Reconstruction of Orthogonal Polygonal Lines

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1 Recostructo of Orthogoal Polygoal Les Alexader Grbov ad Eugee Bodasky Evrometal System Research Isttute (ESRI) 380 New ork St. Redlads CA USA {agrbov Abstract. A orthogoal polygoal le s a le cosstg of adacet straght segmets havg oly two drectos orthogoal to each other. Because of ose ad vectorzato errors the result of vectorzato of such a le may dffer from a orthogoal polygoal le. Ths paper cotas the descrpto of a optmal method for the restorato of orthogoal polygoal les. It s based o the method of restorato of arbtrary groud truth les from the paper []. Specfcty of the algorthm suggested the paper cossts of flterg vectorzato errors usg a pror formato about orthogoalty of the groud truth cotour. The suggested algorthm guaratees that obtaed polygoal les wll be orthogoal ad have mmal devatos from the groud truth le. The algorthm has a low computatoal complexty ad ca be used for restorato of orthogoal polygoal les wth may vertces. It was developed for a rasterto-vector coverso system ArcSca for ArcGIS ad ca be used for teractve vectorzato of orthogoal polygoal les. Keywords: Polygoal le orthogoalty le drawgs maps vectorzato error flterg. Itroducto The term orthogoal polygoal les wll be used to refer to polygoal les cosstg of orthogoal straght segmets. There are oly two permssble drectos for these segmets. These are called cardal drectos. Ay two segmets of a orthogoal polygoal le are ether parallel or perpedcular to each other. Ay two successve segmets are perpedcular to each other. A rectagle s a example of a orthogoal polygoal le. Orthogoal polygoal les ca be see at dfferet le drawgs for example archtectural plas egeerg drawgs ad electrcal schematcs. Fg. shows a fragmet of a cty map. ost of the buldg outles are orthogoal les. The results of vectorzato of les from moochrome mages usually are corrupted. Because of scag ose dscretzato barzato ad vectorzato errors eve straght les are coverted to polygoal les after raw vectorzato. Fg. shows a moochrome mage obtaed by scag a straght le ad the result of raw vectorzato. The umber of segmets a resultg polygoal le ad the devatos of these segmets from the groud truth straght les are sometmes used to evaluate vectorzato error [].. Buke ad A.L. Sptz (Eds.): DAS 006 LNCS 387 pp Sprger-erlag Berl edelberg 006

2 Recostructo of Orthogoal Polygoal Les 463 Fg.. A fragmet of a cty map wth buldgs. ay of the buldg borders are orthogoal les. Fg.. The moochrome mage of straght les ad polygoal les as a result of raw vectorzato Post-processg usually follows raw vectorzato. Oe of the tasks of postprocessg s defragmetato. The goals of defragmetato are data compresso ad creasg precso. I the past data compresso was more mportat. The most wdely used compresso methods solve the problem of data compresso by removg some source polygoal le vertces (see for example the Douglas- Peucker method [3]). The ma crtero for removg a vertex s the dstace from the vertex of the source polygoal les to the polygoal le that s a result of compresso. Because the locato errors of the remag vertces are ot corrected the precso of vectorzato may ot be ehaced. I spte of ths the Douglas-Peucker compresso method s used tll ow for defragmetato ad smplfcato of polygoal les.

3 464 A. Grbov ad E. Bodasky Recetly because of cosderable reducto prce ad creasg capacty of computer memory the problem of data compresso has become less mportat whle the problem of ehacg the precso of vectorzato has become more crtcal. Oe approach to the problem of creasg vectorzato precso of polyles cosstg of geometrc prmtves follows. A source polygoal le obtaed by raw vectorzato s dvded to ooverlappg fragmets such that each could be approxmated wth good precso usg some geometrc prmtve (for example a straght segmet or a crcle arc). By fdg the optmal approxmatos of the prmtves ad the tersectos of adacet prmtves t s possble to buld a sequece of prmtves that s the restorato of groud truth le. The source polygoal le must be dvded such a way that some fuctoal that s a measure of approxmato error wll be mmzed. The vtal mportace of such a approach has a defto of the fuctoal. I [] such a approach s used wth oe restrcto (after dvdg the source le to fragmets they are approxmated oly wth straght segmets). The fuctoal value depeds ot oly o the precso of the approxmato of fragmets of source les but also o the umber of fragmets or the umber of straght segmets of the resultg polygoal le. Ths method uses oly oe parameter the pealty for each segmet of the resultg polygoal le. If the groud truth le s a orthogoal le the method suggested [] does ot guaratee that the resultg polygoal le wll be a orthogoal le. It s possble to resolve the problem by takg to accout geometrcal costrats ( ths case t s a orthogoalty) after a polygozato of the result of raw vectorzato (for example wth a beautfcato method from [4]). But the suggested method resolves the problem wth smultaeous polygozato ad takg to accout geometrc costrats. It provdes the capablty to dramatcally crease the accuracy of resultg polyles. A ew method of le fragmetato suggested ths paper dffers from the method descrbed [] by usg a pror formato that the groud truth le s a orthogoal le. Statemet of the Problem Let p where = 0... be vertces of polygoal le P. Let q -th vertces dvde P to a set of ooverlappg polygoal fragmets ad Q = { q0 = 0 q... qm = } be a set of dexes of the decomposto pots of a source polygoal le where m s a umber of segmets. Suppose that cardal drectos of the sought orthogoal polygoal le are horzotal ( ) ad vertcal ( ) drectos. Let X be oe of the cardal drectos ad X be a drecto perpedcular to X. Let L ad L be les havg cardal drectos X ad X ad mmal X X tegral stadard devatos ε X q q ad X ε q q from the correspodg p q

