An Efficient Image Rectification Method for Parallel Multi-Camera Arrangement

Transcription

1 Y.-S. Kag ad Y.-S. Ho: A Efficiet Image Rectificatio Method for Parallel Multi-Camera Arragemet 141 A Efficiet Image Rectificatio Method for Parallel Multi-Camera Arragemet Yu-Suk Kag ad Yo-Sug Ho, Seior Member, IEEE Abstract I this paper, we preset a image rectificatio method for the multi-view image captured by a parallel multicamera arragemet. Sice most multi-camera arragemets are maually adjusted, there exist misaligmets with the camera positios ad orietatios. These misaligmets cause geometric errors that ca be a serious obstructio to threedimesioal (3D) image processig ad applicatios. Moreover, differet iteral characteristics of cameras icrease geometric errors. I order to miimize those geometric errors, we desig a multi-camera arragemet ad the calculate the multi-view image rectifyig trasform (MIRT). Experimetal results show that the proposed method reduces geometric errors efficietly without ay visual distortio, ad geerates the rectified multi-view image i very short processig time 1. Idex Terms Image rectificatio, multi-view image, multicamera arragemet, 3DTV. I. INTRODUCTION A multi-view image is a collectio of images of the same scee captured by more tha two cameras. We ca experiece three-dimesioal (3D) ad immersive feeligs by the images captured from multiple viewpoits. I additio, after estimatig depth iformatio from the multi-view image, we ca recostruct virtual view images at ovel viewpoits. Hece, the multi-view image ca be widely used for free viewpoit TV (FTV) ad 3DTV [1]. I recet years, ISO/IEC JTC1/SC9/WG11 movig picture experts group (MPEG) has bee iterested i the geeratio ad represetatio of 3D video related to FTV ad 3DTV activities [], [3]. However, it is difficult to hadle the multi-view image sice it is captured by multiple cameras which are maually allocated. These maually located cameras ca cause geometric errors that meas camera misaligmets represeted as the correspodig pixel mismatches i the vertical directio ad the differet horizotal displacemets amog the multiple images. Also, the o-uiform iteral characteristics of the cameras, such as the focal legths ad pricipal poits, icrease geometric errors. These geometric errors make the correspodece matchig such as depth estimatio difficult. Moreover, we caot expect a atural 3D 1 This research was supported by the MKE (The Miistry of Kowledge Ecoomy), Korea, uder the ITRC (Iformatio Techology Research Ceter) support program supervised by the NIPA (Natioal IT Idustry Promotio Agecy). (NIPA-11-(C )). Y. S. Kag ad Y. S. Ho are with the School of Iformatio ad Commuicatios, Gwagju Istitute of Sciece ad Techology (GIST), Gwagju, Korea ( yusuk@gist.ac.kr, hoyo@gist.ac.kr). view i the 3D applicatio based o the multi-view image which has geometric errors, because geometric errors decrease the visual quality. Thus, it is essetial to miimize geometric errors i the multi-view image. Oe of the image processig techiques to reduce geometric errors is image rectificatio that has bee studied for log time i computer visio field [4]. Image rectificatio is the trasform of the origial multiple images to obtai rectified images which satisfy the coditios of a camera cofiguratio. I this camera cofiguratio, all the cameras have the same iteral characteristics, regular orietatios, ad the same distace to the adjacet camera oly alog the horizotal directio. Also, the rectified images have the parallel epipolar lies o the coplaar image plaes. The early versios of image rectificatio are for stereo or triocular camera setups [4]. Especially for stereo image pairs, there have bee may rectificatio algorithms. I geeral, these rectificatio algorithms ca be categorized ito two types; o-calibrated [4], [5] ad calibrated cases [], [7]. No-calibrated rectificatio does ot eed the camera parameters but requires image features. The, the trasform for rectificatio is calculated by the image features. However, for the calibrated came, image rectificatio ad trasform calculatio are performed based o the camera parameters. Both cases are actively utilized as a preprocessig step i some stereoscopic 3D video systems ad applicatios [8], [9], sice image rectificatio ot oly eables the correspodece matchig fast ad accurate [] but also icreases the visual quality of the images as 3D cotets. As the