Ch03 Image+enhancement+in+the+spatial+domain
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Digital Image Processing, 2nd ed.
CH 3 Image Enhancement in the Spatial Domain 3.1 Background g ( x, y ) = T [ f ( x, y ) ] f ( x, y ) : the input image g ( x, y ) : the processed image T [ •] : an operator on f , defined over some neighborhood of ( x, y )
1
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Point Processing
1× 1 neighborhood → s = T (r )
2
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.2 Gray Level Transformations
3
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.2.1 Image Negatives
s = L −1− r
4
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.2.2 Log Transformations
s = c log(1 + r )
5
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.2.3 Power-Law Transformation
s = cr
6
γ
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Gamma Correction
7
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Gamma Correction: Example 3.1
8
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Gamma Correction: Example 3.2
9
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.2.4 Piecewise-Linear Transformation Contrast Stretching
10
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Gray-Level Slicing
11
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Bit-Plane Slicing
12
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.3 Histogram Processing What is Histogram?
Histogram h( rk ) = nk
rk : k - th gray level nk : number of pixels in the image having gray level rk for k = 0,1, , L − 1 Normalized Histogram p( rk ) = nk n
n : total number of pixels in the image rk : k - th gray level nk : number of pixels in the image having gray level rk for k = 0,1, , L − 1
13
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.3.1 Histogram Equalization Fundamental Assumptions on the Transformation s = T ( r ), 0 ≤ r ≤ 1 Two assumptions on T ( r ) : ( i ) single - valued, monotonically increasing ( ii ) 0 ≤ T ( r ) ≤ 1, for 0 ≤ r ≤ 1 The inverse transformation : r = T −1 ( s ) , 0 ≤ s ≤ 1
14
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Histogram Equalization The Continuous Case
From an elementary probability theory, dr ps ( s ) = pr ( r ) , ds Consider a transformation function s = T ( r ) = ∫ pr ( w)dw r
0
ds dT ( r ) d r = p (r) ( ) = = p w dw r r dr dr dr ∫0 dr 1 ps ( s ) = pr ( r ) = pr ( r ) = 1, 0 ≤ s ≤ 1 ds pr ( r )
15
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Histogram Equalization The Discrete Case
rk , k = 0,1,..., L − 1 n k m n j sk = T ( rk ) = ∑ pr ( rj ) = ∑ , k = 0,1,..., L − 1 j =0 j =0 n pr ( rk ) =
16
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.3.2 Histogram Matching (Specification) Development of Method
17
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Histogram Matching (Specification) Implementation
18
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Chapter 3
19
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Chapter 3
20
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Chapter 3
21
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.3.3 Local Enhancement
22
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.3.4 Use of Histogram Statistics for Image Enhancement r ∈ [ 0, L − 1]
pr ( r j ) : normalized histogram of the ith value of r → Probability of occurrence of r j
The nth moment of r n
L −1
L −1
µ n ( n ) = ∑ ( ri − m ) p ( ri ) , where m = ∑ ri p( ri ) i =0
i =0
Since µ 0 = 1, and µ1 = 0, L −1
µ 2 = ∑ ( ri − m ) p( ri ) = σ 2 → variance of r 2
i =0
Local mean and variance mS xy =
∑) r p( r ) s ,t
( s ,t ∈S xy
σ 2 S xt =
∑ [r
( s ,t )∈S xy
s ,t
s ,t
]
2
− mS xy p ( rs ,t ) 23
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
E ⋅ f ( x, y ) if mS xy ≤ k0 M 0 and k1 DG ≤ σ S xy ≤ k 2 DG g ( x, y ) = otherwise f ( x, y )
24
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.4 Enhancement Using Arithmetic/Logic Operations
25
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.4.1 Image Subtraction
26
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Image Subtraction: Mask Mode Radiography
27
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.4.2 Image Averaging
g ( x, y ) = f ( x, y ) + η ( x, y ) where η is zero - mean, uncorrelated 1 g ( x, y ) = K
{
}
K
∑ g ( x, y ) i =1
i
E g( x, y ) = f ( x, y ) , and σ 2 g ( x , y ) =
σ g ( x, y )
1 2 σ η ( x, y ) K
1 = ση ( x , y ) K
28
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
29
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.5 Basics of Spatial Filtering
R = w( − 1,−1) f ( x − 1, y − 1) + w( − 1,0) f ( x − 1, y ) + + w( 0,0) f ( x, y ) + + w(1,0) f ( x + 1, y ) + w(1,1) f ( x + 1, y + 1) g ( x, y ) =
a
b
∑ ∑ w( s, t ) f ( x + s, y + t )
s = −1t = − b
mn
30
R = w1 z1 + + wmn z mn = ∑ wi zi
중앙대학교 첨단영상대학원 i =1
Digital Image Processing, 2nd ed.
