Ying Wu. Professor, ECE department, Northwestern University Research Interests. Computer Vision and Visual Analytics/Understanding; Autonomous Robots and Intelligent Machines; Machine Learning and Pattern Recognition; Multi-modal Perception and Intelligent Interaction;

where dAr is the lighting source surface, x is the location of the source, and r; r is used to represent the lighting direction. Denote illuminated surface by dAi and its foreshortening angle by i, then the solid angle d!r = dAicos i r2: (3) Then Lr(x; r; r) = r2d˚ cos icos rdAidAr: (4) Note the foreshortened area is for the area lighting source, and the solid angle is that one

Ying Wu Electrical & Computer Engineering Northwestern University Evanston, IL 60208 [email protected] ECE432-Advanced Computer Vision 0 Multimedia Human-Computer Interaction Robotics Virtual Enviornments Biomechanics Psychology Computer Vision Graphics Computer Processing Speech Image Processing

O m P v v p m Figure 1: Motion Field i.e., p_ = Z_m^ +Zm^_ i.e., Vp = (VT p k)m^ +Zvm where k is the unit vector of the depth direction. So, vm = 1 Z (Vp (VT p k)m^) which means that the 2D motion eld vm is a function of Vp=Z. 1.3 Optical Flow

Uncalibrated Case: Fundamental Matrix Intrinsic parameters determine the normalized coordinates 3/21

Ying Wu, "Vision and Learning for Intelligent Human-Computer Interaction", Ph.D. Dissertation, 2001, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign. [ PDF ]

Ying Wu: Professor, ECE department, Northwestern University Research Interests: Computer Vision and Visual Analytics/Understanding; Autonomous Robots and Intelligent Machines; Machine Learning and Pattern Recognition; Multi-modal Perception and Intelligent Interaction;

Calibrated Case Assume both cameras are fully calibrated Let’s treat one camera frame as the world frame (e.g., the left one) So we know 1 2R and 1 2t, which describe the relative pose of the two cameras. The two image points in their own frames 1Pˆ 1 = u 1 v 1 1 , 2Pˆ 2 = u 2 v 2 1 Related by a rigid transform (1Pˆ2) = ( 1 2R)( 2Pˆ 2)+( 1 2t) 16/19

Image Formation, Camera Model and Calibration Ying Wu Electrical Engineering and Computer Science Northwestern University, Evanston, IL 60208 [email protected]

YING WU Northwestern University Department of Electrical & Computer Engineering 2145 Sheridan Road, Evanston, IL 60208 Tel: (847)491-2901, Fax: (847) 491-4455