umber of required cameras is icreased for capturig more realistic 3D scee, image rectificatio has bee also developed to hadle the multi-view image. For the multi-view image, most of rectificatio algorithms are belog to the calibrated case sice the camera parameters are very useful iformatio for multi-view 3D systems [1] [18]. Usig the camera parameters also provides the simplicity for algorithm developmet. However, there are some algorithms for o-calibrated multi-view image rectificatio [19], []. I this paper, we propose a efficiet method to rectify the multi-view image captured by a oe-dimesioal (1D) parallel multi-camera arragemet. After obtaiig the camera parameters of the captured multi-view image by camera calibratio [1], we desig a multi-camera arragemet. The multi-view image rectifyig trasform (MIRT) is the calculated as the homography betwee the origial ad desiged multi-camera arragemets. By applyig the MIRT to the origial multi-view image, we obtai the rectified images. The cotributio of this work is that we choose the most appropriate oe of the origial camera coordiate Cotributed Paper Mauscript received 7/15/11 Curret versio published 9/19/11 Electroic versio published 9/19/ /11/$. 11 IEEE

2 14 IEEE Trasactios o Cosumer Electroics, Vol. 57, No. 3, August 11 systems for the basis axes of the multi-camera arragemet. We also calculate the MIRT that miimizes the differece betwee the origial ad multi-camera arragemets. Therefore, the output images have few pixels of vertical mismatches ad uiform horizotal disparities without visual distortio such as image slatig. Therefore, it ca be effectively used for practical 3D applicatios. The remaiig parts of this paper are as follows. We itroduce the characteristics of the multi-camera arragemet ad the briefly review some previous works i Sectio II. After explaiig the proposed method i Sectio III, we show the experimetal results i Sectio IV. We the coclude this paper i Sectio V. II. MULTI-CAMERA ARRANGEMENTS AND MULTI-VIEW IMAGE RECTIFICATION The multi-camera arragemet meas a setup of multiple cameras to capture the multi-view image. The operatio ad features of each camera i the multi-camera arragemet are show i Fig. 1. There are a poit M ad a object aroud M i the 3D space, ad they are projected to multiple image poits m ad D images aroud m o every image plae, respectively. Each camera is located o its camera ceter C ad has its ow coordiate system as the horizotal, vertical, ad optical axes. The cameras also have the iteral characteristics such as the focal legths ad pricipal poits. These operatio ad features of the cameras are modeled by the pihole camera model ad represeted by the camera projectio matrix P as m P M K R t M (1) where meas the -th camera i the multi-camera arragemet. K, R, ad t idicate the itrisic matrix, rotatio matrix, ad traslatio vector, respectively. The compoets of K mea the camera s iteral characteristics. R ad t are called the extrisic parameters. R specifies the camera coordiate system ad t is related to the camera s positio i the world coordiate system. Camera 1 X M Y Z World Coordiate System Camera Camera N Fig. 1. Operatio ad features of multi-camera arragemet. There have bee some multi-view image rectificatio algorithms i this calibrated multi-camera arragemet. W. Matusic et al. proposed the trasform so that the multi-view image becomes parallel to a lie that makes the sum of distaces from the camera ceters to this lie miimum [1]. M. Taimoto et al. also used the lie obtaied by pricipal compoet aalysis (PCA) with the camera ceters ad acquired the rectified images parallel to this lie [13]. I. Feldma et al. developed a multi-view image rectificatio algorithm that defies a ew world coordiate system that has its horizotal axis as the directio of the lie which miimizes the sum of distaces from the camera ceters. The rectified multi-view image is obtaied based o the calculated horizotal axis [14]. E. Ekmekcioglu et al. directly exteded the stereo rectificatio algorithm [7] to the multi-view. They choose two cameras at both eds to geerate the lie through these two camera ceters. The multi-view images are the rectified alog this lie [15]. These methods were proposed for multi-view test sequeces i MPEG 3D video research. Moreover, D. Kim et al. calculated a camera arragemet usig the miimizatio of global re-projectio error betwee the iter-view