4 Recostructo of Orthogoal Polygoal Les 465 fragmet pq pq ) lmted wth q -th ad q -th vertces where ( =... of the source polygoal le (see Fg. 3). I Appedx there s a algorthm for buldg such les. m Fg. 3. Polygoal le P ad horzotal ad vertcal les ( L ad approxmatos of fragmets p q p ) ad q p ) = 0 ( 0 q L = p where 0 ( q The measure of the error of the orthogoal polygoal le approxmato s ( q q q q q q 0 m m ) are X X F Q X m) = m ( ε ε... ε ) () where s a pealty for each straght segmet of the resultg orthogoal polygoal le the secod tem s the sum of the tegral stadard devatos X s the drecto of the frst segmet ad s the drecto of the last segmet of the orthogoal polygoal le. X X Les L ad L are used for buldg orthogoal polygoal les. The vertces of the orthogoal polygoal les are the tersectos of adacet perpedcular les X X L ad L. The begg ad the ed of the orthogoal polygoal les are X proectos of pots p 0 ad p o les L ad L m. The task s to fd such set ˆ {ˆ ˆ... ˆ } Q = q q q ad values Xˆ ad ˆ that do 0 mˆ the value of the fuctoal () mmal. Ths set correspodg orthogoal polygoal le Qˆ drectos Xˆ ad ˆ ad Rˆ are optmal oes.

5 466 A. Grbov ad E. Bodasky 3 Iteratve Algorthm of Decomposto of a Polygoal Le to Fragmets Let k be a mmal error of approxmato of polygoal le P k wth orthogoal R k whe ad are fxed. The correspodg set of dces of P s Q k. The upper dex s a polygoal le decomposto pots of polygoal le k oretato of the last straght segmet of the orthogoal polygoal le. The mmum value of approxmato error of the polygoal le P k f the dex of the ext to last elemet s. The correspodg set ad orthogoal polygoal le k k are deoted Q k ad R. Obvously k = where = 0... k. Therefore k s the mmal value of ε k () for < k k 0. If the oretato of the frst segmet of the sought orthogoal polygoal le ukow ad so ca be horzotal or vertcal the 0 = = Rˆ s. (3) Suppose that all mmal errors of approxmato ad of polygoal le P k ad correspodg sets Q ad Q where =... k are kow. If mmum errors k ad k of P k ad correspodg set Qk ad Qk could be foud a teratve algorthm ca be buld to evaluate a optmal set Qˆ ad optmal orthogoal polygoal le Rˆ. Usg the method of least squares a horzotal or vertcal le ca be bult through the fragmet of the source polygoal le betwee vertces p ad p k ad the stadard devato ε k ad the mmal value of approxmato error k ca be calculated. By aalyzg k where = 0... k the mmum value of k ad correspodg value = ca be foud. Smlarly t ca be foud =. Sets Qk ad Q k ca be foud by addg k to sets Q ad Q respectvely (atteto must be pad to the sequece of superscrpts). Thus the problem s resolved.

