2-d geometry. the image formation pipeline outline vector, matrix basics 2-d point transformations...

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: xT (as opposed to x0)

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

2-D Scaling

1

sx

Horizontal shift proportional to horizontal position

2-D Scaling

1sy

Vertical shift proportional to vertical position

2-D Scaling

Matrix form of 2-D Scaling

2-D Scaling

2-D Rotation

2-D Rotation

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


(Shear factor h is positive for the figure above)

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 P2 is defined as Euclidean space R3 ¡ (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 P2 as rays in R3,

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 formIn homogeneous form

Example: Transformations Don’t Commute

2-D Transformations

Full-generality 3 x 3 homogeneous transformation is called a homography

2-D Transformations


Translation components

2-D Transformations

Full-generality 3 x 3 homogeneoustransformation is called a homography

Scale/rotation components

2-D Transformations


Shear/rotation components

2-D Transformations


Homogeneous scaling factor

2-D Transformations


When these are zero (as they have been so far), H is an affine transformation

2-d geometry. the image formation pipeline outline vector, matrix basics 2-d point transformations...

Documents