Linear Algebra Review Why do we need Linear

Linear Algebra Review

Why do we need Linear Algebra? • We will associate coordinates to – 3 D points in the scene – 2 D points in the image • Coordinates will be used to – Perform geometrical transformations – Associate different coordinate systems (platform, end-effector, camera) – Associate 3 D with 2 D points • Images are matrices of numbers – We will find properties of these numbers 1/30/2022 Octavia I. Camps 2

Matrices Sum: A and B must have the same dimensions Example: 1/30/2022 Octavia I. Camps 3

Matrices Product: A and B must have compatible dimensions Examples: 1/30/2022 Octavia I. Camps 4

Matrices Transpose: If A is symmetric Examples: 1/30/2022 Octavia I. Camps 5

Matrices Determinant: A must be square Example: 1/30/2022 Octavia I. Camps 6

Matrices Inverse: A must be square Example: 1/30/2022 Octavia I. Camps 7

2 D Vector x 2 v x 1 Magnitude: If P Is a UNIT vector , Is a unit vector Orientation: 1/30/2022 Octavia I. Camps 8

Vector Addition v 1/30/2022 V+w w Octavia I. Camps 9

Vector Subtraction v 1/30/2022 V-w w Octavia I. Camps 10

Scalar Product av v 1/30/2022 Octavia I. Camps 11

Inner (dot) Product v w The inner product is a SCALAR! 1/30/2022 Octavia I. Camps 12

Orthonormal Basis P x 2 j v i 1/30/2022 x 1 Octavia I. Camps 13

Vector (cross) Product u w v The cross product is a VECTOR! Magnitude: Orientation: 1/30/2022 Octavia I. Camps 14

Vector Product Computation u w v 1/30/2022 Octavia I. Camps 15

2 D Geometrical Transformations

2 D Translation P’ t P 1/30/2022 Octavia I. Camps 17

2 D Translation Equation ty P P’ t y x 1/30/2022 tx Octavia I. Camps 18

2 D Translation using Matrices ty P P’ t y x 1/30/2022 t tx Octavia I. Camps P 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! 1/30/2022 Octavia I. Camps 20

Back to Cartesian Coordinates: • Divide by the last coordinate and eliminate it. For example, 1/30/2022 Octavia I. Camps 21

2 D Translation using Homogeneous Coordinates ty P P’ t t P y x 1/30/2022 tx Octavia I. Camps 22

Scaling P’ P 1/30/2022 Octavia I. Camps 23

Scaling Equation P’ Sy. y P y x 1/30/2022 Sx. x Octavia I. Camps 24

Scaling & Translating S P’’=T. P P’=S. P T P P’’=T. P’=T. (S. P)=(T. S). P 1/30/2022 Octavia I. Camps 25

Scaling & Translating P’’=T. P’=T. (S. P)=(T. S). P 1/30/2022 Octavia I. Camps 26

Translating & Scaling & Translating P’’=S. P’=S. (T. P)=(S. T). P 1/30/2022 Octavia I. Camps 27

Rotation P P’ 1/30/2022 Octavia I. Camps 28

Rotation Equations Counter-clockwise rotation by an angle Y’ P’ P y X’ 1/30/2022 x Octavia I. Camps 29

Degrees of Freedom R is 2 x 2 4 elements BUT! There is only 1 degree of freedom: The 4 elements must satisfy the following constraints: 1/30/2022 Octavia I. Camps 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 … 1/30/2022 Octavia I. Camps 31

3 D Rotation of Points Rotation around the coordinate axes, counter-clockwise: Y’ P’ g y X’ P x z 1/30/2022 Octavia I. Camps 32

3 D Rotation (axis & angle) 1/30/2022 Octavia I. Camps 33

3 D Translation of Points Translate by a vector t=(tx, ty, tx)T: P’ Y’ z’ z t x’ P x y 1/30/2022 Octavia I. Camps 34