Fourier slide

2
Fourier Analysis and
Image Processing
• History
• Periodic Signals
• Fourier Analysis
– Fourier Series
– Fourier Transform
– Discrete Fourier Transform (DFT)
– Fast Fourier Transform (FFT)
• 2D FFT and Image Processing
– Spatial Frequency in Images
– 2D Discrete Fourier Transform
– 2D FFT Examples
– Applications of FFTs in Image Processing
• Summary

3
History
Jean Baptiste Joseph Fourier
1768-1830
French Mathematician and Physicist
http://de.wikipedia.org/wiki/Kreiszahl
Outlined technique in memoir, On the Propagation of Heat in Solid Bodies, which was
read to Paris Institute on 21 Dec 1807. Controversial then: Laplace and Lagrange
objected to what is now Fourier series: “... his analysis ... leaves something to be desired on
the score of generality and even rigour...” (from report awarding Fourier math prize in 1811)
In La Theorie Analytique de la Chaleur (Analytic Theory of Heat) (1822) Fourier
• developed the theory of the series known by his name, and
• applied it to the solution of boundary-value problems in partial differential equations.
Sources:
www.me.utexas.edu/~me339/Bios/fourier.html and www-gap.dcs.st-and.ac.uk/~history/Biographies/Fourier.html

4
Periodic Signals
A continuous-time signal x(t) is periodic if:
x(t + T) = x(t)
Fundamental period, T0, of x(t) is smallest T satisfying
above equation.
Fundamental frequency: f0 = 1/T0
Fundamental angular frequency: ω0 = 2π/T0 = 2πf0

5
Periodic Signals
x(t + T) = x(t)
Harmonics: Integer multiples of frequency of wave
-1.0
0.0
1.0
t
sin
(
t
)
0 π 2 π 3π 2 2π 3π 4π
-1.0
0.0
1.0
t
sin
(
4t
)
0 π 2 π 3π 2 2π 3π 4π
T0
Fundamental
frequency:
f0 = 1/T0

6
Periodic Signals
Biological time series can be quite complex, and will contain noise.
x(t + T) = x(t)
-2
-1
0
1
2
t
x(t)
0 π 2 π 3π 2 2π 3π 4π
“Biological” Time Series
T0

8
Fourier Analysis
• Fourier Series
Expansion of continuous function into weighted sum of sines and
cosines, or weighted sum of complex exponentials.
• Fourier Transform
Maps one function to another: continuous-to-continuous mapping.
An integral transform.
• Discrete Fourier Transform (DFT)
Approximation to Fourier integral. Maps discrete vector to another
discrete vector. Can be viewed as a matrix operator.
• Fast Fourier Transform (FFT)
Special computational algorithm for DFT.

9
Fourier Series
www.science.org.au/nova/029/029img/wave1.gif
Trigonometric Fourier Series
Expansion of continuous function into weighted sum of sines and cosines.
[ ]
∑
∞
=
⋅
+
⋅
+
=
1
0
0
0
)
(
sin
)
(
cos
2
)
(
k
k
k t
k
t
k
t
x b
a
a ω
ω
∫ ⋅
=
T
dt
t
k
t
x
T
ak
0
)
cos(
)
(
2
0
0
ω
∫ ⋅
=
T
dt
t
k
t
x
T
bk
0
)
sin(
)
(
2
0
0
ω
f
T 0
0
0
2
2
π
π
ω =
=
If x(t) is even, i.e., x(-t) = x(t) like cosine, then bk = 0.
If x(t) is odd, i.e., x(-t) = -x(t) like sine, then ak = 0.
Source: Schaum’s Theory and Problems: Signals and Systems, Hwei P. Hsu, 1995, pp. 211-213

10
Complex Math Review
Solutions to x2 = -1:
Complex Plane
Euler’s Formula:
Operators: +, -, *, /
u + v = (a + ib) + (c + id) = (a + c) + i(b + d)
u × v = (a + ib) (c + id) = (ac – bd) + i(ad + bc)
DeMoivre’s Theorem:
u* = a - ib

11
Fourier Series
f
T 0
0
0
2
2
π
π
ω =
=
Source Schaum’s Theory and Problems: Signals and Systems, Hwei P. Hsu, 1995, pp. 211-213
∑
∞
−∞
=
⋅
⋅
⋅
=
k
t
k
i
k e
c
t
x
ω0
)
(
∫
⋅
⋅
⋅
−
⋅
=
T
dt
t
x e
T
c
t
k
i
k
0
0
)
(
1
0
ω
Notes:
• If x(t) is real, c-k = ck
*.
• For k = 0, ck = average value of x(t) over one period.
• a0/2 = c0; ak = ck + c-k; bk = i · (ck - c-k)
Complex Exponential Fourier Series
Expansion of continuous function into weighted sum of complex exponentials.

