Module 6: Pinhole camera model Lecture 32: Coordinate system conversion, Changing the image/world coordinate system

Transcription

1 The Lecture Contains: Back-projection of a 2D point to 3D 6.3 Coordinate system conversion file:///d /...(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2032/32_1.htm[12/31/ :01:40 PM]

2 Back-projection of a 2D point to 3D Previously, the process of projecting a 3D point onto the 2D image plane was described. We now present how a 2D point can be back-projected to the 3D space and derive the corresponding coordinates. Considering a 2D point p in an image, there exists a collection of 3D points that are mapped and projected onto the same point p. This collection of 3D points constitutes a ray connecting the camera center and. From Equation ( 6.18 ), the ray associated to a pixel can be defined as (6.19) where is the positive scaling factor defining the position of the 3D point on the ray. In the case Z is known, it is possible to obtain the coordinates X and Y by calculating using the relation Where (6.20) The back-projection operation is important for depth estimation and image rendering. For depth estimation, this would mean that an assumption is made for the value of Z and the corresponding 3D point is calculated. With an iterative procedure, an appropriate depth value is selected from a set of assumed depth candidates. file:///d /...(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2032/32_2.htm[12/31/ :01:40 PM]

3 6.3 Coordinate system conversion Sometimes, coordinate systems need to be converted to obtain a more efficient computation procedure. Let us now propose two methods that transform the projection matrix, so that new coordinate systems can be employed. We will also provide applications of those methods. A. Changing the image coordinate system The definition of the coordinate system in 3D image processing is not uniformly chosen. Typically, pixel coordinates are defined such that the origin of the 2D image coordinate system is located at the top left of the image. In this case, the x and y axis point horizontally to the right and vertically downward, respectively ( convention 1 ). However, an alternative convention is to locate the origin of the image coordinate system at the bottom left, with the y image axis pointing vertically upwards. To transform the image coordinate system, it is necessary to flip the y image axis and translate the origin along the y image axis. This can be performed using the matrix denoted (see Equation ( 6.21 )). Additionally, one can distinguish two possible conventions for defining the orientation of the 3D world axis: either a left-handed or a right-handed coordinate system can be adopted. The conversion of a left-handed to a right-handed coordinate system can be performed by flipping the Y world (matrix ). By concatenating the two conversion matrices and with the original projection matrix, one can obtain the converted projection matrix file:///d /...(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2032/32_3.htm[12/31/ :01:40 PM]

4 (6.21) where h corresponds to the height of the image. The obtained converted projection matrix is then defined in an image coordinate system using the convention 1 notation and in a right-handed world coordinate system. Finally, it should be noted that the conversion of the image coordinate system is achieved by modifying the intrinsic parameters while the conversion of the world coordinate system is made by transforming the extrinsic parameters. This is the reason why conversion matrices and are placed as left and right terms in Equation ( 6. 21). B. Changing the world coordinate system A conversion is used for re-specifying depth images into a new world coordinate system. This conversion involves the calculation of the position of a 3D point specified in another camera coordinate system and the projection of this 3D point onto the other image plane. The modification of the location and orientation of the world coordinate system is performed in a way similar to the above-described method. Figure 6.6 illustrates the definition of two world coordinate systems. file:///d /...(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2032/32_4.htm[12/31/ :01:40 PM]

5 Figure 6.6 A 3D point P can be defined in two different world coordinate systems. Defining P in a new coordinate system involves (1) the conversion of its 3D coordinates and (2) the conversion of the extrinsic parameters. The modification of the world coordinate system involves the simultaneous conversion of the projection matrix and the coordinates of the 3D point. Considering a 3D world point P and a camera defined with a projection matrix with intrinsic and extrinsic parameters K, R and C, the coordinatesystem conversion can be carried out in two steps. First, specify the projection matrix in a new world coordinate system, where only the position and orientation of the camera, i.e., extrinsic parameters, should be modified. The extrinsic parameters are converted using the position and orientation of the new coordinate system defined, with respect to the original coordinate system. Second, specify the position of a 3D point P in the new coordinate system. The coordinate-system conversion can be written as (6.) where p represents the projected pixel position and the all-zero element vector is denoted by. For clarity, the image coordinate axes are labeled in lower case and the world coordinate axes are labeled in upper case. file:///d /...(Ganesh%20Rana)/MY%20COURSE_Ganesh%20Rana/Prof.%20Sumana%20Gupta/FINAL%20DVSP/lecture%2032/32_5.htm[12/31/ :01:40 PM]