# Image Analysis - Lecture 2

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```					Review of Linear algebra
Fourier transform

Image Analysis - Lecture 2

Fredrik Kahl

4 September 2009

Fredrik Kahl   Image Analysis - Lecture 2
Review of Linear algebra
Fourier transform

Lecture 2

Contents
◮   Linear Algebra
◮   Linear spaces, bases
◮   Scalar product, orthogonality, ON-bases
◮   Sub-spaces, projections
◮   The Fourier transform
◮   Deﬁnition, examples, 1D and 2D
◮   Discrete Fourier Transform, Fast Fourier Transform
◮   Application to Images
◮   Master thesis proposal of the day

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Linear algebra: Linear spaces

The following linear spaces are well-known:
 
x1
n : all n × 1-matrices, x =  .  ,
◮ R                             .
.     xi ∈ R
xn
 
x1
n : all n × 1-matrices, x =  .  ,
◮   C                             .
.                   xi ∈ C
xn

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Basis

Deﬁnition
e1 , . . . en ∈ Rn is a basis in Rn if
◮   they are linearly independent
◮   they span Rn .

Example (3D space)
e1 , e2 , e3 ∈ R3 is a basis in R3 if they are not located in the
same plane.

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Canonical Basis

Example (canonical basis)
                                                  
1                  0                                 0
0                1                               0
                                                  
e1 =  .  ,       e2 =  .  ,             ...      en =  . 
.
.                .
.                               .
.
0                        0                                1
is called the canonical basis in Rn .
 
x1
.
x =  .  = x1 e1 . . . + xn en .
.
xn

Fredrik Kahl       Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Coordinates

Let e1 , e2 , . . . , en be a basis. Then for every x there is a unique
set of scalars ξi such that
n
x=             ξi e i .
i=1

These scalars are called the coordinates for x in the basis
e1 , e2 , . . . , en .

Fredrik Kahl       Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Image matrix

An M × N image, f , is described by the matrix
                                    
f (0, 0)     ...    f (0, N − 1)
       .        ..            .     
f =       .
.           .          .
.     ,                            f (i, j) ∈ C
f (M − 1, 0) . . . f (M − 1, N − 1)

f (i, ·)=i:th row, f (·, j)=j:th column.

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Row-stacking

Row-stacking:
                    
f T (0, ·)
          f T (1, ·)    
˜=
f                .

 ∈ RMN (CMN )
               .
.        
f T (M − 1, ·)

Properties:
f + g = ˜+ g
f ˜
λf = λ˜,
f        λ∈R
A linear space!

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Canonical basis

             
0 ... 0 ... 0
   .
. 1 . .   
χ(i, j) =    .     .   ,
0 ... 0 ... 0
with the 1 at position (i, j).
Using this canonical basis we can write

f =           f (i, j)χ(i, j) .
i,j

Idea for image transform:
Choose another basis that is more suitable in some sense.
Image matrices can thus be seen as vectors in a linear space.

Fredrik Kahl     Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Scalar product

Deﬁnition
Let A be a (complex) matrix. Introduce
¯
A∗ = (A)T .

Deﬁnition
Let x and y be two vectors in Rn (Cn ). The scalar product of x
and y is deﬁned as

x ·y =             ¯
xi yi = x ∗ y .

Fredrik Kahl     Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Orthogonality
Deﬁnition
The scalar product of two matrices (images) is deﬁned as
M−1 N−1
f ·g =                 ¯(i, j)g(i, j) .
f
i=0 j=0

x, y ∈ R(C) are orthogonal if x · y = 0. This is often written

x⊥y         ⇔      x ·y =0 .

The length or the norm of x is deﬁned as

||x|| =              |xi |2 = (x ∗ x)1/2 .

