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Linear Algebra Review

Linear Algebra Review. Many slides in this review are from O. Camps (Penn Stae University). Why do we need Linear Algebra?. We will associate coordinates to 3D points in the scene 2D points in the image Coordinates will be used to Perform geometrical transformations

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Linear Algebra Review

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  1. Linear Algebra Review Many slides in this review are from O. Camps (Penn Stae University)

  2. Why do we need Linear Algebra? • We will associate coordinates to • 3D points in the scene • 2D points in the image • Coordinates will be used to • Perform geometrical transformations • Associate different coordinate systems (platform, end-effector, camera) • Associate 3D with 2D points • Images are matrices of numbers • We will find properties of these numbers

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

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

  5. Examples: Matrices Transpose: If A is symmetric

  6. Example: Matrices Determinant: A must be square

  7. Example: Matrices Inverse: A must be square

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

  9. v+w v w Vector Addition

  10. Vector Subtraction v-w v w

  11. av v Scalar Product

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

  13. u w  v Orientation: Magnitude: Vector (cross) Product The cross product is a VECTOR!

  14. Orthonormal Basis in3D Standard basevectors: Coordinates of apoint inspace:

  15. u w  v Vector Product Computation

  16. 2D Geometrical Transformations

  17. 2D Translation P’ t P

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

  19. 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!

  20. Back to Cartesian Coordinates: • Divide by the last coordinate and eliminate it. For example,

  21. P’ t ty P y x tx 2D Translation using Homogeneous Coordinates t P

  22. Scaling P’ P

  23. Scaling Equation P’ Sy.y P y x Sx.x

  24. S T Scaling & Translating P’’=T.P’ P’=S.P P P’’=T.P’=T.(S.P)=(T.S).P

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

  26. Translating & Scaling  Scaling & Translating P’’=S.P’=S.(T.P)=(S.T).P

  27. Rotation P P’

  28. P’ Y’  P y x X’ Rotation Equations Counter-clockwise rotation by an angle 

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

  30. 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 …

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

  32. 3D Rotation (axis & angle)

  33. 3D Translation of Points Translate by a vector t=(tx,ty,tx)T: P’ t Y’ x x’ P z’ y z

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