Tabu Split and Merge for the Simplification of Polygonal Curves

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1 Proeedings of the 2009 IEEE Interntionl Conferene on Systems, Mn, nd Cybernetis Sn Antonio, TX, USA - Otober 2009 Tbu Split nd Merge for the Simplifition of Polygonl Curves Gilds Ménier VALORIA, Université de Bretgne Sud Université Européenne de Bretgne Frne gilds.menier@univ-ubs.fr Pierre-Frnçois Mrteu VALORIA, Université de Bretgne Sud Université Européenne de Bretgne Frne pierre-frnois.mrteu@univ-ubs.fr Abstrt A Tbu Move Merge Split (TMMS) lgorithm is proposed for the polygonl pproximtion problem. TMMS inorportes tbu priniple to void premture onvergene into lol minim. TMMS is ompred to optiml, ner to optiml top down Multi-Resolution (TDMR) nd lssil split nd merge heuristis solutions. Experiments show tht potentil improvements for rudest pproximtions n be obtined. The evlution is rried out on 2D geogrphi mps ording to effetiveness nd effiieny mesures. Keywords Polygonl pproximtion, Tbu serh, Splitnd-Merge, top down multi resolution, dynmi progrmming. I. INTRODUCTION Polygonl urve pproximtion hs been widely studied in the pst to sle up time onsuming pplitions suh s grphi disply, ontour detetion or time series dt mining. If we pprehend disrete urve s multidimensionl vetor, polygonl pproximtion n be seen s dimension redution tehnique tht reltes lso to multidimensionl sling. Following polygonl pproximtion is n optimiztion problem tht n be tkled long two ngles: Min-ε problem: Given polygonl urve S hving N segments, find n pproximtion A hving K segments suh tht the mximl pproximtion error ε is minimized. Min-# problem: Given polygonl urve S hving N segments, find n pproximtion A with the minimum number of segments K so tht the mximl pproximtion error does not exeed given tolerne ε. Most of the proposed lgorithms developed to solve these problems belongs either to grph-theoreti pprohes [1,6,8,10,11,12], dynmi progrmming [5,7] or to heuristi pprohes [2,4,12,13,15]. Optiml solutions bsed on dynmi progrmming priniple exist, nevertheless, there omplexities re proved to be O(K.N 2 ) leving spe for muh fster but suboptiml solutions. Finding fst lgorithms s ner-to-optimlity s possible for long input urves is still n open hllenge leding to fruitful pplitions. In [10] for instne, uthor proposed fst nd dirty filtering pproh dedited to time series retrievl whose effiieny is highly orrelted with the qulity of polygonl pproximtions. Reently, we hve proposed top town multiresolution lgorithm (TDMR) [9] tht solves the problem of polygonl urve pproximtion in liner time omplexity O(N). This lgorithm ensures ner to optiml solutions between eh two suessive levels of resolution, but, s we desend the resolution levels, the pproximtions deprt further from optimlity. The only other known lgorithm showing liner omplexity is the Dougls-Peuker lgorithm [2,4]. It is fster thn TDMR but provides pproximtions tht re muh frther to optimlity thn the ones provided by TDMR. In this pper, we explore heuristi strtegies bsed on three elementry opertions (merge, split nd move) ssoited to Tbu serh priniple nd try to evlute how these strtegies ould boost suboptiml solutions suh s TDMR, Merge-L2 [12] or Dougls Peuker [2] heuristis. The seond setion of the pper sttes the problem definitions nd introdues the three elementry opertions t the bsis of the heuristis we will detil into the third setion of the pper. The fourth setion presents nd omments the experiments we hve rried out on set of 2D geogrphi mps; we onlude the pper nd suggest some perspetives in the finl setion of the pper. II. PROBLEM DEFINITIONS AND THE THREE ELEMENTARY OPERATIONS We ttempt to orret n pproximtion urve A of k points pprohing (soure) urve S of n points (k << n) by the men of elementry trnsformtion opertors: move, merge nd split. To simplify the serh spe, nd following the ommon definitions of the Min-ε nd Min-# problems, we tke pproximtion points mong the points of the trget urve: A is therefore sorted list of referenes (index) of points in S. Let S [i] be the i th point of the urve S. Let A [] be the referene to point in S the i th element of A is the A [i] th point of S, i.e. the point S [A [i]] (Fig. 1) /09/$ IEEE 1359