correspodig poits. The, they obtaied the rectified images by applyig the homography matrix [1]. F. Boutarel et al. obtaied the horizotal axis by the least square method of all the camera ceters ad averaged the origial optical axes to geerate the commo optical axes [17]. I our previous work [18], we repeatedly coected each midpoit from the origial camera ceters to calculate the lie as the commo horizotal axis for rectified multi-view image. III. PROPOSED MULTI-VIEW IMAGE RECTIFICATION METHOD I this Sectio, we explai the proposed image rectificatio method for the parallel multi-camera arragemet. The purpose of the proposed method is to reduce geometric errors ad obtai the rectified multi-view image. I other words, the rectified images have the characteristics of the images from the multi-camera arragemet. Based o the camera parameters, we firstly desig the multi-camera arragemets. The, we calculate the MIRT as the homography of each camera betwee the origial ad desiged camera arragemet. By applyig the MIRT, we obtai the rectified multi-view image. The mai differece betwee the previous methods ad the proposed method are as follows. 1. The proposed method does ot employ ay fittig or averagig but chooses the most appropriate oe of the origial camera coordiates for geeratig the commo camera coordiate system for the multi-camera arragemet.. The proposed method miimizes the geometric differece betwee the origial ad multi-camera arragemets. I some previous methods, there is a opportuity that the output images have visual distortio such as a image slat. Figure shows the rectified multi-view image which is slated. These images are rectified based o the lie which miimizes the sum of absolute differeces from each camera ceters. Sice most previous works are based o the fittig or averagig, the uexpected basis axes ca occur i the rectified multi-view image geometry. I this case, although geometric errors are reduced, the visual quality of the rectified multiview image is very low as auto-stereoscopic iput.

3 Y.-S. Kag ad Y.-S. Ho: A Efficiet Image Rectificatio Method for Parallel Multi-Camera Arragemet 143 However, the practical parallel multi-camera arragemet does ot satisfy the coditios. As show i Fig. 5, all the cameras have differet distaces to the adjacet oes, differet iteral characteristics, ad ueve orietatios. Fig.. Slated multi-view image after image rectificatio. Camera Calibratio Camera Calibratio Camera Calibratio Camera Calibratio P 1 P P 3 P N Multi-view Image Capturig Projectio Matrices Camera 3 Camera N Camera 1 Camera Fig. 5. Practical parallel multi-camera arragemet. Multi-view Image Rectifyig Trasform (MIRT): T 1, T, T 3,, T N Basis Axes Calculatio (Commo Rotatio Matrix) Camera Positio Calculatio (New Camera Ceters) Iteral Parameter Calculatio (Commo Itrisic Matrix) P,1 P, P,3 P,N Fig. 3. Procedure of the proposed method. Desig Ideal Multi-Camera Arragemet Ideal Projectio Matrices Multi-view Image Therefore, the multi-view image captured by such a practical multi-camera arragemet has geometric errors. There are several pixel differeces i the vertical coordiates amog the correspodig poits i each viewpoit. Moreover, the disparities i the horizotal directio amog the multiple viewpoits are also differet oe aother. Figure shows the multi-view image ad its sythetic image captured by the practical multi-camera arragemet. The sythetic image is created by overlappig all the images ad adjustig the opacity of each image to observe the differece amog the images. We ca easily otice that there is the large amout of geometric errors i the multi-view image. Therefore, we choose oe of the origial camera coordiate systems. Figure 3 shows the procedure of the proposed method. There are three mai parts of the proposed method. The first part is to estimate the camera parameters by camera calibratio []. I this part, we assume that the accuracy of the estimated parameters is reliable. The ext part is the desig of the multi-camera arragemet based o the camera parameters. The last part is to calculate the MIRT for each camera ad to apply it to the origial images. We the obtai the rectified multi-view image. A. Properties of Parallel Multi-camera Arragemet I the parallel multi-camera arragemet, all the cameras are liearly allocated o a lie i 3D space. They are also equally orieted ad have the same distace to the adjacet cameras. The