Image Processing - Lesson 4
Introduction to Fourier
Transform - Linear Systems
•
Linear Systems
• A linear system T gets an input f(t)
and produces an output g(t):
Linear System Properties
• A linear system must satisfy two conditions:
– Homogeneity:
Linear Systems
• Definitions & Properties
• Shift Invariant Linear Systems
• Linear Systems and Convolutions
• Linear Systems and sinusoids
• Complex Numbers and Complex Exponentials
• Linear Systems - Frequency Response
f(t)
T
g ( t ) = T { f (t )}
– Additivity:
g(t)
T {a f [n ]}= aT { f [n ]}
T{ f1[n]+f2[n]}=T{ f1[n]}+T{ f2[n]}
T
T
T
Homogeneity
• In the discrete caes:
– input : f[n] , n = 0,1,2,…
– output: g[n] , n = 0,1,2,…
g[n] = T[f (n)]
T
Additivity
T
Linear System - Example
• Contrast change by grayscale stretching
around 0:
T{f(x)} = af(x)
– Homogeneity:
T{bf(x)} = abf(x) = baf(x) = bT{f(x)}
– Additivity:
T{f1(x)+f2(x)} = a(f1(x)+f2(x))
= af1(x)+af2(x)
= T{f1(x)}+ T{f2(x)}
Linear System - Example
• Convolution:
Shift-Invariant Linear
System
• AssumeT is a linear system satisfying
T{f(x)} = f*a
– Homogeneity:
T{bf(x)} = (bf)*a = b(f*a) = bT{f(x)}
g (t ) = T { f (t )}
• T is a shift-invariant linear system iff:
g(t − t0 ) = T {f (t − t0 )}
– Additivity:
T{f1(x)+f2(x)} = (f1+ f2)*a
= f1*a+f2*a
= T{f1(x)}+ T{f2(x)}
T
Shift Invariant
T
Shift-Invariant Linear
System - Example
• Contrast change by grayscale
stretching around 0:
T{f(x)} = af(x) = g(x)
– Shift Invariant :
T{f(x-x0 )} = af(x-x0 ) =g(x-x0 )
• Convolution :
T{f(x)} = f(x)*a =g(x)
– Shift Invariant :
T{f(x-x 0)} = f(x-x 0)*a
= ∑ f ( i − x 0 )a( x − i ) = ∑ f( j) a( x − j − x 0 )
i
j
= g(x-x 0)
Matrix Multiplication as a
Linear System
• Assume f is an input vector and T is a
matrix multiplying f:
g = Tf
• g is an output vector.
• Claim: A matrix multiplication is a
linear system:
– Homogeneity T (af ) =a T f
– Additivity T (f1 + f 2 )= Tf1 + Tf 2
• Note that a matrix multiplication is
not necessarily shift-invariant.
Impulse Sequence
• An impulse signal is defined as
follows:
0
d[n − k ] = 
1
where
n≠k
where
n=k
• Any signal can be represented as a
linear sum of scales and shifted
impulses:
∞
f [n ]= ∑ f [ j ]δ [n − j ]
j = −∞
=
+
+
Convolution as a Matrix
Multiplication
Shift-Invariant Linear
System is a Convolution
Convolution Properties
• Commutative:
T1 ∗ T2 ∗ f = T2 ∗T1 ∗ f
Proof:
• The convolution (wrap around):
– f[n] input sequence
– g[n] output sequence
– h[n] the system impulse response:
h[n]=T{δ[n]}
 ∞

g[n] = T {f [n ]}=T  ∑ f [ j ]δ [ n− j]
 j= −∞

∞
= ∑ f [ j ]T {δ [n − j ]} ( from linearity )
j =−∞
∞
= ∑ f [ j ] h[n − j ] ( from shift − inariancce)
[1
2 0 0 −1 − 2]∗[3 2 1] = [6 5 2 −3 − 8 − 2]
can be represented as a matrix multiplication:
Circulant
Matrix
2
1

0

0
0

3
3 0 0
2
1
0
0
0
3
2
1
0
0
0
3
2
1
0
0 1   1  6 
0 0   2   5 
0 0  0  2
  =  
3 0  0  − 3
2 3   − 1  − 8 
   