12
Fourier Series
Complex Exponential Fourier Series
Coefficients can be written as product:
e
c
c k
i
k
k
φ
⋅
=
• ck are known as the spectral coefficients of x(t)
• Plot of |ck| versus angular frequency ω is the amplitude spectrum.
• Plot of φkversus angular frequency is the phase spectrum.
• With discrete Fourier frequencies, k·ω0, both are discrete spectra.

13
Fourier Series
selected
Approximate any function as truncated Fourier series
Given: x(t) = t
Fourier Series:






−
+
−
= ...
3
3
sin
2
2
sin
sin
2
)
(
t
t
t
t
x






−
+
−
= ...
3
3
sin
2
2
sin
sin
2
)
(
t
t
t
t
x
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
t
x(t)

14
Fourier Series
selected
Given: x(t) = t Fourier Series:






−
+
−
= ...
3
3
sin
2
2
sin
sin
2
)
(
t
t
t
t
x






−
+
−
= ...
3
3
sin
2
2
sin
sin
2
)
(
t
t
t
t
x
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Fourier Terms in Expansion of x(t) = t
t
Fourier
Terms
0 π 4 π 2 3π 4 π
1
2
3
4
5 6
Fourier Series Approximation
t
x(t)
First Six Series Terms
0 π 4 π 2 3π 4 π
0
1
2
3

15
Fourier Series
selected
Given: x(t) = t Fourier Series:






−
+
−
= ...
3
3
sin
2
2
sin
sin
2
)
(
t
t
t
t
x






−
+
−
= ...
3
3
sin
2
2
sin
sin
2
)
(
t
t
t
t
x
Fourier Series Approximation
t
x(t)
0 π 4 π 2 3π 4 π
0
1
2
3
100 terms
200 terms

16
Fourier Series
Periodogram
Decomposition
Time Series
Reconstruction
Filtered Series
Remove
High Frequency
Component
0
selected
“Remove” high frequency noise by zeroing a term in series expansion
Noise Removal

17
Fourier Transform
Maps one function to another: continuous-to-continuous mapping.
Fourier transform of x(t) is X(ω):
(converts from time space to frequency space)
∫
∞
∞
−
−
⋅
=
= dt
t
x
t
x
X e
t
iω
ω )
(
)}
(
{
)
( F
Fourier inverse transform of X(ω) recovers x(t):
(converts from frequency space to time space)
∫
∞
∞
−
⋅
=
= ω
ω
π
ω
ω
d
X
X
t
x e
t
i
)
(
2
1
)}
(
{
)
( F
1
-
The Fourier Transform is a special case of the Laplace Transform, s = i ·ω
x(t) and X(ω) form a Fourier transform pair: x(t) ↔ X(ω)

Time Domain and Frequency
Domain
 Time Domain:
 Tells us how properties (air pressure in a sound function,
for example) change over time:
 Amplitude = 100
 Frequency = number of cycles in one second = 200 Hz

Time Domain and Frequency
Domain
 Frequency domain:
 Tells us how properties (amplitudes) change over
frequencies:

18
Fourier Transform
Properties of the Fourier Transform
From http://en.wikipedia.org/wiki/Continuous_Fourier_transform
Also see Schaum’s Theory and Problems: Signals and Systems, Hwei P. Hsu, 1995, pp. 219-223