commo directio of all the optical axes is perpedicular to the lie that passes all the camera ceters. Figure 4 shows these properties. Camera 1 Camera Camera 3 Fig. 4. Ideal parallel multi-camera arragemet. Camera N Fig.. Multi-view image ad its sythetic image. B. Desig of Ideal Multi-camera Arragemet I order to desig the multi-camera arragemet, we propose the followig three-step approach. I this approach, the geometric coditios are obtaied by usig the origial camera parameters. Step 1: I the first step, we defie the basis axes which are used as the commo camera coordiate system i the multi-camera arragemet. I other words, we calculate the rotatio matrix R. As metioed before, we do ot use the aforemetioed fittig, averagig, or optimizatio i the proposed method. Istead of these techiques, we choose oe camera coordiate system that represets the camera arragemet well ad cosider it as the basis axes. We assume that all the vector otatios stad for uit vectors. Let us deote the horizotal, vertical, ad optical axes of -th camera coordiate system as x, y, ad z, where

4 144 IEEE Trasactios o Cosumer Electroics, Vol. 57, No. 3, August 11 =1,, N. I the case of x, we calculate the average directio x avg ad their stadard deviatio σ x. The stadard deviatio is composed of three compoets; σ xx, σ xy, ad σ xz. We the exclude x which is outside the regio bouded by σ x based o x avg, as show i Fig. 7. After that, we calculate x which is the average of the survived avg x, so that we ca avoid the effect of the outliers. I the cases of y ad z, we obtai y avg ad z avg as the same maer. x 1 x C,r = C r. Usig this referece poit, we calculate the ew camera ceter of k-th camera as C, C, r d ( r ) x (4) (=1,,, N, where r) Before that, the umber r for the referece poit is calculated as (5) usig (4). For every m =1,,, N, r argmi m N 1 ( m) C, C Thus, we have the ew camera ceters C, which ot oly satisfy the coditio but also miimize the sum of distaces from the origial camera ceters. Usig these camera ceters, the traslatio vectors for -th camera of the multi-camera arragemet are defied as, C, m (5) t R () x avg Fig. 7. Excludig outliers: x ad x 3 are survived. Fially, we determie the basis axes x, y, ad z as the k-th camera coordiate system, where the umber k is determied as (). For every, k argmi x x y y x 3 z x z Therefore, the commo rotatio matrix R, which is composed of those basis axes, is defied as () T T x xk xk, x xk, y xk, z T T R y y k yk, x yk, y yk, z (3) T T z zk, x zk, y zk, z z k Step : After determiig the basis axes of the commo coordiate system, we fid the camera ceters for the multi-camera arragemet. These camera ceters should be co-liear with the same distace d to the adjacet cameras. Let us deote the ew camera ceter of the -th camera as C,. We firstly choose oe of C, for =r, where 1 r N. This C,r is defied as the referece poit, where Step 3: At the third step, we determie the commo itrisic matrix K for the multi-camera arragemet. We estimate the focal legth f ad pricipal poit p =(p,x, p,y ) T which are the mai compoets of K. As the previous step, we seek to miimize the sum of variatio from the origial values to the ew value. Therefore, f ad p are calculated as (7) ad (8), respectively. p p p f, x, y argmi T ( px, py ) 1 argmi f N 1 f N p, x px p, y p y For calculatig (7) ad (8), rages for f ad p are bouded by their origial values miimum ad maximum. We the obtai the commo itrisic matrix K as (9). Note that the focal legth has its values i two directios as f,x ad f,y. We set the skew parameter s as zero. f (7) (8) f, x s p, x K f, y p, y (9) 1 Usig (3), (), ad (9), we fially obtai the -th camera s projectio matrix of the multi-camera arragemet as (1). The multi-camera arragemet represeted by (1) satisfies the geometric coditios, as show i Fig. 4., K R t, P (1)

