This document is a lecture on digital image processing and enhancement in the frequency domain. It discusses linear and shift invariant systems, and how linear operators allow images to be represented as sums of their parts. It also defines key concepts like impulse response, the delta function, point spread function, filtering, and convolution. Convolution is described as the process of flipping and applying cross-correlation between an image and filter. The document provides examples of discrete linear shift invariant systems and how they relate to convolution.

1. 1
Digital Image Processing
Image Enhancement in the
Frequency Domain,
Part I
Dr Z. Azimifar
Computer Science and Engineering Department
Shiraz University
Winter 2020
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

2. Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 2
An operator 𝐻 (or system) is linear if two properties hold.
Suppose 𝑓1 and 𝑓2 are some functions, 𝑎 is a constant:
1. Superposition (additivity):
𝐻(𝑓1 + 𝑓2) = 𝐻(𝑓1) + 𝐻(𝑓2) (looks like distributive law)
2. Scaling (Homogeneity of degree 1):
𝐻(𝑎∙𝑓1) = 𝑎∙𝐻(𝑓1)
Because it is sums and multiplies, the “filtering”
operation we were doing are linear.
Linear & Shift Invariant Systems

3. 3
Linear & Shift Invariant Systems
H(.)
f(x,y) g(x,y)
g(x,y)=H{f(x,y)}
H(.) is linear if:
H[a f(x,y) + b t(x,y)]=a H[f(x,y)] + b H[t(x,y)]
Define:
H(.) = impulse response
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

4. Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 4
Which of these operators are not linear?
1. SUM
2. SQUARE ROOT
3. AVERAGE
4. MAX
5. SIN
Linear & Shift Invariant Systems

5. Linear & Shift Invariant Systems
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 5
Linearity allows us
1. So we can sum up a bunch of things to make it as
an image.
2. So we apply a linear operator to that whole image
Then
It must be the same as if we sum applying that linear
operator to each piece of the image, separately
The building block of function is what is referred to as
impulse.

6. 6
Shift Invariance
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

7. Shift Invariance
Shift invariant: Operator behaves the same
everywhere, i.e. the value of the output depends
on the pattern in the image neighborhood, not the
position of the neighborhood.
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 7

8. Impulse Function
In the discrete world, and impulse is a very easy signal to
understand: it is just a value of 1 at a single location.
In the continuous world, an impulse is an idealized function
that is very narrow and very tall so that it has a unit area. In
the limit:
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 8

9. Impulse Function
If we have an unknown system and we apply an impulse,
the response is called the impulse response.
If the black box is linear we can describe 𝐻 by h(t). Why?
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 9

10. 10
Delta Function
Impulse Response Function
(Point Spread Function)
H
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

11. 11
Delta Function
n Can be defined as the limit of a sequence of functions
q each member has some area, but getting narrower & higher
n A “sharp spike” at origin
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

12. 12
Delta Function
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

13. 13
Delta Function
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

14. 14
Delta Function
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

15. 15
Point Spread Function (PSF)
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

16. 16
Point Spread Function (PSF)
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

17. 17
Point Spread Function (PSF)
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

18. Filtering an Impulse Signal
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 18
What is the result of filtering the impulse signal (image)
F(x,y) with the arbitrary kernel H(u,v)?

19. Filtering an Impulse Signal
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 19

20. Filtering an Impulse Signal
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 20

21. Filtering an Impulse Signal
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 21

22. Filtering an Impulse Signal
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 22

23. Filtering an Impulse Signal
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 23

24. Filtering an Impulse Signal
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 24

25. Convolution
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020 25
1. Flip where the filter is applied in both dimensions
(bottom to top, right to left)
2. Then apply cross-correlation

26. 26
Convolution
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

27. 27
Convolution
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

28. 28
Convolution
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

29. 29
Convolution
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

30. 30
Discrete LSI Systems
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

31. 31
Discrete LSI Systems
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

32. 32
Discrete LSI Systems
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

33. 33
Discrete LSI Systems
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

34. 34
Discrete LSI Systems
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

35. 35
Discrete LSI Systems
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020

36. 36
Discrete LSI Systems
Shiraz University, Comp. Sc. & Eng Dpt,
Prof. Z. Azimifar, 2020