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Methods in Image Analysis – Lecture 3
Fourier
George Stetten, M.D., Ph.D.
U. Pitt Bioengineering 2630
CMU Robotics Institute 16-725
Spring Term, 2006
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
j1
r1r2e
r e
j 2
2
j 1 2
rotateby 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
jk0t
1
jk 0 t
ak x t e
dt
T0 T0
11
Selected properties of Fourier Series
Fs
xt
ak
Fs
yt
bk
Fs
Axt Byt
Aak Bbk
a k a k
*
for real x t
dx t Fs
jk 0 ak
dt
1
Fs
xt dt jk0 ak
12
Differentiation boosts high frequencies
13
Integration boosts low frequencies
14
Continuous Fourier Transform
F
xt
X
Synthesis
1
x t
2
Analysis
X e d
jt
X
jt
x
t
e
dt
15
Selected properties of Fourier Transform
x t X
1
F
xt yt X Y
F
xt
-
2
1
dt
2
X
2
d
16
Special Transform Pairs
• Impulse has all frequences
xt t X 1
F
• Average value is at frequency = 0
xt 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 xt sampledat s
s
2
is the Nyquist frequency.
21
Discrete Time Fourier Series
xn xn N
assume s N0 2
Sampling frequency is 1 cycle per second, and
fundamental frequency is some multiple of that.
ak ak N
Synthesis
xn
a e
k N
k
Analysis
jk0 t
1
ak
N
xne
jk0 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
j 2
Synthesis
1
xn
2
Analysis
X e e
j
jn
d
X e xn e
j
jn
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
px
f x, y dy
Pu F u, 0
Combine with rotation, have arbitrary projection.
32
Gaussian
F
gx
Gu
g1x g2 x g3 x
G1uG2 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 2qr cos
f r r e
d r dr Fr q
0
34
Elliptical Fourier Series for 2D Shape
Parametric function, usually
with constant velocity.
xt ak
Fs
yt bk
Fs
center a0, b0
Truncate harmonics to smooth.
35
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
36
Quaternions – 3D phasors
a a1 ia2 ja3 ka4
i j k ijk 1
2
2
2
a a1 ia2 ja3 ka4
*
a a a2 a a4
2
1
2
2
3
2
1
2
a b a1 b1 ia2 b2 ja3 b3 k a4 b4
Product is defined such that rotation by arbitrary angles from
arbitrary starting points become simple multiplication.
37
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.
38