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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
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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 Octavia I. Camps
Example: Matrices Sum: A and B must have the same dimensions Octavia I. Camps
A and B must have compatible dimensions Examples: Matrices Product: Octavia I. Camps
Examples: Matrices Transpose: If A is symmetric Octavia I. Camps
Example: Matrices Determinant: A must be square Octavia I. Camps
Example: Matrices Inverse: A must be square Octavia I. Camps
Magnitude: P x2 Is a UNIT vector If , v Orientation: Is a unit vector x1 2D Vector Octavia I. Camps
V+w v w Vector Addition Octavia I. Camps
Vector Subtraction V-w v w Octavia I. Camps
av v Scalar Product Octavia I. Camps
v w Inner (dot) Product The inner product is a SCALAR! Octavia I. Camps
j i P x2 v x1 Orthonormal Basis Octavia I. Camps
u w v Orientation: Magnitude: Vector (cross) Product The cross product is a VECTOR! Octavia I. Camps
u w v Vector Product Computation Octavia I. Camps
2D Translation P’ t P Octavia I. Camps
P’ t ty P y x tx 2D Translation Equation Octavia I. Camps
P’ t ty P t y x tx 2D Translation using Matrices P 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! Octavia I. Camps
Back to Cartesian Coordinates: • Divide by the last coordinate and eliminate it. For example, Octavia I. Camps
P’ t ty P y x tx 2D Translation using Homogeneous Coordinates t P Octavia I. Camps
Scaling P’ P Octavia I. Camps
Scaling Equation P’ Sy.y P y x Sx.x Octavia I. Camps
S T Scaling & Translating P’’=T.P’ P’=S.P P P’’=T.P’=T.(S.P)=(T.S).P Octavia I. Camps
Scaling & Translating P’’=T.P’=T.(S.P)=(T.S).P Octavia I. Camps
Translating & Scaling Scaling & Translating P’’=S.P’=S.(T.P)=(S.T).P Octavia I. Camps
Rotation P P’ Octavia I. Camps
P’ Y’ P y x X’ Rotation Equations Counter-clockwise rotation by an angle 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: 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 … Octavia I. Camps
P’ Y’ g P y x X’ 3D Rotation of Points Rotation around the coordinate axes, counter-clockwise: z Octavia I. Camps
3D Rotation (axis & angle) 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 Octavia I. Camps