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Coordinates:

ξ2                         x

e2

e1             ξ1
Norm:
x2                            x = (x1 , x2 )

||x||

x1
Fredrik Kahl     Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Orthogonal basis (ON-basis)

Deﬁnition
{e1 , . . . , en } is an orthonormal (ON-) basis in Rn (Cn ) if

0 i=j
ei · ej =
1 i=j

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Coordinates in ON-basis

Theorem
Assume that {e1 , . . . , en } is orthonormal (ON) basis and
n
x=             ξi e i .
i=1

Then
n
ξi = ei · x = ei∗ x,            ||x||2 =           |ξi |2
i=1

Fredrik Kahl       Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Illustration

e2 · x                          x

e2

e1        e1 · x

Fredrik Kahl    Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Orthogonal projection

Deﬁnition
Let {a1 , . . . , ak } ∈ Rn , k ≤ n, span a linear subspace, π, in Rn ,
i.e.:
k
π = {w |w =                   xi ai , xi ∈ R} .
i=1

The orthogonal projection of u ∈ Rn on π is a linear mapping
P, such that uπ = Pu and deﬁned by

min ||u − w || = ||u − uπ || .
w ∈π

Fredrik Kahl       Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Orthogonal projection (ctd.)

The orthogonal projection is characterized by
1. uπ ∈ π
2. u − uπ ⊥w for every w ∈ π

u
u − uπ

π            uπ

Fredrik Kahl    Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Example: Matrix-basis

What is the orthogonal projection of f
          
1 2 3
f = 4 5 6
7 7 7

onto the space spanned by (e1 , e2 , e3 )
                                                   
1 1 1                1       1   1              1 0 −1
1                 1                             1
e1 = 1 1 1 , e2 = √  0             0   0  , e3 = √ 1 0 −1
3                  6 −1 −1 −1                    6 1 0 −1
1 1 1

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Example: Matrix-basis (ctd.)

Since (e1 , e2 , e3 ) is orthonormal the coordinates are
√                 √
x1 = f · e1 = 14, x2 = f · e2 = −15/ 6, x3 = f · e3 = −4/ 6.
The orthogonal√                is
projection√ then
ˆ = 14e1 − 15/ 6e2 − 4/ 6e3
f
                  1                        5
1 2 3         1.5 2 6                       26
f = 4 5 6 , ˆ =  4 4 3
f          2
53 ,
1

7 7 7               1                        5
6.5 7 6                       76

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Example: ’Face’-basis
What is the orthogonal projection of f

onto the space spanned by (e1 , e2 , e3 )

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Example: ’Face’-basis (ctd.)

Since (e1 , e2 , e3 ) is orthonormal, the coordinates are
x1 = f · e1 = −2457, x2 = f · e2 = 303, x3 = f · e3 = −603.
The orthogonal projection is then ˆ = −2457e1 + 303e2 − 603e3
f

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Uniqueness of the projection

Let a ∈ π and b ∈ π be two solutions to the minimisation
problem. Set
2
f (t) = u − ta − (1 − t)b                = ...
2
= u−b         + t2 a − b       2
− 2t(a − b) · (u − b),           t ∈ R.
This is a second degree polynomial with minimum in t = 0 and
t = 1 ⇒ f (t) is a constant function and thus ⇒ a = b.

Fredrik Kahl    Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Characterization of the projection

Let f (t) = u − uπ + ta 2 , where a ∈ π. It follows that
f ′ (0) = 2(u − uπ ) · a = 0, i.e. (u − uπ ) ⊥ a.
Conversely: Assume w ∈ π. The property that (u − uπ ) ⊥ a, for
every a ∈ π gives that
2                                 2
u −w            = u − uπ + uπ − w                  =
2                  2                    2
u − uπ           + uπ − w          ≥ u − uπ              ,
i.e. uπ solves the minimization problem.

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

An important result

Let A = [a1 . . . ak ] be a n × k matrix and

π = {w |w = Ax, xi ∈ Rn }

Lemma
If {a1 , . . . , ak } are linearly independent Rn then A∗ A is
invertible.
Proof:    Do it on your own. (Use SVD if you are familiar with it.)

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Projection onto the subspace spanned by A

Theorem
if the columns of A are linearly independent, then the projection
of u on π is given by

uπ = x1 a1 + . . . + xk ak ,        x = (A∗ A)−1 A∗ u .

Proof:   Use the characterization of the projection (above).

a∗ (u − uπ ) = 0
i                        ⇒

A∗ (u − Ax) = 0             ⇒
A∗ u = A∗ Ax           ⇒     x = (A∗ A)−1 A∗ u

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

The pseudo-inverse

Deﬁnition
A+ = (A∗ A)−1 A∗ is called the pseudo-inverse of A.

