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Image Processing - Lesson 6
• 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
Introduction to Fourier Transform - Linear Systems
• A linear system T gets an input f(t)and produces an output g(t):
• In the discrete caes:– input : f[n] , n = 0,1,2,…– output: g[n] , n = 0,1,2,…
f(t) g(t)
( ){ }tfTtg =)(
[ ] [ ]{ }nfTng =
T
• A linear system must satisfy two conditions:
– Homogeneity:
– Additivity:
[ ]{ } [ ]{ }nfTanfaT =
[ ] [ ]{ } [ ]{ } [ ]{ }nfTnfTnfnfT 2121 +=+
T
T
T
Homogeneity
Additivity
T
• Contrast change by grayscale stretching around 0:
– Homogeneity:
– Additivity:
T{bf(x)} = abf(x) = baf(x) = bT{f(x)}
T{f(x)} = af(x)
T{f1(x)+f2(x)} = a(f1(x)+f2(x)) = af1(x)+af2(x)= T{f1(x)}+ T{f2(x)}
• Convolution:
– Homogeneity:
– Additivity:
T{bf(x)} = (bf)*a = b(f*a) = bT{f(x)}
T{f(x)} = f*a
T{f1(x)+f2(x)} = (f1+f2)*a = f1*a+f2*a= T{f1(x)}+ T{f2(x)}
• Assume T is a linear system satisfying
• T is a shift-invariant linear system iff:
T
Shift Invariant
( ){ }tfTtg =)(
( ){ }00 )( ttfTttg −=−
T
• Contrast change by grayscale stretching around 0:
– Shift Invariant:
• Convolution:
– Shift Invariant:
T{f(x-x0)} = af(x-x0) =g(x-x0)
T{f(x)} = af(x) = g(x)
T{f(x-x0)} = f(x-x0)*a
T{f(x)} = f(x)*a =g(x)
� −−� =−−=j
0i
0 )xjx(a)j(f)ix(a)xi(f
= g(x-x0)
• Assume f is an input vector and T is a matrix multiplying f:
• g is an output vector. • Claim: A matrix multiplication is a
linear system:
– Homogeneity– Additivity
• Note that a matrix multiplication is not necessarily shift-invariant.
=
( ) 2121 TfTfffT +=+( ) aTfafT =
• An impulse signal is defined as follows:
• Any signal can be represented as a linear sum of scales and shifted impulses:
[ ]���
=≠
=−knwhereknwhere
kn10
δ
[ ] [ ] [ ]jnjfnfj
−=�∞
−∞=
δ
= + +
Proof: – f[n] input sequence– g[n] output sequence– h[n] the system impulse response:
h[n]=T{δ[n]}
[ ] [ ]{ } [ ] [ ]
[ ] [ ]{ }
[ ] [ ]
hf
nariancceshiftfromjnhjf
linearityfromjnTjf
jnjfTnfTng
j
j
j
∗=
−−=
−=
���
���
−==
�
�
�
∞
−∞=
∞
−∞=
∞
−∞=
)i(
)(δ
δ
The output is a sum of scaled and shifted copies of impulse responses.
Convolution as a sum of impulse responses
*H(n)f(n)
• The convolution (wrap around):
can be represented as a matrix multiplication:
– The matrix rows are flipped and shifted copies of the impulse response.
– The matrix columns are shifted copies of the impulse response.
[ ] [ ] [ ]283256123210021 −−−=∗−−
�������
�
�
�������
�
�
−−−
=
�������
�
�
�������
�
�
−−
�������
�
�
�������
�
�
283256
210021
210003321000032100003210000321100032
CirculantMatrix
• Commutative:
– Only shift-invariant systems are commutative.– Only circulant matrices are commutative.
• Associative:
– Any linear system is associative.
• Distributive:
– Any linear system is distributive.
fTTfTT ∗∗=∗∗ 1221
( ) ( )fTTfTT ∗∗=∗∗ 2121
( )( ) 2121
2121
fTfTffTandfTfTfTT∗+∗=+∗
∗+∗=∗+
The Complex Plane
Real
Imaginary
(a,b)
a
b
• Two kind of representations for a point (a,b) in the complex plane– The Cartesian representation:
12 −=+= iwherebiaZ– The Polar representation:
θReiZ =
Rθ
• Conversions:
– Polar to Cartesian:
– Cartesian to Polar
( ) ( )θsinθcosRe θ iRRi +=( )abiebaiba /tan22 1−
+=+
(Complex exponential)
• Conjugate of Z is Z*:
– Cartesian rep.
