A STUDY ON CALIBRATION OF DIGITAL CAMERA

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1 A STUDY ON CALIBRATION OF DIGITAL CAMERA Ryuji Matsuoka a, *, Kiyonari Fukue a, Kohei Cho a, Haruhisa Shimoda a, Yoshiaki Matsumae a, Kenji Hongo b, Seiju Fujiwara b a Tokai University Research & Information Center -8-4 Tomigaya, Shibuya-ku, Tokyo 11-63, JAPAN ryuji@yoyogi.ycc.u-tokai.ac.j, fku@keyaki.cc.u-tokai.ac.j, kcho@keyaki.cc.u-tokai.ac.j, smd@keyaki.cc.u-tokai.ac.j, jjm@keyaki.cc.u-tokai.ac.j b Kokusai Kogyo Co., Ltd., Rokuban-cho, Chiyoda-ku, Tokyo 1-8, JAPAN kenji_hongo@kkc.co.j, seiju_fujiwara@kkc.co.j Commission III, WG III/ KEY WORDS: Digital, Calibration, Orientation, Simulation, Camera, Distortion, Non-Metric, Method ABSTRACT: Recent increase in number of ixels of images acquired by a non-metric digital camera encourages an amateur to utilize it for 3D measurement. Most of current camera calibration methods are inconvenient and exensive for an amateur to calibrate his digital camera. Therefore we have develoed a method for an amateur to calibrate a non-metric digital camera easily. Our roosed method needs a lot of objects calibration oints) recognizable in an image instead of targets. Eight convergent images are acquired from eight different directions with four different camera frame rotation angles of, +9, +18 and -9. Measurement of image coordinates of calibration oints are in rincile executed by temlate matching. A set of camera arameters is determined by the bundle adjustment. A field exeriment and a numerical simulation were conducted to investigate erformance of our roosed method. The results of the field exeriment and the numerical simulation indicate that the roosed method using a set of calibration oints distributed on the D lane can rovide a reliable set of camera arameters. Error of an image distortion model estimated by the roosed method is exected to be u to aroximately ±3 ixels or ±1 µm on the image. Since our roosed method needs neither secial equiment nor ground survey of control oints, we believe that the roosed method is useful for an amateur to calibrate his digital camera. 1. INTRODUCTION As erformance of a digital camera becomes better and its rice becomes lower in recent years, digital camera images are becoming more oular in diverse fields. Increase in number of ixels of images acquired by a digital camera seems to go on forever without stoing. Some of current oular digital cameras, whose rices are less than US$ 1, can acquire an image of more than million ixels. The increase in number of ixels of images encourages an amateur to utilize a non-metric digital camera for 3D measurement. However the increase in number of ixels causes two roblems: One is radiometric degradation of image quality due to reduction in area of an imaging element Ohmori, 1), and the other is a disclosure of geometric distortion of an image because of smaller interval between imaging elements. In this aer we focus on calibration of geometric distortion of digital camera images. We roose a new calibration method of a non-metric digital camera for an amateur, and indicate effectiveness of the method by a field exeriment and a numerical simulation.. A PROPOSED CALIBRATION METHOD There are several calibration methods of close range cameras Fryer, 1996; Fraser, 1). Otical calibration by using a multi-collimator or a goniometer is oular for metric cameras, but is rarely conducted for a non-metric camera owing to its higher cost. Field calibration methods including self-calibration are usually adoted for a non-metric camera. Since most of field calibration methods need a ground survey for determining recise ositions of controls oints or to lace an array of 3D distributed targets, these methods are inconvenient and exensive for an amateur to calibrate his digital camera. Therefore we have develoed a method for an amateur to calibrate a non-metric digital camera with no secial equiment, no ground survey or no secial targets. Figure 1 shows the rocedure of our roosed calibration method. Start Selection of a calibration field Image acquisition by the target digital camera Selection of calibration oints Measurement of image coordinates of calibration oints Bundle adjustment to determine a set of camera arameters Finish At site In office Figure 1. Procedure of the roosed calibration method

1) Selection of a calibration field Our method needs a lot of objects recognizable in an acquired image instead of targets. These objects are called calibration oints from now on.

