 1: Extract contours from the ector or raster map and conert them nto chan codes. Store the start ponts and the lengths of the chans. Step 2: Create and store the optmal context tree for the chan codes. Step 3: By usng of context tree modelng and any entropy codng, encode the chan codes. Ths scheme s shown n Fgure 1. For smplcty, we store the sze and the begnnng of each chan (BOC) as such wthout any further compresson. Fg. 1. Oerall system dagram of the proposed method 2 Chan Code Representaton Freeman [5], [6] proposed chan codng of dgtal contours drawngs and descrpton. The chan codes represent the dgtal contour by a sequence of lne segments of specfed length and drecton, see Fgure 2. We consder both 8- and 4-drectonal chan codng schemes; see Fgure 3. The chan code representaton s constructed as follows. Step 1: Select a startng pont of the contour. Represent ths pont by ts absolute coordnates n the mage.

3 314 A. Akmo, A. Kolesnko, and P. Fränt Fg connected and 4-connected chan codes and ther dfferental chan codes Fg. 3. An example of chan code constructon: orgnal cure (left), 8-drectonal (center) and 4-drectonal (rght) Step 2: Represent eery consecute pont by a chan code showng the transton needed to go from the current pont to the next pont on the contour. Step 3: Stop f the next pont s the ntal pont, or the end of the contour. Store the lengths of the contours nto the fle. An alternate way for chan code representaton are dfferental chan codes [5]. Each dfferental chan code k s representng by the dfference of the current chan code c and the precedng chan code c -1 : k = c c -1. The chan codes of contours can be extracted from the map mage n two ways. Frstly, f the map s obtaned drectly from the map database, we can extract contours drectly from the ector data. On the other hand, f the map s proded as a color raster mage, we can use color separaton and followng by ectorzaton (must be used to extract the contours). 3 Context Tree Modelng 3.1 Fnte Context Modelng We compress the chan codes sequentally accordng ther order n the nput data. Consder the current symbol x and the strng of the M preous symbols x,..., x M 1 denoted as x. In the context based modelng the probablty of the next symbol x s condtoned on ts context x. The probabltes of the symbols generated n a gen context, are treated as ndependent [17]. Thus, a model becomes a collecton of ndependent sources of random arables. By the assumpton of ndependence, t s easy to assgn probabltes to each new symbol generated at the current context. Let us denote the cardnalty of the alphabet of the encoded data as. If

4 Lossless Compresson of Map Contours by Context Tree Modelng of Chan Codes 315 n ( x ),..., n ( x ) are the counts of all symbols generated at the gen context 1 = s: then the condtonal probablty of the eent x k, k [ 1,.., ] p ( x = k x ) = n ( x j= 1 k n ( x j ) ) x, We consder the encodng of the gen statstcal model by entropy-based encoder. The probablty for the entropy-based coder s estmated as: p ( x = k x ) = j= 1 n ( x k n ( x j ) + δ ) + δ The parameter δ here depends on dfferent arthmetc coders, but t usually equals to1 [8], [14]. 3.2 Context Algorthm Rested Context tree s appled for the compresson n the same manner as the fxed sze context; only the context selecton s dfferent. It s made by traersng the context tree from the root to a termnal node, each tme selectng the branch accordng to the correspondng preous symbol alue. If the correspondng symbol ponts to a non exstng branch, or the current node s a leaf, then we came to a termnal node, whch ponts to the statstcal model that s to be used. The context tree can be constructed beforehand (statc approach) or optmzed drectly for the encoded data (sem-adapte approach). In the second case, the tree structure must be stored n the compressed fle. The process of optmal tree constructon conssts of two man phases: ntalzaton of the context tree, and prunng of the tree. 3.3 Constructon of Intal Context Tree To construct an ntal context tree, we process the mage to collect statstcs for all potental contexts, leaes and nternal nodes. Each node stores nformaton of counts for all symbols generated at the current context. The algorthm of the context tree constructon s: Step 1: Create a root of the tree. Step2: For all = 1 to n, traerse the tree along the path defned by the past strng x. If some ndces of the symbols n x are less than one, then set these symbols to zero. If some node, sted accordng the correspondent symbol of the strng x, does not hae a consequent branch (for transton to the next symbol of x ), then create the necessary chld node and process t. Each new node has counts, whch are ntally set to zero. In all sted nodes, ncrease the count of x by 1. (1) (2)

