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Linear Algebra Review

This article reviews the importance of linear algebra in various applications, such as associating coordinates to points, performing geometrical transformations, analyzing matrices, and more.

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Linear Algebra Review

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  1. Linear Algebra Review

  2. Why do we need Linear Algebra? • We will associate coordinates to • 3D points in the scene • 2D points in the CCD array • 2D points in the image • Coordinates will be used to • Perform geometrical transformations • Associate 3D with 2D points • Images are matrices of numbers • We will find properties of these numbers Octavia I. Camps

  3. Example: Matrices Sum: A and B must have the same dimensions Octavia I. Camps

  4. A and B must have compatible dimensions Examples: Matrices Product: Octavia I. Camps

  5. Examples: Matrices Transpose: If A is symmetric Octavia I. Camps

  6. Example: Matrices Determinant: A must be square Octavia I. Camps

  7. Example: Matrices Inverse: A must be square Octavia I. Camps

  8. Magnitude: P x2 Is a UNIT vector If , v Orientation:  Is a unit vector x1 2D Vector Octavia I. Camps

  9. V+w v w Vector Addition Octavia I. Camps

  10. Vector Subtraction V-w v w Octavia I. Camps

  11. av v Scalar Product Octavia I. Camps

  12. v  w Inner (dot) Product The inner product is a SCALAR! Octavia I. Camps

  13. j i P x2 v  x1 Orthonormal Basis Octavia I. Camps

  14. u w  v Orientation: Magnitude: Vector (cross) Product The cross product is a VECTOR! Octavia I. Camps

  15. u w  v Vector Product Computation Octavia I. Camps

  16. 2D Geometrical Transformations

  17. 2D Translation P’ t P Octavia I. Camps

  18. P’ t ty P y x tx 2D Translation Equation Octavia I. Camps

  19. P’ t ty P t y x tx 2D Translation using Matrices P Octavia I. Camps

  20. Homogeneous Coordinates • Multiply the coordinates by a non-zero scalar and add an extra coordinate equal to that scalar. For example, • NOTE: If the scalar is 1, there is no need for the multiplication! Octavia I. Camps

  21. Back to Cartesian Coordinates: • Divide by the last coordinate and eliminate it. For example, Octavia I. Camps

  22. P’ t ty P y x tx 2D Translation using Homogeneous Coordinates t P Octavia I. Camps

  23. Scaling P’ P Octavia I. Camps

  24. Scaling Equation P’ Sy.y P y x Sx.x Octavia I. Camps

  25. S T Scaling & Translating P’’=T.P’ P’=S.P P P’’=T.P’=T.(S.P)=(T.S).P Octavia I. Camps

  26. Scaling & Translating P’’=T.P’=T.(S.P)=(T.S).P Octavia I. Camps

  27. Translating & Scaling  Scaling & Translating P’’=S.P’=S.(T.P)=(S.T).P Octavia I. Camps

  28. Rotation P P’ Octavia I. Camps

  29. P’ Y’  P y x X’ Rotation Equations Counter-clockwise rotation by an angle  Octavia I. Camps

  30. Degrees of Freedom R is 2x2 4 elements BUT! There is only 1 degree of freedom:  The 4 elements must satisfy the following constraints: Octavia I. Camps

  31. Scaling, Translating & Rotating Order matters! P’ = S.P P’’=T.P’=(T.S).P P’’’=R.P”=R.(T.S).P=(R.T.S).P R.T.S  R.S.T  T.S.R … Octavia I. Camps

  32. P’ Y’ g P y x X’ 3D Rotation of Points Rotation around the coordinate axes, counter-clockwise: z Octavia I. Camps

  33. 3D Rotation (axis & angle) Octavia I. Camps

  34. 3D Translation of Points Translate by a vector t=(tx,ty,tx)T: P’ t Y’ x x’ P z’ y z Octavia I. Camps

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