6 Recostructo of Orthogoal Polygoal Les 467 Repeatg ths procedure tmes sets Q ad the mmum values of fuctoal () for gve values of ad. Q wll be obtaed that provde If the last segmet of the resultg orthogoal le must be horzotal or vertcal the the optmal sets are Q or Q respectvely. If there s o such requremet for the last segmet of the resultg orthogoal le the the optmal set s ˆ Q < ; = Q otherwse. Q (4) Fg. 4 shows the source polygoal le ad orthogoal le obtaed usg ths method. Fg. 4. Orthogoal polygoal le obtaed usg the descrbed method for = 30 (a - the source polygoal le b - orthogoal polygoal le c - the source ad the result le together) 4 Optmzato of the Iteratve Algorthm The algorthm descrbed above has the square calculatg complexty. It ca be used whe the source polygoal le does ot have may vertces for example a outle of buldgs maps of mddle scale. I the case whch the source le has may vertces the suggested algorthm ca cause a essetal delay. It s especally admssble for teractve modes. A techque to reduce the calculatg complexty of the descrbed algorthm s suggested. There are equaltes that ca be used for defg f a gve part of the polygoal le P k ca cota the ext to last pot of decomposto that mmzes k. Ths makes t uecessary to aalyze every vertex p where = 0... k of the polygoal le whle fdg a ew optmal pot. Ths techque s based o two obvous equaltes.

7 468 A. Grbov ad E. Bodasky The frst arses from the fact that a error of the optmal approxmato of a polygoal le wth two straght segmets s ot greater tha the error of the optmal approxmato of the same polygoal le wth oe straght segmet. ε where q q q X X X q ε q q ε q q q (5) 3 The secod equalty arses from the fact that the mmum error of the approxmato of some part of the polygoal le s ot greater tha the mmum error of the approxmato of the whole polygoal le. m{ q q } q (6) 0 q q ad s a drecto orthogoal to the drecto. where From equaltes (5) ad (6) t follows (see a Appedx ) that for q q < q k : 0 ~ q k q q k (7) ~ q k q k (8) where ~ { } q q k = m q q q k ε (9) P Deote ~ { } q k m q q q k ˆ = ε. (0) ~ ~ { q q k q k } q q k max =. () Suppose the value of the fuctoal of some decomposto of the polygoal le k where the ext to last pot of the decomposto does ot belog to the halfterval [ ) q was calculated. Deote ths fuctoal q q of decomposto for whch becomes mmal ca be located sde halfterval [ q q ) oly f q k F. The ext to last pot ˆ q q k F. () If ths codto s ot met the ext to last pot of decomposto does ot belog to [ q q ) ad ths half-terval ca be skpped. Usg ths t s possble to accelerate a search at each step of the descrbed teratve algorthm.

8 Recostructo of Orthogoal Polygoal Les Buldg a Close Orthogoal Polygoal Le The task of aalyzg a case whe the source polygoal le s closed as for example the borders of buldgs or other area obects ca be resolved by reducg t to the prevous oe. Frst t s ecessary to ope a source polygoal le other words to fd the begg. The frst pot ad the ed pot of the source le cocde p 0 = p. Let the frst pot be the upper-left vertex of the source le. Because of such choce of the begg of the polygoal le the error of approxmato s ot mmal but the addtoal error s small. Reorder the vertces so that a ew source polygoal le passes aroud the area obect a clockwse drecto. I ths case the frst segmet of the orthogoal polygoal le s horzotal. Therefore the last segmet must be vertcal because the le s closed. Whle the dervg the above algorthm to buld a orthogoal polygoal le t s assumed that the frst segmet of the orthogoal le ca be ether horzotal or vertcal. Substtutg 0 = 0 (3) stead of codto (3) the orthogoal polygoal le wth horzotal frst segmet X = s obtaed. Because the last segmet must be vertcal the codto Q ˆ = Q s used stead of codto (4). 0 = Fg. 5. A buldg ad approxmatos of ts border wth orthogoal polygoal les (a - raster mage of the buldg wth ose ad b-f - approxmatos of buldg borders obtaed wth accordgly) =

9 470 A. Grbov ad E. Bodasky Fg. 6. Image of three buldgs ad correspodg orthogoal polygoal les Fg. 5 shows a moochrome mage of a buldg (wth ose) ad orthogoal polygoal les obtaed wth dfferet values of. Fg. 6 shows a fragmet of a scaed map wth three buldgs ad orthogoal polygoal les obtaed wth the suggested method. 6 ow to Fd Cardal Drectos Usually cardal drectos are ot kow advace. Sometmes dfferet obects have dfferet cardal drectos (see for examples the borders of buldgs Fg. 6). I these cases the orthogoal polygoal les are bult N tmes wth oe of the cardal drectos α = h where h = 90 o / N ; = 0... N. (4) The coordate system s rotated to the agle α ad the task s resolved wth horzotal ad vertcal cardal drectos. The orthogoal polygoal le wth mmal error of approxmato s the desred soluto. The t s ecessary oly to tur t back through agle α. The value of N depeds o the requred precso. Usg dchotomy t s possble to crease the precso wth the same N. 7 Cocluso I ths paper the optmal method s suggested to recostruct orthogoal polygoal les after vectorzato. Ths method s based o the dyamc programmg techque. Because of errors caused by scag barzato vectorzato ad other processes eve straght les become polygoal les. Oe of the goals of postprocessg s ose flterg. I [] a ew method was suggested for flterg errors of vectorzato.