5 316 A. Akmo, A. Kolesnko, and P. Fränt Ths completes the constructon of the context tree for all possble contexts. The tme complexty of ths algorthm s O(n). 3.4 Constructon of Optmal Context Tree The ntal context tree needs to be pruned by comparng the parent node and ts chldren nodes for fndng the optmal combnaton of sblngs. Let us denote by c(t ) the number of bts, requred to store the tree structure n the compressed fle. For dfferent strateges of the tree constructon t wll be dfferent: 0, statc approach c ( T ) = K, semadapte approach, complete tree (3) K, semadapte approach, ncomplete tree, where K s the cardnalty of the tree T. We wll denote the set of all termnal nodes as S (T ). Let us denote as n (s), s S(T), the count of the symbol, encoded by the statstcal model, ponted by the termnal node s. By the cost of a termnal node s here we understand the followng expresson [7], [13]: c ( n ( s), n ( s),..., n ( s) ) 1 2 0, f n ( s) = n ( s) =... = n ( S) = n ( s) 1 ( j + δ) = = 1 j= 0 log, otherwse. 2 n0 ( s) + n1 ( s) n ( s) 1 ( + ) j δ j= 0 Ths defnton corresponds algorthmcally to the use of a one pass arthmetc codng wthout the update of the statstcal model [8]. By the cost of the context tree T, we wll denote the followng expresson: L ( T ) = c ( T ) + c ( n ( s ), n ( s ),..., n ( s )) 1 2 (5) s S ( T ) The problem of the tree prunng s to modfy the structure of the full context tree so that the expresson (5) wll be mnmzed. For solng ths problem, we use a bottomup algorthm [15]. The man prncple of ths algorthm s that the optmal tree conssts of optmal sub-trees. For any node t from the tree T, let us denote the ector of counts as n ( t) = ( n ( t), n ( t),..., n ( t) ), the chld nodes as t 1 2, and the node confguraton ector as = ( 1,..., ), {0,1 }. The ector defnes whch of the node branches wll reman: f = 0, then the th branch wll be deleted from the node. Then the prncple of sub optmalty for any gen sub tree Tˆ, startng from the gen node t can be represented as follows: the optmal cost L (Tˆ opt ) for any gen sub tree T ˆ T can be expressed by the followng recurse equaton: (4)

6 Lossless Compresson of Map Contours by Context Tree Modelng of Chan Codes 317 where 0, f Tˆ s null L ( Tˆ) = c ( n ( t) ) + α, f Tˆ hase no chlds opt (6) mn{ L ( Tˆ, ) }, otherwse, ( L ( Tˆ ) + α opt L ( Tˆ, ) = c n ( t)! n ( t ) + (7) The tree Tˆ Tˆ s a sub tree oftˆ, startng from ts chld node t and the constant α s the amount of bts requred for descrbng a sngle node. In general, the cost calculaton of an optmal context tree T can be descrbed as follows: Step 1: If T has no chld nodes, then return the accumulated code length of ts root accordng to (4). Step 2: For all sub trees T T, startng from the chld nodes of T root, calculate ther optmal costs L opt ( T ). Step 3: Accordng to the found L opt ( T ), the ectors of counts n ( t ), and n ( t ),..., n ( t ), fnd the optmal ector = arg mn L ( T, ). 1 Step 4: Prune out the chldren sub trees accordng the ector. Step 5: Return the alue L ( T, ). The algorthm recursely prunes out all unnecessary sub trees, and fnally gets the optmal structure of the context tree, see Fgure 3. Fg. 3. An example of prunng of the context tree 4 Experments We proded two dfferent seres of experments. The frst one llustrates the effcency of the optmal context tree encodng of the chan codes. The second one llustrates the ablty of the ndependent chan encodng to ncrease the compresson

7 318 A. Akmo, A. Kolesnko, and P. Fränt performance of the map mage compresson n general. Images of Fgure 4, whch we use, are ector maps, rasterzed wth the resoluton of pxels. The statstcs for all mages are shown n Table 1. The frst three mages are contours of geographcal objects, and the last mage s a collecton of eleaton lnes. The absolute chan codes were transformed nto dfferental chan codes before compresson. As the entropy coder we used range-coder [14]. Tables 1 and 2 show the results for dfferent depth of the context model n the case of 8-connected and 4-connected chan code representatons. The numbers n Table 1 are the estmated bt rate accordng (6). The numbers n Table 2 are the real bt rate, resulted after the range coder. For comparng the compresson effcency, there are results of chan codes compresson PPMd algorthm wth the maxmum context order 8 [17]. Hgher context order n PPM leads us to the context dluton problem and, consequently, decreasng the compresson performance. For the test mages the effcent range of the context tree depth s from 4 to 10. Fg. 4. The set of test mages Table 1. Test mage propertes and estmated bt rate (bts per symbol) 4-connected chan codes umber of chan Depth codes Image # Image # Image # Image # Aerage connected chan codes umber of chan Depth codes Image # Image # Image # Image # Aerage The second seres of experments were amed to estmate the effcency of chan codes compresson n comparson to an effcent raster mage compresson algorthm, namely the JBIG. Table 3 summarzes compressed fle szes, when compressed by the optmal context tree algorthm (CTC 4 and CTC 8), and for correspondng raster

