2-D Geometry

The Image Formation Pipeline

Outline Vector, matrix basics 2-D point transformations Translation, scaling, rotation, shear Homogeneous coordinates and transformations Homography, affine transformation

Notes on Notation (a little different from book) Vectors, points: x, v (assume column vectors) Matrices: R, T Scalars: x, a Axes, objects: X, Y, O Coordinate systems: W, C Number systems: R, Z Specials Transpose operator: x T (as opposed to x 0 ) Identity matrix: Id Matrices/vectors of zeroes, ones: 0, 1

Block Notation for Matrices Often convenient to write matrices in terms of parts Smaller matrices for blocks Row, column vectors for ranges of entries on rows, columns, respectively E.g.: If A is 3 x 3 and :

2-D Transformations Types Scaling Rotation Shear Translation Mathematical representation

2-D Scaling

1 sxsx Horizontal shift proportional to horizontal position

2-D Scaling 1 sysy Vertical shift proportional to vertical position

2-D Scaling

Matrix form of 2-D Scaling

2-D Scaling

2-D Rotation

µ

µ

Matrix form of 2-D Rotation µ (this is a counterclockwise rotation; reverse signs of sines to get a clockwise one)

Matrix form of 2-D Rotation µ

2-D Shear (Horizontal)

Horizontal displacement proportional to vertical position

2-D Shear (Horizontal) (Shear factor h is positive for the figure above)

2-D Shear (Horizontal)

2-D Shear (Vertical)

2-D Translation

Representing Transformations Note that we ’ ve defined translation as a vector addition but rotation, scaling, etc. as matrix multiplications It ’ s inconvenient to have two different operations (addition and multiplication) for different forms of transformation It would be desirable for all transformations to be expressed in a common, linear form (since matrix multiplications are linear transformations)

Example: “ Trick ” of additional coordinate makes this possible Old way: New way:

Translation Matrix We can write the formula using this “ expanded ” transformation matrix as: From now on, assume points are in this “ expanded ” form unless otherwise noted:

Homogeneous Coordinates “ Expanded ” form is called homogeneous coordinates or projective space Technically, 2-D projective space P 2 is defined as Euclidean space R 3 ¡ (0, 0, 0) T Change to projective space by adding a scale factor (usually but not always 1):

Homogeneous Coordinates: Projective Space Equivalence is defined up to scale, (non-zero for finite points) Think of projective points in P 2 as rays in R 3, where z coordinate is scale factor All Euclidean points along ray are “ same ” in this sense

Leaving Projective Space Can go back to non-homogeneous representation by dividing by scale factor and dropping extra coordinate: This is the same as saying “ Where does the ray intersect the plane defined by z = 1 ” ?

Homogeneous Coordinates: Rotations, etc. A 2-D rotation, scaling, shear or other transformation normally expressed by a 2 x 2 matrix R is written in homogeneous coordinates with the following 3 x 3 matrix: The non-commutativity of matrix multiplication explains why different transformation orders give different results — i.e., RT  TR In homogeneous form In homogeneous form

Example: Transformations Don ’ t Commute

2-D Transformations Full-generality 3 x 3 homogeneous transformation is called a homography

2-D Transformations Full-generality 3 x 3 homogeneous transformation is called a homography Translation components

2-D Transformations Full-generality 3 x 3 homogeneous transformation is called a homography Scale/rotation components

2-D Transformations Full-generality 3 x 3 homogeneous transformation is called a homography Shear/rotation components

2-D Transformations Full-generality 3 x 3 homogeneous transformation is called a homography Homogeneous scaling factor

2-D Transformations Full-generality 3 x 3 homogeneous transformation is called a homography When these are zero (as they have been so far), H is an affine transformation