Linear Algebra Review

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 9/22/2018 Octavia I. Camps

Matrices Sum: A and B must have the same dimensions Example: 9/22/2018 Octavia I. Camps

Matrices Product: A and B must have compatible dimensions Examples: 9/22/2018 Octavia I. Camps

Matrices Transpose: If A is symmetric Examples: 9/22/2018 Octavia I. Camps

Matrices Determinant: A must be square Example: 9/22/2018 Octavia I. Camps

Matrices Inverse: A must be square Example: 9/22/2018 Octavia I. Camps

2D Vector P x1 x2  v Magnitude: If , Is a UNIT vector Orientation: 9/22/2018 Octavia I. Camps

Vector Addition v w V+w 9/22/2018 Octavia I. Camps

Vector Subtraction V-w v w 9/22/2018 Octavia I. Camps

Scalar Product v av 9/22/2018 Octavia I. Camps

Inner (dot) Product v  w The inner product is a SCALAR! 9/22/2018 Octavia I. Camps

Orthonormal Basis P x1 x2  v i j 9/22/2018 Octavia I. Camps

Vector (cross) Product w v  u The cross product is a VECTOR! Magnitude: Orientation: 9/22/2018 Octavia I. Camps

Vector Product Computation w v  u 9/22/2018 Octavia I. Camps

2D Geometrical Transformations

2D Translation P’ t P 9/22/2018 Octavia I. Camps

2D Translation Equation P x y tx ty P’ t 9/22/2018 Octavia I. Camps

2D Translation using Matrices P x y tx ty P’ t t P 9/22/2018 Octavia I. Camps

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! 9/22/2018 Octavia I. Camps

Back to Cartesian Coordinates: Divide by the last coordinate and eliminate it. For example, 9/22/2018 Octavia I. Camps

2D Translation using Homogeneous Coordinates P x y tx ty P’ t t P 9/22/2018 Octavia I. Camps

Scaling P’ P 9/22/2018 Octavia I. Camps

Scaling Equation P’ Sy.y P y x Sx.x 9/22/2018 Octavia I. Camps

Scaling & Translating P’’=T.P’ S P’=S.P T P P’’=T.P’=T.(S.P)=(T.S).P 9/22/2018 Octavia I. Camps

Scaling & Translating P’’=T.P’=T.(S.P)=(T.S).P 9/22/2018 Octavia I. Camps

Translating & Scaling  Scaling & Translating P’’=S.P’=S.(T.P)=(S.T).P 9/22/2018 Octavia I. Camps

Rotation P P’ 9/22/2018 Octavia I. Camps

Rotation Equations Counter-clockwise rotation by an angle  P’ Y’  P x Y’ P’  X’ y 9/22/2018 Octavia I. Camps

Degrees of Freedom R is 2x2 4 elements BUT! There is only 1 degree of freedom:  The 4 elements must satisfy the following constraints: 9/22/2018 Octavia I. Camps

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 … 9/22/2018 Octavia I. Camps

3D Rotation of Points Rotation around the coordinate axes, counter-clockwise: P x Y’ P’ g X’ y z 9/22/2018 Octavia I. Camps

3D Rotation (axis & angle) 9/22/2018 Octavia I. Camps

3D Translation of Points Translate by a vector t=(tx,ty,tx)T: P’ t Y’ x x’ P z’ y z 9/22/2018 Octavia I. Camps