CHAPTER 3 BACKGROUND AND RELATED WORK

Transcription

1 CHAPTER 3 BACKGROUND AND RELATED WORK Ths chpter wll provde the ckground knowledge requred for the rest of ths thess. It wll strt wth the defnton of the nput sgnl functon nd termnology relted to the trngulton mesh. Ths wll e followed y dscusson of the trngulr representton, the mge reconstructon method nd ts error metrc. The defntons of Deluny nd dt dependent trngulton wll lso e presented n ths chpter. Fnlly, ths chpter wll end wth dscusson of grph theory for trngulr mesh pplctons. 3. Input nd Output Dt Usully nturl gry-scle mge contns lot of redundnt nformton. Ths redundncy n dt cn e used for fle compresson so tht less storge or trnsfer cpcty wll e requred. The smlrty n ntensty nd rte of chnge t dfferent scles re some of the duplctons n dt tht cn e used n compresson. Snce t s possle to segment some regons nto trngles wthout sustntlly degrdng the qulty of the mge, trngulr mesh s good pproch for dt compresson for storge reducton nd dt processng. 3.. Gry-Scle Imge A gry-scle mge s usully presented n the two-dmensonl domn. It cn e vewed s set of fnte ntensty (or mgntude) vlues eng smpled t fnte ntervls wth tme (or spce) domn. Ths cn e mthemtclly expressed s: { (, ),,...,,,..., j j j } V = v v = f x y Ω = N j = N2 (3.) Ω N 2 where represents the prmeter domn nd nd N represent the numer of smples long ech dmenson of the mge. Menwhle, the ntensty of the mge, whch cn e found t ech of the smple ponts, cn e defned s: 22

2 (, ) f x y = z j,,..., N nd j =,..., N (3.2) j = 2 where represents the quntzed vlue of the ntensty t smple locton ( x, y. Normlly z j ths vlue s n n nteger tht rnges from 0 to 255. j ) 3..2 Trngulton A trngulton s prttonng of the 2D mge plne nto set of trngles. It my e represented usng set of pproprte vertces, whch re locted on the orgnl grd of smple ponts, nd set of edges tht connects those vertces. A vertex cn e represented y three prmeter vrles s (,, ) V = x y I N (3.3) 2 3 V where s the numer of vertces found n the trngulton. Note tht ( x, y s the NV 2 smpled locton of the vertex V nd I s the quntzed ntensty t tht locton. ) In ddton to vertex nformton, trngulton requres vertex connecton nformton. Ths reltonshp s usully defned n the form of edges. An edge s lne segment tht connects two vertces nd t cn e expressed y E ( V, V ) =, N (3.5) 2 2 E A trngle s plnr ptch, whch s determned y three non-collner ponts tht re connected y three lne segments. Ths ptch cn e defned y T ( V, V, V ) =, N (3.4) T where the three vertces V,V nd V re locted t three dfferent smple ponts n the twodmensonl x y plne nd N s the numer of trngles. The trngulton, Γ, n fnte prmetrc domn, Ω 2 R j T k, must stsfy four condtons: () Let V e the set of ll vertces found n Γ such tht V V j = φ. V V () E E s ether φ or vertex f j. j N T () Ω= T, where N s the numer of trngles n gven trngulton. = T (v) T T j s ether φ or common edge E, or common vertex Vk f j. k 23

3 Therefore the trngulton Γ V cn e defned s Γ = { T =,..., N } (3.6) V T Snce t s common to defne n opertor for the numer of elements n set s, we defne N =Γ V s the numer of trngles n the trngulton, whch covers the prmeter T Γ V domn 2 Ω R nd NV = V s the numer of vertces n the trngulton Γ V. Fgure 3. nd 3.2 llustrte some of the regulr smpled grd trngultons nd how the smplng cn ffect the qulty of the mge. The orgnl mge of crter lke s shown n Fgure 3. () whle () shows n exmple of the loctons of ponts on the 6x6 regulr smple grd. Fgure 3.2 (-c) show the trngultons tht re smpled t dfferent regulr fnte ntervls of 4, 8 nd 6 respectvely. Snce the mge sze s 256x256, the grd szes re 64x64, 32x32 nd 6x6 respectvely whle the reconstructed mges re shown n Fgure 3.2 (d-f). These mges re reconstructed usng the Gourud shdng technque, whch wll e explned n Secton 3.3. Although denser trngulton gves etter pproxmton of the orgnl mge, t requres more vertces nd trngles for mge representton. Γ V 3.2 Deluny nd Dt-Dependent Trngultons There hs een much reserch concernng the mnmzton of error usng refnement methods. Some of these methods re greedy nserton [De Florn et l. 85], feture-sed [Scrltos nd Pvlds 92], nd herrchcl sudvson [De Florn et l. 84]. The greedy nserton lgorthm scns for vertex locton where totl error reducton s cheved. Ths method produces good pproxmton of the orgnl dtset. However t s tme-consumng to serch for n optml pont. Ths drwck motvtes the serch for fster lgorthm. A feture-sed method s proposed usng the fetures to determne the vertces or edges of the trngulton [Chen nd Schmtt 93]. Edge detectors nd Lplcn flters hve een proposed to detect some of these fetures for vertex nserton n mesh refnement [Southrd 9]. 24

