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Chapter4 Image Enhancement
• Preview• 4 1 General introduction and Classification4.1 General introduction and Classification• 4.2 Enhancement by Spatial Transforming(contrast enhancement)• 4 3 Enhancement by Spatial Filtering (image smoothing)4.3 Enhancement by Spatial Filtering (image smoothing)• 4.4 Enhancement by Frequency Filtering (image sharpening)• 4 5 Color Enhancement• 4.5 Color Enhancement• Summary
Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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4 1 G l I t d ti d Cl ifi ti4.1 General Introduction and Classification4 1 1 Purposes4.1.1 Purposes
• improve the visual effectsimprove the visual effects• easy to edge extracting
4.1.2 Methods
• spatial domain: point operations, local operationsp p p , p• frequency domain: DFT Filter IDFT
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4.1 General Introduction and Classification4.1.3 contents • contrast enhancement: linear transform• contrast enhancement: linear transform
non-linear transform,histogram equalizationg qhistogram matching local enhancement
i hi i k• image smoothing: averaging mask, order-statistics filterlowpass filterlowpass filter.
• image sharpening: derivatives, highpass filterg p
• color image enhancement: pseudo color processing, full color processing
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4.2 Contrast Enhancement4.2.1. Introduction: Histogram
Histogram gives an estimate of the probability ofHistogram gives an estimate of the probability of the occurrence of gray levels
( ) / 0,1, , 1k kp s n n k L= = −
Where s is the the kth gray level and the n is the number ofWhere sk is the the kth gray level and the nk is the number ofPixels in the image having gray level sk
tlapparently1
( ) 1L
kp s−
=∑0
kk=∑
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4.2 Contrast Enhancement4.2.1. Introduction: Histogram
H i t l i l l lHorizontal axis: gray level valuesVertical axis: probability of gray level values
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4.2 Contrast Enhancement4.2.1. Introduction: Histogram
Hi t d i litHistograms and images quality
Dark image
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4.2 Contrast Enhancement4.2.1. Introduction: Histogram
Hi t d i litHistograms and images quality
Bright image
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4.2 Contrast Enhancement4.2.1. Introduction: Histogram
Hi t d i litHistograms and images quality
Low-contrast image
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4.2 Contrast Enhancement4.2.1. Introduction: Histogram
Hi t d i litHistograms and images quality
Double-peaks image
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4.2 Contrast Enhancement4.2.1. Introduction: Histogram
Hi t d i litHistograms and images quality
Equalized image
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4.2 Contrast Enhancement4.2.1. Introduction: Classification
• Direct gray level transformationsg y• Histogram processing• Operations among images
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations: Linear transformations
( )s T r ar b= = + [ , ] [ , ]r A B s C D∈ ∈
D C BC AD
Expression:
D C BC ADs rB A B A− −
= +− −
Formulation:
255
Example:C 0C=0 D=255
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4.2 Contrast Enhancement
example: A= 69 B=213 C=0 D=255
4.2.2 Direct gray level transformations : Linear transformations
1 7 122 2s r= −example: A= 69, B=213, C=0, D=255 1.7 122.2s r=
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Piecewise-linear transformations
[0, )c r r aa
⎧ ∈⎪Formulation:
[ , )
ad cs r c r a bb a
⎪⎪
