Elastic Conformal Transformation of Digital Images

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1 Lubmír SOUKUP, Ja HAVRLANT, Odre BOHM, ad Mila TALICH, Czech Republic Key wrds: Cartgraphy, Geifrmati, Egieerig survey, Cadastre, Image Prcessig, Data quality, Accuracy aalysis, Bayesia apprach SUMMARY A vel trasfrmati mdel fr registrati f gemetrically distrted digital images is prpsed i the ctributi. The registrati methd is based a set f grud ctrl pits (GCPs) whse crdiates have bee captured with limited accuracy. The spatial iaccuracy f the GCPs iflueces precisi f trasfrmati betwee iput ad referece images. Quality f the trasfrmati is als affected by ukw liear elastic distrtis f the iput image. Simultaeus impact f the bth surces f iaccuracy results i spatial imprecisi f the trasfrmed image. Crrect estimati f the resultig spatial imprecisi is sigificat cstituet f the prpsed registrati methd. Theretical priciple f the prpsed methd stems frm thery f Gaussia prcesses (cllcati, krigig) ad is wrked ut with the aid f Bayesia apprach. The verall sluti embdies advatages f -parametric ad parametric estimati - it is bth data-drive ad tuable by a simple set f parameters. The prpsed methd f image registrati was implemeted as a web applicati by meas f up-t-date sftware stadards f Iteret techlgy. The applicati is freely available at fr ay Iteret user. The prpsed registrati methd ca be easily applied i may areas f gedesy ad cartgraphy, ad remte sesig, e.g. matchig maps, gereferecig f satellite r aerial images, spatial data quality maagemet i GIS, cadastre surveyig, defrmati mdelig etc. 1/10

2 Lubmír SOUKUP, Ja HAVRLANT, Odre BOHM, ad Mila TALICH, Czech Republic 1. INTRODUCTION Crdiate trasfrmati is frequet task i gedesy ad cartgraphy, amely whe a GIS is created r updated by a cmpsiti f digital images. Such digital images ca rigiate frm miscellaeus surces, e.g. aerial r satellite cameras, digitized aalgue maps, ifrared cameras, radar scees etc. Oe imprtat techique t cmpse differet digital images is image registrati. Applicati width f image registrati spreads ut widely ver the brach f gedesy ad cartgraphy. It has bee exteded t a umber f ther braches, amely medical imagig, rbt visi, micrscpy, vide ad multimedia prcessig, defrmati aalysis etc. All the applicatis f image registrati ca be divided i tw mai classes: chage detecti ad msaicig. The bth applicati areas are very prmisig tday. Cmprehesive verview f the image registrati methds ffers [5]. Trasfrmati mdels which have bee mstly used i image registrati are usually set up by parametric estimati. Typical example f such a parametric trasfrmati mdel is affie, plymial, perspective r splie trasfrmati. These trasfrmati mdels are easy t implemet, but accuracy aalysis f resultig registrati ca be misleadig whe the chse trasfrmati mdel is crrupted by sme ukw irregular factrs. This disadvatage ca be disslved by -parametric estimati methds which are based rather measured data tha sme artificial presumptis as plymial apprximati. Usage f -parametric methds is t s straightfrward ad therefre less ppular. Furthermre, cmputatial demads f parametric methds are higher. The prpsed methd f image registrati has advatageus features f the bth appraches. It is tuable by a explicit set f gemetrical ad statistical parameters. Simultaeusly, it is als data-drive sice the statistical parameters allw feasible matchig f the trasfrmati mdel t the measured data. The methd stems frm cllcati methd. Statistical prperties f cllcati (see [2]) are emphasized i this ctributi. Bayesia apprach [1] is applied t estimati f the statistical parameters. 2. PROBLEM FORMULATION 2.1 Required result Trasfrmati prcedure betwee tw digital images has t be desiged. The required trasfrmati has t cicide apprximately at sme grud ctrl pits (GCPs). The cicidece has t be as tight as precise the grud ctrl pits are. The required trasfrmati eed t be strictly liear (i.e. slight elastic distrtis are allwed), but has t be cfrmal. Spatial accuracy f ay trasfrmed pit has t be estimated as well. 2/10