1 2  − 2   − 2 
j =−∞
= f ∗h
The output is a sum of scaled and shifted
copies of impulse responses.
– The matrix rows are flipped and shifted
copies of the impulse response.
– The matrix columns are shifted copies
of the impulse response.
– Only shift-invariant systems are commutative.
– Only circulant matrices are commutative.
• Associative:
(T1∗T2 )∗ f = T1 ∗(T2 ∗ f )
– Any linear system is associative.
• Distributive:
(T1 +T2 )∗ f = T1 ∗ f +T2 ∗ f
and T ∗( f1 + f2 )= T ∗ f1 + T ∗ f2
– Any linear system is distributive.
Complex Numbers
Algebraic operations :
• Conjugate of Z is
Imaginary
• addition/subtraction:
(a,b)
b
– Cartesian rep. (a + ib) = a − ib
(a+ ib)+ (c+ id)= (a + c) + i(b + d)
∗
(Re ) = Re
R
The Complex Plane
Aeia Bei ß = ABei (a+ß)
a + bi = Re iθ
b
R
where i 2 = − 1
• Norm:
θ
– The Polar representation:
-θ
(Complex exponential)
a
Real
– Polar to Cartesian: Rei θ = R cos(θ ) + iR sin(θ )
−1
(b / a )
a + ib = ( a + ib) (a + ib) = a + b
a − bi = Re− iθ
∗
2
iθ 2
R
-b
• Conversions:
a + bi = a2 + b2 e i tan
• multiplication:
(a + ib)( c + id) = (ac −bd) + i( bc + ad)
Imaginary
– The Cartesian representation:
– Cartesian to Polar
−i θ
Real
• Two kind of representations for a point
(a,b) in the complex plane
Z = a + bi
iθ ∗
– Polar rep.
θ
a
Z = Reiθ
Z*:
Re
(
= Re
2
) Re
iθ ∗
iθ
2
= Re−iθ Reiθ = R2
The (Co-) Sinusoid
– Changing Amplitude:
A sin (2πωx )
1
The (Co-) Sinusoidfunction
ix
e
sin(x)
• The (Co -)Sinusoid as complex exponential:
A
x
eix + e− ix
cos( x ) =
2
ix
e −e
2i
sin(x ) =
cos(x)
1
x
−ix
sin (2πω x )
Or
cos( x ) = Real (e
ix
)
– Changing Phase:
A sin (2πω x + ϕ )
1
x
sin( x ) = Imag(e ix )
1
– The wavelength of sin(2πω x) is
eix
sin(x)
x
cos(x)
– The frequency is ω .
1
1
ω
x
.
Scaling and shifting can be represented as a
multiplication with Ae i ϕ
A sin (2πω x + ϕ ) = Imag( Ae iϕ ei 2πω x )
Frequency Analysis
Every function equals a sum of scaled and shifted Sines
Frequency
Method
• If a function f(x) can be expressed as a
linear sum of scaled and shifted sinusoids:
it is possible to predict the system
response to f(x):
3 sin(x)
g ( x) = T { f (x )} = ∑ H (ω ) F (ω )e
+ 1 sin(3x)
i 2πωx
+
+
ω
+ 0.8 sin(5x)
• The Fourier Transform :
It is possible to express any signal as a
sum of shifted and scaled sinusoids at
different frequencies.
f (x) = ∑ F (ω )e
Or
i 2πωx
f ( x ) = ∫ F (ω ) e i 2πω x d ω
ω
Input Signal
space/time
Method
=
f ( x) = ∑ F (ω )ei2πω x
ω
Linear System Logic
Express as
sum of scaled
and shifted
sinusoids
Express as
sum of scaled
and shifted
impulses
Calculate the
response to
each sinusoid
Calculate the
response to
each impulse
Sum the
sinusoidal
responses to
determine the
output
Sum the
impulse
responses to
determine the
output
+
+ 0.4 sin(7x)
G(ω) = F(ω) H(ω)
g (x) = f (x) ∗ h(x)