ScienceDirect. The Influence of Subpixel Corner Detection to Determine the Camera Displacement

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1 Avalable onlne at ScenceDrect Proceda Engneerng ( ) 8 8 th DAAAM Internatonal Symposum on Intellgent Manufacturng and Automaton, DAAAM The Influence of Subpxel Corner Detecton to Determne the Camera Dsplacement Lukas Sroba*, Rudolf Ravas, Jan Grman Unversty of Technology n Bratslava, Faculty of Electrcal Engneerng and Informaton Technology, Insttute of Electrcal Engneerng, Ilkovcova, 89 Bratslava, Slovak Republc Abstract Ths paper deals wth drect comparson of tradtonal pxel and chosen subpxel corner detecton approaches. For ths reason the Harrs corner detector and two subpxel algorthms used to mprove the detecton accuracy were consdered. Frst one s based on the orthogonal vectors theory and teraton process and the other one s refnng the found pxel coordnates by fttng quadratc curve to the cornerness map n both x and y drectons separately. As the procedure to verfy the performance of these approaches the real dsplacement determnaton between two postons of the same camera usng nformaton comng from mage stereo par was chosen. To fulfll ths task requres employng the several computer vson and mage processng areas whch are descrbed n presented paper. The approprate sets of mage pars were created and used for further testng and comparson. There were also the statstcal analyss and evaluaton performed and the obtaned results were lsted n correspondng tables. The Authors. Publshed by Elsever by Elsever Ltd. Ths Ltd. s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of DAAAM Internatonal Venna. Peer-revew under responsblty of DAAAM Internatonal Venna Keywords: pxel corner detecton; subpxel detectors; mage stereo par; eppolar geometry; statstcal analyss. Introducton The area of corner pont detecton s well known and very often used n many practcal tasks, for llustraton the moton trackng, object detecton and recognton, robot navgaton, stereo matchng, D modelng and many others. It s possble to magne the corner pont as a pont, where at least two edges are ntersected, pont around whch s * Correspondng author. Tel.: ; fax: E-mal address: lukas.sroba@gmal.com The Authors. Publshed by Elsever Ltd. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of DAAAM Internatonal Venna do:.6/j.proeng...8

2 Lukas Sroba et al. / Proceda Engneerng ( ) hgh change of brghtness ntensty n all drectons or pont havng the smallest radus of curvature for example. But there are stuatons when pxel resoluton s not suffcent enough. For that reason the mathematcal technques such as approxmatons and nterpolatons can be used to fnd the chosen features n subpxel accuracy [] []. One of the applcatons where localzaton of correspondng corner ponts s crucal s D scene nvestgaton and dstance measurng usng mage stereo par []. But the problem of determnng the dsplacement between two cameras s qute complex and conssts from several partcular tasks. The prncples and theory behnd camera calbraton, feature detecton and matchng, mage undstorton, eppolar geometry, essental matrx decomposton, mage rectfcaton and mage dsparty have to be employed to get the proper and vald results. All these procedure steps together wth the corner detecton algorthms are also brefly descrbed n the followng sectons.. Corner detecton Many corner detectors were nvented over the years and the Harrs corner detector [] s one of the most famous. The man dea s to fnd the mnmum of brghtness ntensty dfference between chosen parts of mage (marked as W) and shfted parts of mage n all drectons. There s frst-order Taylor seres for that purpose used. The frst step s the determnaton of matrx A accordng the followng formula: I x A I x I y () W I y The elements nsde represent the dfferences (as the approxmatons of dervatons) n horzontal and vertcal drecton. Next step s the calculaton of cornerness map usng prevously found matrces and t gves us the nformaton how much the current pont can be consdered as the corner. After the global and local thresholdng the fnal set of corner s found. The frst mentoned subpxel algorthm [] s based on the orthogonal vector theory and teraton process. Due to the theory, a vector from the corner (marked q) to any part of ts adjacent area (marked p) should be perpendcular to the gradent of the pont p. If we use the ponts from the chosen adjacent area W and after applyng some mathematcal treatment the fnal formula looks lke ths: W T T ( I ( p ) I( p ) ) q ( I( p ) I( p ) p ) () W The other elements nsde stand for the gradents of partcular ponts consst of x and y component. When ths equaton s solved and q pont poston s found, ths pont s taken as the new center of the searchng wndow W and the whole process s solvng through the teratons agan. It should be notced, that ths algorthm was desgned only for x-corner ponts detecton (typcally the corners appearng n the chessboard patterns). After the pxel detecton to get the ntal corner ponts, the algorthm s used to refne the corner poston n subpxel precson. The second method also uses the prevously found pxel corner pont as ntal step. Once ths pont was detected, ts postons accordng of ths approach [6] s refned to subpxel accuracy by fttng the quadratc curve to the corner strength functon (cornerness map) n x and y drecton separately. The approxmatng functon could be followng: h( x, y) ax bx c () When the coeffcents are calculated, the assumpton that the maxmum of corner functon corresponds to the frst dervaton of ths functon could lead us to the fnal subpxel coordnates very easly. In the contrary to prevous algorthm, ths one does not necessarly requre the x-corner ponts and bascally works wth any knd of corners.