2 S [0] Figure 1. The referene urve S nd n pproximtion A Let S [,b] be the sub urve between the th point nd the b th point of the urve S. Let A [,b] be the sub urve between the th point nd the b th point of the urve A. A. move opertor With the im of minimizing the Min-ε riterion, the move opertor slides potentilly mispled point thru the set of urve s indexes ording to lest surfe error grdient diretion. The error is omputed s follows: For the pproximtion point i (oding the i th element of the urve A ), n evlution of the surfe between the sub urve S [A [i -1], A [i ]] nd the sub urve of A [i -1, i ] is omputed s E left (S, i ) summing the Eulidin distnes between the points of S nd the pproximtion line provided by A[i -1, i ] (Figure 2). E left ( i ) = j< A[ i ] deul j= A[ i 1 ] ( A [ i 1,i ],S [ j ]) d eul (D,P) stnds for the Eulidin distne between point P nd line D. Likewise, A [0] E left S [1] A [i -1] S [2] A [1] A [2] = 7 S [A [2]] A [i ] S [j] E right A [2] S [7] A [i +1] A [3] Figure 2. Grdient of the errors for the move opertion A [k-1] S [n-1] A S E right ( i ) = j< A[ i + 1] deul j= A[ i ] ( A [ i,i + 1],S [ j ]) Let E left/right (i) be the surfe error blne supported by the i th point: E left/right (i )= E left (i )- E right (i ). The min ide here is to slide the i th point (in the S indexes spe) to blne the left nd right errors. If E right (i )> E left (i ) the i th point is moved by one position right in the S indexes spe (respetively left if E right (i )< E left (i ) ). The displement is onstrined by the segment boundry points i - 1 nd i +1 (Alg. 1). OPERATOR move foreh point i in A ompute E left/right (i) end foreh selet the point i mx hving the mximum E left/right (i) if E left (i mx ) < E right (i mx ) if (A [i mx -1]<A [i mx ] -1) A [i mx ] := A [i mx ] -1 // slides i to the left else if E left (i mx ) > E right (i mx ) if (A [i mx +1] > A [i mx ] +1) A [i mx ] = A [i mx ] +1 // slides i to the right Algorithm 1 One exeution of the opertor move slides only one point - the point of A hving the worse E left/right (i) (thus being the most unblned point on the A urve regrding the lol surfe error). B. split opertor The split opertor (Alg. 2) finds one of the point i in S tht mximises d eul (A, S [i]) nd inserts new point in A to orret this error thus deresing (eventully not stritly) the Min-ε vlue. 1360

3 OPERATOR split foreh point i in S ompute d eul (A, S [i]) end foreh selet the point i mx with the mximum distne insert new point i new in A tht minimizes d eul (A [i new -1, i new +1], S [i mx ]) Algorithm 2 C. merge opertor The merge opertor removes n element from A, thus merging the two djent pproximting segments. Sine the rtio k/error is importnt, the point to remove should be the point ontributing the lest to the globl error ie, it is importnt to disrd first the points whih removl inreses the lest the globl error. A [i -1] A [i ] A [i +1] A S OPERATOR merge foreh element i in A ompute R p (i ) end foreh selet the element i hving the higher removl priority R p remove it Algorithm 3 D. Refining the urve As introdued bove, we ttempt to orret n existing pproximtion urve A of k points pprohing (soure) urve S of n points by the men of elementry trnsformtion opertors move, merge nd split : FUNCTION Refine (A, S ) // Refines the pproximtion A (of k elements) of the urve S (n points) repet 2*k loops of move(); merge(); split(); end repet Algorithm 4 A [i -1] Figure 