3 2.2 Give assumptis Bth give images have their w crdiate systems. Crdiates f pits i the iput image are called iput crdiates, crdiates f pits i the referece image are called referece crdiates. A regi f iterest is give i the verlappig area f the give images. The required trasfrmati ca be expressed as mappig where x, y iput crdiates, XY, referece crdiates. Apprximate similarity trasfrm hlds betwee bth crdiate systems i the give regi f iterest. X p1 q1, q2 x, (1.1) Y p q, q y where p, p, q, q trasfrmati cefficiets Give quatities Crdiates f grud ctrl pits (GCPs) x, y iput crdiates f -th GCP, J, X, Y referece crdiates f -th GCP, J, J a idex set f GCP's idetifiers, e.g. J {1,2,, }, Accuracy f grud ctrl pits (GCPs) xy, stadard deviati f iput crdiates f -th GCP, J, XY, stadard deviati f referece crdiates f -th GCP, J, Nrmal distributi with rtatially symmetric prbability desity fucti is assumed abut psitis f GCPs i bth crdiate systems. 3/10

4 3. PROBLEM SOLUTION Tw pricipal prblems ccur while apprpriate trasfrmati mdel is searched fr. Firstly, suitable trasfrmati mdel has t be chse t express the basic relatiship betwee iput ad referece crdiates f crrespdig pits. Such a basic trasfrmati mdel shuld be chse with respect t physical circumstaces f capturig the give images, e.g. psiti f the camera, its ier cstructi r uter cditis f bradcast f electrmagetic waves. Sme simple apprximate trasfrmati mdel is usually applied istead f a rigrus cmplicated e. Secdly, irregular defrmatis f the give images ca egatively ifluece suitability f the chse basic trasfrmati mdel. Such defrmatis have t be embdied i the trasfrmati mdel althugh they are ukw. They ca be treated as a result f sme radm errrs whe sufficiet umber f ctrl pits is available. The bth prblems ca be slved simultaeusly by meas f cllcati methd. 3.1 Cllcati methd Cllcati methd is well kw amg gedesists sice the early 70's (see [3]), but its rigi is much lder. The methd f cllcati rigiates frm the thery f stchastic prcesses ad time series. Similar methd was als itrduced i 1951 by Dr. Krige ad therefre it is called krigig, amely i gestatistics. It is almst equivalet t cllcati. The mai priciple f cllcati is decmpsiti f the psiti f a cmm pit i the referece crdiate system it tw cmpets: tred ad sigal. These tw cmpets crrespds t the tw abve metied pricipal prblems. Thus tred meas the basic trasfrmati mdel that apprximately describes the relatiship betwee iput ad referece crdiate systems. Sigal stads fr the irregular defrmatis f the give images. The sigal actually represets crrecti f the tred t btai better cicidece f GCPs tha the basic trasfrmati mdel ca ffer. The sigal is treated as radm prcess. The basic trasfrmati mdel has t be similarity trasfrm due t requiremet (1.1). The similarity trasfrm ca be ccisely frmulated with the aid f cmplex represetati f crdiate pairs X, Y, resp. x, y. where i stads fr imagiary uit, i 1, ad is set f the all cmplex umbers. Hece, similarity trasfrm ca be expressed as a simple equati: W p q w. (1.2) Variables are trasfrmati parameters p traslati f the bth crdiate systems (cmplex umber), q scale ad rtati (cmplex umber). 4/10