3 86 Lukas Sroba et al. / Proceda Engneerng ( ) 8 8. The dsplacement determnaton The whole process of gettng the dstance between two cameras n SI unts was also descrbed n [7] and conssts of several steps... Camera calbraton Camera calbraton [8] s prmarly lookng for quanttes that affect the magng process, such as focal length, prncpal pont, skew factor and lens dstorton. All these data are used later n dstorton removal and eppolar geometry theory to provde us the better results... Feature detecton and matchng Feature detecton and matchng are one of the most crucal and mportant components of many computer vson algorthms. Bascally the meanng of ths step s the searchng for nterested parts of an mage (features) whch n case of two mages contan the same scene can be correspondngly matched usng so called feature descrptors. Amongst the common procedures can be counted SIFT [9] or SURF [] algorthm for example... Image undstorton By the defnton the term dstorton [] s n photography generally representng the deformaton and bendng physcally straght lnes what makes them appear curly n mages. These effects are very often n form of radal and tangental dstorton and are mostly caused by lens and manufacturng errors. Because the dstorton can be descrbed as mathematcal functon usng the nformaton comng from camera calbraton, the reversed mathematcal operaton called mage undstorton s usually appled. The found correspondng ponts (n our case the corners) are because of that reason undstorted and processed n ths form for the rest of the procedure... Essental matrx In case of properly matched the correspondng ponts, the theory behnd the eppolar geometry [] n employed to extract the relatve rotaton and relatve translaton between both cameras (mages) n stereo par. As the frst step the fundamental matrx has to be stated. Ths matrx descrbes the relaton between the ponts n frst mage and so called eppolar lnes n second mage on whch the set of correspondng ponts must le. Usng agan the nformaton from camera calbraton and ths matrx, the essental matrx can be computed... Decomposton of essental matrx There s possblty due ts character to decompose the essental matrx nto rotaton matrx and translaton vector what can fully descrbe the relatve movement from one camera to another one n D Cartesan coordnate system. For that reason the SVD (sngular value decomposton) technque [] s usually used. But there s an ssue. Due to fact that the objects havng the dfferent sze and dfferent scene dstance can also look the same n mage, the translaton only relatve up to scale can be obtaned..6. Image rectfcaton Image rectfcaton [] s bascally a transformaton process used to project two or more mages onto a common mage plane. The found translaton vector as the nformaton about mutual cameras poston s used n rectfyng process. It s not necessary to rectfy the whole mage due to fact that the rectfyng transformaton only for detected corners s for our purposes suffcent enough. Because of rectfcaton both mage planes are parallel and the depth