4 () () Fgure 3. Regulr smple grd. () Orgnl mge of crter lke of sze 256x256. () Loctons of smple ponts tken y the regulr smplng grd of sze 6x6. () () (c) (d) (e) (f) Fgure Approxmton of n mge usng regulrly smpled trngulr mesh. Trngulr meshes wth regulr smplng grd sze of () 64x64, () 32x32 nd (c) 6x6. Reconstructed mges usng Gourud shdng on the regulr smplng grd of (d) 64x64, (e) 32x32 nd (f) 6x6 respectvely. Another pproch s herrchcl sudvson [Scrltos nd Pvlds 92], whch recursvely sudvdes trngle nto smller su-trngles to represent etter pproxmton of the orgnl dtset. The dvntge of ths method s ts fst computton nd use of multresoluton modelng. Its reconstructed mge qulty, however, s no etter thn tht of the greedy nserton method. Besdes these refnement processes, one of the eqully mportnt decsons for trngle representton s to determne the choce of trngles wthout consderng error metrcs. One of the proposed methods for choosng trngles s Deluny trngulton [Kroptsch nd Bschof 0]. Generlly, ths type of trngulton s known s the dul of the Vorono 25

5 dgrm, whch segments the prmeter domn nto regons n such wy tht ponts lyng n the sme regon re locted nerest to the vertex of ts regon. A Vorono polygon s defned y κ ( ) (, p H p p j) j ) (3.7) where p s dscrete pont (or vertex) of nterest nd H( p, p s hlf-plne tht contns 2 the set of ponts tht le n R nd re locted nerer to thn to p. Therefore, f there re N vertces n the domn, ech Vorono polygon cn result from the ntersecton of t most N hlf-plnes. These polygons re convex wth no more thn N sdes. p j j Deluny trngulton cn e cheved from Vorono dgrm y connectng vertces, whose regons n the Vorono dgrm ntersect. The result of usng the Deluny scheme s the trngulton tht mxmzes the mnmum ngles of ll trngles. In other words, n Deluny trngulton, the crcle tht crcumscres three vertces of ny trngle contns no other vertces. Fgure 3.3 elow shows n exmple of Vorono dgrm wth ten vertces of nterest nd ts correspondng Deluny trngultons, whle Fgure 3.4 shows n exmple of Deluny trngulton nd non-deluny trngulton. Fgure Exmple of Vorono dgrm nd ts Deluny trngulton. () Vorono dgrm nd ts ponts (or vertces) of nterest. () Correspondng Deluny trngulton. 26

6 Fgure 3.4 Concept of Deluny trngulton. () In Deluny trngulton, no other vertces le n the crcumscrng crcle of ech trngle. Crcle C does not nclude vertex V 4 nd crcle C 2 does not contn vertex V. () Non-Deluny trngulton. Crcle C contns more thn three vertces. It ncludes vertces V, V, V nd V Although the Deluny trngulton method s fst n regulrzng the trngulton, t does not gurntee n optml soluton to the pproxmton. In mny pplctons, the choce of Deluny trngulton cn sgnfcntly ncrese the pproxmton error. It s found tht non-deluny trngles cn produce etter result n most cses ecuse long, slver trngles cn gve etter pproxmton, especlly long the edges, where lrge chnges n ntensty occur. Fgure 3.5 compres the results of reconstructed mges of Len usng oth Deluny trngulton nd non-deluny trngulton. An edge swp operton, whch wll e dscussed n Secton 5.4., s performed on the 6x6 regulr grd trngulton to mprove the reconstructon mge qulty. It cn e oserved tht the Deluny trngulton tends to produce lot of npproprte trngles long the edge n the mge. Ths s ecuse t does not llow the thn, slver trngles n the set. However, the dt-dependent trngulton, whch permts slver trngles, yelds etter rough pproxmton to the mge dt. The reson s tht chnge n ntensty cn e represented more ccurtely y thn, slver trngles y short, ft trngles. However t s possle to comne these two mportnt prmeters to cheve etter result. The choce of comnng Deluny nd dt-dependent trngles wll e dscussed n Secton