3. Applition of the merge opertion on element i. The removl priority R p for n element i is omputed s follows : The error riterion Error (Min-ε for instne) is omputed on the segments [A [i -1], A[i ]], [A [i ], A[i +1]] (Figure 3) nd if the element A [i ] were to be disrded, on the segment [A [i -1], A[i +1]], the two segments [A [i -1], A[i ]] nd [A [i ], A[i +1]] re merged. For instne, if the Min-ε error is used: R p (i ) = Mx(Error min-e [i -1, i ], Error min-e [i, i +1]) / Error min-e [i -1, i +1] Alg. 3 desribes the proess : A [i +1] Unfortuntely, sequene of move, split nd merge does not ensure tht the error riterion will stritly derese. In ft, the experiments (see Fig. 4) show sequene of globl error hnges similr to those existing in inrementl lerning (suh s Bk Propgtion Networks, or to lest mount in Geneti Algorithm) ie in stbiliztions nd step breks: this suggests tht the optimiztion of the lotions of the points is not grdient proess but rther omplex orgnizing mehnism neessry to explore the prmeters spe. It is neessry to keep side the best solution found nd to expet the next itertions to eventully improve this solution. In the simple lgorithm desribed bove (Alg. 4), we deided to perform number of loops proportionl to the k elements of the pproximtion sine we think tht, in the idel onfigurtion, eh element of the pproximtion should undergo lest one - move/merge/split opertor. Experimentlly, we observe tht k loops is minimum limit to hieve the best expetble error: experimentlly, more thn 2*k loops doesn t inreses the performnes so fr. 1361

4 During its explortion of the prmeter s spe (s vetor of k indexes in the S urves), this lgorithm seems to fll in grdient wells: Fig. 4 exhibits osillting behviours tht prevents the serh to go on: it seems tht the sequene of move/merge/split n led to onfigurtions tht our sequentilly fter t loops. Figure 5. Tbu with 5 elements (best solution min-e = 0.412) For Tbu list of size t b elements, not only this strtegy seems to prevent the rpid osillting behviour under t b loops, but lso seems to inrese the serh speed. Figure 4. Curve of n = points with k = 50 (best solution min-e = 0.716) No Tbu involved. This suggests tht the sme onfigurtion of elements in A my be snned more thn one. Keeping the n th lst onfigurtions of A n nd preventing (temporry) the lgorithm to re-sn these previous solutions is somehow relted to serh spe strtegy lled Tbu Serh [8]. In this exmple, we ssumed tht the ost to store nd ompre new solution of A (of k points) to the t previous solutions would be importnt. Mostly, it is the reurring hoie of the points to move, merge nd split tht leds to n endless evlution of sequenes of lredy-seen onfigurtions. Therefore, we propose to introdue some Tbu serh priniples through the mngement list of reently points hosen to perform move/merge/split nd prevent the opertor from piking (gin) one of the tgged element t lest s long s these elements re enqueued in this FIFO list. Restriting the serh by the hoie of one index point segments the serh spe in lsses of pproximtion urves ontining (or not) speifi point: this ould be very restritive t first glne, but this strtegy speeds up the serh by seleting the urve ontining the best / worse point in regrd to the Min-ε riterion. Figure 6. Tbu of size 20 (best solution min-e = 0.409) Inresing the size of the Tbu list bove elements dereses the performne of the lgorithm without signifint better results (mostly under 10-4 for the mximum error). III. APPROACHES A. RSDP: Redued Serh Dynmi Progrmming The referene lgorithms for urve pproximtions for the Min-ε riterion re mostly bsed on dynmi progrmming [12]: they usully provide the miniml error t omputtionl ost of O(n 2 ). It is possible to redue this omplexity, onstrining the serh when ner-optiml solutions re eptble lowering the omputtionl ost to O(n 2 /k) : in the following experiments, RSDP stnds for Redued Serh Dynmi Progrmming [7]. B. MR : Multi Resolution In [9], we introdued top-down multi resolution lgorithm TDMR designed to ompute itertively nested pproximtions with omplexity (t the best se) of O(n): it fetures sequentil proessing of RSDP-like proessing nd outputs multiresolution solution to the pproximtion problem. 1362

