Image Processing

Image Processing Prof. Christian Bauckhage

outline additional material for lecture 05

more on affine transformations

summary

note

in lecture 05 of our course on image processing, we briefly discussed planar affine transformations T : R2 → R2 however, affine transformations are not restricted to points in R2 but also apply to higher dimensional spaces

general affine transformations

affine transformations T : Rm → Rm are given by

u = T(x) = M x + t where vectors x, u, t ∈ Rm and matrix M ∈ Rm×m ⇔ affine transformations combine a linear map (M x) and a translation (+t)

composing affine transformations

composing two or more affine transformations T = Tn ◦ . . . ◦ T2 ◦ T1 produces yet another affine transformation


this is, because u = T2 T1 (x) amounts to
u = M 2 M 1 x + t1 + t2 = M 2 M 1 x + M 2 t1 + t2 = M x + t | {z } | {z } =M

=t

composing affine transformations

however, since matrix multiplication does not generally commute, the order in which affine transformations are applied usually matters ⇔ we generally (but not always) have T2 ◦ T1 6= T1 ◦ T2

example

a 2D shape

first rotated then sheared

first sheared then rotated

note

compositions of affine transformations play a crucial role when computing rotations about an arbitrary point

rotations about arbitrary points

to rotate about a point p different from the origin O, we translate by −p using T1 (x) = x − p apply a rotation R using T2 (x) = R x translate by +p using T3 (x) = x + p

⇔ we compute
u = T3 ◦ T2 ◦ T1 (x) = R x − p + p

example

p

a 2D shape and point p

p

rotation about p

rotations about arbitrary points

looking at
u = R x − p + p = Rx − Rp + p and substituting M=R t = p − Rp we indeed find u = Mx+t and recognize an affine transformation

inverse affine transformations

if the matrix M in an affine transformation u = T(x) = M x + t is invertible, the inverse transformation is x = T −1 (u) = M−1 u − t




inverse affine transformations

looking at
x = M−1 u − t = M−1 u − M−1 t and substituting M0 =

M−1

t 0 = −M−1 t we find that x = M 0u + t 0 and recognize that the inverse of an invertible affine transformation is yet another affine transformation

the affine group

in lecture 05 of our course on image processing, we briefly discussed the notion of a group and saw that certain matrices form groups given our discussion above, we now observe that the set of all invertible affine transformations T : Rm → Rm forms the affine group Aff (m, R)

summary

we now know about

general affine transformations, their composition, and their inverses the fact that invertible affine transformations form yet another group