Observe that if A is quadratic and invertible then A+ = A−1 .
Theorem
If {a1 , . . . , ak } are orthonormal, then the projection of u on π is
given by
uπ = y1 a1 + . . . + yk ak , yi = a∗ u .
i

Proof:    This follows from A∗ A = I.

Fredrik Kahl   Image Analysis - Lecture 2
Linear space, basis
Review of Linear algebra
Scalar product
Fourier transform
Orthogonal projection

Illustration

u

y2 a2                   uπ
y1 a1
π

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Fourier transform

Deﬁnition
Let f be a function from R to R. The Fourier transformen of f is
deﬁned as
+∞
(Ff )(u) = F (u) =                  e−i2πxu f (x)dx .
−∞

Theorem
Under the right assumptions on f , the following inversion
formula
+∞
f (x) =                ei2πux F (u)du
−∞

holds.
Fredrik Kahl     Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Examples

Example

δ(x) → 1(u)
sin(2πu)
rect(x) → 2          = 2 sinc(2πu)
2πu

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Illustrations

F
1

x                                       u
F
1                                        2

−1              1      x                                       u

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Properties

c1 f1 (x) + c2 f2 (x) → c1 F1 (u) + c2 F2 (u) (linearity)
1     u
f (λx) →       F( )              (scaling)
|λ| λ
f (x − a) → e−i2πua F (u)                    (translation)
e−i2πxa f (x) → F (u + a)                        (modulation)
f (x) → F (−u)                             (conjugation)
df
→ 2πiuF (u)                          (differentiation I)
dx
dF
−2πixf (x) →                                    (differentiation II)
du
Example: δ(x − 1) → e−i2πu
Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Review of Distributions

∞
Test functions: ϕ ∈ C0 .
A distribution u is a functional (with certain constraints) acting
on the space of test functions. This action is written

(u, ϕ).

E.g. (δ, ϕ) = ϕ(0), Dirac delta function. Had u been a ordinary
function, then
(u, ϕ) =            u(x)ϕ(x)dx,

i.e. the scalar product.

Fredrik Kahl     Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Properties of Distributions

Properties
◮   Differentiation: (u ′ , ϕ) = −(u, ϕ′ ).
◮   Convolution: u ∗ ϕ(x) = (u, ϕ(x − ·)).
E.g. δ ∗ ϕ(x) = (u, ϕ(x − ·)) = ϕ(x).
◮                        ˆ           ˆ
Fourier transform: (u , ϕ) = (u, ϕ).
E.g.
ˆ          ˆ     ˆ
(δ, ϕ) = (δ, ϕ) = ϕ(0) =

ϕ(x)dx = (1, ϕ)

ˆ
i.e. δ = 1.

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

The discrete Fourier transform (DFT)
Sample f (x) and F (u).

x                                   u
0                     N −1 0                              N −1
∆x                                  ∆u
N−1
F (n∆u ) ∼             e−i2πk ∆x n∆u f (k∆x )∆x ,
k =0
N−1
f (k∆x ) ∼             ei2πk ∆x n∆u F (n∆u )∆u .
n=0

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

The discrete Fourier transform (DFT) (ctd.)

1
This works particularly well if ∆x ∆u =              N:

N−1
1
F (n∆u ) ∼                 e−i2πkn/N f (k∆x ),
∆x
k =0

N−1
1                        1
f (k∆x ) ∼                  ei2πkn/N      F (n∆u ).
N                        ∆x
n=0

F and f are extended to periodic functions with periods N∆u
and N∆x respectively.

Fredrik Kahl      Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Deﬁnition of the discrete Fourier transform
Let the vector
(f (0), f (1), . . . , f (N − 1)) .
represent the discretized version of f (x).
Deﬁnition
The discrete Fourier Transformen (DFT) of f is
N−1
ku
F (u) =            f (k)ωN , u = 0, . . . , N − 1 ,
k =0

where ωN = e−i2π/N .
Represent the sequence F (u) with the vector

(F (0), F (1), . . . , F (N − 1)) .

