Image Processing

Image Processing Prof. Christian Bauckhage

outline lecture 05 recap

linear (in)dependence and bases

coordinate systems and coordinate transformations affine transformations affine transformations and matrices

summary

recap our long term goal is to understand image transformations

recap

vector space, inner product space, and normed space over a field K V, +, · V, +, ·, h i V, +, ·, k k where +:V ×V →V · :K×V →V hi:V ×V →K kk:V→R

recap

distance of x, y ∈ V ⇔ a function d(x, y) ∈ R, such that d(x, y) > 0 d(x, y) = 0 if and only if x = y d(x, y) = d(y, x) d(x, z) 6 d(x, y) + d(y, z)

recap

linear coordinate system ⇔ origin, axes, and unit Cartesian coordinate system ⇔ axes are pairwise perpendicular and positively oriented and the unit length is 1

qˆ

yˆ

u ˆ

p

rˆ

sˆ

v ˆ tˆ

x ˆ

recap if B = b1 , b2 , . . . , bm is a set of linearly independent vectors in an m-dim vector space V over K, then the linear combinations v = v1 b1 + v2 b2 + . . . + vm bm span the space V and the set B forms a basis of V the vi ∈ K are the coordinates of vector v and we write   v1  v2    v= .   ..  vm

linear (in)dependence and bases

example: vectors in R2

since b2 = α b1 the problem b1

0 = v1 b1 + v2 b2 = v1 b1 + v2 α b1 = v1 + v2 α b1 can be solved for v1 , v2 6= 0

b2


since b2 = α b1 the problem b1

0 = v1 b1 + v2 b2 = v1 b1 + v2 α b1 = v1 + v2 α b1

b2

can be solved for v1 , v2 6= 0

⇒ b1 and b2 are not linearly independent and do not form a basis


in both cases, b1 and b2 are linearly independent

b2

they are oblique and not of the same length but still form valid bases for R2

b1

b2

b1


here, b1 and b2 are linearly independent moreover kb1 k = 1

b2

kb2 k = 1 and hb1 , b2 i = 0

⇒ b1 and b2 form an orthonormal basis of R2

b1

note if B = b1 , b2 , . . . , bm is a basis of an m-dim space V such that kbi k = 1 for all i, the co-variant coordinates of v ∈ V with respect to B are given by vi = hv, bi i

v

v v2

v

v2

v2 b2

b2 v1 b1

b2 b1 v1

b1

v1

note

in this case, we have

v=

m X i=1

m m X m X X vi bi = hv, bi i bi = vj bji bi i=1

i=1 j=1

coordinate systems and coordinate transformations

affine transformations

an affine transformation translates a coordinate system O, ˆx, ˆy into another one O 0 , ˆ x 0 , ˆy 0

affine transformations

an affine transformation translates a coordinate system O, ˆx, ˆy into another one O 0 , ˆ x 0 , ˆy 0 the axes of O 0 , xˆ 0 , ˆy 0 may have different unit lengths and may include different angles

types of affine transformations

affine transformations include identity mappings rotations scalings shears reflections translations

compositions of affine transformations are affine, too

2D examples

original

identity

rotation

scaling x

scaling y

scaling xy

2D examples (continued)

shearing x

shearing y

shearing xy

reflection x

reflection y

reflection xy

2D examples (continued)

translation

general affine transformation

affine transformations of a digital image

original

first scaled . . .


then sheared . . .

then translated . . .


then rotated . . .

and finally mirrored

rotations in the plane

consider a point p in a Cartesian system ˆ, ˆv Suv = O, u p v ˆ

O

u ˆ


consider a point p in a Cartesian system ˆ, ˆv Suv = O, u p

consider a rotated Cartesian system Sxv = O, ˆ x, ˆy

yˆ

v ˆ

x ˆ O

u ˆ


consider a point p in a Cartesian system ˆ, ˆv Suv = O, u consider a rotated Cartesian system Sxv = O, ˆ x, ˆy consider the coordinates of p in either system

pv yˆ

p

v ˆ

py px x ˆ O

u ˆ pu


consider a point p in a Cartesian system ˆ, ˆv Suv = O, u consider a rotated Cartesian system Sxv = O, ˆ x, ˆy consider the coordinates of p in either system consider the rotation angle ϕ

pv yˆ

p

v ˆ

py px x ˆ O

u ˆ pu

ϕ

question | | if we knew pu , pv and ϕ, could we compute px , py ?

