60-520 Presentation
Image Filters
Student: Xiaoliu Chen
Instructor: Dr. I. Ahmad
School of Computer Science
University of Windsor
November 2003
Outline
Introduction
 Spatial Filtering

– Smoothing
– Sharpening

Frequency-Domain Filtering
– Low pass
– High pass

Summary
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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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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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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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Spatial Filters

The general processes can be denoted
by the expression:
g ( x, y )  T [ f ( x, y )]
– 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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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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Spatial Filters
Example:
3×3 neighborhood about a point (x,y) in an image
x
(x,y)
(x-1,y-1)
(x-1,y)
(x,y-1) (x+1,y-1)
(x,y)
(x+1,y)
Image f(x,y)
(x-1,y+1) (x,y+1) (x+1,y+1)
y
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x
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
Image f(x,y)
y
f(x-1,y-1)
f(x-1,y)
f(x,y-1) f(x+1,y-1)
f(x,y)
Mask coefficients
f(x+1,y)
f(x-1,y+1) f(x,y+1) f(x+1,y+1)
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Pixels under
mask
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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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Spatial Filters – convolution

In general, linear filtering of an image is
given by the expression:
g ( x, y ) 
a
b
  w(s, t ) f ( x  s, y  t )
s   at   b
– 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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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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Spatial Filters – smoothing, linear
1 1 1

Mean filters
– example:
1

9
Gaussian
noise
Original
1 1 1
1 1 1
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5×5
3×3 mean
mean filter
filter
13
Spatial Filters – smoothing, linear
1 1 1

Mean filters
– example:
1

9
Salt and pepper
1 1 1
1 1 1
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5×5
3×3 mean filter
14
Spatial Filters – smoothing, linear

Weighted average filters
1 2 1
1

16
– example:
1 2 1
– general expression:
a
g ( x, y ) 
2 4 2
b
  w(s, t ) f ( x  s, y  t )
s   at   b
a
b
  w(s, t )
s   at   b
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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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Spatial Filters – smoothing, nonlinear

Median filters
– replace a pixel value with the median of its
neighboring pixel values
– example:
23 25 26 30 40
Neighborhood values:
15, 19, 20, 23,
24, 25, 26, 27, 50
22 24 26 27 35
Median value: 24
11 16 10 20 30
18 20 50 25 34
19 15 19 23 33
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Spatial Filters – smoothing, nonlinear

Median filters
– have excellent noise-reduction capabilities
V.S.
Gaussian noise removed
Gaussian noise removedImage Filters
By 3×3 median filter
by 3×3 mean filter
18
Spatial Filters – smoothing, nonlinear

Median filters
– are particularly effective in salt & pepper
V.S.
Salt & pepper removed
Salt & pepper removedImage Filters
By 3×3 median filter
by 3×3 mean filter
19
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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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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Spatial Filters – sharpening

For one dimensional function f(x)
– first order derivative
f
 f ( x  1)  f ( x)
x
– second order derivative
2 f
 f ( x  1)  f ( x  1)  2 f ( x)
2
x
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Spatial Filters – sharpening
– A sample
(a)
(b) 5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7
(c)
-1 -1 -1 -1 -1 0 0 6 -6 0 0 0 1 2 -2 -1 0 0 0 7 0 0 0
(d)
-1 0 0 0 0 1 0 6
-12
(a) a scan line
(c) first derivative
6 0 0 1 1 -4 1 1 0 0 7 -7 0 0
(b) image strip
(d) second derivative
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Spatial Filters – sharpening

The Laplacian
– second derivative of a two dimensional
function f(x,y)
2
2

f

f
2
 f  2  2
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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Spatial Filters – sharpening

The Laplacian
– use a convolution mask to approximate
0 1 0
1 1 1
-1 2 -1
1 -4 1
1 -8 1
2 -4 2
0 1 0
1 1 1
-1 2 -1
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Spatial Filters – sharpening

The Laplacian
– example:
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Spatial Filters – sharpening

The Laplacian
– example:
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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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Frequency Filters – Fourier transform

Discrete Fourier transform (DFT)
1
F (u, v) 
MN

M 1 N 1

f ( x, y)e  j 2 (ux / M vy / N )
x 0 y 0
Inverse DFT
f ( x, y ) 
M 1 N 1
j 2 ( ux / M  vy / N )
F
(
u
,
v
)
e

u 0 v 0
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Frequency Filters – Fourier transform
– example:
FT
(log)
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Frequency Filters

Basic steps for filtering in the frequency
domain
Filter
function
DFT
f(x,y)
Input image
F(u,v)
Inverse
DFT
H(u,v)F(u,v)
g(x,y)
Processed image
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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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Frequency Filters

Lowpass filters
– attenuate high frequencies while “passing”
low frequencies

Highpass filters
– attenuate low frequencies while “passing”
high frequencies
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Frequency Filters – lowpass filters

Ideal lowpass filters (ILPF)
1 if D(u, v)  D0
H (u, v)  
0 if D(u, v)  D0
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Frequency Filters – lowpass filters

Butterworth lowpass filters (BLPF)
1
H (u, v) 
1  [ D(u, v) / D0 ]2 n
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Frequency Filters – lowpass filters

Gaussian lowpass filters (GLPF)
H (u, v)  e
 D2 (u ,v ) / 2 2
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Frequency Filters – highpass filters

Highpass filters
H hp (u, v)  1  H lp (u, v)
– Ideal higpass filters (IHPF)
– Butterworth highpass filters (BHPF)
– Gaussian highpass filters (GHPF)
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Frequency Filters – bandpass filters

Bandpass filters
– attenuate very low frequencies and very
high frequencies
– H bp  H hp (u , v)  H lp (u , v)
– enhance edges while reducing the noise at
the same time
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Frequency Filters

Examples: (lowpass filters)
Gaussian
noise
Original
ILPF with
with
BLPF
ILPF
cut-off frequency
frequency of
of 1/2
1/3
cut-off
1/3
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Frequency Filters

Examples: (highpass filters)
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Frequency Filters

Relationship and comparison with
spatial filters
– spatial filtering
g ( x, y )  h ( x, y ) f ( x, y )
– frequency filtering
G (u, v)  H (u, v) F (u, v)
–
h( x, y )  H (u, v)
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Frequency Filters

Comparison with spatial filters
– more computational efficient
– more intuitive
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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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Summary

Filtering techniques are very useful in
image analysis and processing
– Noise removal
– Edge detection
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Thank you
The&end
Questions ?
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