4 Lukas Sroba et al. / Proceda Engneerng ( ) trangulaton s therefore much easer. Another parameter needed for D depth or camera dsplacement calculaton s pxel dstance between correspondng ponts..7. Image dsparty If we assume calbrated and rectfed stereo par, dsparty [] by defnton measures the dsplacement of a pont between two mages n pxels. It s obvous that n case of close ponts n scene there s large dsplacement and small dsplacement for far ponts respectvely. The depth Z calculaton usng dsparty d nformaton s computed lke ths: f B Z () d The sgn f s stand for focal length n pxels we found durng camera calbraton process. The B as the baselne also represents the dsplacement between both stereo par cameras n SI unts. Due to the mentoned fact that objects havng dfferent sze and scene dstance can look smlarly or even the same n mage, we need to know some ntrnsc pror nformaton lke focal length f and depth Z (ths nformaton may easly be obtaned usng another sensors for example) n rectfed D coordnate system to be able successfully determne real dsplacement B between cameras n SI unts. In case ths procedure s used for navgaton purposes, the Z pror nformaton s needed only n frst step, for further steps (applyng descrbed procedure repeatedly) s possble to calculate another depth coordnates by usng data (rotaton, translaton and so on) gathered durng the process.. Expermental tests For the purpose to demonstrate the advantage of subpxel detecton unlke the usual pxel one, the mage stereo par confguratons usng calbrated camera were prepared. All partcular presented steps together wth pxel and subpxel algorthms were followed and the am was to determne the dstance (baselne) between the postons of the same camera. It s mportant to notce that these camera postons were mutually both translated and rotated. Except the camera ntrnsc matrx also the real scene depth Z (n rectfed coordnate system) n cm unts was known for all chosen ponts. As the llustratve example you can see the scene structure and matched ponts correspondences between two mages n Fg.. Because of easy detected corners n case of both pxel and subpxel algorthms the chessboard patterns were used. Another advantage of these patterns s that corners havng the same locaton even f scene vew s shfted and rotated. Ths s not guaranteed n case of real scene because corners may change ther poston or even dsappear from multple vews. Fg.. Correspondng ponts between both mages.

5 88 Lukas Sroba et al. / Proceda Engneerng ( ) 8 8 The detected chessboard corners (by methods we prevously descrbed) were used to obtan guess of essental matrx (translaton vector and rotaton matrx) and 9 pont of them ( n each chessboard) were further nvestgatng to fnd dsplacement through dsparty computaton. Three mage resolutons were processed: 6x9, 8x96 and 6x8 pxel sze. To make the analyss more robust and accurate, multple mutual camera dstances were tested. There were 7 mage pars for each,, cm, mage par for each, 6, 7 cm and mage pars for each 8, 9, cm shfts set. The found results were statstcally analyzed and are descrbed n followng secton.. Expermental results Because the whole process s relatvely complex, the robustness of procedure s based mostly on the rght ponts correspondences poston. In the other words, the better corner ponts localzaton, the better results and less change of algorthm falure. Table. Stablty results analyss. Shft between cameras [cm] Number of vald result stereo pars Resoluton 6 x 9 8 x 96 6 x 8 Algorthm P A B P A B P A B Total The frst comparson was dealng wth stablty results analyss. The ground truth (real dsplacement) nformaton was known and therefore the smple flter was appled to confrm or neglect the results as satsfactory or not. Just for llustraton the rato between found and real camera shft for every consdered pont correspondences and correct translaton and rotaton decomposton were taken nto account. The obtaned results are lsted n Table. Fg.. Averaged relatve error of found dsplacements.

6 Lukas Sroba et al. / Proceda Engneerng ( ) The pxel approach s n table marked as P and subpxel algorthms n same order as they were descrbed are represented by sgn A and B respectvely. As t s possble to see n table last row, probablty of not faled stereo par s between and tmes hgher for subpxel detecton than n case of tradtonal pxel one. The number of all tested mage pars for every resoluton was (as was explaned n prevous secton) and t gves us roughly % of success rate for subpxel algorthm. These results prove the fact that employng the subpxel detecton s advantageous and sgnfcantly ncreases the algorthm robustness and the chance of vald results. The second comparson s about relatve errors of found dsplacements. These errors were computed usng followng formula: () B * B B B The varable B represents real shft and B * stands for found shft. These values belongng to partcular shft and resoluton were averaged and are shown n Fg.. The same mage pars as t was n frst comparson were nvestgated and accordng to data n Table, there are also the shfts where no results were obtaned (mssng legend sgns). As t s obvous, the subpxel algorthms produce agan better results for most of cases n every resoluton. Moreover, the accuracy of subpxel results tends to decrease n case of longer camera shfts. One of the explanatons could be that mage rectfcaton (part of presented procedure) s workng mostly for qute smlar mages wth small mutual rotaton and translaton. For large rotatons and translaton these algorthms are losng precson as well. Concluson Ths paper has dealt wth comparson of tradton pxel and two algorthms of subpxel corner detecton. As the subject of ths comparson the determnaton of dsplacement between two postons of the same camera usng mage stereo par was chosen. In frst secton the consdered corner detectors were explaned. Specfcally the well known Harrs corner detector, the detector based on orthogonal vector theory and the one usng quadratc curve over cornerness map. Second secton s explanng the theory behnd dsplacement determnaton from sngle mage par and the whole process. The mage processng and computer vson areas lke camera calbraton, feature detecton and matchng, mage undstorton, eppolar geometry, mage rectfcaton and mage dsparty were brefly mentoned. The expermental test whch was performed accordng the descrbed procedure s presented n next secton. All partcular procedure steps together wth pxel and subpxel detectons were followed to determne the shft between two cameras. Also the ntrnsc nformaton such as scene depth coordnates for nvestgated ponts were known. The example of used scene contanng the chessboard patterns was shown n Fg.. The multple mutually shfted (and rotated) mage were tested and statstcally analyzed. The frst test was about comparng the results stablty. Because of knowng the real dsplacements (ground truth), only certan results were consdered as satsfactory enough. In Table the numbers of these successful confguratons were lsted. As t s obvous, the probablty of not faled stereo par was between and tmes hgher n case of subpxel detecton n the contrary of pxel one and the success rate was roughly % from all tested pars. It proves the advantage of subpxel detecton paradgm and sgnfcant ncreasng of procedure robustness. Second analyss was comparng the relatve errors computed from found and real dsplacements and the results were dsplaced n Fg.. As t was possble to see, the subpxel approaches produce agan better results for most of cases n every resoluton. Moreover, n case of relatvely longer mutual camera shfts the accuracy s decreasng, what can be caused by larger mutual translaton and rotaton. There are many ways how to contnue to ths research. It would be nterestng to examne how to subpxel detecton affects the accuracy of poston determnaton n case of navgaton purposes f ths procedure s appled repeatedly.