5 T imprve flexibility f equati (1.2), radm crrecti f similarity trasfrm, say ( w ), has t be added. W ( w) p q w, (1.3) where sigal - radm crrecti (radm cmplex fucti). Equati (1.2) has t be fulfiled fr the ctrl pits t. Hece W ( w ) p q( w ), J, (1.4) measuremet errr f iput crdiates f -th GCP (cmplex radm variable), measuremet errr f referece crdiates f -th GCP (cmplex radm variable), ( w ) sigal at a cmm pit, ( w ) sigal at the -th GCP. Equatis (1.3), (1.4) fr ukw parameters W, p, q cstitute system f equatis that has t be adusted by methd f cllcati. These equatis have t be liearized t separate the ukw parameters frm measured quatities. where p q w W ( w ) (1.5) p q w W W q ( w ), J ; W W W p p p q q q W p q w W p q w, J. Prbability distributi f radm vectrs [ 1,, ], [ 1,, ], [ ( w), ( w1 ),, ( w )] ca be characterized by their cvariace matrices C, C, C. If these cvariace matrices are give i advace, ukw parameters W, p, q ca be estimated by rdiary leastsquares methd. The, after mittig ukw parameters p, q, the required crdiates f a trasfrmed pit ca be expressed as a cmplex umber : where (1.6) 5/10

6 c w first rw f matrix C withut the first elemet f the rw, P weight matrix, P =, submatrix f C after mittig first rw ad first clum f C, W cmplex vectr, W [ W1, W2,, W ] T, W cmplex vectr f apprximate crdiates, [ W 1, W 2,, W ] T A desig matrix, A [ 1, w ], 1 = [1,1,,1] T, w cmplex vectr, w [ w1, w2,, w ] T, W, =, T cmplex cugate f A,, a w similarity trasfrm peratr, a w = [1, w ], h apprximate cefficiets f similarity trasfrm, p q. T h = [, ] Real cmpets, f cmplex umber cmputed by (1.6) are the required referece crdiates f a trasfrmed pit. The resultig trasfrmati mdel is as fllws. Trasfrmati t is cfrm because frmula (1.6) defies cmplex fucti f cmplex argumet. Such a fucti (s called hlmrphic fucti) has bee prved t represet cfrmal mappig (see [4], therem 8.2). 3.2 Image registrati Cllcati methd described i the previus secti ca be easily applied t registrati f digital images. Trasfrmati frmula (1.6) ca be evaluated fr each pixel f the iput image. This straightfrward applicati brigs prblem with assigmet f clrs t pixels f the trasfrmed image, sice the trasfrmed pixels create irregular grid. Therefre prper assigmet f clrs eeds additial iterplati i the irregular grid, especially i case f sigificat liear defrmati f images. T avid the iterplati, methd f earest eighbr ca be applied istead. This methd assigs t a pixel [ XY, ] f the iput image the clr f pixel 1 1 t ( XY, ) frm the iput image. It meas that iverse mappig t has t be cmputed t trasfrm the iput image. Iversi f cmplicated frmula (1.6) eed t be 1 cmputed sice much simpler way exists t btai t. It is mre suitable t simply exchage iput ad referece crdiate systems ad apply cllcati methd by the same maer as i the frward case. (1.7) 6/10

7 3.3 Precisi f the registrati Psitial precisi f the registrati is characterized by stadard deviati ( X XY, Y ) which ca be cmputed fr ay pit [ X, Y ] i the referece image. Stadard deviati ( X, Y ) depeds tw statistical parameters. These parameters XY ctrl fittig degree f GCPs. The bth parameters ifluece cvariace matrix C thrugh cvariace fucti. Oe f the parameters,, characterizes prbability distributi f differece betwee trasfrmati t ad similarity trasfrmati. This prbability distributi is assumed t be rmal with variace 2. Optimal values f the statistical parameters ca be ptially etered by the user r estimated by Bayesia apprach. T evaluate frmula (1.8) fr sme give pit [ X, Y ] i the referece image, crrespdig iput crdiates [ xy, ] have t be cmputed first. (1.8) 1 [ x, y] t ( X, Y), w x i y. 1 Cmputati f iverse trasfrmati t is described i secti 3.2. After the iverse trasfrmati ad after settig up a ptimal value f parameter, frmula (1.8) ca be evaluated. 3.4 Sftware implemetati The prpsed methd f image registrati was implemeted as a web applicati by meas f up-t-date sftware stadards f Iteret techlgy. The mai prcedure which evaluates frmulae (1.7) ad (1.8) is writte i C++. Special library fr cmplex arithmetics was used t cde frmulae (1.7) ad (1.8) easily. Other server-side mdules were prgrammed i Pyth laguage with the aid f web framewrk Dag. Cliet-side sftware is based HTML ad SVG stadards, JavaScript supprt is utilized as well. The user ca lad his w images it the web applicati r use Web Map Services (WMS). Precisi f the registrati ca be shw glbally by islies f fucti r lcally by a circle f radius ( X, Y ) at pit where the user has clicked by his muse. XY XY 7/10

