however, affine transformations are not restricted to points in R2 but also apply to ... to rotate about a point p different from the origin O, we translate by âp using ...
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
