# Module 6: Pinhole camera model Lecture 30: Intrinsic camera parameters, Perspective projection using homogeneous coordinates

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1 The Lecture Contains: Pinhole camera model 6.1 Intrinsic camera parameters A. Perspective projection using homogeneous coordinates B. Principal-point offset C. Image-sensor characteristics file:///d /...0(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2030/30_1.htm[12/31/2015 1:23:40 PM]

2 Pinhole camera model In this section, we describe the image acquisition process known as the pinhole camera model, which is widely used. More specifically, we first discuss the model that integrates the internal or intrinsic camera parameters, such as the focal length and the lens distortion. Secondly, we extend the presented simple camera model to integrate external or extrinsic camera parameters corresponding to the position and orientation of the camera. 6.1 Intrinsic camera parameters The pinhole camera model defines the geometric relationship between a 3D point and its 2D corresponding projection onto the image plane. When using a pinhole camera model, this geometric mapping from 3D to 2D is called a perspective projection. We denote the center of the perspective projection (the point in which all the rays intersect) as the optical center or camera center and the line perpendicular to the image plane passing through the optical center as the optical axis (see Figure 6.2 ). Additionally, the intersection point of the image plane with the optical axis is called the principal point. The pinhole camera that models a perspective projection of 3D points onto the image plane can be described as follows. file:///d /...0(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2030/30_2.htm[12/31/2015 1:23:40 PM]

3 A. Perspective projection using homogeneous coordinates Let us consider a camera with the optical axis being collinear to the center being located at the origin of a 3D coordinate system (see Figure 6.1). - axis and the optical Figure 6.1 The ideal pinhole camera model describes the relationship between a 3D point and its corresponding 2D projection. file:///d /...0(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2030/30_3.htm[12/31/2015 1:23:40 PM]

4 The projection of a 3D world point onto the image plane at pixel position can be written as (6.3) where f denotes the focal length. To avoid such a non-linear division operation, the previous relation can be reformulated using the projective geometry framework, as (6.4) This relation can be the expressed in matrix notation by (6.5) where is the homogeneous scaling factor. file:///d /...0(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2030/30_4.htm[12/31/2015 1:23:41 PM]

5 B. Principal-point offset Most of the current imaging systems define the origin of the pixel coordinate system at the top-left pixel of the image. However, it was previously assumed that the origin of the pixel coordinate system corresponds to the principal point, located at the center of the image (see Figure 6.2(a)). A conversion of coordinate systems is thus necessary. Using homogeneous coordinates, the principal-point position can be readily integrated into the projection matrix. The perspective projection equation becomes now (6.6) C. Image-sensor characteristics To derive the relation described by Equation ( 6.6 ), it was implicitly assumed that the pixels of the image sensor are square, i.e., aspect ratio is 1 : 1 and pixels are not skewed. However, both assumptions may not always be valid. First, for example, an NTSC TV system defines non-square pixels with an aspect ratio of 10 : 11. In practice, the pixel aspect ratio is often provided by the image-sensor manufacturer. Second, pixels can potentially be skewed, especially in the case that the image is acquired by a frame grabber. file:///d /...0(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2030/30_5.htm[12/31/2015 1:23:41 PM]

6 In this particular case, the pixel grid may be skewed due to an inaccurate synchronization of the pixelsampling process. Both previously mentioned imperfections of the imaging system can be taken into account in the camera model, using the parameters? and t, which model the pixel aspect ratio and skew of the pixels, respectively (see Figure 6.2(b) ). The projection mapping can be now updated as (6.7) with being a 3D point defined with homogeneous coordinates. In practice, when employing recent digital cameras, it can be safely assumed that pixels are square and nonskewed. The projection matrix that incorporates the intrinsic parameters is denoted as K. The all zero element vector is denoted by. Figure 6.2 (a) The image and camera coordinate system. (b) Non-ideal image sensor with non-square, skewed pixels. file:///d /...0(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2030/30_6.htm[12/31/2015 1:23:41 PM]

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