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Important assumption

All sequences are assumed to be period with period N.
0 N −1
21
3

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Properties of DFT
There are similar formulas for the discrete Fourier Transform
(as compared to that of the continuous Fourier Transform), e.g.
(f (−k0 ), f (1 − k0 ), . . . , f (N − 1 − k0 )) →
N−1
→           f (k − k0 )ω ku = [l = k − k0 ]
k =0
N−1−k0
=             f (l)ω (l+k0 )u =
l=−k0
N−1−k0                                                N−1
k0 u                    lu                            k0 u
=ω                   f (l)ω = [f periodic] = ω                          f (l)ω lu =
l=−k0                                                l=0
k0 u               −i2πk0 /N
=ω          F (u) = e                F (u)

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

DFT in matrix form
Let                                                              
f (0)                             F (0)
         
.
.                               .
.    
f =         ,
.                F =            .     .
f (N − 1)                             F (N − 1)
Deﬁnition
The Fourier Matrix FN is given by
                                                             
1    1        1                  ...              1
1 ω           ωN2                 ...         ωNN−1          
       N                                                     
       2        4                             2(N−1)         
FN = 1 ωN
.
ωN                  ...        ωN               .

.     .
.        .
.                  ..             .
.           
.     .        .                     .           .           
N−1              2(N−1)                (N−1)(N−1)
1 ωN   ωN                       . . . ωN

Fredrik Kahl     Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Computation of DFT

Theorem

F = FN f

Proof: Use the deﬁnition of DFT and of the Fourier Matrix
Consequence: DFT can be computed using matrix
multiplication.

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Properties of the Fourier matrix

Lemma

−1                    1
F N FN = NI ⇐⇒ FN =                     FN
N

Proof:   Multiply F N with FN and use
N−1                   Np
p        1 − ωN
ωN ωN = 1,               (ωN )j =           =0 .
1 − ωN
j=0

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

The inversion formula

This lemma gives us the following inversion formula
Theorem

N−1
1                1                      −ku
f =      F F ⇐⇒ f (k) =                  F (u)ωN ,          k = 0, . . . , N − 1
N                N
u=0

Proof:
1      1
F = Ff ⇒          F F = F Ff = If = f .
N      N

Fredrik Kahl    Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Example

Example
N = 2, ω = −1:
1 1
F2 =                       .
1 −1
1                   4
f =            ⇒F =                     .
3                   −2

Fredrik Kahl     Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Example

Example
N = 4, ω = −i:
       
1 1 1  1
1 −i −1 i 
F4 =            
1 −1 1 −1 .
1 i −1 −i

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Factorization of the Fourier matrix
N = 22 = 4, ω = ω4 , τ = ω 2
                              
1 1 1                    1
1 ω ω 2                   ω3
F4 = 1 ω 2 ω 4
=
ω6
1 ω3 ω6                  ω9
                                                  
1 1 1                    1      1      0    0     0
1 ω 2 ω                     3 
ω  0        0    1     0
=1 ω 4 ω 2
=
ω 6  0      1    0     0
1 ω6 ω3                  ω9     0      0    0     1
                                        
1 1                1 1
 1 τ                   ω ωτ  
=                              P
 1 1                  −1 −1  4
1 τ               −ω −ωτ

Fredrik Kahl     Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Factorization of the Fourier matrix (ctd.)

P4 denotes a 4 × 4 permutation matrix (a matrix with zeros and
ones, where each row and each column only contains one one).
                  
1 0
F2      0 ω
F2 
F2 0
F4 =                     P4 = I D2                    P4
         1 0               I −D2       0 F2
F2 −           F2
0 ω

where
D2 = diag(1, ω) .

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

The Fast Fourier Transform (FFT)

Theorem
The Fourier Matrix can be factorized as
I DN            FN        0
F2N =                                         P2N ,
I −DN            0       FN

where
2             N−1
DN = diag(1, ω2N , ω2N , . . . , ω2N ) .
and P2N is a permutation matrix of order 2N × 2N that maps

(x(0), x(1), . . . , x(2N − 1)) −→
(x(0), x(2), . . . , x(2N − 2), x(1), x(3), . . . , x(2N − 1)) .

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

The Fast Fourier Transform (FFT) (ctd.)

Corollary
Calculation of FN f thus involves two calculations of FN/2 f ,
which involves 4 calculations of FN/4 f , etc.
This algorithm is called the Fast Fourier Transform.