7 () () (c) (d) (e) Fgure 3.5 Comprson of Deluny trngulton nd non-deluny trngulton. () Orgnl mge of Len. (-c) A Deluny trngulton nd ts reconstructed mge. (d-e) A non-deluny trngulton nd ts reconstructed mge. 28

8 3.3 Gourud Shdng There re mny technques to perform shdng of trngle. The choce of renderng not only helps the trngulton-sed mge look more relstc, ut lso smooths the edges etween djcent trngles. The shdng technque used n ths thess s sed on the Gourud shdng technque [Wtt 93]. Gven sequence of three vertces (V,V nd V ) nd ther ntensty 2 3 vlues (, nd I ), t s possle to perform Gourud shdng on trngle y lner I I2 3 nterpolton. Fgure 3.6 shows the concept of how Gourud shdng cn e performed. To render trngle, frst, ts mxmum nd mnmum vlues ( y nd y ) n the mx mn y (vertcl) drecton re determned. These two vlues re used for the vertcl scn lmtton. Next, for ech row strtng from y to y, ts horzontl oundres ( x y s nd x ) nd ther ntenstes ( nd I ) re clculted y nterpoltng necessry pr of I vertces, whose edges gve the lmtton of the horzontl oundres. ( ) ( x ) 2 x s ( y y ) x = y y + 2 ( ) ( x ) 3 x s ( y y ) x = y y + 3 ( ) ( I ) 2 I s ( y y ) I = y y + 2 ( ) ( I ) 3 I s ( y y ) I = y y + 3 mx mn x (3.8) x (3.9) I (3.0) I (3.) These nterpolted vlues not only smooth the ntensty etween two vertex ponts ut lso smooth the connecton etween two connected trngles. Ths s ecuse the two connected trngles shre the two vertces of sme locton nd ntensty. Fnlly, n the sme wy, the ntensty nsde the trngle cn e clculted y the horzontl scn of xs from x to x, the ntensty nsde the trngle, I, cn e clculted s ( ) ( I ) I ( x x ) I = x x + s s s x s. Wth ech vlue of I (3.2) 29

9 y mx (,, ) V x y I Vertcl Scn of s y ( x, ys, I ) ( x, ys, I ) V ( x y I ),, c c c y (,, ) V x y I x y mn (,, ) V x y I Fgure 3.6 Gourud shdng lner ntensty nterpolton. For computtonl effcency, ncrementl clculton s preferle: x I = s ( ) ( I ) I x x (3.3) Isn, = Isn, + Is (3.4) where s the ncremented chnge n ntensty nd n s the vlue from x to x. Ths I s method sgnfcntly mproves the clculton tme y replcng multplcton opertor wth ddton. However one of ts drwcks s ts mprecson. Snce there s round off n error tends to get lrger s n ncreses. I s, the One of the dsdvntges of employng Gourud shdng s vsul sde effect known s Mch ndng. The chnge n ntensty trggers the humn vsul system to perceve rghter or drker nd of ntensty. Ths s due to the hgh senstvty n the humn vsul system to the frst dervtve of ntensty, whch s used to detect nd enhnce edges [Wtt 93]. Fgure 3.7 shows the Mch nd phenomenon. Notce the overshoot tht occurrng n the perceved sgnl reltve to the ctul rmp sgnl. 30

10 Fgure 3.7 Mch nd phenomenon. () Actul sgnl ntensty. () Sgnl ntensty perceved y humn vsul system. 3.4 Error Metrc Snce reconstructed mge s n pproxmton of the orgnl, there wll e n error etween the orgnl nd the reconstructon. Therefore t s necessry to defne n error metrc to mesure the qulty of the reconstructed mge. Let e plnr ptch tht pproxmtes trngulr regon, T, n n mge. For =,2,..., NT, f T must stsfy plne equton defned y ( ) P x, y = x+ y+ c, whch cn e determned y the three vertces formng the trngle T. In the prmeter domn, ths pproxmted ptch cn e defned s:, xy, Ω T ft ( x, y ) = (3.5) 0 otherwse f T P( x y ) ( ) ( P ( x, y ), = g( z ) =, 2, 3 (3.6) ( ) Ω T 2 where defned the oundry of the prmeter domn of the trngultons nd, nd re the vertces of trngle. The functon g z s used for etter pproxmton 3 T ( ) when nose or hgh frequency component s present n the dt. Snce the wvelet trnsform yelds oth pproxmton nd detl of the orgnl dt, the functon ) g( z) cn tke 3