10 Recostructo of Orthogoal Polygoal Les 47 A polygoal le obtaed as a result of the raw vectorzato s dvded to o overlappg fragmets. The method guaratees the mmum value of the fuctoal that depeds o precso of approxmato of resultg parts wth straght segmets ad o the umber of parts or the umber of segmets of the result polygoal le. Ths method uses oe parameter a pealty for each straght segmet of the resultg polygoal le. The error of approxmato s calculated as tegral stadard error. It s possble to modfy the method usg aother measure of the error. By fdg tersectos of straght les obtaed as a optmal approxmato of fragmets a ew polygoal le ca be bult. Fg. 7. A fragmet of the cty map from Fg. wth borders of orthogoal buldgs (the result of processg by ArcSca for ArcGIS) The method descrbed ths paper s a modfcato of the method from []. odfyg ts method wth a pror formato that the sought polygoal le s a orthogoal le provdes the method descrbed ths paper. Ths method guaratees that the resultg polygoal le wll be a orthogoal le wth almost mmal error compared to the source orthogoal polygoal le. The method s a combatoral oe ad has the quadratc computato complexty. There was a suggested optmzato that reduces the umber of aalyzed solutos whch essetally creases the speed of resolvg the task. After optmzato the algorthm ca be used for recostructo of the orthogoal polygoal les from the source polygoal les wth a large umber of vertces whch s commo for polygoal les obtaed wth vectorzato. The method has bee geeralzed for closed orthogoal polygoal les for example borders of buldgs maps. The polygoal le must have segmets roughly equal sze to a pxel; otherwse t s ecessary to perform desfcato.

11 47 A. Grbov ad E. Bodasky The method ca also be geeralzed for the followg cases: Decomposto of the source polygoal les to fragmets some of whch are sgular (wth zero legth) Polygoal les wth a fxed agle betwee adacet segmets dfferg from 90 Polygoal les wth the arbtrary umber of permssble drectos -dmesoal polygoal les where > 0 The method has bee mplemeted ArcSca for ArcGIS. Examples Fg. 4-7 show the results obtaed wth the suggested method. Refereces. Grbov A. Bodasky E.: A New ethod of Polyle Approxmato. Structural Sytactc ad Statstcal Patter Recogto Portugal LNCS 338 Sprger (August 004) Phllps I.T. Chhabra A.K.: Emprcal Performace Evaluato of Graphcs Recogto Systems. IEEE Trasactos o Patter Aalyss ad ache Itellgece ol. No. 9 (September 999) Davd. Douglas ad Thomas K. Peucker: Algorthms for the Reducto of the Number of Pots Requred to Represet a Dgtzed Le or Its Carcature. Caada Cartographer ol. 0 No. (December 973) - 4. Pavlds T. awyk C.J.: A Automatc Beautfer for Drawgs ad Illustratos. Computer Graphcs ol. 9 No. 3 AC Press (July 985) 5-34 Appedx : orzotal ad ertcal Les Approxmatg Some Part of the Polygoal Le Let t be a parameter equal to the dstace from the begg of the polygoal le tll the curret pot alog ths le. Let l ad l be values of the parameter defg the begg ad the ed of the aalyzed part of the polygoal le. The horzotal le approxmatg a gve part of the polygoal le ca be defed as The vertcal le ca be foud smlarly y x l y = where y = () l l y t l dt. l x = where x = () l l x t l dt.

12 Recostructo of Orthogoal Polygoal Les 473 Itegral stadard devatos of these straght les are defed as l = y y l () t dt ( l l ) ε l = x x l () t dt ( l l ) ε. Appedx : Dervato of Iequaltes (7) ad (8) 0 Let q q < q k. From equaltes () ad (6) ad obvous equalty possble to obta { } q k q q q k expresso (9). From equaltes () ad (5) follows ε q k q k ε t s m ε other words q k q q q q k ε. ε q q q m q q From a obvous equalty { } follows q k m { q q } q k ε t ε or expresso (0).