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Image Processing
3. Area Processes
Computer Engineering, Sejong University
Dongil Han
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Area Processing
uses the input pixel as well as the pixels around it to generate a new output pixel
Some area processing techniques
• Convolution
• Smoothing
• Sharpening
• Edge detection
• Derivative operators
• Median filtering
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Convolution
Mask• usually 2-D matrix• m x n dimension &
usually m, n values are odd number
Convolution• centers on each pixel in
an input image and generates new output pixels
• new pixel is computed by multiplying each pixel value in the neighborhood with the corresponding weight in the mask and summing these products
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1-D convolution example : 1x3 mask
1
1),(),0(),(
ttyxitmyxg
)2,0()1,0()1,0()0,0()0,0()1,0()1,0( imimimg
)3,0()1,0()2,0()0,0()1,0()1,0()2,0( imimimg
i
m
g
0 1 0
0 1 0
Convolution
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1-D convolution example : 1x3 mask
i
m
g
-1 2 -1
-1 2 -1
1
1),(),0(),(
ttyxitmyxg
)2,0()1,0()1,0()0,0()0,0()1,0()1,0( imimimg
)3,0()1,0()2,0()0,0()1,0()1,0()2,0( imimimg
Convolution
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2-D convolution example : 3x3 mask
11
1),(),(
1),(
ttysxitsm
syxg
)2,2()1,1()1,2()0,1()0,2()1,1()2,1()1,0()1,1()0,0()0,1()1,0()2,0()1,1()1,0()0,1()0,0()1,1()1,1(
imimimimimimimimimg
)9,4()1,1()8,4()0,1()7,4()1,1()9,3()1,0()8,3()0,0()7,3()1,0()9,2()1,1()8,2()0,1()7,2()1,1()8,3(
imimimimimimimimimg
Convolution
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ba
bttysxitsm
asyxg ),(),(),(
2-D convolution example : mxn mask
here, a = (m-1)/2, b = (n-1)/2.
If we let the resolution of an image be MxN, then
x= 0, 1, 2, …, M-1,y= 0, 1, 2, …, N-1
Convolution
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convolution in image borders• zero padding : treat the empty cells in the convolution
window as zeros• start convolving with the pixel at (1,1) instead of the pixel
at (0,0). And also to end at (M-2, N-2) instead of (M-1,N-1)
• duplicate the border edges• wrap the image
Convolution
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Graph of mask size vs. number of computations
Convolution
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Convolution
Properties of mask coefficients• The bigger the sum of mask coefficients, the brighter the output
image• To preserve the brightness of output image, the sum of mask
coefficients should be ‘1’• If the sum of mask coefficients is less than 1, the output image
gets darker• The sum of coefficients is mostly 1 or 0
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Blurring
Blurring• removes the finer details of an image• used to simulate a camera out of focus or de-emphasize a
background• accomplished through convolution• the bigger the mask, the greater the blurring effect and
the greater the computation time• effective way of reducing Gaussian noise• several kinds of blurring filters
- average filter- weighted average filter- low-pass filter
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Average filter• generates neighborhood averaging in mask• 3x3 mask : all the coefficients equal 1/9• 5x5 mask : all the coefficients equal 1/25• removes extreme values in an image• effective way of reducing Gaussian noise in an image
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Blurring
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Average filter
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mask
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mask
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Image
* = 50
40
20
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30
90
60
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50
10
Image
* = 50
Blurring
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Average Filter Example
Original Image 3X3 mask
7X7 mask5X5 mask
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Average Filter Example
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Weighted average filter• approximates more closely to a Gaussian profile• the sum of coefficients is 1• well known weighted average filter : Gaussian smoothing
filter• controls the effective spread
22
22
)22(
),(
yx
eyxw
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Blurring
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Weighted average filter
mask
mask
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Image
* = 50
40
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80
30
90
60
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50
10
Image
* = 48
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Blurring
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Weighted Average Filter Example
Original Image Gaussian smooth Filtering
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Sharpening
Sharpening• has the opposite effect of blurring• also referred to as crispening• emphasizes the details in an image• increases local contrast• is based on the high-pass filter• several kinds of sharpening filters
9
-1
-1
-1
-1
-1
-1
-1
-1
5
-1
-1
-1
0
0
-1
0
0
5
-2
-2
-2
1
1
-2
1
1
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Sharpening