mn
R = w1 z1 + + wmn z mn = ∑ wi zi i =1
31
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.6 Smoothing Spatial Filters 3.6.1 Smoothing Linear Filters
1 9 R = ∑ zi 9 i =1 a
g ( x, y ) =
b
∑ ∑ w( s, t ) f ( x + s, y + t )
s = − at = − b
a
b
∑ ∑ w( s, t )
s = − at = − b
32
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Averaging Filter: Hubble Image
33
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.6.2 Order Statistics Filters
34
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.7 Sharpening Spatial Filters 3.7.1 Foundation
The 1D, first - order derivative ∂f = f ( x + 1) − f ( x ) ∂x The 1D, second - order derivative ∂2 f = f ( x + 1) − 2 f ( x ) + f ( x − 1) ∂x 2
35
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.7.2 Second Derivatives: Laplacian
The 2D, second - order derivative : Laplacian ∂2 f ∂2 f ∇ f = 2 + 2 ∂x ∂x ∂2 f = f ( x + 1, y ) − 2 f ( x, y ) + f ( x − 1, y ) ∂x 2 ∂2 f = f ( x, y + 1) − 2 f ( x, y ) + f ( x, y − 1) ∂y 2 2
∇ 2 f = [ f ( x + 1, y ) + f ( x − 1, y ) + f ( x, y + 1) + f ( x, y − 1)] − 4 f ( x, y )
36
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Laplacian
f ( x, y ) − ∇ 2 f ( x, y ) if the center is negative g ( x, y ) = 2 f ( x, y ) + ∇ f ( x, y ) if the center is positive
37
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Laplacian Enhancement: Simplification
g ( x, y ) = f ( x, y ) − ∇ 2 f ( x, y ) 5 f ( x, y ) − [ f ( x + 1, y ) + f ( x − 1, y ) + f ( x, y + 1) + f ( x, y − 1)]
38
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Unsharp Masking and High-Boost Filtering
Unsharp masking f s ( x, y ) = f ( x , y ) − f ( x, y ) High - boost filtering f hb ( x, y ) = Af ( x, y ) − f ( x, y )
39
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.7.3 First Derivatives: The Gradient
40
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.8 Combining Spatial Enhancement Methods
41
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Homework #2 • Problems in Chapter 3 – 3.2(a), 3.8, 3.22, 3.27
42
중앙대학교 첨단영상대학원
CH 3 Image Enhancement in the Spatial Domain 3.1 Background g ( x, y ) = T [ f ( x, y ) ] f ( x, y ) : the input image g ( x, y ) : the processed image T [ •] : an operator on f , defined over some neighborhood of ( x, y )
1
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Point Processing
1× 1 neighborhood → s = T (r )
2
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.2 Gray Level Transformations
3
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.2.1 Image Negatives
s = L −1− r
4
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.2.2 Log Transformations
s = c log(1 + r )
5
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.2.3 Power-Law Transformation
s = cr
6
γ
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Gamma Correction
7
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Gamma Correction: Example 3.1
8
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Gamma Correction: Example 3.2
9
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.2.4 Piecewise-Linear Transformation Contrast Stretching
10
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Gray-Level Slicing
11
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Bit-Plane Slicing
12
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.3 Histogram Processing What is Histogram?
Histogram h( rk ) = nk
rk : k - th gray level nk : number of pixels in the image having gray level rk for k = 0,1, , L − 1 Normalized Histogram p( rk ) = nk n
n : total number of pixels in the image rk : k - th gray level nk : number of pixels in the image having gray level rk for k = 0,1, , L − 1
13
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.3.1 Histogram Equalization Fundamental Assumptions on the Transformation s = T ( r ), 0 ≤ r ≤ 1 Two assumptions on T ( r ) : ( i ) single - valued, monotonically increasing ( ii ) 0 ≤ T ( r ) ≤ 1, for 0 ≤ r ≤ 1 The inverse transformation : r = T −1 ( s ) , 0 ≤ s ≤ 1
14
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Histogram Equalization The Continuous Case
From an elementary probability theory, dr ps ( s ) = pr ( r ) , ds Consider a transformation function s = T ( r ) = ∫ pr ( w)dw r
0
ds dT ( r ) d r = p (r) ( ) = = p w dw r r dr dr dr ∫0 dr 1 ps ( s ) = pr ( r ) = pr ( r ) = 1, 0 ≤ s ≤ 1 ds pr ( r )
15
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Histogram Equalization The Discrete Case
