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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
Answer
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
reader/solver
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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