ICP Algorithm for Alignment of Stars from Astronomical Photographic Images

Transcription

1 Interntionl Conference on Computer Systems nd echnologies - CompSysech 010 ICP Algorithm for Alignment of Strs from Astronomicl Photogrphic Imges Alexnder Mrinov, Ndezhd Zltev Abstrct: his rticle proposes method for the lignment of strs from stronomicl photogrphic imges to strs from stndrd stronomicl ctlogue. Given the corse celestil coordintes of the imge centre nd the field of view, we re looking for mpping between the strs extrcted from the imge nd the strs from the ctlogue. he method is bsed on the Itertive Closest Point (ICP) lgorithm, which is robust enough to the presence of noise nd the ppernce of flse strs or such tht re missing from the imge. Due to the purely geometricl chrcter of the lgorithm, the intensity noises of the imge pixels hve no impct. he result when pplying the lgorithm is tht the originl imge is trnslted nd rotted so tht the strs from the imge re ligned with their exct coordintes, given by the str ctlogue. Key words: ICP lgorithm, lignment of strs, stronomicl photogrphic imges. INRODUCION he registrtion of photogrphed sky imge is clssicl problem nd regulr tsk in every observtory nowdys. he contemporry photogrphic instruments produce high qulity imges with high fidelity mesurements of their observtions, which fcilitte the imge registrtion problem. But there re still lrge number of photogrphic pltes from the observtories rchives tht suffer from the qulity of the pst instruments. hese rchives my contin pltes with vrious kinds of defects, such s obscured surfce res due to physicl dmge, lost informtion regrding the imge celestil coordintes, imge noisertefcts like the réseu grid, introduced to ssist the process of visul mesurement, etc. he Institute of Astronomy t the Bulgrin Acdemy of Sciences hs prepred list of such photogrphic plte rchives "Ctlogue of Wide-Field Plte Archives [7] enumerting over,000,000 observtions obtined since the end of 19th century from observtories worldwide. he problem rising with the registrtion of such pltes involves one or more of the following seprte tsks: 1. Filtering the regulr noise nd rtefcts from the input imge. Extrction of the centres of the strs 3. Serching the strs in str ctlogue to determine the imge loction on the sky [4] 4. Alignment of the str set from the imge with the str set from the ctlogue In our work we ssume tht the first three tsks re lredy solved by other methods, so we only focus on solving the lst tsk. We ssume tht tsk 3 reduces the ctlogue only to the re contining the imge. hen we model the problem by considering the str mps from the imge nd from the ctlog s two point sets in the Euclidin -dimensionl spce: A, B E, A n, B m (1) nd we denote with A nd b B the elements from the corresponding sets. his work ws supported by following grnts of the Institute of Informtion echnologies (II) t Bulgrin Acdemy of Sciences (BAS): Grnt # DO-0-75/008, Grnt # BG 051PO /13 of the Ntionl Science Fund t Bulgrin Ministry of Eduction & Science, nd Grnt # /007 of BAS

2 We wnt to mtch the strs from the input imge with the strs from the ctlogue in order to estimte their mutul correspondence nd lign more precisely their coordintes, including the imge centre. his lignment cn be chieved by moving the strs from the input imge uniformly to optimlly djust them to the corresponding strs from the ctlogue. hus we consider two functions μ nd λ s defined below: : A B; b () R, t:a E ; R t ), (3) t E, R SO() -specil orthonorml x mtrix, i.e. det(r)=1, RR=I. We mesure the djustment between the two point sets using the sum of squred distnces between corresponding points (4) which we re interested to minimize. hus, our problem is to find such λ, μ which stisfy the criterion: min, IERAIVE CLOSES POIN ALGORIHM Since the introduction of the bsic concepts of Itertive Closest Point lgorithm (ICP) by Chen [3] nd Besl [], there hve been developed multiple vrints of the lgorithm, the clssifiction of which we cn find in Rusinkiewicz [6]. An nlyticl solution finding n optiml rottion mtrix R nd trnsltion vector t is developed by Arun [1]. he implementtion of the lgorithm is bsed on the work of the uthors bove with the optimiztion improvements by Kpoutsis [5]. Bsiclly ICP is expressed by the following lgorithm: : I,0 1. Initiliztion: min. Mtching: min 3. rnsforming: 4. Loop to step until convergence or exceeding the itertions limit Mtching step: he mtching lgorithm must select the closest point b B for ech point A. rivil implementtion my exhust ll point pirs. Further optimiztions cn be found in the more efficient lgorithm vrints reviewed in [6, 5]. Although this step my look simplest to understnd, this is the slowest step in the whole lgorithm. rnsformtion step: In this step we wnt to find the orthonorml mtrix R nd vector t tht for given μ minimizes (4). (5)