2 1) Selection of a calibration field Our method needs a lot of objects recognizable in an acquired image instead of targets. These objects are called calibration oints from now on. It is requested that calibration oints are imartially distributed in a calibration field. Calibration oints are not necessary to be laced satially. A lane with calibration oints laced at regular intervals, for instance a tiled avement or a brick wall of a building, is suitable for a calibration field. Figure shows a suitable calibration field. ) Bundle adjustment to determine a set of camera arameters A set of camera arameters is determined by the bundle adjustment. In the roosed method image distortion x, y) of a oint x, y) is reresented as 4 6) 3 ) 4 6) 3 ) x = x + x k + k r + k r + k r + r + x + xy y = y + y k + k r + k r + k r + xy+ r + y r = x + y x = x x y = y y 1) ) Figure. A suitable calibration field ) Image acquisition by the target digital camera Eight convergent images are acquired from eight different directions with four different camera frame rotation angles of, +9, +18 and -9 around the otical axis of the camera as shown in Figure 3. An aroriate inclination angle α deends on the field of view of the target camera. 3) Selection of calibration oints More than hundred calibration oints are selected on a PC dislay, and a temlate image of each calibration oint is automatically created by rectification of the image used in the selection to the calibration field lane. 4) Measurement of image coordinates of calibration oints Measurement of image coordinates of calibration oints are in rincile executed by temlate matching on the calibration field lane. If temlate matching of a oint fails, the osition of the oint is manually measured on a PC dislay. where x, y is the offsets from the rincial oint to the center of the image frame, k, k 1, k, k 3 are the coefficients of balanced radial distortion, and 1, are the coefficients of decentering distortion. The first order coefficient k of balanced radial distortion is also exressed as k c = 3) c using the correction c to the assumed rincial distance c. Both balanced radial distortion r and the rofile of decentering distortion d referred in the next chater are functions of a r from the rincial oint. They are exressed as r = k r + k r + k r + k r ) d = + r 4) S4 S S6 α S h calibration field Z Y l S3 S7 X Y S X S1 S8 calibration oint Camera frame rotation angle around the otical axis of the camera at each exosure station as follows: [T] S1 and S4: no rotation) [L] S3 and S6: +9 left sideways) [B] S and S8: +18 uside down) [R] S7 and S: -9 right sideways) Figure 3. Convergent image acquisition from eight different directions

3. EXPERIMETNS A field exeriment and a numerical simulation were conducted to investigate erformance of the roosed method. 3.

The dimensions of each tile are.1 m by.1 m. The digital camera used in the field exeriment is Olymus CAMEDIA E-1.

4 Olymus lens. Each image was acquired with 9 mm of the focal length at the widest view) and recorded as a 4 ixels by 168 ixels image.

The inclination angle α at image acquisition is aroximately 4 h 1. m, l 1. m).

and are sets of eight convergent images acquired according to our roosed method. is a set of eight images from eight different directions with no camera frame rotation.

