Image Enhancement in Spatial Domain
Histogram
The histogram of a digital image with gray levels in
the range [0,L-1] is a discrete function
h ( rk ) = n k
where rk is the kth gray level and nk is the number of pixels
in the image having gray levels rk.
Image Enhancement in Spatial Domain
Histogram
Image Enhancement in Spatial Domain
Histogram
Image Enhancement in Spatial Domain
Histogram
It is common practice to normalise a histogram by
Dividing each of its values by the total number of
Pixels in the image, denoted by n. Thus, a normalised
histogram is given by
nk
p(rk ) =
n
for k = 0,1,…, L-1.
p(rk) gives an estimate of the probability of occurrence
of gray level rk.
Image Enhancement in Spatial Domain
Histogram Processing
to extract and enhance the image intensity histogram
Image Enhancement in Spatial Domain
Histogram Equalisation
s = T (r )
Image Enhancement in Spatial Domain
Histogram Equalisation
s = T (r )
dr
p s ( s ) = pr ( r )
ds
r
s = T ( r ) = ∫ pr ( w)dw
0
Image Enhancement in Spatial Domain
Histogram Equalisation
1
dr
ps ( s ) = pr ( r )
= pr ( r ) ⋅
ds
ds
dr
1
1
= pr ( r ) ⋅
= pr ( r ) ⋅
=1
r
pr ( r )


d  ∫ pr ( w)dw 
0

dr
Image Enhancement in Spatial Domain
Histogram Equalisation
k
sk = T (rk ) = ∑ pr (rj )
j =0
k
nj
j =0
n
=∑
nj = the number of pixels with intensity rj
n = the number of total pixels
Image Enhancement in Spatial Domain
Histogram Equalisation
Intensity
# pixels
Accumulative Sum of Pr
0
20
20/100 = 0.2
1
5
(20+5)/100 = 0.25
2
25
(20+5+25)/100 = 0.5
3
10
(20+5+25+10)/100 = 0.6
4
15
(20+5+25+10+15)/100 = 0.75
5
5
(20+5+25+10+15+5)/100 = 0.8
6
10
(20+5+25+10+15+5+10)/100 = 0.9
7
10
(20+5+25+10+15+5+10+10)/100 = 1.0
Total
100
1.0
Image Enhancement in Spatial Domain
Histogram Equalisation
Intensity
(r)
No. of Pixels
(nj)
Acc Sum
of Pr
Output value
Quantized
Output (s)
0
20
0.2
0.2x7 = 1.4
1
1
5
0.25
0.25*7 = 1.75
2
2
25
0.5
0.5*7 = 3.5
3
3
10
0.6
0.6*7 = 4.2
4
4
15
0.75
0.75*7 = 5.25
5
5
5
0.8
0.8*7 = 5.6
6
6
10
0.9
0.9*7 = 6.3
6
7
10
1.0
1.0x7 = 7
7
Total
100
Image Enhancement in Spatial Domain
Histogram Equalisation
Image Enhancement in Spatial Domain
Histogram Equalisation
Image Enhancement in Spatial Domain
Histogram Equalisation
Image Enhancement in Spatial Domain
Histogram Equalisation
Image Enhancement in Spatial Domain
Histogram Matching (Specification)
original image
after histogram equalisation
Image Enhancement in Spatial Domain
Histogram Matching (Specification)
v = G(z)
s = T(r)
2
1
3
z = G-1(v)
4
Image Enhancement in Spatial Domain
Histogram Matching (Specification)
r
s = T ( r ) = ∫ pr ( w)dw
0
z
v = G ( z ) = ∫ pz (u )du
0
−1
G
T
r → s 
→z
Image Enhancement in Spatial Domain
Histogram Matching (Specification)
We assign
Intensity
(s)
# pixels
Intensity
(z)
# pixels
0
20
0
5
1
5
1
10
2
25
2
15
3
10
3
20
4
15
4
20
5
5
5
15
6
10
6
10
7
10
7
5
Total
100
Total
100
Image Enhancement in Spatial Domain
Histogram Matching (Specification)
r
(nj)
ΣPr
s
z
(nj)
ΣPz
v
0
20
0.2
1
0
5
0.05
0
1
5
0.25
2
1
10
0.15
1
2
25
0.5
3
2
15
0.3
2
3
10
0.6
4
3
20
0.5
4
4
15
0.75
5
4
20
0.7
5
5
5
0.8
6
5
15
0.85
6
6
10
0.9
6
6
10
0.95
7
7
10
1.0
7
7
5
1.0
7
sk = T(rk)
vk = G(zk)
Image Enhancement in Spatial Domain
Histogram Matching (Specification)
r à s
v à z
s à v
v
z
r
z
z
# Pixels
1
0
0
0
1
0
0
1
2
1
1
1
2
1
20
2
3
2
2
2
2
2
30
3
4
4
3
3
3
3
10
4
5
5
4
4
4
4
15
5
6
6
5
5
5
5
15
6
6
7
6
6
5
6
10
7
7
7
7
7
6
7
0
r
s
0
zk = G-1(vk)
Image Enhancement in Spatial Domain
Histogram Matching (Specification)
Image Enhancement in Spatial Domain
Histogram Matching (Specification)
Original
image
After
histogram
equalization
After
histogram
matching
Image Enhancement in Spatial Domain
Local Enhancement: Local Histogram
Equalisation
Original image
After histogram
equalisation
After local
histogram
equalisation
in 7x7
neighbourhood
Image Enhancement in Spatial Domain
Basic Statistics
Let p(ri) denote the normalised histogram component
corresponding to the ith value of r.
