1. Outline Linear Shift-invariant system Linear filters
Fourier transformation
Time and frequency representation
Filter Design

2. Visual Perception Modeling
Source Separation
Mixed signal
Music and speech
Separated signals
Music
Speech
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Visual Perception Modeling

3. 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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Visual Perception Modeling

4. Visual Perception Modeling
Fourier Transforms
Fourier transform
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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Visual Perception Modeling

5. Inverse Fourier Transform
It recovers a signal from its Fourier transform
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Visual Perception Modeling

6. Some Fourier Transform Pairs
Step function
Window function
sinc function
Gaussian function
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Visual Perception Modeling

7. 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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Visual Perception Modeling

8. Visual Perception Modeling
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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Visual Perception Modeling

9. Visual Perception Modeling
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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Visual Perception Modeling

10. 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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Visual Perception Modeling

11. 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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Visual Perception Modeling

12. PCA for Face Recognition
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Visual Perception Modeling

13. PCA for Face Recognition – cont.
First 20 principal components
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Visual Perception Modeling

14. PCA for Face Recognition – cont.
Components with low eigenvalues
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Visual Perception Modeling

15. PCA for Face Recognition – cont.
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Visual Perception Modeling

16. 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)
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Visual Perception Modeling