3 3. EXPERIMETNS A field exeriment and a numerical simulation were conducted to investigate erformance of the roosed method. 3.1 Field exeriment The field exeriment was aimed at investigating stability of estimation results of the roosed method and assessing influence of camera frame rotation on camera calibration Outline of the field exeriment: The field exeriment was conducted at an oen sace aved with square tiles. The dimensions of each tile are.1 m by.1 m. The digital camera used in the field exeriment is Olymus CAMEDIA E-1. Its image icku element is /3-inch rimary-color interlaced CCD 4 million ixels total; 3.9 million ixels effective), and its lens is 9-36 mm equivalent to 3-14 mm zoom in 3 mm film format), F - F.4 Olymus lens. Each image was acquired with 9 mm of the focal length at the widest view) and recorded as a 4 ixels by 168 ixels image. Hence, the radial distance from the center to each corner of the image frame is 14 ixels, and the interval between ixels is aroximately 3.9 µm. The inclination angle α at image acquisition is aroximately 4 h 1. m, l 1. m). Figure 4 shows some images acquired in the field exeriment and Figure shows a screen for measuring image coordinates of calibration oints. Six sets of images are investigated. and are sets of eight convergent images acquired according to our roosed method. is a set of eight images from eight different directions with no camera frame rotation., and SetB are sets of eight images from eight different directions with the fixed camera frame rotation angles of +9, -9 and +18 resectively Results and discussion: Root mean square errors RMSEs) of image coordinates of calibration oints are shown in Table 1. Table 1 shows sets of the calibrated rincial distance c and the offsets x, y from the calibrated rincial oint to the center of image frame obtained in the field exeriment. The offsets x, y are also illustrated in Figure 6. Figure 7 shows grahs of the balanced radial distortions r and the rofiles of decentering distortions d obtained in the field exeriment. RMSE Set a) Image at S1 [T] b) Image at S [B] c mm) x y Table 1. Camera arameters obtained in the field exeriment 1 ixels radius -1 d) Image at S [R] Figure 4. Images acquired in the field exeriment y c) Image at S6 [L] x 1 Figure 6. Offsets x, y) of the rincial oint obtained in the field exeriment RMSEs of all six sets are less than.8 ixels as shown in Table 1. This means that image coordinates of calibration oints were measured recisely. Left: temlate image for matching Right: target image Figure. Measurement of image coordinates of a calibration oints The estimated rincial oints of six sets are distributed widely as Figure 6 shows. The rincial oints estimated by and are located nearly in the center of the distribution of six estimated rincial oints. Moreover the distance between these two rincial oints is 3.6 ixels, which does not seem so long. On the other hand, the rincial oints estimated by the

4 radial distortion decentering distortion a) Balanced radial distortion r b) Profile of decentering distortion d Figure 7. Balanced radial and decentering distortion grahs obtained in the field exeriment other,, and are located far from the center of the distribution, and the disersion of these four rincial oints from the center exceeds ixels, which cannot be allowed. The results on the estimated balanced radial distortion give the same indications as the results on the estimated rincial oint give. The balanced radial distortion curves estimated by all six sets are distributed widely as shown in Figure 7 a). Two curves of and are located nearly in the center of the distribution of all six curves. The maximum difference between the balanced radial distortions estimated by and is 1.9 ixels. On the other hand, the curves of the other,, and are far from the curves of and Set-, and the disersion of these four curves amounts to aroximately 1 ixels at the corners of the image frame where r is 14 ixels. The estimated decentering distortions of all six sets are considerably smaller than the estimated radial distortions as Figure 7 b) shows. The difference between the rofiles of Set- 1 and is small, while the disersion of the rofiles of the other,, and from those of and is rather large. From the field exeriment results shown in Table 1, Figure 6 and Figure 7, it can be concluded that a set of different camera frame rotations has a great influence on camera calibration. And furthermore, sets of images acquired with the same camera frame rotation cannot rovide a reliable set of camera arameters. Though reliability of our roosed method will be investigated in the numerical simulation mentioned in the next section, deviation of image distortions estimated from different sets of images may be u to ±3 ixels ±1 µm) on the image. 3. Numerical simulation The numerical simulation was aimed at investigating reliability of estimation results of the roosed method, assessing influence of an inclination angle at image acquisition on camera calibration, and verifying effectiveness of a set of calibration oints distributed on the lane Outline of the numerical simulation: Conditions of image acquisition in the numerical simulation were almost equal to those of the field exeriment mentioned in the revious section. values of the rincial distance c and the offsets x, y from the rincial oint to the center of the image frame were 9.7 mm, -. ixels, and -. ixels resectively. Four values of the inclination angle α at image acquisition in Table were set u. Two sets of calibration oints were set u: One was a set of calibration oints distributed on the D lane Z = m, and the other was a set of calibration oints distributed in the 3D sace h %) Z +h %), that is.3 m Z +.3 m. Accordingly, eight cases D-C, D-N, D-M, D- F,, 3D-N, 3D-M and 3D-F, which were combinations of two calibration oints sets D and 3D) and four inclination angles C, N, M and F), were investigated. Random Gaussian errors with. ixels of standard deviation were added each image coordinate of calibration oints. Case D-C D-N D-M D-F 3D-N 3D-M 3D-F h 1. m 1. m 1. m 1. m l. m 1. m 1. m. m α Table. Inclination angle α in the simulation 3.. Results and discussion: Table 3 shows sets of the calibrated rincial distance c and the offsets x, y from the calibrated rincial oint to the center of the image frame obtained in the numerical simulation. The offsets x, y are also illustrated in Figure 8. Figure 9 shows grahs of the balanced radial distortion r obtained in the numerical simulation. Case c x y mm) D-C D-N D-M D-F D-N D-M D-F Table 3. Camera arameters obtained in the simulation