−⎪= + ∈⎨ −⎪255 ( ) [ , 255)255
b ad r b d r bb
⎪−⎪ − + ∈⎪ −⎩
cd
⎩
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Piecewise-linear transformations
( , ) (10,50), ( , ) (210,150)a c b d= =
Original image
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : image negatives
1s L r= − −Formulation:
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Log transformations
l (1 )log(1 )s c r= +Formulation:
i t t d it ic is a constant and it is assumed that 0r ≥
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Log transformations
Example: display of DFT spectrum
Direct display After log operation
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Power transformations
s cr γFormulation: s cr γ=Formulation:
Where and are positive constants.γc
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Power transformations
G tiGamma correction
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Power transformations
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Gray-lever slicing
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
If a transform has the form:( )s T r= 0 1r≤ ≤
If a transform has the form:
We hope the Probability Density Function (PDF) of s isWe hope the Probability Density Function (PDF) of s is
( ) 1sp s = Inverse transform exist( )sp
satisfies the following conditions:( )T r
ve se s o e sand, 1 2 1 2if then r r s s< <
0 1r≤ ≤(a) is single-valued and monotonically in the interval (b) for 0 1r≤ ≤0 1s≤ ≤
s has the same range as the r
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
C ti diti( ) ( )s r
drp s p rds
=( )s T r=Continuous condition
( )( )
rp rdsdr p s
⎫= ⎪⎬ ( )
rs p x dx= ∫( )
( ) 1s
s
dr p sp s
⎬⎪= ⎭
0( )rs p x dx∫
namely0
( ) ( )r
rT r p x dx= ∫0∫
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4.2 Contrast Enhancement
Di t diti
4.2.3 Histogram processing : Histogram equalization
Discrete condition
k k n
0 0( ) ( ) 0,1, 2,..., 1
k kj
k k r jj j
ns T r p r k L
n= =
= = = = −∑ ∑
I iIn practice
( ) ( 1) ( )k
k ks T r L p r= = − ∑0
( ) ( 1) ( )k k r jj
s T r L p r=∑
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
Example:page 78p e:p ge 78
Suppose that a 64*64, 3bits image has the gray-level distributionas show in the first row the calculate steps are:
7,...,1,0, =ksk
序号
运 算 步骤和结果
1 列出原始图灰度级 0 1 2 3 4 5 6 7
as show in the first row, the calculate steps are:
k
kn2 统计原始直方图各灰度级象素 790 1023 850 656 329 245 122 81
3 计算原始直方图 0.19 0.25 0.21 0.16 0.08 0.06 0.03 0.02
]5.0)1int[( +−= kk tNt
)( kk ts → 10 → 31→ 52 → 64,3 → 77,6,5 →
4 计算累计直方图 0.19 0.44 0.65 0.81 0.89 0.95 0.98 1.00
5 取整 1 3 5 6 6 7 7 7
6 确定映射对应关系10 → 31→
kn7 统计新直方图各灰度级象素 790 1023 850 985 448
8 用计算新直方图 0.19 0.25 0.21 0.24 0.11
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
0.3 0 25 0 40.6
0.8
1
0.44
0.650.81
0.890.95 0.98 1
Histograms
0.1
0.20.19
0.250.21
0.160.08
0.060.03 0 02
0 1 2 3 4 5 6 7
0.20
0.40.19
0 1 2 3 4 5 6 70
0.02
(a)
(b)
0 3 0 25
transformation function(a)
0.1
0.2
0.3
0.19
0.250.21
0.24
0.11original histogram
00 1 2 3 4 5 6 7
(c)
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equalized histogram
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
Experimental resultspe e esu s
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
Experimental results
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
Experimental results
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
Experimental results
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
Experimental results
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram equalization
question: page 99 4 2
Why the discrete histogram
question: page 99, 4.2
y gequalization technique does not yield a flat histogram?
