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

2. Visual Perception Modeling
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?
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Visual Perception Modeling

3. Visual Perception Modeling
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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Visual Perception Modeling

4. 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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Visual Perception Modeling

5. Visual Perception Modeling
Filtering
Closely related to convolution
Filter examples
Smoothing by averaging
Smoothing by Gaussian
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Visual Perception Modeling

6. 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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Visual Perception Modeling

7. Visual Perception Modeling
Gaussian Pyramid
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Visual Perception Modeling

8. Why Gaussian Smoothing?
Scale space
If we convolve a Gaussian with a Gaussian, it will also be a Gaussian
Efficiency
A small kernel is generally enough
Separable
Central limit theorem
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Visual Perception Modeling

9. 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

10. 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

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

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

13. 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

14. 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

15. 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

16. 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

17. PCA for Face Recognition
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Visual Perception Modeling

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

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