1. 60-520 Presentation Image Filters
Student:
Xiaoliu Chen
Instructor:
Dr. I. Ahmad
School of Computer Science
University of Windsor
November 2003

2. Outline Introduction Spatial Filtering Frequency-Domain Filtering
Smoothing
Sharpening
Frequency-Domain Filtering
Low pass
High pass
Summary
Image Filters

3. Introduction
Filtering is the process of replacing a pixel with a value based on some operations or functions.
The operations/functions used on the original image are called filters.
or masks, kernels, templates, windows…
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4. Introduction In digital image processing, filters are usually used to
suppress the high frequencies in an image
i.e., smoothing the image
suppress the low frequencies in an image
i.e., enhancing or detecting edges in the image
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5. Introduction Image filters fall into two categories: Spatial domain
Filters are based on direct manipulation of pixels on an image plane.
Frequency domain
Filters are based on modifying the Fourier transform (FT) of an image.
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6. Spatial Filters
The general processes can be denoted by the expression:
f(x,y) is the input image
g(x,y) is the processed image
T is an operator on f, defined over some neighborhood of (x,y)
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7. Spatial Filters
The principal approach in defining a neighborhood about a point (x,y)
use a subimage area centered at (x,y)
shapes of the neighborhood
circle
square
rectangular
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8. Spatial Filters Example:
3×3 neighborhood about a point (x,y) in an image x
(x,y)
Image f(x,y)
(x+1,y+1)
(x,y+1)
(x-1,y+1)
(x+1,y)
(x-1,y)
(x+1,y-1)
(x,y-1)
(x-1,y-1) y
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9. x Image f(x,y) y Mask Mask coefficients Pixels under mask f(x+1,y+1)
w(1,1)
w(0,1)
w(-1,1)
w(1,0)
w(0,0)
w(-1,0)
w(1,-1)
w(0,-1)
w(-1,-1)
Mask
Mask coefficients x y
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10. Spatial Filters – linear filters
For linear spatial filtering, the result, R, at a point (x,y) is
R=w(-1,-1)f(x-1,y-1) + w(0,-1)f(x,y-1) +
…+ w(0,0)f(x,y) +…
+ w(0,1)f(x,y+1) + w(1,1)f(x+1,y+1)
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11. Spatial Filters – convolution
In general, linear filtering of an image is given by the expression:
The image f is of size M×N
The filter mask is of size m×n
m=2a+1, n=2b+1
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12. Spatial Filters – smoothing
Smoothing filters are used for blurring and for noise reduction.
Smoothing, linear spatial filter
average filters
reduce “sharp” transitions
side effect
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13. Spatial Filters – smoothing, linear1
Mean filters
example:
Original
Gaussian noise
5×5 mean filter
3×3 mean filter
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14. Spatial Filters – smoothing, linear1
Mean filters
example:
Salt and pepper
5×5 mean filter
3×3 mean filter
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15. Spatial Filters – smoothing, linear
Weighted average filters
example:
general expression: 1 2 4
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16. Spatial Filters – smoothing, nonlinear
Order-statistic filters
nonlinear spatial filters
order/rank the pixels contained in the image area encompassed by the filter
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17. Spatial Filters – smoothing, nonlinear
Median filters
replace a pixel value with the median of its neighboring pixel values
example: 23 25 26 30 40 22 24 27 35 18 20 50 34 19 15 33 11 16 10
Neighborhood values:
15, 19, 20, 23,
24, 25, 26, 27, 50
Median value: 24
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18. Spatial Filters – smoothing, nonlinear
Median filters
have excellent noise-reduction capabilities
V.S.
Gaussian noise removed
by 3×3 mean filter
Gaussian noise removed
By 3×3 median filter
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19. Spatial Filters – smoothing, nonlinear
Median filters
are particularly effective in salt & pepper
V.S.
Salt & pepper removed
by 3×3 mean filter
Salt & pepper removed
By 3×3 median filter
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20. Spatial Filters – smoothing, nonlinear
Max filters
maximum of neighboring pixel values
useful for finding the brightest points in an image
Min filters
minimum of neighboring pixel values
useful for finding the darkest points in an image
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21. Spatial Filters – sharpening
Principal objective
highlight fine detail in an image
enhance detail that has been blurred
Sharpening can be accomplished by spatial differentiation
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22. Spatial Filters – sharpening
For one dimensional function f(x)
first order derivative
second order derivative
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23. Spatial Filters – sharpening
A sample
(a) a scan line (b) image strip
(c) first derivative (d) second derivative
(a)
(b)
(c)
(d) 7 1 3 6 2 4 5 -1 -2 -6 -7 -4
-12
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24. Spatial Filters – sharpening
The Laplacian
second derivative of a two dimensional function f(x,y)
= [f(x+1,y)+f(x-1,y)+f(x,y+1)+f(x,y-1)]
-4f(x,y)
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25. Spatial Filters – sharpening
The Laplacian
use a convolution mask to approximate 1 -4 1 -8 -1 2 -4
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26. Spatial Filters – sharpening
The Laplacian
example:
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27. Spatial Filters – sharpening
The Laplacian
example:
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28. Frequency Filters – Fourier transform
Fourier transform (FT)
decompose an image into its sine and cosine components
transform real space images into Fourier or frequency space images
In a frequency space image, each point represents a particular frequency contained in the real domain image.
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29. Frequency Filters – Fourier transform
Discrete Fourier transform (DFT)
Inverse DFT
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30. Frequency Filters – Fourier transform
example: FT
(log)
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31. Frequency Filters Basic steps for filtering in the frequency domain
function
DFT
Inverse
f(x,y)
Input image
g(x,y)
Processed image
F(u,v)
H(u,v)F(u,v)
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32. Frequency Filters
Frequencies in an image correspond to the rate of change in pixel values
High frequencies
rapid changes of gray level values
Low frequencies
slow changes of gray level values
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33. Frequency Filters Lowpass filters Highpass filters
attenuate high frequencies while “passing” low frequencies
Highpass filters
attenuate low frequencies while “passing” high frequencies
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34. Frequency Filters – lowpass filters
Ideal lowpass filters (ILPF)
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35. Frequency Filters – lowpass filters
Butterworth lowpass filters (BLPF)
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36. Frequency Filters – lowpass filters
Gaussian lowpass filters (GLPF)
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37. Frequency Filters – highpass filters
Ideal higpass filters (IHPF)
Butterworth highpass filters (BHPF)
Gaussian highpass filters (GHPF)
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38. Image Filters

39. Frequency Filters – bandpass filters
attenuate very low frequencies and very high frequencies
enhance edges while reducing the noise at the same time
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40. Frequency Filters Examples: (lowpass filters) BLPF with
cut-off frequency of 1/3
BLPF with
cut-off frequency of 1/2
ILPF with
cut-off frequency of 1/2
ILPF with
cut-off frequency of 1/3
Original
Gaussian noise
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41. Frequency Filters
Examples: (highpass filters)
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42. Frequency Filters Relationship and comparison with spatial filters
spatial filtering
frequency filtering
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43. Frequency Filters Comparison with spatial filters
more computational efficient
more intuitive
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44. Summary
Filtering is the operation of applying a transform on an image in order to enhance it.
Filtering techniques can be subdivided into two types
Spatial domain filtering
Frequency domain filtering
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45. Summary
Filtering techniques are very useful in image analysis and processing
Noise removal
Edge detection
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46. Thank you & Questions ? The end
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