Extending FT in 2D
• Forward FT
• Inverse FT

Example: 2D rectangle function
• FT of 2D rectangle function
2D sinc()
top view

19
Discrete Time Signal
-1.0
-0.5
0.0
0.5
1.0
sinθ
θ
sinθ 0 π 2 π 3π 2 2π 3π 4π
A discrete-time signal x[n] is periodic if:
x[n + N] = x[n]
Fundamental period, N0, of x[n] is smallest integer N
satisfying above equation.
Fundamental angular frequency: Ω0 = 2π/N0
N0

20
Discrete Fourier Transform (DFT)
The Eight Eighth Roots of Unity
http://math.fullerton.edu/mathews/
c2003/ComplexAlgebraRevisitedMod.html
e
N
kn
i
N
n
n
x
n
x
k
X
)
/
2
(
1
0
]
[
]}
[
{
]
[
π
−
−
=
∑ ⋅
=
= DFT
Sources: Schaum’s Theory and Problems: Signals and Systems, Hwei P. Hsu, 1995, pp. 305
and http://en.wikipedia.org/wiki/Fast_Fourier_transform
Given discrete time sequence, x[n], n = 0, 1, …, N-1
k = 0, 1, …, N-1
Nth root of unity
Inverse Discrete Fourier Transform (IDFT)
e
N
kn
i
N
n
k
X
N
k
X
n
x
)
/
2
(
1
0
]
[
1
]}
[
{
]
[
π
∑
−
=
⋅
=
= IDFT
• One-to-one correspondence between x[n] and X[k]
• DFT closely related to discrete Fourier series and the Fourier Transform
• DFT is ideal for computer manipulation
• Share many of the same properties as Fourier Transform
• Multiplier (1/N) can be used in DFT or IDFT. Sometimes 1/SQRT(N) used in both.

21
For N = 4, the DFT becomes:




























=














−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
⋅
−
x
x
x
x
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
X
X
X
X
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
3
2
1
0
2
/
9
2
/
6
2
/
3
2
/
0
2
/
6
2
/
4
2
/
2
2
/
0
2
/
3
2
/
2
2
/
1
2
/
0
2
/
0
2
/
0
2
/
0
2
/
0
3
2
1
0
π
π
π
π
π
π
π
π
π
π
π
π
π
π
π
π


























−
−
−
−
−
−
=














x
x
x
x
X
X
X
X
i
i
i
i
3
2
1
0
3
2
1
0
1
1
1
1
1
1
1
1
1
1
1
1
e
N
kn
i
N
n
n
x
n
x
k
X
)
/
2
(
1
0
]
[
]}
[
{
]
[
π
−
−
=
∑ ⋅
=
= DFT
n
k
k
k = 0, 1, …, N-1

22
For N = 4, the DFT is:


























−
−
−
−
−
−
=














x
x
x
x
X
X
X
X
i
i
i
i
3
2
1
0
3
2
1
0
1
1
1
1
1
1
1
1
1
1
1
1
x = [1, 0, 1, 0] X = [2, 0, 2, 0]
x = [0, 3, 0, 3] X = [6, 0, -6, 0]
x = [1, 1, 1, 1] X = [4, 0, 0, 0]
x = [0, 0, 0, 0] X = [0, 0, 0, 0]
x = [0, 0, 1, 1] X = [2, -1+i, 0, -1-i]
x = [1, 1, 0, 0] X = [2, 1- i, 0, 1+i]
X[0]/N = mean

27
Fast Fourier Transform (FFT)
e
N
kn
i
N
n
n
x
n
x
k
X
)
/
2
(
1
0
]
[
]}
[
{
]
[
π
−
−
=
∑ ⋅
=
= DFT
Source: http://en.wikipedia.org/wiki/Fast_Fourier_transform
k = 0, 1, …, N-1
• The FFT is a computationally efficient algorithm to compute the
Discrete Fourier Transform and its inverse.
• Evaluating the sum above directly would take O(N2) arithmetic
operations.
• The FFT algorithm reduces the computational burden to
O(N log N) arithmetic operations.
• FFT requires the number of data points to be a power of 2
(usually 0 padding is used to make this true)
• FFT requires evenly-spaced time series

29
Software
www.fftw.org
FFTW is a C subroutine library for computing the discrete Fourier transform
(DFT) in one or more dimensions, of arbitrary input size
IDL (see Signal Processing Demo for Fourier Filtering)
R
MatLab: Signal Processing/Image Processing Toolboxes
Mathematica: Perform symbolic or numerical Fourier analysis

