60-520 Presentation
Image Filters
Student: Xiaoliu ChenInstructor: Dr. I. Ahmad
School of Computer ScienceUniversity of Windsor
November 2003
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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:
– 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)
)],([),( yxfTyxg
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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
Spatial Filters
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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,y)(x-1,y)
(x+1,y-1)(x,y-1)(x-1,y-1)
y
Spatial Filters
Image Filters 9Pixels under mask
Image f(x,y)
f(x+1,y+1)f(x,y+1)f(x-1,y+1)
f(x+1,y)f(x,y)f(x-1,y)
f(x+1,y-1)f(x,y-1)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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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:
– The image f is of size M×N
– The filter mask is of size m×n
m=2a+1, n=2b+1
a
as
b
bt
tysxftswyxg ),(),(),(
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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
Mean filters– example: 111
111
111
9
1
OriginalGaussian noise 3×3 mean filter 5×5 mean filter
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Spatial Filters – smoothing, linear
Mean filters– example: 111
111
111
9
1
Salt and pepper 3×3 mean filter 5×5 mean filter
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Spatial Filters – smoothing, linear
Weighted average filters– example:
– general expression: 121
242
121
16
1
a
as
b
bt
a
as
b
bt
tsw
tysxftswyxg
),(
),(),(),(
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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
22 24 26 27 35
18 20 50 25 34
19 15 19 23 33
11 16 10 20 30
Neighborhood values:15, 19, 20, 23,
24, 25, 26, 27, 50
Median value: 24
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Spatial Filters – smoothing, nonlinear
Median filters– have excellent noise-reduction capabilities
Gaussian noise removedby 3×3 mean filter
Gaussian noise removed By 3×3 median filter
V.S.
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Spatial Filters – smoothing, nonlinear
Median filters– are particularly effective in salt & pepper
Salt & pepper removedby 3×3 mean filter
Salt & pepper removed By 3×3 median filter
V.S.
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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
– second order derivative
)()1( xfxfx
f
)(2)1()1(2
2
xfxfxfx
f
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Spatial Filters – sharpening
– A sample
(a) a scan line (b) image strip
(c) first derivative (d) second derivative
7777000013100006000123455
0007000-1-221000-6600-1-1-1-1-1
00-770011-411006-126010000-1
(a)
(b)
(c)
(d)
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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)
2
2
2
22
y
f
x
ff
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Spatial Filters – sharpening
The Laplacian– use a convolution mask to approximate
010
1-41
010
111
1-81
111
-12-1
2-42
-12-1
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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)
Inverse DFT
1
0
1
0
)//(2),(1
),(M
x
N
y
NvyMuxjeyxfMN
vuF
1
0
1
0
)//(2),(),(M
u
N
v
NvyMuxjevuFyxf
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Frequency Filters – Fourier transform
– example:
FT
(log)
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Basic steps for filtering in the frequency domain
Frequency Filters
Filterfunction
FilterfunctionDFTDFT Inverse
DFT
InverseDFT
f(x,y)Input image
g(x,y)Processed image
F(u,v) H(u,v)F(u,v)
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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
Frequency Filters
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Lowpass filters– attenuate high frequencies while “passing”
low frequencies
Highpass filters– attenuate low frequencies while “passing”
high frequencies
Frequency Filters
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Ideal lowpass filters (ILPF)
Frequency Filters – lowpass filters
0
0
),( if0
),( if1),(
DvuD
DvuDvuH
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Butterworth lowpass filters (BLPF)
Frequency Filters – lowpass filters
nDvuDvuH
20 ]/),([1
1),(
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Gaussian lowpass filters (GLPF)
Frequency Filters – lowpass filters
22 2/),(),( vuDevuH
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Highpass filters
– Ideal higpass filters (IHPF)
– Butterworth highpass filters (BHPF)
– Gaussian highpass filters (GHPF)
Frequency Filters – highpass filters
),(1),( vuHvuH lphp
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Bandpass filters– attenuate very low frequencies and very
high frequencies
–
– enhance edges while reducing the noise at the same time
Frequency Filters – bandpass filters
),(),( vuHvuHH lphpbp
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Frequency Filters
Original Gaussian noise ILPF withcut-off frequency of 1/3
ILPF withcut-off frequency of 1/2
BLPF withcut-off frequency of 1/3
BLPF withcut-off frequency of 1/2
Examples: (lowpass filters)
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Examples: (highpass filters)
Frequency Filters
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Frequency Filters
Relationship and comparison with spatial filters– spatial filtering
– frequency filtering
–
),(),(),( yxfyxhyxg
),(),( vuHyxh
),(),(),( vuFvuHvuG
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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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The end
Thank you
&
Questions ?