5 Y.-S. Kag ad Y.-S. Ho: A Efficiet Image Rectificatio Method for Parallel Multi-Camera Arragemet 145 C. Multi-view Image Rectifyig Trasform (MIRT) The rectified multi-view image is acquired by applyig the MIRT to each viewpoit image i the origial multi-camera arragemet. Figure 8 shows two multi-camera arragemets. Oe is the origial camera arragemet ad the other is the camera arragemet composed of the cameras that have their projectio matrices as (1). Sice the epipolar geometry exists i each viewpoit, there is a poit-to-poit mappig called D homography H from the origial image plae to the image plae [7]. We employ this homography as the MIRT T for -th camera as (11), where P + is the pseudo iverse matrix of P. (a) Trai T H, P, P (11) H,1 m 1 m 1 Camera 1 Camera 1 Camera 3 Camera Camera 3 Camera Camera N Camera N Fig. 8. Homography betwee origial ad multi-camera arragemets. IV. EXPERIMENTAL RESULTS I order to test the proposed method, we set up two parallel multi-camera arragemets. The first oe is composed of four cameras with 8cm distace. Usig this camera arragemet, we captured Trai sequece. The other is composed of ie cameras apart with 5.5cm distace. Drum sequece was captured by the secod camera arragemet. The image resolutio is The multicamera arragemet we used is show i Fig. 9(a), ad he multi-view image sequeces were captured i the idoor studio show i Fig. 9(b). The distaces from the cameras to the backgroud of each scee were approximately.m for Trai ad 5.m for Drum. Figure 1 shows the captured multi-view images ad their sythetic images. As show i Fig. 1(c) ad Fig. 1(d), we otice that there exist geometric errors. There are vertical pixel mismatches ad o-uiform horizotal disparities betwee the eighborig views. (c) Sythetic image for Trai (d) Sythetic image for Drum Fig. 1. Captured multi-view images. By performig the proposed method, we obtaied the rectified multi-view images show i Fig. 11. From Fig. 11(a) ad Fig. 11(b), we observed that the hole-regios are geerated at image boudaries by ot oly the movemet ad rotatio of each image plae, but also the relocatio of each camera ceter. However, there is o visual distortio i all the output images. The proposed method also reduces the vertical pixel mismatches ad o-uiformity of the disparity i both test sequeces, as show i Fig. 11(c) ad Fig. 11(d). (a) Trai (b) Drum (a) Multi-camera arragemet (b) Idoor studio Fig. 9. Capturig eviromet. (c) Sythetic image for rectified Trai (d) Sythetic image for rectified Drum Fig. 11. multi-view images.

6 14 IEEE Trasactios o Cosumer Electroics, Vol. 57, No. 3, August 11 I order to evaluate the performace of the proposed method, we measured the reductio of the umber of mismatched pixels i the vertical directio ad also the regularizatio of irregular disparities i the horizotal directio. Thus, we used two kids of check-pattered plaar boards ad extracted the coordiates of a umber of corer poits, as show i Fig. 1. Note that we assumed that all the extracted poits o the board have the same or very similar depth values. We the measured the vertical mismatches ad horizotal disparities i the pixel uit betwee the correspodig poits of the adjacet view images. TABLE I TOTAL AVERAGE OF VERTICAL PIXEL MISMATCHES Sequece () () Trai Drum Figure 14 shows the average disparity values of all the extracted poits betwee two adjacet views. Before applyig the MIRT to the origial images, the disparity variatio was large. However, the proposed method provided almost uiform disparities for both image sequeces. These uiform disparities ad miimized vertical pixel mismatches represet the characteristics of the multi-camera arragemet (a) 48 poits from Trai (b) 19 poits from Trai Fig. 1. Corer poit extractio Figure 13 shows average vertical pixel mismatches of all the extracted poits. We cosidered the first viewpoit as the referece view, ad we calculate the vertical coordiate differece betwee the correspodig poits for the other viewpoits. As show i Fig. 13, the proposed method reduced the vertical mismatches for both sequeces. The total average values of the vertical pixel mismatch from all the corer poits are show i Table I. These values also idicate that the proposed method efficietly decreases the vertical mismatches i the origial images ~ ~3 3~4 Viewpoits (a) Trai ~ ~3 3~4 4~5 5~ ~7 7~8 8~9 Viewpoits Fig. 14. Average disparity variatios Viewpoits (compared to view 1) (a) Trai However, we eed to evaluate the vertical pixel mismatches ad horizotal disparities at the other positios which have the differet depth values because the plaar boards have the same or very similar depth values, as show i Fig. 1. Therefore, we select eight feature poits for each sequece, which are marked as the red dots show i Fig. 15. These poits are visible for all viewpoits ad have the differet depth values oe aother Viewpoits (compared to view 1) Fig. 13. Average vertical pixel mismatches. Fig. 15. Extractio of eight feature poits.