8 Lossless Compresson of Map Contours by Context Tree Modelng of Chan Codes 319 mages compressed by JBIG. The raster map mages are obtaned by rasterzaton from 4-connected chan codes. The experments show that the runnng tme of the CTC algorthm s than that of the JBIG. Ths s because of much smaller amount of encoded nformaton: JBIG encodes pxels at each mage, when CTC encodes only chan codes. Table 4 represents the structure of the compressed fle: the percentage of all three types of the data n the fle: begnnng of the chans (BOC), structure of the context tree (CT), and the encoded chan codes (Chan Codes). The most used contexts n Image#4 for CTC 4 and for CTC 8 compresson are shown n Tables 5 and 6 consequently. The most used contexts descrbe horzontal, ertcal or dagonal straght lnes. All the experments were proded on computer P3 500MHz, 256 Mb RAM, Wndows T. Table 2. Real bt rate (bts per symbol) 4-connected chan codes PPM CTC 4 Depth Image # Image # Image # Image # Aerage connected chan codes PPM CTC 8 Depth Image # Image # Image # Image # Aerage Table 3. Comparson of the CTC-encoded chans and JBIG encoded raster mages Compressed fle sze (bytes) Compresson tme (sec) JBIG CTC 4 CTC 8 JBIG CTC 4 CTC 8 Image # Image # Image # Image # Aerage Table 4. The proportons of dfferent parts n the compressed fle BOC CT Chan codes Image #1 0.2% 1.5% 98.3% Image #2 0.1% 0.6% 99.3% Image #3 1.8% 0.3% 97.9% Image #4 7.9% 1.0% 91.1%

9 320 A. Akmo, A. Kolesnko, and P. Fränt Table 5. Three most used context for Image #4 n CTC 4 Context n0 n1 n2 n3 total Table 6. Three most used context for Image #4 n CTC 8 Context n0 n1 n2 n3 n4 n5 n6 n7 total Conclusons We hae proposed context tree algorthm for encodng chan codes of contours n map mages. The proposed algorthm ncreased the compresson performance oer the PPM algorthm by 2-3%. The use of chan codes, nstead of the compresson of rasterzed contours, mproes the compresson by 25%, on aerage. The results could be mproed up to the theoretcal lmts by usng a more sutable entropy encoder, nstead of sub-optmal range coder. References [1] Bossen, F., Ebrahm, T.: Regon shape codng, Techncal Report M0318, ISO/IEC JTC1/SC29/WG11, oember 1995 [2] Cleary, J., Wtten, I.: Data compresson usng adapte codng and partal strng matchng, IEEE Trans. on Communcatons, 32(4), Aprl 1984, [3] Eden, M., Kocher, M.: On performance of a contour codng algorthm n the context of mage Codng Part 1: Contour Segment Codng, Sgnal Processng, 1985, 8, [4] Estes, R., Algaz, R.: Effcent error free encodng of bnary documents, In: Proc. of IEEE Data Compresson Conference, March 1995, [5] Freeman, H.: Computer processng of lne drawng mages, ACM Computng Sureys, 6, March 1974, [6] Freeman, H.: Applcaton of the generalzed chan codng scheme to map data processng, In: Proc. of IEEE Pattern Recognton and Image Processng, May 1978, [7] Helfgott, H., Cohn, M.: Lnear-tme constructon of optmal context trees, In: Proc. of the IEEE Data Compresson Conference, Aprl 1998, [8] Howard, P., Vtter, J.: Analyses of arthmetc codng for data compresson, In: Proc. of the IEEE Data Compresson Conference, 1991, 3-12 [9] JBIG: Progresse b-leel mage compresson, ISO/IEC Internatonal Standard 11544, 1993 [10] Kaneko, T., Okudara, M.: Encodng of arbtrary cures based on chan code representaton, IEEE Trans. on Communcatons, July 1985, 33, [11] Lu, Y.K., Zalk, B.: An effcent chan code wth Huffman codng, Pattern Recognton, 38(4), 2005,

10 Lossless Compresson of Map Contours by Context Tree Modelng of Chan Codes 321 [12] Lu, C.C., Dunham, G.: Hghly effcent codng schemes for contour lnes based on chan code representatons, IEEE Trans. on Communcatons, 39(10), October 1991, [13] Martns, B., Forchhammer, S.: Tree codng of b-leel mages, IEEE Trans. on Image Processng, 7(4), Aprl 1998, [14] Martn, G.: An algorthm for remong redundancy from a dgtzed message, Presented at: Vdeo and Data Recordng Conference, July 1979 [15] orhe, R.: Topcs n descrpte complexty, PhD Thess, Unersty of Lngköpng, Sweden, 1994 [16] Rssanen, J.: A unersal data compresson system, IEEE Transactons on Informaton Theory, 29(5), September 1983, [17] Shkarn, D.: PPM: one step to practcalty, In: Proc. of the IEEE Data Compresson Conference, Aprl 2002, [18] Wenberger, M., Rssanen J.: A unersal fnte memory source, IEEE Trans on Informaton Theory, 41(3), May 1995, [19] Wenberger, M., Rssanen, J., Arps, R.: Applcaton of unersal context modelng to lossless compresson of grey-scale mages, IEEE Transactons on Image Processng, 5, Aprl 1996,