7 8 Lukas Sroba et al. / Proceda Engneerng ( ) 8 8 Acknowledgements Ths work was supported by the Slovak Research and Development Agency under the contract No. APVV-69- and also by the project VEGA /96/. References [] Y. Qao, Y. Tang, J. L, Improved Harrs sub-pxel corner detecton algorthm for chessboard mage, Conference on Measurement, Informaton and Control (ICMIC) Volume () 8-. [] N. Chen, J. Wang, L. Yu, Ch. Su, Sub-pxel Edge Detecton of LED Probes Based on Canny Edge Detecton and Iteratve Curve Fttng, Internatonal Symposum on Computer, Consumer and Control (ISC) () -. [] V. Flaretov, A. Zuev, An Advanced Approach to Automatc Machnng of Composte Parts Wthout Ther Rgd Fxng by Means of Multlnk Manpulators wth Stereo Vson System, DAAAM Internatonal Scentfc Book () -6. [] Ch. Harrs, M. Stephens, A combned corner and edge detector, In Alvey Vson conference (988) 7-. [] Z. Wexng, M. Changhua, X. Lbng, L. Xncheng, A fast and accurate algorthm for chessboard corner detecton, CISP nd Internatonal Congress on, Image and Sgnal Processng (9) -. [6] M. Rea, D. McRobbea, D. Elhawary, Z. Tse, M. Lamperth, I. Young, Sub-pxel localzaton of passve mcro-col fducal markers n nterventonal MRI, MAGMA (9). [7] L. Sroba, R. Ravas, Determnaton of dsplacement between two cameras usng correspondng mage stereo par, 6 th Conference of Doctoral Students ELITECH (). [8] D. C. Brown, Close-Range Camera Calbraton, Photogrammetrc Eng. Vol. 7 (97) [9] D. G. Lowe, Dstnctve Image Features from Scale-Invarant Keyponts, Internatonal Journal of Computer Vson Volume 6 Issue () 9-. [] H. Bay, A. Ess, T. Tuytelaars, L. Van Gool, Speeded-Up Robust Features (SURF), Computer Vson and Image Understandng Volume Issue (8) 6-9. [] D. C. Brown, Decenterng dstorton of lenses, Photogrammetrc Eng. Vol. (966) -6. [] R. Hartley, A. Zsserman, Multple Vew Geometry n Computer Vson, Cambrdge Unversty Press () 9-9. [] W. Wang, H. Tsu, A SVD decomposton of essental matrx wth eght solutons for the relatve postons of two perspectve cameras, th Internatonal Conference on Pattern Recognton () 6-6. [] D. Oram, Rectfcaton for any eppolar geometry, Brtsh Machne Vson Conference () [] N. Navab, Ch. Unger, Rectfcaton and dsparty, Computer Aded Medcal Procedures n Techncal Unversty Munch.

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