Chapter 09.00D Physical Problem for Optimization Computer Engineering

Transcription

1 hapte 9.D Phsal Poblem o Optmaton ompte Engneeng Poblem Statement An mage s a olleton o ga level vales at a set o pedetemned stes knon as pels hh ae aanged n an aa. hese ga level vales ae also knon as mage ntenstes. Fge 1: On the let s a tpal mage olleted sng a amea. On the ght s a thee dmensonal shape data o the ae. On the let o Fge 1 s an eample o an mage that e ae amla th. Eah pont n that mage an be ndeed sng to oodnates the o and olmn nmbe o and v. It s to dmensonal o 2D o shot. Fo eah v e have an ntenst o olo vale. Gven jst an mage t s not possble to eate ho an objet ll appea veed om anothe deton. Fo nstane gven jst the ontal ve o the ae t s not possble to geneate a sde o ttled ve o the ae. Hoeve n addton to the 2D mage ve e have thee dmensonal 3D shape data then t s possble to do ths. hs s hat e ll eploe n ths poblem. An eample o the 3D shape s sho n Fge 1 on the ght. Eah pont n that data epesents an atal 3D pont n spae ndeed b oodnates. aken 9.D.1

2 9.D.2 hapte 9.D togethe the epesent the shape o the ae. What s mssng s the tete o the olo nomaton. he 2D mage gves s that tete nomaton. Note that o dspla pposes e have shaded the sae th atal lghtng sng Matlab s path nton.. he atal data s basall a set o 3D ponts as shon n the 3d plot Fge 2. hee ae man deent 3D ameas on the maket. We sed the Kona-Mnolta amea o olleton o the ae data. Fge 2: hee-dmensonal ponts on the ae. Solton Phss o the Poblem We stat b onsdeng the geomet o mage omaton. Ho ae the 3D oodnates n the old elated to the 2D pel loatons n the mage? Lght om a 3D pont P passes thogh the amea lens and egstes on a pont P n the mage as depted on the let n Fge 3. We have abstated the amea to be a pn-hole amea. he eqatons get omplated hen onsdeng the atal lens sstem bt the essene o the geometal elatonshp s apte b the pn-hole model o the amea. he ndelng geomet an be depted as n Fge 3. A pont P n the old s pojeted onto a pont p n the mage. he lne pp passes thogh the lens ente.

3 Phsal Poblem o Optmaton: ompte Engneeng 9.D.3 Fge 3: Pojetve geomet. oodnates o a plane th espet to the mage ae elated to the old pont oodnates. We have to oodnate sstems one ooted n the 3D old hh e ll ee to as the old oodnate sstem and the othe ooted at the ente o the mage alled the amea oodnate sstem. he amea-based and old-based oodnates o an gven 3D ponts P ae P [ ] and P [ ] espetvel. hese to oodnate vales ae elated va a gd otaton R and tanslaton : P RP + o epandng n mat notaton e have he otaton mat an be epessed as a podt o thee otaton mates eah aptng otaton along one o the thee oodnate aes.

4 9.D.4 hapte 9.D R R R R o n mat om 1 os sn sn os os sn 1 sn os os sn sn os 1 R he mage oodnate o the pojeted pont v p ll be elated to the amea-based oodnates o the 3D pont P ] [ sng hat s knon as the pespetve pojeton eqatons. v hee s the oal length o the lens apted b the dstane beteen the pn-hole opt ente and the mage plane. hese eqatons an be deved sng ato-based elatonshps n smla tangle geomet. Gven these geometal elatonshps one an detl elate the old-based oodnates o a pont th ts mage oodnates. It s a nonlnea elatonshp nvolvng a ato o sn and osne ntons th the oal length 3 otaton angles and the 3 tanslaton vales as the paametes. ; ; V v U O task s to egste the gven 2D tete map th the 3D shape data. Fst e ll have to mak oespondng ponts beteen the to data. Fo nstane note the oodnates o the nose tp n the mage and the 3D data. Othe aal eates that an be easl oesponded ae the nsde and otsde ones o the ees ponts on the nostl bonda makng on the oehead and so on. hese oespondenes ll gve s a lst o N paed pont sets: } { N v. Gven ths e have to nd the best otaton and tanslaton vales that ll egste the gven ponts. Ate e have estmated the otaton and tanslaton vales then e an easl egste est o the ponts n the mages sng the pespetve pojeton eqatons that e have otlned eale. hs o optmaton poblem at these ponts s gven b ; ; mn ag V v U N + Note that e ae optmng the dstane beteen the mage loaton o the 3D ponts based on the estmate otaton tanslatons and the atal obseved loaton n the mage.

5 Phsal Poblem o Optmaton: ompte Engneeng 9.D.5 Woked Ot Eample We selet the oodnates o the aal eates as shon n Fge 4. he oespondng ponts ae gven n the ollong able 1. able 1: Vales o pojeted pont at oodnates. v

6 9.D.6 hapte 9.D Fge 4: Faal eates that ee seleted to estmate the tansomaton beteen and the mage and 3D ae data ae maked th ble les. he oespondng ponts ae also seleted om the 3D ae data. Sne the optmaton om s a sm o sqaes e sed the MALAB nton lsqnonlne to nd the optmm otaton and tanslaton vales. Fo ths optmaton e kept the oal length ed at 1. O ntal estmates o the otaton and tanslaton vales ee all eo. Fge 5 shos the hange n the optmed vale th teaton. We see that the eo stables n 25 teatons o ths eample. he nal esdal as he estmated otaton angles ee degees and the tanslatons ee

7 Phsal Poblem o Optmaton: ompte Engneeng 9.D.7 a b Fge 5: o-dmensonal sles thogh the s dmensonal eo nton. a Angles and ae vaed hle the est ae held at the ond solton vales. b Angles and ae vaed hle the est ae held onstant at the ond solton vales. Some eamples o the tpal objetve nton ae vsaled n Fge 5. he objetve nton s s-dmensonal so t s had to vsale. We have shon hee some todmensonal sles thogh the nton. In othe ods to o the vales ae held onstant hle the othes ae held ed at the ond optmal vales. Note that the vaables plotted hee ae angles hene the aes ap aond. We see that the sae tsel s qte smooth; hoeve thee ae mltple soltons. heeoe t s mpotant that the ntal ondton be hosen to all n the bket oespondng to the solton. he otnate haatest s that the bkets ae a apat. Usng these estmates e an map the est o the mage tete onto the 3D ae data. Fge 6 shos some eample ves o the nal mapped data. We an mage the ves o the ae as t old appea om an angle. We ae done!

8 9.D.8 hapte 9.D Fge 6: Vaaton o the optmed eo th teaton. Fge 7: Mltple ves o the egsteed tete and shape data. We an no geneate ves o the ae om an angle. he ollong les ae avalable on the eb. ae mage VRML omatted 3D data o the ae. Matlab le to ead the VRML data ode sed to geneate the eample eslts.

9 Phsal Poblem o Optmaton: ompte Engneeng 9.D.9 Qestons and Assgnments 1. In o eample e had kept the oal length ed dng the optmaton. What happens one also delaes that as an nknon nto the optmaton? 2. Whh paametes ae the most senstve ones? Rotaton? anslaton? O oal length? 3. What s the senstvt o the hoe o the ntal vale? 4. Ae e alas gaanteed to get the global optmm vale? Unde hat ondtons? OPIMIAION op Phsal optmaton poblem o ompte engneeng Smma A phsal poblem o olo and tete o 3-D mages Majo ompte Engneeng Athos Sdeep Saka Date Deembe Web Ste