CS6825: Linear Algebra Overview

2 Why do we need Linear Algebra? We will associate coordinates to We will associate coordinates to 3D points in the scene3D points in the scene 2D points in the CCD array2D points in the CCD array 2D points in the image2D points in the image Coordinates will be used to Coordinates will be used to Perform geometrical transformationsPerform geometrical transformations Associate 3D with 2D pointsAssociate 3D with 2D points Images are matrices of numbers Images are matrices of numbers We will find properties of these numbersWe will find properties of these numbers

3 2D Vector P x1 x2  v

2D Geometrical Transformations Use Vectors and Matrices to represent instead of multiple equations. Can save time especially when have multiple transformations and only have 1 vector matrix equation instead of applying multiple sets of equations.

5 Translation Recall the equations used to translated an Image by in the x direction and in the y direction would be. Recall the equations used to translated an Image by in the x direction and in the y direction would be. P’ = translated location of old pixel P P’ = translated location of old pixel P = (x’, y’) = (x’, y’) where P = (x, y) where P = (x, y) x’ = x + x’ = x + y’ = y + y’ = y + txtxtxtx tytytyty tytytyty txtxtxtx

6 2D Translation t P P’

7 2D Translation Equation P x y txtxtxtx tytytyty P’ t x’ = x + y’ = y + txtxtxtx tytytyty

8 2D Translation using Matrices P x y txtxtxtx tytytyty P’ t tP x’ = x + y’ = y + txtxtxtx tytytyty                            'y x t t ty tx y x y x P                            'y x t t ty tx y x y x P

9Rotation Recall equations below to determine new position when rotating a pixel. You must apply the laws of trigonometry to rotate an image around its center point by the angle desired. The following equations govern the relationship between the old position (x,y) and the new position (x',y'): Recall equations below to determine new position when rotating a pixel. You must apply the laws of trigonometry to rotate an image around its center point by the angle desired. The following equations govern the relationship between the old position (x,y) and the new position (x',y'): x' = x*cos(angle) + y*sin(angle) y' = y*cos(angle) - x*sin(angle) y' = y*cos(angle) - x*sin(angle)

10 Rotation P P’

11 Rotation Equations Clockwise rotation by an angle  x' = x*cos(angle) + y*sin(angle) y' = y*cos(angle) - x*sin(angle) P x Y’ P’  X’ y                     y x y x   cossin cos ' '

12 Scaling P P’

13 Scaling Equation P x y S x.x P’ S y.y

14 Scaling & Translating P P’=S.P P’’=T.P’ P’’=T.P’=T.(S.P)=(T.S).PS T

15 Scaling & Translating P’’=T.P’=T.(S.P)=(T.S).P                                                                    '' yy xx yy xx y x y x tys txs y x ts ts y x s s t t PSTP