question | | if we knew pu , pv and ϕ, could we compute px , py ?

answer let’s see . . .


we have ˆ pu + ˆv pv p=ˆ x px + ˆy py = u

pv yˆ

p

v ˆ

py px x ˆ O

u ˆ pu

ϕ



pv yˆ

“dotting” with ˆ x yields

p

v ˆ

py px

î pu + hˆ hˆ x, ˆ xi px + hˆ x, ˆyi py = hˆ x, u x, ˆvi pv

x ˆ O

u ˆ pu

ϕ



pv yˆ


p

v ˆ

py px

î pu + hˆ hˆ x, ˆ xi px + hˆ x, ˆyi py = hˆ x, u x, ˆvi pv since ˆ x ⊥ ˆy and kˆ xk = 1, this simplifies to î pu + hˆ px = hˆ x, u x, ˆvi pv

x ˆ O

u ˆ pu

ϕ



pv yˆ


p

v ˆ

py px

î pu + hˆ hˆ x, ˆ xi px + hˆ x, ˆyi py = hˆ x, u x, ˆvi pv since ˆ x ⊥ ˆy and kˆ xk = 1, this simplifies to î pu + hˆ px = hˆ x, u x, ˆvi pv and similarly î pu + h ˆy, ˆvi pv py = h ˆy, u

x ˆ O

u ˆ pu

ϕ

note

the resulting system of linear equations î pu + h ˆ px = h ˆx, u x, ˆvi pv î pu + h ˆy, ˆvi pv py = h ˆy, u can be written in matrix-vector notation î h ˆ px h ˆx, u x, ˆvi pu = î h ˆy, ˆvi pv py h ˆy, u

note

we can write this even more compact, namely î hˆ px hˆx, u x, ˆvi pu = î hˆy, ˆvi pv py hˆy, u cos ϕ sin ϕ pu = − sin ϕ cos ϕ pv î = kˆ ˆ, and ˆv are unit vectors, we have recall that hˆ x, u xk kˆ uk cos ϕ and since ˆ x, ˆy, u î = kˆ hˆ x, u xk kˆ uk cos ϕ = cos ϕ π π hˆ x, ˆvi = kˆ xk kˆvk cos − ϕ = cos −ϕ 2 2 π π î = kˆyk kˆ hˆy, u uk cos + ϕ = cos +ϕ 2 2 hˆy, ˆvi = kˆyk kˆvk cos ϕ = cos ϕ which together with cos(α ± β) = cos α cos β ∓ sin α sin β yields the above result

note

to reduce clutter, we substitute x = pxy u = puv so as to obtain x = Ru

note

to reduce clutter, we substitute x = pxy u = puv so as to obtain x = Ru

for which we note that matrix R ∈ R2×2 is rather special

group G,

∀ a, b ∈ G : a b ∈ G ∀ a, b, c ∈ G : (a b) c = a (b c)

closure associativity

∃I ∈G:∀a∈G:I a=a I =a

identity element

∀ a ∈ G : ∃ a−1 ∈ G : a a−1 = a−1 a = I

inverse element

Abelian group G,

a group where is commutative, that is where ∀ a, b ∈ G : a b = b a is called an Abelian group

note

groups are a dime a dozen, too e.g. R \ {0}, ·, I = 1 ,

Q \ {0}, ·, I = 1 ,

Z, +, I = 0 , . . .

many groups, however, are not commutative certain kinds of (square!) matrices form groups

examples

the general linear group, the orthogonal group, and the special orthogonal group of m × m matrices over R