– Polar rep.
( ) ibaiba −=+ ∗
( ) θθ ReRe ii −∗ =
Reala
b
θ
-θ
-b
θReiiba =+
θRe iiba −=−
R
R
Imaginary
Algebraic operations:
• addition/subtraction:
• multiplication:
• Norm:
)()()()( dbicaidciba +++=+++
( )( ) ( ) ( )adbcibdacidciba ++−=++( )βαβα += iii ABeBeAe
( ) ( ) 222 baibaibaiba +=++=+ ∗
( ) 2θθθθ2θ ReReReReRe Riiiii === −∗
• The (Co-)Sinusoid as complex exponential:
Or
( )2eexcos
ixix −+=
( )i2eexsin
ixix −−=
( ) ( )ixex Realcos =
( ) ( )ixex Imagsin =
x 1
1
cos(x)
sin(x)eix
x
( )xπω2sin
ω1
– The wavelength of sin(2πωx) is 1/ω .– The frequency is ω .
1
x 1
1
cos(x)
sin(x)eix
The (Co-) Sinusoid- function
x
( )xA πω2sin
A
– Changing Amplitude:
– Changing Phase:
x
( )φπω2sin +xA
( )φsinA
( ) )(Imagφπω2sin πω2φ xii eAexA =+
Scaling and shifting can be represented as a multiplication with φiAe
• If we add a Sine wave to a Cosine wave with the same frequency we get a scaled and shifted (Co-) Sine wave with the same frequency:
( ) ( ) ( )φωωω +=+ xRxbxa sin cossin
��
���
�=+= −
abbaR 122 tanandwhere φ
( ) ( ) cossin xbxa ωω +
( )φω +xi
Two alternatives to represent a scaled and shifted Sine wave.
a2+b2
Combining Sine and Cosine - Proof
Linear Combination of sin(ωx) and cos(ωx) produces sin(ωx) with a change in Phase and Amplitude.
Proof:
a sin(ωx) + b cos(ωx) = R sin(ωx + θ)
a sin(ωx) + b cos(ωx) =
[ sin(ωx) + cos(ωx) ]ײa2+b2
a2+b2ײa
a2+b2ײb
Since
a2+b2ײa
ײb+
22= 1
there exists θ such that :
a2+b2ײa = cos(θ)
a2+b2ײb = sin(θ)
*
*Thus from we obtain:
[ cos(θ) sin(ωx) + sin(θ) cos(ωx) ]ײa2+b2
sin(ωx + θ)ײa2+b2=
AmplitudeR = ײa2+b2
θ = tg-1(b/a) Phase
The response of Shift-Invariant Linear System to a Sine wave.• The response of a shift-invariant linear system
to a sine wave is a shifted and scaled sine wave with the same frequency.
• The frequency response (or Transfer Function) of a system:
( )x2sin πω ( ) ( )( )ωφπω2sinω +xA
ω ω
A(ω) φ(ω)
x2ie πω ( ) ( ) x2ii eeA πωωφω
The response of Shift-Invariant Linear System to a Sine wave.• The response of a shift-invariant linear system
to a sine wave is a shifted and scaled sine wave with the same frequency.
• The frequency response (or Transfer Function) of a system:
ω ω
A(ω) φ(ω)
x2ie πω ( ) ( ) x2ii eeA πωωφω
( ) ( ) ( )ωω ωφ HeA i =
xie πω2
( ) ( ) ( )ωω ωφ HeA i =
( ) xieH πω2ω
)(δ x ( )xh
Impulse response
Frequency response
• 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):
• The Fourier Transform:It is possible to express any signal as a sum of shifted and scaled sinusoids at different frequencies.
( ) xieFxf πω
ωω 2)( �=
( ){ } ( ) ( ) xieFHxfTxg πω
ωωω 2)( �==
( )�=ω
πω2ω)( xieFxf
( ) ωω)(ω
πω2 deFxf xi�=
Or
3 sin(x)
+ 1 sin(3x)
+ 0.8 sin(5x)
+ 0.4 sin(7x)
=
Every function equals a sum of scaled and shifted Sines
+
+
+
Linear System Logic
Input Signal
Express as sum of scaled
and shifted impulses
Express as sum of scaled
and shifted sinusoids
Calculate the response to
each impulse
Calculate the response to
each sinusoid
Sum the impulse
responses to determine the
output
Sum the sinusoidal
responses to determine the
output
Frequency Method
space/time Method
)()()( xhxfxg ∗=)ω()ω()ω( HFG =