Outline
• Linear Shift-invariant system
• Linear filters
• Fourier transformation
– Time and frequency representation
• Filter Design
Linear System Theory
• What is a system?
– A system is anything that accepts an input and
produces an output in response
y[n] = T{x[n]}
where x[n] is the input sequence and y[n] is the output
sequence in responses to x[n]
– How to represent a sequence?
x[n] 

 x[k ] [n  k ]
k  
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Linear System
• Linearity
–
–
–
–
y1[n] = T{x1[n]}
y2[n] = T{x2[n]}
Then
y1[n]+y2[n] = T{x1[n]+x2[n]}
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Shift-Invariant System
• Shift invariance
– y[n] = T{x[n]}
– y[n-T] = T{x[n-T]}
• LSI system
– A LSI system is completely characterized by its
impulse response h[n]
– For any other input, we can obtain the response
through convolution
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Filtering
• Closely related to convolution
• Filter examples
– Smoothing by averaging
– Smoothing by Gaussian
2
2

1
(x  y ) 

G ( x, y) 
exp  
2
2
2
2


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Multi-scale Representation
• Scale in the Gaussian function
–  is the standard deviation of the Gaussian
distribution
– When  is small, no smoothing or very little
– When  is large, the noise will be largely
disappear. However, the image detail will
disappear along with the noise
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Gaussian Pyramid
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Why Gaussian Smoothing?
• Scale space
– If we convolve a Gaussian with a Gaussian, it
will also be a Gaussian
G 1 * G 2  G
– Efficiency
 12  22
• A small kernel is generally enough
• Separable
– Central limit theorem
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Spatial Frequency Analysis
• Filter response analysis
– For example, why does smoothing reduce noise?
– What is the difference between the discrete
image representation and a continuous surface
representation?
– Is there any way we can design the best filter for
a certain task?
• For smoothing, how can we have the best smoothing
kernel?
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Fourier Transforms
• Fourier transform
 
F ( g ( x, y ))(u , v) 
  g ( x, y )e
 j 2 ( ux  vy )
dxdy
  
– The transformation takes a complex valued
function x, y and returns a complex valued
function of u, v
– U and v determine the spatial frequency and
orientation of the sinusoidal component
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Inverse Fourier Transform
• Inverse Fourier transform
 
g ( x, y ) 
  F ( g ( x, y))(u, v)e
j 2 ( ux  vy )
dudv
  
– It recovers a signal from its Fourier transform
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Some Fourier Transform Pairs
• Step function
• Window function
• sinc function
x
sinc ( x) 
sin( x)
• Gaussian function
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Filter Design
• Design filters to accomplish particular goals
• Lowpass filters
– Reduce the amplitude of high-frequency
components
– Can reduce the visible effects of noise
– Box filter
– Triangle filter
– High-frequency cutoff
– Gaussian lowpass filter
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Filter Design – cont.
• Bandpass and bandstop filters
• Highpass filters
• Optimal filter design
– In some sense, optimal of doing a particular job
– Establish a criterion of performance and then
maximize the criterion by proper selection of the
impulse response
– Wiener estimator
– Wiener deconvolution
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Other Transformations
• Fourier transform is one of a number of linear
transformations that are useful in image
processing
• Basis functions
– How to represent an image by weighted sum of
some functions of our choice?
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Principal Component Analysis
• Optimal representation with fewer basis
functions
– We want to design a set of basis functions such
that we can reconstruct the original image with
smallest possible error with a given number of
basis functions
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PCA for Face Recognition
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PCA for Face Recognition – cont.
First 20 principal components
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PCA for Face Recognition – cont.
Components with low eigenvalues
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PCA for Face Recognition – cont.
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