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram matching (specification)
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram matching (specification)
Hi t li ti
( ) ( ) 0,1, 2 1k
k k r js T r p r k L= = = −∑
Histogram equalization: r s
0j=
k
Histogram equalization: z v
0
( ) ( ) 0,1, 2 1k
k k z ii
v G z p z k L=
= = = −∑
1 1 1
k kv s=let then r z
1 1 1( ) ( ) ( ( ))k k k kz G v G s G T r− − −= = =
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram matching (specification)
PzG(zk)
zv, ,v
z,,Pv T(rk)
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram matching (specification)
Example:page 80Suppose that a 64*64, 3bits image has the gray-level distribution as show in the first row, and the 5th row is the
p e:p ge 80
序 运 算 步骤和结果
,specified gray-level distribution. The calculate steps are:
7,...,1,0, =ksk
kn
序号
运 算 步骤和结果
1 列出原始图灰度级 0 1 2 3 4 5 6 7
2 统计原始直方图各灰度级象素 790 1023 850 656 329 245 122 81
3 计算原始直方图 0 19 0 25 0 21 0 16 0 08 0 06 0 03 0 02
4096,/)( == nnnup kku
3 计算原始直方图 0.19 0.25 0.21 0.16 0.08 0.06 0.03 0.02
4 计算原始累计直方图 0.19 0.44 0.65 0.81 0.89 0.95 0.98 1.00
5 规定直方图 0 0 0 0.2 0 0.6 0 0.2
6 计算规定累计直方图 0 0 0 0 2 0 2 0 8 0 8 1 0
31,0 → 54,3,2 → 77,6,5 →
6 计算规定累计直方图 0 0 0 0.2 0.2 0.8 0.8 1.0
7s SML映射 3 3 5 5 5 7 7 7
8s 确定映射对应关系
9s 变换后直方图 0 0 0 0 44 0 0 45 0 0 11
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9s 变换后直方图 0 0 0 0.44 0 0.45 0 0.11
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram matching (specification)
0.80 6
0.6
0.2
0.40.2
0.6
0.2
0.2
0.3
0.19
0.250.21
0 16
0 1 2 3 4 5 6 7
(b)
0
0 1 2 3 4 5 6 7
0
0.1
0.16
0.080.06
0.030.02
specified histogram
0.6
0.8
0 44 0.450 1 2 3 4 5 6 7
(a)
i i l hi t 0 5
0.2
0
0.4
0.44
0.11
original histogram 0 1 2 3 4 5 6 7
(c)
resulting histogramDigital Image Processing
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resulting histogram
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram matching (specification)
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram matching (specification)
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4.2 Contrast Enhancement4.2.3 Histogram processing : Histogram matching (specification)
Example
i i l equalized matchedoriginal equalized matched
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Bit-plane slicing
b b b b b b b b1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 0
eg: 255b7 b6 b5 b4 b3 b2 b1 b0
1 1 1 1 1 1 1 0
1 0 0 0 0 0 0 1
254
129B7
B6
B5
一个8bit的字节位面7
B5B4
B3
B2
.
..
.B2B1
B0....
位面0
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4.2 Contrast Enhancement4.2.2 Direct gray level transformations : Bit-plane slicing
b7 b6
b b bb5 b4 b3
b2 b1 b0
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4.3 Image Smoothingg g
• Introduction • Smoothing linear filters• Order- statistic filters• Low-pass filter in frequency domain
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4.3 Image Smoothing4.3.1 Introduction
g g
P i iPurposes: removing noiseRequest: keeping edges and details
Noise Edges Details Gray levels change greatl in neighborhoodgreatly in neighborhood
Having the same texture in a regionHaving the same texture in a region
Having the similar grads directions in a segment
Random in gray levels and spatial locationRandom in gray levels and spatial location
Salt-pepper noiseGaussian noise
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Gaussian noise
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4.3 Image Smoothing4.3.1 Introduction: Noise models
g g
PDF of Gaussian noise:2 2( ) / 21( ) zp z e μ σ− −=PDF of Gaussian noise: ( )
2p z e
πσ=
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4.3 Image Smoothing4.3.1 Introduction: Noise models
g g
Gaussian noise ( , ) ( , ) ( , )g x y f x y x yη= +G uss o se ( ) ( ) ( )g y f y yη
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4.3 Image Smoothing4.3.1 Introduction: Noise models
g g
PDF of salt-pepper noise: aP z a=⎧⎪PDF of salt-pepper noise:
( )0
bp z P z botherwise
⎪= =⎨⎪⎩⎩
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4.3 Image Smoothing4.3.1 Introduction: Noise models
