What is the center of the image?

People need to calibrate camera systems in order to determine the relationship between the positions of features in object space and their corresponding positions in the image. Part of camera calibration is the determination of image center. But, what is the image center? Ideally, the image center is considered to be the point of intersection of the camera's optical axis with the camera's sensing plane. In fact there are many possible de nitions of image center, and in real lenses most do not have the same coordinates. In addition, the image centers move as lens parameters are changed. In this paper we examine why image centers are not necessarily the same for di erent image properties and why they vary with lens parameters. We then provide a taxonomy of 16 di erent image centers and describe procedures for measuring them. Finally we examine the calibration of image center for a variable parameter lens. Several techniques are applied to a precision automated zoom lens and experimental results are shown. We conclude that the accuracy of the image center can be an important factor in the accuracy of the overall camera calibration, and that the large variation in the position of the image center across di erent de nitions and di erent lens settings makes the calibration problem much more complex than is conventionally believed. With proper modeling, by using correct de nitions for all image centers in a system, we can improve the accuracy of our camera calibration. 1 Camera Calibration and Image Center Camera calibration involves modeling the relationship between the positions of features in 3D object space and their corresponding positions in the 2D image. Precise camera models have many terms that describe the properties of the imaging process, and some of these terms are used to account for properties that vary with distance from the center of the image. To model such properties we need to know where their image center is. Naturally, the accuracy of the model depends on the accuracy of the center that we use. In an ideal lens there would be one image center which could be used in modeling any of the radially varying imaging properties. In practice the manufacturing tolerances for lenses result in di erent imaging properties having centers in di erent places, as shown in Figure 1 for our camera and lens. So, image centers are not necessarily interchangeable. Indeed, to fully model a camera we may need several di erent image centers. The situation becomes even more complex for an adjustable lens. When camera parameters such as focus or zoom are varied, the position of the camera's eld of view and image centers will also vary. Figure 2 shows how the position of a xed point at the center of our camera's eld of view shifts as a function of the focus and zoom motors of the camera lens. We start this paper by examining why di erent image properties do not necessarily have the same image center in real lens systems. We also discuss why the image centers move in variable focus and variable focal length camera lenses. We then present a taxonomy of image center de nitions based on the number of lens settings that are required to determine the image center. Procedures for measuring 16 di erent image centers are given and experimental results are then presented for ten of the methods. We conclude by examining how image center and eld of view move in a variable parameter zoom lens. Camera calibration in machine vision has traditionally paid little attention to the issue of image center. Typically the image center used to model one imaging property is obtained by measuring a completely di erent property, if a measurement is made at all. Such approaches can reduce the overall accuracy of the camera calibration. By using the proper image center for each image property that we are trying to model and by calibrating the image centers over the appropriate ranges of lens parameters we can signi cantly improve the accuracy of our camera models. 1 576 pixels 38 4 pi xe ls 1 2 3 4 5 6 7 8 9 10 1. Numerical image center 2. Sensor center 3. COE zoom 4. COE focus 5. COE aperture 6. Radiometric falloff 7. Center of vignetting 8. Autocollimated laser 9. Radial lens distortion 10. Perspective projection Figure 1: Di erent image centers for the same camera system. 205.00 206.00 207.00 208.00 266.00 267.00 268.00 269.00 204.00 203.00 202.00 X coordinate (pixels) 100 500 90