11 dvntge of ths computton. Wth the defnton of the pproxmton of trngles, the pproxmton of the trngulton cn e defned s f t = Γ V = whch ccounts for the pproxmton of f T N T (3.7) non-overlppng trngles. The error metrc, whch s shown n Fgure 3.8, cn e denoted y f fγ V. It s normlly mesured y the verge of the sum of squred errors, whch s normlly clled men-squred error (MSE). MSE cn e defned s = Ths s computtonlly equvlent to ( ) ( ) 2,, 2 f x y f x y dxdy x y (3.8) Γ v y x M N MSE = f x y f x y MN Γ V y= x= ( ) ( ) 2,, (3.9) where M nd N denote the heght nd wdth of the mge respectvely. Another populr error metrc s the pek-sgnl-to-nose rto ( PSNR ). It s defned y PSNR mx = (3.20) 20log 0 I MSE where I mx s the possle mxmum ntensty vlue n of the mge. Usully, ths vlue s 255 for gry-scle mge. In reconstructon, PSNR, tht s lrger thn 30 db, wll often pper to hve lttle or no vsle degrdton. However ths stll depends on the mge tself. 3.5 Grph Theory Snce the pproxmton lgorthm n ths thess s sed on trngulton, the twodmensonl mge hs to e segmented nto mny trngulr regons, whch re represented y three vertces. One of the solutons to present ths type of dt structure s to use n undrected grph. A grph G = ( V, E) conssts of set of vertces, V, nd set of edges, E, whch defnes reltonshp etween the vertces. 32

12 ( x, y, I ) Dt ( x, y, I) I Error d z (,, ) I x y I c c c c ( x, y, I ) ( x, y, I ) y x Fgure 3.8 Error clculton for the Gourud shdng reconstructed mge. To construct vld trngulton, there re mny cutons tht should e mde. Frst, nsted of generl grph, the dt structure should preserve the smple plnr grph. Otherwse, t voltes the fourth condton of the defnton of trngulton, whch sttes tht the ntersecton of two trngles should e equl to zero or n edge. Fgure 3.9 shows exmple of generl grph nd smple plnr grph. Notce self-loop, z, n edge tht connects vertex to tself, s found n generl grph. Ths self-loop lso must not exst n trngulton. B A w u s r v t C D A F B E x E y () z D () C Fgure 3.9 Generl nd smple plnr grphs. () Generl grph. () Smple plnr grph. 33

13 Snce the grph s mthemtclly defned y ts vertces nd ther nry relton, the postons of the vertces nd the curvture of the edge re not fctors used to consder the equvlence of the two grphs. Fgure 3.0 shows two equvlent grphs, whch hve dfferent representtons. z A v B P N w x t s q C y D O r M u () Fgure 3.0 Equvlent grphs. () Orgnl grph. () Equvlent grph wth dfferent representton. () It s possle to show tht these two grphs re equvlent y the jecton mppng of the vertex nd edge nmes. The followng shows the equvlences to show tht they re the sme grphs. Vertex Equvlent A M B N C O D P Edge Equvlent v q w r x s y t z u Two grphs G ( V, E nd G = ( V, E re somorphc f there exsts t lest one jecton f : G G, such tht ( UV ) E f nd only f f ( U), f ( V) E. Ths vertex nd edge jecton cn e defned y: = ) ) ( ) f : V V nd f : E E (3.2) v G G e Furthermore, the composte functon f lso nherts the somorphsm property from the two somorphc functons. The composte functons of two somorphc functons cn e denoted y ( ) 2 2 G G f = f f = f f (3.22) 34

14 In the specl cse of smple grph, f there exsts vertex jecton f : V V such tht v G G f ( x ) s djcent to f ( y ) f nd only f x s djcent to y for ll ( xy, ), the two grphs re somorphc. Therefore only vertex jecton s suffcent to show somorphsm of the two smple grphs. Ths property s essentl to desgnng templtes for ntl trngulton, whch wll e dscussed n Secton V G 35