Outline
• Linear Shift-invariant system
• Linear filters
• Fourier transformation
– Time and frequency representation
• Filter Design
Source Separation
• Mixed signal
– Music and speech
• Separated signals
– Music
– Speech
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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
sin( x)
sinc ( x) 
x
• Gaussian function
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Properties of Fourier Transform
• There are nice properties of Fourier
transforms
– Convolution theorem
F(f(x,y) * g(x,y)) = F(f(x,y)) F(g(x,y))
• Can be used to speed up convolution especially for
large filters
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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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Wavelet Transformations
• Transient signal components
– Nonzero only during a short interval
– Many important features in images are highly
localized
• Wavelets
– Given a real-valued function (s)
1  x b
a ,b ( x) 


a  a 
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