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 datadrive ad tuable by a simple set f parameters. The prpsed methd f image registrati was implemeted as a web applicati by meas f uptdate 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 datadrive 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 uptdate 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 serverside mdules were prgrammed i Pyth laguage with the aid f web framewrk Dag. Clietside 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. (Nuifrm 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] KarlRudlf Kch. Bayesia Iferece with Gedetic Applicatis, vlume 31 f Lecture Ntes i Earth Scieces. SprigerVerlag, [2] E. J. Krakiwski ad Z. F. Biacs. Least squares cllcati ad statistical testig. Bulleti Gedesique, 64(1):7387, [3] Mritz H. Leastsquares 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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