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Calculational complexity
Let µn be the number of multiplications needed for calculating
DFT of order 2n . Factorization gives

µn = 2µn−1 + 2n−1 .
A solution to this recursion formula is
n2n   N log2 N
µn =       =                      om N = 2n .
2        2

Example
N = 1024 = 210
◮   FFT gives µ ∼ 104 multiplications.
◮   DFT gives N 2 ∼ 106 multiplications.

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Two-dimensional Fourier Transform

Deﬁnition
Let f (x, y) be a function from R2 to R. The Fourier transform of
f is deﬁned as
+∞
Ff (u, v) = F (u, v) =                  e−i2π(ux+vy ) f (x, y)dxdy .
−∞

This can be written (using u = (u, v), x = (x, y)):
+∞
F (u) =                  e−i2πu·xf (x)d x .
−∞

Fredrik Kahl      Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

The inversion formula in 2D

Theorem
Under certain conditions on f , the following inversion formula
+∞
f (x, y) =             ei2π(ux+vy ) F (u, v)dudv
−∞

holds.

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Properties of the 2D Fourier transformation
Properties: (in addition to those for the 1-D Fourier Transform)

f1 (x)f2 (y) → F1 (u)F2 (v) (separability)
f (Qx) → F (Qu)                  (rotation)

where Q denotes an orthogonal matrix.
Example

rect(x) rect(y) → 4 sinc(2πu) sinc(2πv)
δ(x)1(y) → 1(u)δ(v)
δ(x) rect(y) → 1(u)2 sinc(2πv)
f (x − 1) + f (x + 1) → (e−i2πu + ei2πu )F (u) =
= 2 cos(2πu)F (u)

Fredrik Kahl     Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

A useful fact

If f real (usual case for images):
◮   even f → real F
◮   odd f → imaginary F
◮   F (u) = F (−u)
Observe: F (u, v) is in general complex valued. It is common to
illustrate the transform with |F (u, v)|.

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

DFT and FFT in two dimensions
The discrete Fourier Transform (DFT) of f is deﬁned as
M−1 N−1
ux   vy
F (u, v) =                f (x, y)e−i2π( M + N ) =
x=0 y =0
M−1 N−1
ux vy
=                 f (x, y)ωM ωN =
x=0 y =0
                 
1
         ωNv       
                   
u               2u               (M−1)u                  2v
ωN        
= 1 ωM              ωM . . . ωM                    f                   ,
          .
.        
          .        
(N−1)v
ωN
x = 0, . . . , M − 1,            y = 0, . . . N − 1 .
Fredrik Kahl    Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

DFT in Matrix form

Let the matrix F represent the Fourier transform of the image
f (x, y):
F = FM f FN
or
F = FM (FN f T )T .
i.e. the DFT in two dimensions can be calculated by repeated
use of the one-dimensional DFT, ﬁrst for the rows, then for the
columns.

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

f                   DFT

DFT

F
Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

FFT on images

Let the M × N-matrix f represent an image f (x, y).
Extend the image periodically

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

FFT on images (ctd.)

FFT gives a double periodic function

fftshift

It is common to move the origin to the middle of the image for
illustration purposes.

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Interpretation of the Fourier Transform

◮   Usually, the gray-levels of the Fourier Transform images
are scaled using c log(1 + |F (u, v)|).
◮   The middle of the Fourier image (after fftshift) corresponds
to low frequencies.
◮   Outside the middle high components in F corresponds to
higher frequencies and the direction corresponds to
"edges"in the images with opposite orientation.

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Example

What does the original image look like if this is the Fourier
transform?

Left: Magnitude, Right: Phase

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Masters thesis suggestion of the day: Crossword

Construct a system for automatic crossword scanning and
solution. Idea: Take an image, ﬁnd squares with text and
without, interpret the text. Is it possible to solve crosswords
automatically?

Fredrik Kahl   Image Analysis - Lecture 2
Continuous Fourier transform
Review of Linear algebra
Discrete Fourier Transform (DFT,FFT)
Fourier transform
Two-dimensional Fourier Transform

Review - Lecture 2

◮   Linear Algebra
◮   Subspaces
◮   Projections, Pseudo-inverse
◮   Image matrix
◮   Fourier Transform in 1 and 2 dimensions
◮   Discrete Fourier Transform in 1 and 2 dimensions
◮   Fast Fourier Transform (FFT)

Fredrik Kahl   Image Analysis - Lecture 2

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