5 C. I-TMMS : Refine with equidistnt initiliztion This proessing involves the refine funtion (desribed bove) strting with first urve of k points to orret: eh k i point is initilly set t equidistnt position on the S urve (rounded to the nerest index). 2*k loops re performed. D. SPLT : Split Strting with n initil urve of k 0 =2 points (the first nd lst point of S ), this lgorithm performs k-2 split opertions to reh the finl k elements for A. RSDP, tht is ner-optiml, is used s referene solution. Fig.7 shows the Fidelity for ll the experimented methods. Bsilly, the TMMS proedure boosts the experimented methods for low k vlues. ISM performs quite well for vlues of k << n: it is only outperformed by MR nd MR/TMMS : for k>64, ISM gives the worst results. MR/TMMS seems to improve mrginlly the error of MR. The TMMS proedure introdues time ost tht is mesurble for ll experimented methods in Fig.8. E. MRG : Merge Likewise, this lgorithm strts with the omplete urve S s the full olletion of indexes for A nd deimtes itertively the points (by the men of merge opertors) until the number of remining elements rehes k elements in A. F. MR/TMMS A multiresolution proess (MR) is performed [9], refined by the tbu move/merge/split (TMMS) sequene of opertors , , ,000 10,000 MR MR/TMMS SPLT SPLT/TMMS MRG MRG/TMMS RSDP/OPT G. SPLT/ TMMS The SPLT (split) is performed, followed by the refine (tbu move/merge/split) TMMS proess. H. MRG/TMMS The MRG (merge) is performed, followed by the refine (tbu move/merge/split) TMMS proess. IV. EXPERIMENTS The experiments hve been performed on 10 urves S depiting the ostl mps of Western Europe. The 10 urves S hve t lest n = 8192 points. We mesure the fidelity of n pproximtion using the following formul: where E is the Min-ε error. F method = E RDSP /E method 1,000 0,100 0, Figure 8. Evlution for ll experimented methods of omputtion time (in se.) for different vlues of k. V. CONCLUSION We hve introdued the use of move/merge/split opertors using Tbu-like seletion to refine n existing pproximtion urve. We ompred this pproh with other sub optiml lgorithms, nmely top down multiresolution, split lgorithm nd merge lgorithm. The TMMS proedure offers some boosting pbility for the rudest pproximtions nd ould probbly be used diretly inside multiresolution pproh to improve the overll fidelity of the provided pproximtions low level of resolution. 1,0000 0,9500 0,9000 0,8500 0,8000 0,7500 MR MR/TMMS I-TMMS 0,7000 SPLT SPLT/TMMS MRG 0,6500 MRG/TMMS 0,6000 0,5500 0, Figure 7. F method for different vlues of k (number of points in A ) REFERENCES [1] Chen D.Z., Desu O.. Spe-effiient lgorithms for pproximting polygonl urves in two-dimensionl spe. In Int. Conf. on Computing nd Combintoris, Leture Notes in Computer Siene, volume 1449, pp 4554, [2] Dougls D., Peuker T., "Algorithms for the redution of the number of points required to represent digitized line or its riture", The Cndin Crtogrpher 10(2), pp , [3] Glover F., Lgun M. Tbu Serh. Kluwer, Norwell, MA, [4] Hershberger J., Snoeyink J., "Speeding Up the Dougls-Peuker Line-Simplifition Algorithm", Proeedings 5th Symp on Dt Hndling, UBC Teh Report TR-92-07, [5] Horng J.-H. Improving fitting qulity of polygonl pproximtion by using the dynmi progrmming tehnique. Pttern Reognition Letters, 23:pp , [6] Imi H., Iri. M. Polygonl pproximtions of urve - formultions nd lgorithms. Computtionl Morphology, pp 71-86,

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