3 Let nd b be the respective point set centroids - 1 1, b b A B A bb nd the point coordintes reltive to them - Hence,, b b b (6) Rt b ( R b) ( R b t). (7) From here we cn derive t directly by setting the second member of the expression to 0: t R b. (8) Now the minimiztion of (4) reduces to minimizing: R b ( R R b b R R b R b b b R), b rce RH (9) R b b R rce( Rb (10) nd H b. (11) Hence the problem of minimizing (4) is reduced to finding: mx rcerh. R (1) Arun [1] proves tht (1) is mximized by R VU (13) H UV (14) is the singulr vlue decomposition of H. EXPERIMENAL RESULS For our experiment we used scnned wide-field photogrphic plte nd extrcted 1975 strs with stellr mgnitudes up to 14, which we tret s the model. From these strs we rndomly selected 63 nd dded rottionl, trnsltionl nd point position noise to them. Our gol is to find the best fit of these 63 strs onto the model. Fig. 1 shows the initil stge of the lgorithm, the strs from the model re displyed in gry points nd the noised strs - in blck crosses. After the mtching phse, the dt to model mpping is discovered nd is mrked with drk lines on the sme figure. he trget strs to which the dt strs point re mrked in blck. he squred distnce (4) monotoniclly decreses until (5) 3

4 convergence fter 154 itertions (Fig. ). Fig. 3 shows n intermedite stge of ICP, while the finl result of the lgorithm is displyed in Fig. 4. Fig. 1 initil ICP stge Fig. squre distnce between point sets on ech itertion Fig. 3 intermedite ICP stge Fig. 4 finl ICP stge CONCLUSION AND FUURE WORK We illustrted the Itertive Closest Point lgorithm pplied for stronomicl imge strs lignment, nd successful experiment restoring shuffled set of strs to their originl position. In the common cse ICP converges to locl minimum, but this is good enough in the cse of stronomicl imges with known celestil centre, even when it is not quite precise. he present lgorithm does not benefit from the stellr mgnitudes, which re usully known. A direction for future reserch is the modifiction of the suggested lgorithm to use ssigned weights to the points, which will improve the mtching function μ. he uthors believe tht such improvements will skip inpproprite locl minim nd decrese the number of itertions until convergence occurs. 4

5 REFERENCES [1] Arun K. S., Hung. S. nd Blostein S. D. "Lest-Squres Fitting of wo 3-D Point Sets", IEEE rns. PAMI, Vol. 9, No. 5, 1987 [] Besl P. nd McKy N. A Method for Registrtion of 3-D Shpes, IEEE rns. PAMI, Vol. 14, No., 199. [3] Chen Y. nd Medioni G. Object Modeling by Registrtion of Multiple Rnge Imges, Proc. IEEE Conf. on Robotics nd Automtion, [4] Dustin L., Hogg D., Mierle K., Blnton M., Roweis S. "Mking the Sky Serchble" [5] Kpoutsis C. A., Vvoulidis C. P. nd Pits I Morphologicl Itertive Closest Point Algorithm, IEEE rns. IP, Vol. 8, No. 11, 1999 [6] Rusinkiewicz S. nd Levoy M. Efficient Vrints of the ICP Algorithm, IEEE 3DIM, 001 [7] svetkov M. K. Wide-Field Plte Dtbse: Decde of Development, Proc. of the Interntionl Workshop on Virtul Observtory, 006, pp.10. ABOU HE AUHOR Alexnder Mrinov, Ph.D. Student, Institute of Informtion echnologies, Bulgrin Acdemy of Sciences, Sofi, Phone: , Е-mil: mrinov@iinf.bs.bg. Ndezhd Zltev, Ph.D. Student, Institute of Informtion echnologies, Bulgrin Acdemy of Sciences, Sofi, Phone: , Е-mil: nzltev@iinf.bs.bg. 5