g g
salt-pepper noise:
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4.3 Image Smoothing4.3.1 Introduction: Smoothing filters
g g
i hb i(1) S thi li filt neighbor averaging(1) Smoothing linear filters:out range pixel smoothingMaximum homogeneity smoothing
(2)Order- statistic filters: Max filters
Mid i filMin filtersMidpoint filtersMedian filtersAlpha-trimmed mean filter p
(3)Low-pass filters: Idea low-pass filter (ILPF)Butterworth low-pass filter (BLPF)Gaussian low-pass filter (GLPF)
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4.3 Image Smoothing Maskg g4.3.2 Smoothing linear filters: neighbor averagingSpatial filter: mask, kernel, template, window
Maskcoefficients
Sp e : s , e e , e p e, w dow
w(-1,-1) w(-1,0) w(-1,1)
w(0,-1) w(0,0) w(0,-1)
w(1,-1) w(1,0) w(1,1)
maskf( 1 1)
f(x,y)f(x-1,y-1) f(z-1,y) f(x-1,y+1)
f(x y-1) f(x y) f(x y+1)f(x,y 1) f(x,y) f(x,y+1)
f(x+1,y-1) f(x+1,y) f(x+1,y+1)
Pixels undermask
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4.3 Image Smoothingg g4.3.2 Smoothing linear filters: neighbor averaging
Spatial filter: operating steps: (page 83)(1) Moving the filter mask from point to point in an image
Spatial filter: operating steps: (page 83)
(2) Multiplying the filter coefficient and the corresponding image pixels
(3) Adding all the products(3) Adding all the products (4) The sum is the response of the filter at a given point
( , ) ( , ) ( , )a b
i j bg x y w i j f x i y j= + +∑ ∑
i a j b=− =−
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4.3 Image Smoothingg g4.3.2 Smoothing linear filters: neighbor averaging
T i l kTypical mask:
1 1 1
1 1 1
1 2 1
2 4 21×
116
×1 1 1
1 1 1
2 4 2
1 2 1
9 16
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4.3 Image Smoothingg g4.3.2 Smoothing linear filters: neighbor averaging
Experimental results: different masks
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4.3 Image Smoothingg g4.3.2 Smoothing linear filters: neighbor averaging
Experimental results:different sizes
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4.3 Image Smoothingg g4.3.2 Smoothing linear filters: out range pixel smoothing
( , ) ( , ) ( , )a b
i a j bg x y w i j f x i y j
=− =−
= + +∑ ∑formulate
i a j b
( ) ( ) - ( )g x y g x y f x y T⎧ >( , ) ( , ) - ( , )ˆ ( , )( , ) others
g x y g x y f x y Tg x y
f x y⎧ >
= ⎨⎩
Advantage: keep details in the image with salt-pepper noise
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4.3 Image Smoothingg g4.3.2 Smoothing linear filters: out range pixel smoothing
Experimental results
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4.3 Image Smoothingg g4.3.2 Smoothing linear filters: Maximum homogeneity smoothing
Advantage: keep edgesDigital Image Processing
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Advantage: keep edges
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4.3 Image Smoothingg g4.3.3 Order- statistic filters: Max filters
( , ) { ( , )}maxg x y f x i y j= + +formulate( , 0, 1 )maxi j= ±
o u e Advanttage:reem
oves pepperr noise
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4.3 Image Smoothing
( , ) { ( , )}ming x y f x i y j= + +
g g4.3.3 Order- statistic filters: Min filters
formulate( , 0, 1 )
( , ) { ( , )}mini j
g y f y j= ±
o u e Advanttage:reem
oves salt nooise
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4.3 Image Smoothingg g4.3.3 Order- statistic filters: Midpoint filter
1( , ) [ { ( , )} { ( , )}]max ming x y f x i y j f x i y j= + + + + +formulate( , 0, 1 )( , 0, 1 )
( , ) [ { ( , )} { ( , )}]2 max min
i ji jg y f y j f y j
= ±= ±o u e A
dvanttage:reemoves G
ausiaan noise
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e
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4.3 Image Smoothingg g4.3.3 Order- statistic filters: Median filter1-D: replace the signal value by the median value of neighborhood
L=5 X=[4 5 9 5 4]
Adv
[4 4 5 5 9]
vantage
[4 5 5 5 4]
e:keep edges
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4.3 Image Smoothingg g4.3.3 Order- statistic filters: Median filter
