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Unformatted text preview: Assignment 3 By Assignment 3 Honor Pledge: I pledge on my honor that I have not given or received any unauthorized assistance on this assignment/examination. 1. Question 1 Let us assume the rotation matrix and translation matrix has the following form R = r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 T = t 1 t 2 t 3 Hence, we have 12 parameters to estimate in total. For any given 3-D point coordinate with respect to the world coordinate, C = X Y Z , and the corresponding coordinate with respect to camera coordinate system is c = x y z . The transformation between C and c is x y z = R X Y Z + T To be more explict, x = r 11 X + r 12 Y + r 13 Z + t 1 y = r 21 X + r 22 Y + r 23 Z + t 2 z = r 31 X + r 32 Y + r 33 Z + t 2 We do not know the exact coordinate ( x,y,z ) , but we have the 2-D coordinate ( x ,y ) after projection in the image plane. x f = x z y f = y z Hence, x f = r 11 X + r 12 Y + r 13 Z + t 1 r 31 X + r 32 Y + r 33 Z + t 3 y f = r 21 X + r 22 Y + r 23 Z + t 2 r 31 X + r 32 Y + r 33 Z + t 3 Therefore, for every calibration point, we could be able to construct two equations. However, due to the orthonormality properties of the rotation matrix, i.e., R T = R- 1 and det R = 1 , we have another 4 equations. Therefore, we need at least four calibration points to determine the transformation....
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This note was uploaded on 01/12/2012 for the course CMSC 733 taught by Professor Staff during the Spring '08 term at Maryland.
- Spring '08