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High-pass filter• removes the low frequency components• show only high detail• the sum of coefficients is 0• original image is added back to generate sharpened
image• several kinds of high-pass filters
- (common) high-pass filter- unsharp masking- high-boost filter
Sharpening
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High-pass filter• removes the low frequency components – e.g. mean value
of image• preserves the high frequency components – e.g. edge
component of image• used to emphasize the high image detail
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Sharpening
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High-pass filter
mask
mask
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Image
* = 0
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Image
* = 62
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Sharpening
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Unsharp masking• to subtract a low-pass filtered image from its original
• widely used because of its noise insensitive property
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0
0
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0
0
),(),(),( yxpasslow
fyxorgfyxsharp
f
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Sharpening
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3x3 Mean
Filter
Saturator+
++
-
Lum. In Lum. Out
Edge Signal
Mean Filtering
Result
Difference signal
Resulting Signal
Sharpening
Unsharp masking example
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High-boost filter• computed by increasing the intensity of the original and
subtracting a low-pass filtered image
• when A = 1 : unsharp masking• when A > 1: a fraction of the original is added
w/5
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0
0
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0
0
),(),(),(_
yxpasslow
fyxorgfAyxboosthigh
f
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w = 9A-1w = 5A-1
Sharpening
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High-boost filter
mask
mask
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Image
* = 11
Image
* = 80
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Sharpening
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Original Image Sharpening
A =1.2 High-boostHigh-pass
Sharpening
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Sharpening
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Sharpening vs. blurring
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Filtering
1-D Filter mask(ideal case)
f
Low-Pass Filter
f
High-Pass Filter
f
Band-Pass Filter
f
Band-Stop Filter
cutoff frequencyH(f)
H(f) H(f)
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frequency sweep signal input
after high-pass filtering
after band-stop filtering
Properties of 1-D Filter mask
Filtering
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Zone-plate image
Filtering
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Zone-plate image
• center region : 0 frequency component
• moves to the outer region : frequency components goes to high
• widely used to test the filters
• widely used to test characteristics of display equipments
• an equation for making zone-plate
))cos(1(5.127),( 22 yH
xV
yxf
Filtering
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Low-pass filteredimage
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Originalimage
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High-pass filteredimage
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Originalimage
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Laplacianfilteredimage
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Edge• set of connected pixels that lie on the boundary between
two regions • (두 영역의 경계 면에 위치한 연결된 화소들의 집합)• holds much of the information in the image• tells where objects are, their shape, size, and texture• the difference between an edge and a boundary
- edge : local concept- boundary : more global idea
Edge Detection
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Edge detection• detects meaningful discontinuities in gray-level image• first step in image segmentation(영역 분할)• image segmentation
- a field of image analysis- used to group pixels into regions to determine an
image’s composition• examples of edge profile
roof edge line edge step edge ramp edge
Edge Detection
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Ideal edge and its model
Edge Detection
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Simple edge detectors• Homogeneity operator
- the simplest edge detector
• Difference operator- the quickest edge detector
Edge Detection
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Edge Detection
Homogeneity operator• determines the maximum value from a series of pixel
subtraction• subtracts each 8 surrounding pixels from the center
pixel of a 3x3 window• output : maximum of the absolute value of each
difference
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Edge Detection
Difference operator • requires 4 subtractions per pixel
upper left – lower rightmiddle left – middle right lower left – upper right top middle – bottom middle
• output : maximum of the absolute value of each difference
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(a) Original image (b) Edge map of homogeneity operator (c) Edge map of difference operator
Edge Detection
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Derivative Operators