8 Figure 1: Registrati f cadastral map it rthpht Typical use f the web applicati is shw Figure 1. The white rectagle is part f cadastral map that is registered it rthpht. GCPs are marked by red pits, the blue curves are islies f same psitial accuracy. The applicati is freely available at fr ay Iteret user. The user has t register at 4. CONCLUSION The prpsed registrati methd has several advaced features that make it uique amg ther existig methds, amely: 1. Psitial precisi f ay trasfrmed pixel i the registered image ca be estimated withut eed f grud truth. 2. Trasfrmati betwee images is cfrmal (preserves agles). 8/10

9 3. All the tuable parameters f the trasfrmati mdel have real iterpretati: gemetrical r statistical. 4. Psitial biases at GCPs are ptimally spread ut i the area f iterest t avid verfittig. (Immderate distrti f iput image caused by frced fit f GCPs is restraied.) 5. Smthess f the trasfrmati is rbust t cfigurati f GCPs. (N-uifrm distributi f GCPs i the area f iterest des t matter.) The prpsed methd f image registrati was implemeted as a web applicati which is freely available at fr ay Iteret user. REFERENCES [1] Karl-Rudlf Kch. Bayesia Iferece with Gedetic Applicatis, vlume 31 f Lecture Ntes i Earth Scieces. Spriger-Verlag, [2] E. J. Krakiwski ad Z. F. Biacs. Least squares cllcati ad statistical testig. Bulleti Gedesique, 64(1):73-87, [3] Mritz H. Least-squares cllcati. Techical Reprt A 75, DGK, [4] H. A. Priestley. Itrducti t Cmplex Aalysis. Oxfrd Uiversity Press, [5] Barbara Zitvá ad Ja Flusser. Image registrati methds: a survey. Image ad Visi Cmputig, 21(11): , BIOGRAPHICAL NOTES Lubmír Sukup (*1963) was graduated frm the Czech Techical Uiversity i Prague, Faculty f Civil Egieerig, Departmet f Gedesy ad Cartgraphy, specializati Remte sesig. Nwadays he wrks applicati f prbability thery ad mathematical statistics i gedetic measuremets ad image prcessig. Ja Havrlat (*1978) was graduated frm the Czech Techical Uiversity (ČVUT) i Prague, Faculty f Civil Egieerig, Departmet f Gedesy ad Cartgraphy. Nwadays he wrks defrmati aalysis, 3D mdelig ad implemetati f web applicatis. Odře Böhm (*1979) was graduated frm the Czech Techical Uiversity i Prague, Faculty f Civil Egieerig, Departmet f Gedesy ad Cartgraphy, specializati Remte sesig. Nwadays he wrks prcessig f image data, creati f web applicatis ad studies f usig ISAR data fr defrmatis. 9/10

10 Mila Talich (*1961) was graduated frm the Czech Techical Uiversity (ČVUT) i Prague, Faculty f Civil Egieerig, Departmet f Gedesy ad Cartgraphy. Sice 1987 he was wrkig at gedetic etwrks prcessig ad gedyamic prblems. At preset he is fcused ifrmati systems rieted t web applicatis. All f the authrs are staff f the Research Istitute f Gedesy, Tpgraphy, ad Cartgraphy (VÚGTK). CONTACTS Dr. Lubmír Sukup Mila Talich, Ph.D. Ja Havrlat, Ph.D. Odře Böhm Research Istitute f Gedesy, Tpgraphy, ad Cartgraphy Ústecká 98, Zdiby, CZECH REPUBLIC Tel Fax Web site: 10/1 0

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