Replaces the value of a pixel by the median of the
( , ) { ( , )}g x y f x i y jmedian= + +gray levels in the neighborhood of that pixel
( , 0, 1 )i j= ±
Shapes of 2-D filter
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4.3 Image Smoothingg g4.3.3 Order- statistic filters: Median filter
Experiment resultExperiment result
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4.3 Image Smoothingg g4.3.3 Order- statistic filters: alpha-trimmed mean filter
Delete the lowest and highest gray-level valuesα αDelete the lowest and highest gray level values of a sub-image, averaging the remaining pixels as output
α α
0 1 1NA A A −≤ ≤Let the pixels in a sub-image:
1( )N
g x y Aα−
= ∑Then:
( , )2 i
ig x y A
N αα =
=− ∑
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Idea low-pass filter (ILPF)
formulate ( , ) ( , )* ( , )g x y h x y f x y=
D0: cutoff frequency( , ) ( , ) ( , )G u v H u v F u v=
01 ( , )( )
if D u v DH u v
≤⎧= ⎨where
0
( , )0 ( , )
H u vif D u v D
= ⎨ >⎩
2/122 ])2/()2/[(),( NvMuvuD −+−=
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Idea low-pass filter(ILPF)
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Idea low-pass filter (ILPF)
Properties: total image powerope es: o ge powe
2003 Digital Image Processing Prof.Zhengkai Liu Dr.Rong Zhang
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Idea low-pass filter (ILPF)
Properties: blurring and ringing
h(x,y)
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Idea low-pass filter (ILPF)Experiment result cutoff frequencies set at radii values of 5、30、80Experiment result cutoff frequencies set at radii values of 5、30、80
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Butterworth low-pass filter
12
0
1( , )1 [ ( , ) / ] nH u v
D u v D=
+formulate
h2/122 ])2/()2/[(),( NvMuvuD −+−=
where
])()[()(
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Butterworth low-pass filter
H( ) 0 5 h D( ) DH(u,v)=0.5 when D(u,v)=D0
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Butterworth low-pass filter
Properties: blurring and ringing
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Butterworth low-pass filterExperiment result cutoff frequencies set at radii values of 5、30、80Experiment result cutoff frequencies set at radii values of 5、30、80
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Gaussian low-pass filter
2 20( , ) / 2( ) D u v DH u v e−=
formulate
( , )H u v e=
h2/122 ])2/()2/[(),( NvMuvuD −+−=
where
])()[()(
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Gaussian low-pass filter
Inverse Fourier transform of the Gaussian lowpass filter l i G ialso is Gaussian
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Gaussian low-pass filter
Applications: machine perceptionApplications: machine perception
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4.3 Image Smoothing4 3 4 L filt G i l filt4.3.4 Low-pass filters: Gaussian low-pass filter
Applications: printing and publishing industry
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4.3 Image Smoothing
Experiment result cutoff frequencies set at radii values of 5、30、80
g g4.3.4 Low-pass filters: Gaussian low-pass filterExperiment result cutoff frequencies set at radii values of 5、30、80
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Comparisons (cutoff 5)
Ideal filter Butterworth filter Gaussian filterOriginDigital Image Processing
Prof.Zhengkai Liu Dr.Rong Zhang81
Ideal filter Butterworth filter Gaussian filterOrigin
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Comparison (cutoff 30)
Ideal filter Butterworth filter Gaussian filterOrigin Digital Image Processing
Prof.Zhengkai Liu Dr.Rong Zhang82
g
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4.3 Image Smoothingg g4.3.4 Low-pass filters: Comparison (cutoff 80)
Ideal filter Butterworth filter Gaussian filterOriginDigital Image Processing
Prof.Zhengkai Liu Dr.Rong Zhang83
Origin
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The End
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