Derivative Operators• detects the edges using the first-order derivative or
second-order derivative of an image
• digital approximation is used for 1-, 2-order derivatives
• first-order partial derivative
• second-order partial derivative)
)()1( xfxfx
f
)()1( yfyfy
f
)(2)1()1(2
2xfxfxf
x
f
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Derivative Operators
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First-order derivative operators• the gradient of an image f(x,y) at location (x,y) is defined
as
• the magnitude of the gradient is
• the direction angle of the gradient (x,y) at (x,y)
y
fx
f
yGxG
f
)(1tan),(x
y
G
Gyx
||||22)( yx GGoryGxGmagf f
Derivative Operators
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Properties of first-order derivative operators• consists of row detector Gx and column detector Gy
• the sum of mask coefficients is ‘0’
• several kinds of first-order derivative operators
1
0
0
0
-1
0
0
0
0
Gx
Roberts operator
Gy 1
0
0
0
0
0
0
-1
0
Prewitt operator
0
-1
1
0
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1
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1
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0
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Sobeloperator
0
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2
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1
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0
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-1
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2
1
1
Frei-chenoperator
0
-
0
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1
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-1
1
0
0
0
-
-1
-1
1
1
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2
2
2
Derivative Operators
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Properties of first-order derivative operators• Roberts operator
- smaller mask size- susceptible to noise
• Prewitt operator- widely used- sensitive to vertical and horizontal edges
• Sobel operator- widely used- sensitive to vertical and horizontal edges- more sensitive to diagonal edges than Prewitt operator- better noise suppression property than Prewitt operator
Derivative Operators
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Prewitt operator
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vertical edge
* = 0
horizontal edge
* = 18020
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0
-1
1
0
-1
1
0
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0
0
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-1
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1
1
Gx
Gy
Derivative Operators
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Sobel operator
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* = 0
* = 24020
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Gx
Gy
Derivative Operators
vertical edge
horizontal edge
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Prewitt operator
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* = 40
* = 16020
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-1
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Gx
Gy
Derivative Operators
vertical edge
horizontal edge
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Sobel operator
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60
80
* = 60
* = 22020
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Gx
Gy
Derivative Operators
vertical edge
horizontal edge
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original
|||| yx GG xG
yG
Derivative Operators
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original Roberts Operator
Sobel Operator
Prewitt Operator
Derivative Operators
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Compass gradient operators
Compass gradient operators• find edges in eight different orientations • convolve an image with eight different convolution masks• output : maximum of all eight convolutions• several kinds of compass gradient operators
- Prewitt- Kirsh- Robinson 3-level- Robinson 5-level
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Prewitt
-2
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west
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1
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north
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-1
1
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-1
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1
-1
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-2
1
-1
1
1
1
-1
1
-1
northeast
-2
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1
1
1
1
-1
-1
1
-2
1
1
1
1
1
-1
-1
-1
southwest south southeast east
Compass gradient operators
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Kirsh
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west
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north
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-3
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-3
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northeast
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-3
5
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-3
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-3
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0
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-3
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5