7 Y.-S. Kag ad Y.-S. Ho: A Efficiet Image Rectificatio Method for Parallel Multi-Camera Arragemet 147 For these eight feature poits, we also calculated the average vertical pixel mismatches ad disparity deviatios. The average vertical pixel mismatches were calculated as the same method of the previous case. For each feature poit, the differeces of vertical coordiates betwee the first viewpoit ad the other viewpoits were averaged. As show i Fig. 1, the proposed method reduced the vertical mismatches for all the extracted poits of both sequeces. The pixel mismatches are aroud or less tha oe pixel for almost poits. The total average of the vertical pixel mismatches for the eight feature poits are show i Table II Feature poits (a) Trai Feature poits (a) Trai Feature poits Fig. 1. Average vertical pixel mismatch for feature poits. TABLE II TOTAL AVERAGE OF VERTICAL PIXEL MISMATCHES FOR FEATURE POINTS Sequece () () Trai Drum I order to evaluate the disparity values at each feature poit, we used the disparity deviatio. We firstly calculated the disparity betwee two adjacet viewpoits ad the average them for each feature poit. The, the disparity deviatio is acquired as the absolute differece betwee the each viewpoit s disparity ad the average disparity for each feature poit. Figure 17 shows the average disparity deviatios of two sequeces. This disparity deviatio meas the degree of disparity variatio i the multiple viewpoits. The results showed that the average disparity deviatios were decreased for all the feature poits of both sequeces. I other words, the proposed method provided the uiform disparity values i the rectified multi-view images Feature poits Fig. 17. Average disparity deviatio for feature poits. We checked the processig time of the proposed method. After scee capturig ad camera calibratio, the processig time to calculate the MIRT is.9 secods for four views ad.19 secods for ie views. The required time for applyig MIRT to the origial image ad savig the result as a image file was.4 secods for oe view. This results show that the proposed method provides more tha 4 frames of the rectified images at each viewpoit per oe secod. V. CONCLUSIONS I this paper, we proposed a efficiet image rectificatio method for a parallel multi-camera arragemet. The proposed method desiged a multi-camera arragemet ad calculated the multi-view image rectifyig trasform (MIRT). I order to avoid visual distortio, such as a slatig image, we chose the most appropriate oe of the origial camera coordiate systems as the basis axes for the multi-camera arragemet. By applyig the MIRT to the origial images, we efficietly reduced geometric errors ad obtaied rectified multi-view images without visual distortio. Also, the proposed method performed i very short period of processig time. We expect that the proposed method could provide advatages of efficiecy ad accuracy i 3D image processig ad applicatios usig the multi-view image as well as a high quality of 3D sese. REFERENCES [1] C. Feh, R. Barre, ad S. Pastoor, Iteractive 3-DTV-cocepts ad key techologies, Proceedig of the IEEE, vol. 94, o. 3, pp , Mar.. [] ISO/IEC JTC1/SC9/WG11 N5877, Applicatios ad requiremets for 3DAV, July 3.

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