rk , k = 0,1,..., L − 1 n k m n j sk = T ( rk ) = ∑ pr ( rj ) = ∑ , k = 0,1,..., L − 1 j =0 j =0 n pr ( rk ) =
16
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.3.2 Histogram Matching (Specification) Development of Method
17
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Histogram Matching (Specification) Implementation
18
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Chapter 3
19
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Chapter 3
20
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Chapter 3
21
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.3.3 Local Enhancement
22
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.3.4 Use of Histogram Statistics for Image Enhancement r ∈ [ 0, L − 1]
pr ( r j ) : normalized histogram of the ith value of r → Probability of occurrence of r j
The nth moment of r n
L −1
L −1
µ n ( n ) = ∑ ( ri − m ) p ( ri ) , where m = ∑ ri p( ri ) i =0
i =0
Since µ 0 = 1, and µ1 = 0, L −1
µ 2 = ∑ ( ri − m ) p( ri ) = σ 2 → variance of r 2
i =0
Local mean and variance mS xy =
∑) r p( r ) s ,t
( s ,t ∈S xy
σ 2 S xt =
∑ [r
( s ,t )∈S xy
s ,t
s ,t
]
2
− mS xy p ( rs ,t ) 23
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
E ⋅ f ( x, y ) if mS xy ≤ k0 M 0 and k1 DG ≤ σ S xy ≤ k 2 DG g ( x, y ) = otherwise f ( x, y )
24
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.4 Enhancement Using Arithmetic/Logic Operations
25
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.4.1 Image Subtraction
26
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Image Subtraction: Mask Mode Radiography
27
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.4.2 Image Averaging
g ( x, y ) = f ( x, y ) + η ( x, y ) where η is zero - mean, uncorrelated 1 g ( x, y ) = K
{
}
K
∑ g ( x, y ) i =1
i
E g( x, y ) = f ( x, y ) , and σ 2 g ( x , y ) =
σ g ( x, y )
1 2 σ η ( x, y ) K
1 = ση ( x , y ) K
28
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
29
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.5 Basics of Spatial Filtering
R = w( − 1,−1) f ( x − 1, y − 1) + w( − 1,0) f ( x − 1, y ) + + w( 0,0) f ( x, y ) + + w(1,0) f ( x + 1, y ) + w(1,1) f ( x + 1, y + 1) g ( x, y ) =
a
b
∑ ∑ w( s, t ) f ( x + s, y + t )
s = −1t = − b
mn
30
R = w1 z1 + + wmn z mn = ∑ wi zi
중앙대학교 첨단영상대학원 i =1
Digital Image Processing, 2nd ed.
mn
R = w1 z1 + + wmn z mn = ∑ wi zi i =1
31
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.6 Smoothing Spatial Filters 3.6.1 Smoothing Linear Filters
1 9 R = ∑ zi 9 i =1 a
g ( x, y ) =
b
∑ ∑ w( s, t ) f ( x + s, y + t )
s = − at = − b
a
b
∑ ∑ w( s, t )
s = − at = − b
32
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Averaging Filter: Hubble Image
33
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.6.2 Order Statistics Filters
34
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.7 Sharpening Spatial Filters 3.7.1 Foundation
The 1D, first - order derivative ∂f = f ( x + 1) − f ( x ) ∂x The 1D, second - order derivative ∂2 f = f ( x + 1) − 2 f ( x ) + f ( x − 1) ∂x 2
35
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.7.2 Second Derivatives: Laplacian
The 2D, second - order derivative : Laplacian ∂2 f ∂2 f ∇ f = 2 + 2 ∂x ∂x ∂2 f = f ( x + 1, y ) − 2 f ( x, y ) + f ( x − 1, y ) ∂x 2 ∂2 f = f ( x, y + 1) − 2 f ( x, y ) + f ( x, y − 1) ∂y 2 2
∇ 2 f = [ f ( x + 1, y ) + f ( x − 1, y ) + f ( x, y + 1) + f ( x, y − 1)] − 4 f ( x, y )
36
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Laplacian
f ( x, y ) − ∇ 2 f ( x, y ) if the center is negative g ( x, y ) = 2 f ( x, y ) + ∇ f ( x, y ) if the center is positive
37
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Laplacian Enhancement: Simplification
g ( x, y ) = f ( x, y ) − ∇ 2 f ( x, y ) 5 f ( x, y ) − [ f ( x + 1, y ) + f ( x − 1, y ) + f ( x, y + 1) + f ( x, y − 1)]
38
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Unsharp Masking and High-Boost Filtering
Unsharp masking f s ( x, y ) = f ( x , y ) − f ( x, y ) High - boost filtering f hb ( x, y ) = Af ( x, y ) − f ( x, y )
39
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.7.3 First Derivatives: The Gradient
40
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
3.8 Combining Spatial Enhancement Methods
41
중앙대학교 첨단영상대학원
Digital Image Processing, 2nd ed.
Homework #2 • Problems in Chapter 3 – 3.2(a), 3.8, 3.22, 3.27
42
중앙대학교 첨단영상대학원