30
1D FFT in IDL Software
IDL Run Demo, Data Analysis, Signal Processing, Filtering Demo

32
2D FFT and Image Processing
• Spatial Frequency in Images
• 2D Discrete Fourier Transform
• 2D FFT Examples
• Applications of FFT
– Noise Removal
– Pattern / Texture Recognition
– Filtering: Convolution and Deconvolution

Applications: Frequency
Domain In Images
 Spatial frequency of an image refers to the
rate at which the pixel intensities change
 In picture on right:
 High frequences:
 Near center
 Low frequences:
 Corners

34
2D Discrete Fourier Transform
Source: Seul et al, Practical Algorithms for Image Analysis, 2000, p. 249, 262.
2D FFT can be computed as two discrete Fourier transforms in 1 dimension
I[m,n] F[u,v]
Fourier
Transform
Spatial Domain Frequency Domain
(0,0)
(0,0)
(0,N/2)
(0,-N/2)
(-M/2,0) (M/2,0)
(M,N)
∑ ∑
−
=






+
−
−
=
⋅
=
1
0
2
1
0
]
,
[
1
]
,
[
M
m
N
vn
M
um
i
N
n
e
n
m
I
MN
v
u
F
π
M pixels
SM units
N
pixels
S
N
units

36
2D FFT Example
FFTs Using ImageJ
ImageJ Steps: (1) File | Open, (2) Process | FFT | FFT
(0,0) Origin (0,0) Origin

37
2D FFT Example
FFTs Using ImageJ
ImageJ Steps: Process | FFT | Swap Quadrants
(0,0) Origin
(0,0) Origin
Regularity in image manifests itself in the degree of order or randomness in FFT pattern.
Default display is to swap quadrants

38
2D FFT Example
FFTs Using ImageJ
Overland Park Arboretum and Botanical Gardens, June 2006
ImageJ Steps: (1) File | Open, (2) Process | FFT | FFT
Regularity in image manifests itself in the degree of order or randomness in FFT pattern.

Why is FT Useful?
• Easier to remove undesirable frequencies in the
frequency domain.
• Faster to perform certain operations in the frequency
domain than in the spatial domain.

Example: Removing undesirable frequencies
remove high
frequencies
reconstructed
signal
frequencies
noisy signal
To remove certain
frequencies, set their
corresponding F(u)
coefficients to zero!

How do frequencies show up in an image?
• Low frequencies correspond to slowly varying pixel
intensities (e.g., continuous surface).
• High frequencies correspond to quickly varying pixel
intensities (e.g., edges)
Original Image Low-passed

Example of noise reduction using FT
Input image
Output image
Spectrum (frequency domain)
Band-reject
filter

Frequency Filtering: Main Steps
1. Take the FT of f(x):
2. Remove undesired frequencies:
3. Convert back to a signal:
We’ll talk more about these steps later .....

Filtering in Frequency Domain
 one reason for using Fourier transform in image processing is
due to convolution theorem
 Spatial convolution can be performed by element‐wise
multiplication of the Fourier transform by suitable “filter
matrix”

Ideal Low Pass Filtering
 Low pass filter: Keep frequencies below a certain frequency
 Low pass filtering causes blurring
 After DFT, DC components and low frequency components
are towards center
 May specify frequency cutoff as circle c

 Multiply Image Fourier Transform F by some filter matrix m

 Low pass filtered image is inverse Fourier Transform of
product of F and m
 Example: Consider the following image and its DFT
Image DFT

Applying
low pass filter
to DFT
Cutoff D = 15
Image after
inversion
low pass filter
Cutoff D = 5
low pass filter
Cutoff D = 30
Note: Sharp filter
Cutoff causes
ringing

Ideal High Pass Filtering
 Opposite of low pass filtering: eliminate center (low
frequency values), keeping others
 High pass filtering causes image sharpening
 If we use circle as cutoff again, size affects results
 Large cutoff = More information removed
DFT of Image after
high pass Filtering
Resulting image
after inverse DFT

