1 / 36

Fundamentals of Linear Algebra in Geometric Transformations

Explore why Linear Algebra is essential in associating coordinates, performing transformations, and understanding matrices. Learn key operations like matrix sum, product, determinant, inverse, and vector manipulations in various coordinate systems. Dive into detailed examples and equations for 2D and 3D transformations with homogeneous coordinates.

dlarson
Download Presentation

Fundamentals of Linear Algebra in Geometric Transformations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related