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Unformatted text preview: Course overview 1. Geometry 2. Low & Midlevel vision 3. High level vision Course overview 1. Geometry 2. Low & Midlevel vision 3. High level vision How to extract 3d information?  Which cues are useful?  What are the mathematical tools? Linear Algebra & Geometry why is linear algebra useful in computer vision? Some of the slides in this lecture are courtesy to Prof. Octavia I. Camps, Penn State University References:Any book on linear algebra![HZ] chapters 2, 4 Why is linear algebra useful in computer vision? Representation 3D points in the scene 2D points in the image Coordinates will be used to Perform geometrical transformations Associate 3D with 2D points Images are matrices of numbers Find properties of these numbers Agenda 1. How did you like the movie? 2. Basics definitions and properties 3. Geometrical transformations 4. Application: removing perspective distortion P = [x,y,z] Vectors (i.e., 2D or 3D vectors) Image 3D world p = [x,y] Vectors (i.e., 2D vectors) ) , ( 2 1 x x = v P x1 x1 x2 x2 v Magnitude: Magnitude: 2 2 2 1   x x + = v Orientation: Orientation: = 1 2 1 tan x x =   ,     2 1 v v v v x x Is a unit vector Is a unit vector If If 1   = v , , v Is a UNIT vector Is a UNIT vector Vector Addition ) , ( ) , ( ) , ( 2 2 1 1 2 1 2 1 y x y x y y x x + + = + = + w v v w v+w v+w Vector Subtraction ) , ( ) , ( ) , ( 2 2 1 1 2 1 2 1 y x y x y y x x = = w v v w vw Scalar Product ) , ( ) , ( 2 1 2 1 ax ax x x a a = = v v av Inner (dot) Product v w 2 2 1 1 2 1 2 1 y x y x ) y , y ( ) x , x ( w v + = = The inner product is a The inner product is a SCALAR! SCALAR! cos  w   v  ) y , y ( ) x , x ( w v 2 1 2 1 = = ? w v , w v if = = Orthonormal Basis 1   1   = ) , ( 2 1 x x = v = j i j i v 2 1 x x + = P x1 x1 x2 x2 v i j ) 1 , ( ) , 1 ( = = j i = j i ? = i v 2 2 1 2 1 x 1 . x . x ) x x ( = + = + = j j i j v 1 2 1 2 1 x x 1 x ) x x ( = + = + = i j i Vector (cross) Product w v u = The cross product is a The cross product is a VECTOR! VECTOR! w v u ) ( ) ( = = = = w w v w u w u v w v v u v u Orientation: Orientation: sin  w  v   w v   u  = = Magnitude: Magnitude: ? w // v if u = Vector Product Computation ) , , ( ) , , ( 3 2 1 3 2 1 y y y x x x = = w v u ) 1 , , ( ) , 1 , ( ) , , 1 ( = = = k j i 1   1   1   = = = k j i = = = k j k i j i k ) 1 y j ( ) 2 1 3 1 y x y x + i ) 2 y ( 1 3 y x + ( 2 x 2 x 3 x 3 y = Matrices...
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 Fall '09
 Savarese

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