Ideal High Pass Filtering: Effect of Cutoffs
Low cutoff
frequency removes
Only few lowest
frequencies
High pass filtering
of DFT with filter
Cutoff D = 5
High pass filtering
of DFT with filter
Cutoff D = 30
High cutoff
frequency removes
many frequencies,
leaving only edges

39
Application of FFT in Image Processing
Noise Removal
Source: www.mediacy.com/apps/fft.htm, Image Pro Plus FFT Example. Last seen online in 2004.
FFT Inverse
FFT
Edit FFT
Four Noise
Spikes Removed
Noise Pattern
Stands Out as
Four Spikes

40
Application of FFT
Pattern/Texture Recognition
Source: Lee and Chen, A New Method for Coarse Classification of Textures and Class Weight Estimation
for Texture Retrieval, Pattern Recognition and Image Analysis, Vol. 12, No. 4, 2002, pp. 400–410.

41
Application of FFT
Source: http://www.rpgroup.caltech.edu/courses/PBL/size.htm
Could FFT of Drosophila eye be used to identify/quantify subtle phenotypes?
The Drosophila eye is a great example a cellular crystal with
its hexagonally closed-packed structure. The absolute
value of the Fourier transform (right) shows its hexagonal
structure.
Pattern/Texture Recognition

42
Application of FFT
Filtering in the Frequency Domain: Convolution
Source: Gonzalez and Woods, Digital Image Processing (2nd ed), 2002, p. 159
I[m,n]
Raw Image
I’[m,n]
Enhanced Image
Fourier Transform
F[u,v]
Filter Function
H[u,v]
Inverse
Fourier Transform
Pre-
processing
Post-
processing
F[u,v] H[u,v] · F[u,v]
FFT{ I[u,v] } FFT-1{ H[u,v] · F[u,v] }

43
Application of FFT
Filtering: IDL Fourier Filter Demo
IDL Run Demo, Data Analysis, Image Processing, Image Processing Demo

44
Application of FFT
Filtering: IDL Fourier Filtering Demo
IDL Run Demo, Data Visualization, Images, Fourier Filtering

45
The Point Spread Function (PSF) is the Fourier transform of a filter.
(the PSP says how much blurring there will be in trying to image a point).
Source: http://www.reindeergraphics.com/index.php?option=com_content&task=view&id=179&Itemid=127
Hubble image and measured PSF
Dividing the Fourier transform of the PSF into
the transform of the blurred image, and
performing an inverse FFT, recovers the
unblurred image.
Application of FFT
Deblurring: Deconvolution
FFT(Unblurred Image) * FFT(Point Spread Function) = FFT(Blurred Image)
Unblurred Image = FFT-1[ FFT(Blurred Image) / FFT(Point Spread Function) ]

46
The Point Spread Function (PSF) is the Fourier transform of a filter.
(the PSP says how much blurring there will be in trying to image a point).
Source: http://www.reindeergraphics.com/index.php?option=com_content&task=view&id=179&Itemid=127
Hubble image and measured PSF
Deblurred image
Dividing the Fourier transform of the PSF into
the transform of the blurred image, and
performing an inverse FFT, recovers the
unblurred image.
Application of FFT
Deblurring: Deconvolution

School of Engineering
History of Computer Tomography
• Johan Radon (1917) showed how a reconstruction from
projections was possible.
•Cormack (1963,1964) introduced Fourier transforms into the
reconstruction algorithms.
•Hounsfield (1972) invented the X-ray Computer scanner for
medical work, (which Cormack and Hounsfield shared a Nobel
prize).
•EMI Ltd (1971) announced development of the EMI scanner
which combined X-ray measurements and sophisticated
algorithms solved by digital computers.

47
Summary
• Fourier Analysis is a powerful tool even when periodicity is not
directly a part of the problem being solved.
• Discrete Fourier Transforms (DFT) are well-suited for computation
by computer, especially when using Fast Fourier Transform (FFT)
algorithms.
• Fourier Analysis can be used to remove noise from a signal or
image.
• Interpretation of the complex Fourier Transform is not always
straightforward.
• Convolution and Deconvolution are “simple” in Fourier transform
space to restore or enhance images.
• There are many other image processing uses of Fourier Analysis,
such as image compression [JPGs use the Discrete Cosine
Transform (DCT), which is a special kind of DFT]

Fourier slide

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