GL(m) = M ∈ R

m×m

det M 6= 0

O(m) = M ∈ Rm×m det M 6= 0 ∧ M| M = MM| = I

SO(m) = M ∈ Rm×m det M = 1 ∧ M| M = MM| = I

note

R ∈ SO(2) it is square its columns are pairwise orthonormal its rows are pairwise orthonormal its transpose equals its inverse R| = R−1 ⇔ R| R = RR| = I its determinant is det R = cos2 ϕ + sin2 ϕ = 1

note

R ∈ SO(2) is a 2 × 2 rotation matrix it is square its columns are pairwise orthonormal its rows are pairwise orthonormal its transpose equals its inverse R| = R−1 ⇔ R| R = RR| = I its determinant is det R = cos2 ϕ + sin2 ϕ = 1

note

rotations R ∈ SO(m) are unitary operators ⇔ they preserve inner products and norms

note

rotations R ∈ SO(m) are unitary operators ⇔ they preserve inner products and norms for example, let q, p ∈ Rm and r = R q, s = R p, then

|

r, s = r| s = R q R p = q| R| R p = q| p = q, p

note

rotations R ∈ SO(m) are unitary operators ⇔ they preserve inner products and norms for example, let q, p ∈ Rm and r = R q, s = R p, then

|

r, s = r| s = R q R p = q| R| R p = q| p = q, p

and

2

r = r| r = R q | R q = q| R| R q = q| q = q 2

convention

if a transformation matrix T is applied to (the points in) the domain of a function g(u) to produce a function ˜g(x), we write ˜g x = ˜g Tu = g u

convention

if a transformation matrix T is applied to (the points in) the domain of a function g(u) to produce a function ˜g(x), we write ˜g x = ˜g Tu = g u

y

v

u

g u

x

˜ g x =˜ g Tu

scaling, shear, reflection (a.k.a. flipping)

other (planar) affine transformations, too, can be realized by means of (2 × 2) matrices


other (planar) affine transformations, too, can be realized by means of (2 × 2) matrices scaling in x−, y− and x, y−direction a 0 1 0 a 0 , , 0 1 0 b 0 b


other (planar) affine transformations, too, can be realized by means of (2 × 2) matrices scaling in x−, y− and x, y−direction a 0 1 0 a 0 , , 0 1 0 b 0 b shearing in y−, x− and x, y−direction 1 c 1 0 1 c , , 0 1 d 1 d 1


other (planar) affine transformations, too, can be realized by means of (2 × 2) matrices scaling in x−, y− and x, y−direction a 0 1 0 a 0 , , 0 1 0 b 0 b shearing in y−, x− and x, y−direction 1 c 1 0 1 c , , 0 1 d 1 d 1 reflecting about the x- and y-axis and about the origin −1 0 1 0 −1 0 , , 0 1 0 −1 0 −1

composing affine transformations

we may compose several affine transformation by applying them in succession

composing affine transformations

we may compose several affine transformation by applying them in succession the corresponding result can be accomplished in a single step, if the corresponding matrices are combined into a single operation to this end, we simply multiply them together

note: matrices do not generally commute

original g(u)

rotation ˜ g(Ru)

shearing ˜ g(Su)

˜ g(SRu)

˜ g(RSu)

linear transformations

affine transformations that can be realized through matrix multiplications are linear transformations

linear transformations

affine transformations that can be realized through matrix multiplications are linear transformations a mapping T is a linear mapping, if, for any two vectors u and v and a scalar a, it is characterized by T u+v =T u +T v T a u) = a T u

translations are not linear

a translation can not be realized by means of matrix multiplication for instance, in 2D, we have px p a = u + py pv b

general affine transformations

in 2D, a general affine transformation is therefore given by px a b pu e = + py c d pv f

or equivalently x = Mu + t

general affine transformations

in 2D, a general affine transformation is therefore given by px a b pu e = + py c d pv f

or equivalently x = Mu + t

⇔ general affine transformations consist of a linear map and a translation

note

to see that affine transformations are not necessarily linear, let T u) = Mu + t be an affine transformation

note

to see that affine transformations are not necessarily linear, let T u) = Mu + t be an affine transformation, then T u+v =M u+v +t = Mu + Mv + t 6= Mu + t + Mv + t = T u + T v

note

every linear transformation is affine not every affine transformation is linear

summary

we now know about

affine transformations types of affine transformations, especially rotations the relationship between linear and affine transformations

exercises

recall that a mapping T is linear, if T u+v =T u +T v T au = aT u in the lecture, we applied property (1) to show that an affine transformation T u = Mu + t is not linear if t 6= 0 now, use property (2) to show this can you think of a way of turning affine transformations like the one in (3) into linear transformations?

(1) (2)

(3)