Image Registration - Agenda. Image Registration II. Optical Flow. Estimating Optical Flow. Dr. Yossi Rubner

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1 mage egisraio - geda mage egisraio Dr. Yossi ber ossi@rber.co.il licaios Comoes of image regisraio Oical flow Lcas-Kaade algorihm Similari measres ad oimali Bods o regisraio accrac 4/8/7 Oical low Esimaig Oical low Cosa brighess assmio Time Time d ( ) ( d d) ( ) ( d d d) Pierre Korrobs's Demo * rom Lihi Zelik-Maor Calech 4

2 Oical low Eqaio ( ) ( d d d) Saial coherece mose addiioal cosrais ssme he iel s eighbors hae he same () B firs order Talor easio (D): where Shoo! Oe eqaio wo eloci () kows d d d d M ( ) ( ) ( ) ( ) ( ) ( ) M M ( ) ( ) ( ) b b b ( ) b mage egisraio - geda Lcas-Kaade Eqaio licaios Comoes of image regisraio Oical flow Lcas-Kaade algorihm Similari measres ad oimali Bods o regisraio accrac b This is a sigle se i ewo-ahso

3 Lcas-Kaade lgorihm : image ad referece O: rasformaio T bewee ad.come mari for image.t 3.eea il coergece:.come ecor b.id b solig Lcas-Kaade eqaio 3.T T 4.War owards sig T Sm of Sqared Differeces SSD id ( ) ( ( ) ( ) ) mi SSD ( ) SSD is oimal i he sese of ML whe. Cosa brighess assmio. i.i.d. addiie Gassia oise eisiig he small moio assmio edce he resolio! s his moio small eogh? Probabl o i s mch larger ha oe iel ( d order erms domiae) How migh we sole his roblem? * rom Khrram Hassa-Shafiqe CP545 Comer Visio 3 * rom Khrram Hassa-Shafiqe CP545 Comer Visio 3 3

4 Pramid Creaio Coarse-o-fie oical flow esimaio.5 iels filer mask.5 iels Gassia Pramid Lalacia Pramid Creaed from Gassia ramid b sbracio L l G l ead(g l ) * rom ichard Szeliski Saford CS3B 5 5 iels image - iels image Gassia ramid of image - Gassia ramid of image Coarse-o-fie oical flow esimaio Oical low esls r ieraie L-K r ieraie L-K war & samle... image J - image Gassia ramid of image - Gassia ramid of image * rom Khrram Hassa-Shafiqe CP545 Comer Visio 3 4

5 Oical low esls Whe is LK Solable? b shold be ierible shold o be oo small de o oise eigeales λ ad λ of shold o be oo small shold be well-codiioed λ / λ shold o be oo large (λ larger eigeale) is solable whe here is o aerre roblem * rom Khrram Hassa-Shafiqe CP545 Comer Visio 3 The erre Problem The erre Problem Differe moios classified as similar Similar moios classified as differe sorce: a Eshel sorce: a Eshel 5

6 Classificaio of large λ small λ large λ large λ small λ small λ Α Oical low for ffie Trasformaio f c e d b a f e d c b a ) ( ) ( f e d c b a M M Oical low for ffie Trasformaio b f e d c b a mage egisraio - geda licaios Comoes of image regisraio Oical flow Lcas-Kaade algorihm Similari measres ad oimali Bods o regisraio accrac

7 7 SSD Oimali SSD is oimal i he sese of ML whe. Cosa brighess assmio. i.i.d. addiie Gassia oise E SSD SSD Oimali or each iel: e P π SSD Oimali co. or all iels i area E: cos e log logp P P π { } mi logp ma ormalized Cross-Correlaio CC is oimal i he sese of ML whe. liear relaioshi bewee he images. i.i.d. addiie Gassia oise CC

8 Eamle - CC 5 The Joi Hisogram CC Oimm YaXb SSD Oimm YX esi of Trasformed Targe re locaio CC SSD esi of eferece The Joi Hisogram Mal formaio esi of Trasformed Targe Mal formaio oimm -- Tighl clsered hisogram or mli-modal regisraio Usefl for medical imagig ssmes fcioal relaioshi bewee iesiies of images id he bes regisraio b maimizig he iformaio ha oe image roides abo he oher eqires o a riori model of he relaioshi esi of eferece efereces Shao mahemaical heor of commicaio 948 Viola ad Wells ligme b maimizaio of mal iformaio 995 Plim e al. Mal formaio Based egisraio of Medical mages: Sre 3 8

9 H Ero ( ) ( a) log ( a) a Joi Ero ( B) B ( a b) log B ( a b) H b a B H7.435 H ifreqe ee roides more iformaio ha a freqe ee Ero is a measre of hisogram disersio B ( B) H Joi Ero ( B) B( a b) log B( a b) H a bb Colligo [995] Bes regisraio miimizes he joi ero Mal formaio Defied i erms of eroies ( B) H ( ) H ( B) H ( B) H ( ) H ( B) H ( B) H ( B ) Problem - backgrod ca lead o low ero (B) is high whe is well-elaied b B (or ice ersa) Maimizig (B) is beer ha miimizig H(B) backgrod ca lead o low ero H() ad H(B) forces sigifica coe as well as low joi ero 9

10 Proeries of M (B) (B) (B) () H() (B) H() (B) H(B) f B are ideede he (B) f B are correlaed he (B) H() H(B) H() H(B) Esimaig Probabili Desiies Parze Widow Desi esimae P*(z) ( z) P*( z) ) ( z z j z j is he mber of rials i he samle is a widow fcio ha iegraes o Ofe a Gassia desi fcio Parze Widowig Eamle - M CC M

11 mage egisraio - geda licaios Comoes of image regisraio Oical flow Lcas-Kaade algorihm Similari measres ad oimali Bods o regisraio accrac egisraio Precisio How accrae ca we esimae ()? ssme i.i.d. addiie Gassia oise Use Cramer-ao lower bod (CLB) m Cramer-ao Lower Bod or biased esimaors he lower bod for esimaig he ariace of arameer ecor m is Where is he isher iformaio mari: r ois daa m arameers T r r E m m m m m log Pr log Pr [ ] m ˆ ii m i m i E D Shif Esimaio Bod isher iformaio mari: Lower bods for shif esimaio: () de ar de ar

12 iio ssme Higher oise lower regisraio accrac Higher deriaies higher regisraio accrac Bigger area higher regisraio accrac () ar ar mage deriaies racice we hae ol ad o he real where D is he -deriaie oeraor. Similarl: Ed Thak o CC Oimali or each iel: or all iels i : cos e log logp P P π { } mi logp ma

13 3 CC Oimali id oimal o logp logp eraie Solio Lcas Kaade (98) Use Talor easio of id where SSD mi mi Lcas Kaade co. Solig for : Solig for : racice we hae ol ad o he real where D is he -deriaie oeraor. Similarl: D D