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-3
southwest south southeast east
Compass gradient operators
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Robinson 3-level
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west
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0
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north
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-1
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-1
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0
-1
1
0
1
-1
1
1
0
-1
0
-1
northeast
0
-1
1
1
0
1
-1
-1
0
0
0
0
1
1
1
-1
-1
-1
southwest south southeast east
Compass gradient operators
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Robinson 5-level
0
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0
-2
-1
-1
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1
1
west
0
1
-1
-1
0
-2
1
2
0
northwest
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-1
1
-1
-2
0
1
0
2
0
2
-2
0
1
-1
0
1
-1
north
0
-2
2
0
-1
1
0
-1
1
0
1
-1
1
2
0
-1
0
-2
northeast
0
-1
1
1
0
2
-1
-2
0
0
0
0
2
1
1
-2
-1
-1
southwest south southeast east
Compass gradient operators
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0
1
-1
-1
0
-2
1
2
0
0
-1
1
-1
-2
0
1
0
2
Compass gradient operators
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Derivative Operators
Laplacian• definition
• digital appriximation: x-directional 2-order partial derivative
• digital appriximation:
• final form
),(2),1(),1(2
2yxfyxfyxf
x
f
),(2)1,()1,(2
2yxfyxfyxf
y
f
2
2
2
22
y
f
x
ff
),(4)]1,()1,(),1(),1([2 yxfyxfyxfyxfyxff
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Properties of Laplacian• produces sharper edges than first order derivatives • highlights edges in all directions• unacceptably sensitive to noise• produces double edges• unable to detect edge direction• several kinds of Laplacian operators
-4
1
1
1
0
0
1
0
0
4
-1
-1
-1
0
0
-1
0
0
-8
1
1
1
1
1
1
1
1
8
-1
-1
-1
-1
-1
-1
-1
-1
Derivative Operators
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Derivative Operators
67/80original 1차미분 2차미분
Derivative Operators
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LoG(Laplacian of Gaussian)• Laplacian is not used in the original form for edge detection• Gaussian smoothing filter is used before application of the
Laplacian• Gaussian smoothing filter example
• LoG(Laplacian of Gaussian)
deviationstandardisandyxrwhereerhr
222,)(2
2
2
2
2
24
22)(2
r
er
rh
Derivative Operators
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Derivative Operators
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LoG(Laplacian of Gaussian)• due to its shape, sometimes called the Mexican hat function• the larger the value of , the smaller the peak value and
the wider the convolution mask• the smaller the value of , the higher the peak value and
the narrower the convolution mask• the wider the mask, wider edge will be detected • the narrower the mask, sharp edge will be detected• insensitive to noise• similar to human visual perception model
2
2
24
22)(2
r
er
rh
Derivative Operators
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sobeloriginal
gaussian smoothing
laplacian
LoGZero
crossing
Derivative Operators
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Median Filtering
Median Filtering• replaces the value of a pixel by the median of the gray
levels in the neighborhood of that pixel• median : choose the center pixel after the sorting the
pixels in the window in ascending order• (마스크 내에 존재하는 화소들을 정렬(sorting) 한 후 중간에
위치하는 화소 값을 출력으로 사용) • the best-known order-statistics filter, nonlinear operator• reduces noise with preserving sharp edges and detail• particularly efficient for reducing impulse noise
- impulse noise : also known as salt-and-pepper noise- white and black dots in an image
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5
6
3
4
5
5
13
29
7
6
26
27
9
5
8
6
5
5
9
45
9
28
27
31
7
6
15
32
32
33
5 6 267
8
6
9
6
28
27
15
9
A
B
Median Filtering• operation A : median {3, 5, 5, 4, 6, 5, 5, 5, 45}
= median {3, 4, 5, 5, 5, 5, 5, 6, 45} = 5• operation B : median {6, 5, 6, 5, 45, 6, 9, 9, 7}
= median {5, 5, 6, 6, 6, 7, 9, 9, 45} = 6
Median Filtering
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Several kinds of Median Filters
Median Filtering
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Median Filtering
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Salt & pepper 잡음 Median filter 처리 결과 Mean filter 처리 결과
Median Filtering
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Gaussian 잡음 Median filter 처리 결과 Mean filter 처리 결과
Median Filtering
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Max filter
• Maximum window value is used
• removes dark spikes, pepper noise
• overall intensity of output image is increased
Min filter
• Minimum window value is used
• removes white spikes, salt noise
• overall intensity of output image is reduced
)},({max),(ˆ),(
tsgyxfxySts
Min/Max Filtering
)},({min),(ˆ),(
tsgyxfxySts
79/80
Midpoint filter
• simply computes the midpoint between the maximum and minimum values in the window
• works best for randomly distributed noise, like Gaussian and uniform noise
)}],({min)},({max[2
1),(ˆ
),(),(tsgtsgyxf
xyxy StsSts
Min/Max Filtering
80/80
pepper noise 추가 salt noise 추가
max filter 결과 min filter 결과
Min/Max Filtering
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