5 y D-C 3D-M 3D-N 3D-F D-F D-M x D-N ixels radius Figure 8. Offsets x, y ) of the rincial oint obtained in the simulation The estimated rincial oints of all eight cases are distributed within a slightly small area as shown in Figure 8. The estimated rincial oints of two cases D-C and whose inclination angle is the smallest one are located without the circle of ixels radius from the setu osition. The disersion of the rincial oints from the setu osition estimated in all cases excet D-C and is less than ixels. This disersion is nearly equal to that of and in the field exeriment stated in the revious section. An inclination angle at image acquisition seems to have a slight influence on estimation of the osition of the rincial oint. A set of calibration oints on the D lane may be as effective in estimation of the osition of the rincial oint as a set of calibration oints in 3D sace. The balanced radial distortion curves estimated in two cases D-C and are slightly away from the setu curve as Figure 9 shows. The balanced radial distortion curves of all cases excet D-C and are found close to the setu curve. Deviations of these six curves from the setu curve are less than 1 ixel all over the image. A set of calibration oints on the D lane may be as effective in estimation of the balanced radial distortion as a set of calibration oints in 3D sace. Based on the numerical simulation results shown in Table 3, Figure 8 and Figure 9, it may be concluded that our roosed method using a set of calibration oints distributed on the D lane can rovide a reliable set of estimated camera arameters when images are acquired at an aroriate inclination angle α, for instance α CONCLUSION The results of the field exeriment and the numerical simulation indicate that an image distortion model estimated by our method is reliable enough. Error of an image distortion model estimated by the roosed method is exected to be u to aroximately ±3 ixels or ±1 µm on the image. The rimary comonent of the estimation error is the estimation error of the osition of the rincial oint, which may be aroximately ± ixels. The secondary comonent is the estimation error of the balanced radial distortion, which may be less than ±1 ixel. On the other hand, since our roosed calibration method of a non-metric digital camera needs neither secial equiment nor ground survey of control oints, the roosed method is inexensive and convenient for an amateur to calibrate his digital camera. Additionally the fact that a set of calibration oints distributed on the D lane is as effective as a set of calibration oints distributed in the 3D sace makes our method more convenient. We believe that the calibration method roosed in this aer is useful for some non-rofessional fields. REFERENCES Fraser, C. S., 1. Photogrammetric Camera Comonent Calibration in Calibration and Orientation of Cameras in Comuter Vision, A. Gruen and T. S. Huang Eds., Sringer- Verlag, Berlin, Fryer, J. G., Camera Calibration in Close Range Photogrammetry and Machine Vision, K.B. Atkinson Ed., Whittler Publishing, Caithness, Ohmori, T., 1. Digital Camera of 4 million / million Pixels. Nikkei Byte, December 1, in Jaanese) radial distortion D-C D-N D-M D-F D-C radial distortion 1 1-3D-N 3D-M 3D-F a) D distributed calibration oints b) 3D distributed calibration oints Figure 9. Balanced radial distortion grahs obtained in the simulation

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