1. Linear Algebra Review

2. 6/26/2015Octavia I. Camps2 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

3. 6/26/2015Octavia I. Camps3 Matrices Sum:Example: A and B must have the same dimensions

4. 6/26/2015Octavia I. Camps4 Matrices Product:Examples: A and B must have compatible dimensions

5. 6/26/2015Octavia I. Camps5 Matrices Transpose: If A is symmetric Examples:

6. 6/26/2015Octavia I. Camps6 Matrices Determinant: Example: A must be square

7. 6/26/2015Octavia I. Camps7 Matrices Inverse: A must be square Example:

8. 6/26/2015Octavia I. Camps8 2D Vector P x1 x2  v Magnitude: Orientation: Is a unit vector If, Is a UNIT vector

9. 6/26/2015Octavia I. Camps9 Vector Additionv w V+w

10. 6/26/2015Octavia I. Camps10 Vector Subtraction v w V-w

11. 6/26/2015Octavia I. Camps11 Scalar Product vav

12. 6/26/2015Octavia I. Camps12 Inner (dot) Product v w  The inner product is a SCALAR!

13. 6/26/2015Octavia I. Camps13 Orthonormal Basis P x1 x2  v i j

14. 6/26/2015Octavia I. Camps14 Vector (cross) Product The cross product is a VECTOR! w v  u Orientation: Magnitude:

15. 6/26/2015Octavia I. Camps15 Vector Product Computation w v  u i=(1,0,0)j=(0,1,0)k=(0,0,1) i.j=0, i.k=0, j.k=0

16. 2D Geometrical Transformations

17. 6/26/2015Octavia I. Camps17 2D Translation t P P’

18. 6/26/2015Octavia I. Camps18 2D Translation Equation P x y txtxtxtx tytytyty P’ t

19. 6/26/2015Octavia I. Camps19 2D Translation using Matrices P x y txtxtxtx tytytyty P’ t tP

20. 6/26/2015Octavia I. Camps20 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!

21. 6/26/2015Octavia I. Camps21 Back to Cartesian Coordinates: Divide by the last coordinate and eliminate it. For example,

22. 6/26/2015Octavia I. Camps22 2D Translation using Homogeneous Coordinates P x y txtxtxtx tytytyty P’ t t P

23. 6/26/2015Octavia I. Camps23 Scaling P P’

24. 6/26/2015Octavia I. Camps24 Scaling Equation P x y S x.x P’ S y.y

25. 6/26/2015Octavia I. Camps25 Scaling & Translating P P’=S.P P’’=T.P’ P’’=T.P’=T.(S.P)=(T.S).PS T

26. 6/26/2015Octavia I. Camps26 Scaling & Translating P’’=T.P’=T.(S.P)=(T.S).P

27. 6/26/2015Octavia I. Camps27 Translating & Scaling  Scaling & Translating P’’=S.P’=S.(T.P)=(S.T).P

28. 6/26/2015Octavia I. Camps28 Rotation P P’

29. 6/26/2015Octavia I. Camps29 Rotation Equations Counter-clockwise rotation by an angle  P x Y’ P’  X’ y

30. 6/26/2015Octavia I. Camps30 Degrees of Freedom R is 2x2 4 elements BUT! There is only 1 degree of freedom:  The 4 elements must satisfy the following constraints:

31. 6/26/2015Octavia I. Camps31 Scaling, Translating & Rotating Order matters! P’ = S.P P’’=T.P’=(T.S).PP’’’=R.P”=R.(T.S).P=(R.T.S).P R.T.S  R.S.T  T.S.R …

32. 6/26/2015Octavia I. Camps32 3D Rotation of Points Rotation around the coordinate axes, counter-clockwise: P x Y’ P’  X’ y z

33. 6/26/2015Octavia I. Camps33 3D Rotation (axis & angle)

34. 6/26/2015Octavia I. Camps34 3D Translation of Points Translate by a vector t=(t x,t y,t x ) T : P x Y’ P’ x’ y z z’ t