Transcript Document 7311519

Methods in Image Analysis – Lecture 3
Fourier
George Stetten, M.D., Ph.D.
CMU Robotics Institute 16-725
U. Pitt Bioengineering 2630
Spring Term, 2004
1
Frequency in time vs. space
• Classical “signals and systems” usually temporal
signals.
• Image processing uses “spatial” frequency.
• We will review the classic temporal description first,
and then move to 2D and 3D space.
2
Phase vs. Frequency
• Phase,  , is angle, usually represented in radians.
• 2 radians  360 (circumference of unit circle)
• Frequency,  , is the rate of change for phase.
 t
• In a discrete system, the sampling frequency,  s ,
is the amount of phase-change per sample.
  s n
3
Euler’s Identity
e
j
 cos   j sin 

4
Phasor = Complex Number
5
multiplication = rotate and scale
j
z  x  jy  re
x1  jy1 x2  jy2 
 r1e
j1
 r1r2 e
r e 
j 2
2
j 1  2 
rotate by  and scale by r.
6
Spinning phasor
  2 f
7
8
9
10
Continuous Fourier Series
0
is the Fundamental Frequency
Synthesis
x t  
Analysis

a e
k
k  
jk0t
1
 jk 0 t
ak   x t e
dt
T0 T0
11
Selected properties of Fourier Series
Fs
x t 
ak
Fs
y t  
bk
Fs
Axt   By t  
Aak  Bbk
a k  a k
*
for real xt 
dx t  Fs
 jk 0 ak
dt
1
Fs
 xt dt  jk0 ak
12
Differentiation boosts high frequencies
13
Integration boosts low frequencies
14
Continuous Fourier Transform
F
x t 
X  
Synthesis
1
xt  
2

jt


X

e
d


Analysis
X   

 jt


x
t
e
dt


15
Selected properties of Fourier Transform
 
x t  X  
  
1
F
xt   y t  X  Y  
F


-
x t 
2
1
dt 
2


X   d
2

16
Special Transform Pairs
• Impulse has all frequences
xt   t   X    1
F
• Average value is at frequency = 0
xt   1  X    2 
F
• Aperture produces sync function
1,
x t   
0
t  T1
2 sin T1 
F
 X   
t  T1

17
Discrete signals introduce aliasing
Frequency is no longer
the rate of phase change
in time, but rather the
amount of phase change
per sample.
18
Sampling > 2 samples per cycle
19
Sampling < 2 samples per cycle
20
Under-sampled sine
For xt  sampled at s
s
2
is the Nyquist frequency.
21
Discrete Time Fourier Series
xn  xn  N 
assume s  N0  2
Sampling frequency is 1 cycle per second, and
fundamental frequency is some multiple of that.
ak  ak  N
Synthesis
xn 
a e
k N
k
Analysis
jk0 t
1
ak 
N
 xne
 jk0 t
k N
22
Matrix representation w  e
2
j
N
w 1
N
23
Fast Fourier Transform
• N must be a power of 2
• Makes use of the tremendous
symmetry within the F-1 matrix
• O(N log N) rather than O(N2)
24
Discrete Time Fourier Transform
assume s  2
Sampling frequency is still 1 cycle per second, but now
any frequency are allowed because x[n] is not periodic.
X e
j
  X e
Synthesis
1
xn 
2


Analysis
 X e  e

j   2 
j
jn
d
X e j  

 jn


x
n
e

n  
25
The Periodic Spectrum
26
Aliasing Outside the Base Band
1
4
Perceived as  sin s
27
2D Fourier Transform
F u,v  
 
Analysis
  f x, y  e
 j 2  uxvy 
dx dy
 
or separating dimensions,


 j 2  ux
 j 2  vy
F u,v     f x, y  e
dx e
dy

 
 
f x, y  
 
Synthesis
  F u,v  e
 
j 2  uxvy 
du dv
28
Properties
•
•
•
•
Most of the usual properties, such as linearity, etc.
Shift-invariant, rather than Time-invariant
Parsevals relation becomes Rayleigh’s Theorem
Also, Separability, Rotational Invariance, and
Projection (see below)
29
Separability
if
f  x , y   f1  x  f 2  y 
f  x, y   F u, v 
then
F
f1  x  f 2  y   F1 u F2 v   F u, v 
F
f1  x   F1 u 
F
f 2  y   F2 v 
F
30
Rotation Invariance
 x  cos 
 y   sin 
  
sin    x 



cos    y 
F
f x cos   y sin  ,  x sin   y cos   
F u cos   v sin  ,  u sin   v cos  
31
Projection
px  

 f x, y  dy

P u   F u, 0
Combine with rotation, have arbitrary projection.
32
Gaussian
F
gx 

Gu
g1 x   g2 x   g3 x 
G1 uG2 u  G3 u
g3 x 
G3 u
F

seperable

e
 x2  y2
2 2

e
 x2
2 2
e
 y2
2 2
Since the Fourier Transform is also
separable, the spectra of the 1D
Gaussians are, themselves, separable.
33
Hankel Transform
For radially symmetrical functions
f  x, y   f r r , r  x  y
2
2
F u, v   Fr q , q  u  v
2
F u, v  
  

f  x, y  e
2
2
2
 j 2 ux  vy 
dx dy 
  


0
2
  j 2qr cos 
f r r   e
d  r dr  Fr q 
0

34
Variable Conductance Diffusion (VCD)
• Attempt to get around the global nature of Fourier.
• Smoothing with a Gaussian in the spatial domain yields
multiplication by a Gaussian in the frequency domain,
i.e., a low pass filter.
• This lowers noise, but also blurs boundaries.
• Gaussian smoothing simulates uniform heat diffusion.
• VCD makes conductance an inverse function of
gradient, so that “heat” does not flow well across
boundaries. This homogenizes already homogenious
regions while preserving boundaries.
35
Elliptical Fourier Series for 2D Shape
Parametric function, usually
with constant velocity.
x t   ak
Fs
y t   bk
Fs
center  a 0 , b0 
Truncate harmonics to smooth.
36
Fourier shape in 3D
• Fourier surface of 3D shapes (parameterized on
surface).
• Spherical Harmonics (parameterized in spherical
coordinates).
• Both require coordinate system relative to the
object. How to choose? Moments?
• Problem of poles: singularities cannot be avoided
37
Quaternions – 3D phasors
a  a1  ia2  ja3  ka4
i  j  k  ijk  1
2
2
2
a  a1  ia 2  ja3  ka4
*

a  a  a 2  a3  a 4
2
1
2
2
2

1
2
a  b  a1  b1   i a2  b2   j a3  b3   k a4  b4 
Product is defined such that rotation by arbitrary angles from
arbitrary starting points become simple multiplication.
38
Summary
• Fourier useful for image “processing”, convolution
becomes multiplication.
• Fourier less useful for shape.
• Fourier is global, while shape is local.
• Fourier requires object-specific coordinate system.
39