lec5 - Cameras and Radiometry Computer Vision I CSE 252A...

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1 CS252A, Fall 2010 Computer Vision I Cameras and Radiometry Computer Vision I CSE 252A Lecture 5 CS252A, Fall 2010 Computer Vision I Last lecture in a nutshell CS252A, Fall 2010 Computer Vision I Conversion Euclidean -> Homogenous -> Euclidean In 2-D Euclidean -> Homogenous: (x, y) -> k (x,y,1) Homogenous -> Euclidean: (x, y, z) -> (x/z, y/z) In 3-D Euclidean -> Homogenous: (x, y, z) -> k (x,y,z,1) Homogenous -> Euclidean: (x, y, z, w) -> (x/w, y/w, z/w) X Y (x,y) (x,y,1) 1 Z CS252A, Fall 2010 Computer Vision I Affine Camera Model Take perspective projection equation, and perform Taylor series expansion about some point (x 0 ,y 0 ,z 0 ). Drop terms that are higher order than linear. Resulting expression is affine camera model Appropriate in Neighborhoo About (x 0 ,y 0 ,z 0 CS252A, Fall 2010 Computer Vision I Simplified Camera Models Perspective Projection Scaled Orthographic Projection Affine Camera Model Orthographic Projection CS252A, Fall 2010 Computer Vision I Coordinate Changes: Rigid Transformations Euclidean Homogeneous
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2 CS252A, Fall 2010 Computer Vision I Some points about SO(n) SO(n) = { R nxn : R T R = I, det(R) = 1} SO(2): rotation matrices in plane 2 SO(3): rotation matrices in 3-space 3 Forms a Group under matrix product operation: – Identity – Inverse – Associative – Closure Closed (finite intersection of closed sets) Bounded R i,j [-1, +1] Does not form a vector space. Manifold of dimension n(n-1)/2 Dim(SO(2)) = 1 Dim(SO(3)) = 3 CS252A, Fall 2010 Computer Vision I SO(3) Parameterizations of SO(3) 3-D manifold, so between 3 parameters and 2n +1 parameters (Whitney’s Embedding Thm.) – Roll-Pitch-Yaw – Euler Angles – Axis Angle (Rodrigues formula) – Cayley’s formula – Matrix Exponential – Quaternions (four parameters + one constraint) CS252A, Fall 2010 Computer Vision I Camera parameters • Issue World units (e.g., cm), camera units (pixels) camera may not be at the origin, looking down the z-axis extrinsic parameters one unit in camera coordinates may not be the same as one unit in world coordinates intrinsic parameters - focal length, principal point, aspect ratio, angle between axes, etc.
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