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# tsaiexplain - Tsai’s camera calibration method revisited...

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Tsai’s camera calibration method revisited Berthold K.P. Horn Copright © 2000 Introduction Basic camera calibration is the recovery of the principle distance f and the princi- ple point (x 0 ,y 0 ) T in the image plane or, equivalently, recovery of the position of the center of projection (x 0 0 ,f) T in the image coordinate system. This is referred to as interior orientation in photogrammetry. A calibration target can be imaged to provide correspondences between points in the image and points in space. It is, however, generally impractical to position the calibration target accurately with respect to the camera coordinate system using only mechanical means. As a result, the relationship between the target coordinate system and the camera coordinate system typically also needs to be recovered from the correspondences. This is referred to as exterior orientation in photogrammetry. Since cameras often have appreciable geometric distortions, camera calibra- tion is often taken to include the recovery of power series coefﬁcients of these distortions. Furthermore, an unknown scale factor in image sampling may also need to be recovered, because scan lines are typically resampled in the frame grabber, and so picture cells do not correspond discrete sensing elements. Note that in camera calibration we are trying to recover the transforma- tions, based on measurements of coordinates, where one more often uses known transformation to map coordinates from one coordinate system to another. method for camera calibration recovers the interior orientation, the exterior orientation, the power series coefﬁcients for distortion, and an image scale factor that best ﬁt the measured image coordinates corresponding to known target point coordinates. This is done in stages, starting off with closed form least- squares estimates of some parameters and ending with an iterative non-linear optimization of all parameters simultaneously using these estimates as starting values. Importantly, it is error in the image plane that is minimized. Details of the method are different for planar targets than for targets occu- pying some volume in space. Accurate planar targets are easier to make, but lead to some limitations in camera calibration, as pointed out below.

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2 Interior Orientation Camera to Image Interior Orientation is the relationship between camera-centric coordinates and image coordinates. The camera coordinate system has its origin at the center of projection, its z axis along the optical axis, and its x and y axes parallel to the x and y axes of the image. Camera coordinates and image coordinates are related by the perspective projection equations: x I x 0 x C y I y 0 y C = and = f z C f z C where f is the principle distance (distance from the center of projection to the image plane), and (x 0 ,y 0 ) is the principle point (foot of the perpendicular from the center of projection to the image plane). That is, the center of projection is at (x 0 0 ,f) T , as measured in the image coordinate system.
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## This note was uploaded on 10/06/2009 for the course ECE Vision taught by Professor Bertholdhorn during the Spring '04 term at MIT.

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tsaiexplain - Tsai’s camera calibration method revisited...

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