The mean value (average gray level) of r is
L −1
m = ∑ ri p (ri )
i =0
Image Enhancement in Spatial Domain
Basic Statistics
The nth moment of r about m is defined as
L−1
µn (r) = ∑(ri − m)n p(r i )
i =0
µ 0 = 1, µ 1 = 0
The second moment is
L−1
µ2 (r) = ∑(ri − m)2 p(r i ) = σ 2 (r)
i =0
σ 2 (r) is the variance of r.
σ (r) is the standard deviation of r.
Image Enhancement in Spatial Domain
Histogram Statistics for Image Enhancement
Image Enhancement in Spatial Domain
Histogram Statistics for Image Enhancement
Let (x,y) be the coordinates of a pixel in an image,
let Sxy denote a neighbourhood (subimage) of
specified size, centred at (x,y).
The mean value mS xy of the pixel in S xy is computed using
mS xy =
∑
( s ,t )∈S xy
rs ,t p (rs ,t )
where rs,t is the gray level at coordinates (s,t) in the
neighbourhood, and p(rs,t) is the neighbourhood
normalised histogram component corr. to that value
of gray level.
Image Enhancement in Spatial Domain
Histogram Statistics for Image Enhancement
The gray-level variance of the pixel in S xy is computed
using
σ
2
S xy
=
∑
( s ,t )∈S xy
( rs ,t − mS xy ) p (rs ,t )
2
Image Enhancement in Spatial Domain
Histogram Statistics for Image Enhancement
 E ⋅ f ( x, y ) when msxy ≤ k0 M G and k1DG ≤ σ sxy ≤ k2 DG
g ( x, y ) = 
 f ( x, y ) otherwise
where MG is the global mean,
and DG is the global standard deviation
Image Enhancement in Spatial Domain
Histogram Statistics for Image Enhancement
Image Enhancement in Spatial Domain
Histogram Statistics for Image Enhancement
Image Enhancement in Spatial Domain
Logic Operations
AND
OR
Original
image
Image mask
Region of Interest
(ROI)
Image Enhancement in Spatial Domain
Arithmetic Operation: Subtraction
Image Enhancement in Spatial Domain
Arithmetic Operation: Subtraction
Application: Mask mode radiography
Image Enhancement in Spatial Domain
Arithmetic Operation: Subtraction
g ( x, y ) = f ( x, y ) − h ( x , y )
The difference image g(x,y) may be negative. There are
two ways to solve this:
1. Add 255 to every pixel and then divide by 2.
2. Find the value of the minimum difference and add
its negative to every pixel. Then find the maximum
pixel value (Max) in the modified difference image and
multiply each pixel by 255/Max.
Image Enhancement in Spatial Domain
Arithmetic Operation: Averaging
g ( x, y ) = f ( x, y ) + η ( x , y )
(noise)
}
Image averaging
1
g ( x, y ) =
K
K
∑ g ( x, y )
i =1
i
Image Enhancement in Spatial Domain
Arithmetic Operation: Averaging
Image Enhancement in Spatial Domain
Spatial Filtering
Image Enhancement in Spatial Domain
Spatial Filtering
filter = mask = kernel = template = window
The values in a filter subimage are called coefficients.
Image Enhancement in Spatial Domain
Spatial Filtering
9
R = ∑ wi zi
i =1
Image Enhancement in Spatial Domain
Spatial Filtering
Pixels at boundary
• accept a smaller filtered image
• padding by replicating rows or columns
• padding by adding rows and columns of 0’s
(or other constant gray level)
Image Enhancement in Spatial Domain
Smoothing Spatial Filters
1. Blurring
• remove small details in order to extract large objects
in an image.
• bridge small gaps in lines or curves.
2. Noise reduction
Image Enhancement in Spatial Domain
Smoothing Linear Filters
sometimes called averaging filters or lowpass filters
Examples
average or
box filter
weighted
average
Image Enhancement in Spatial Domain
Smoothing Linear Filters
Image Enhancement in Spatial Domain
Smoothing Linear Filters
Image Enhancement in Spatial Domain
Order-Statistics Filters
subimage
Statistic parameters:
Mean, Median, Mode,
Min, Max, etc.
Moving
window
Output image
Image Enhancement in Spatial Domain
Order-Statistics Filters: median filter
Image Enhancement in Spatial Domain
Sharpening Spatial Filters
to highlight or enhance fine detail in an image
Applications
• Medical imaging
• Printing
• Industrial inspection
• Autonomous guidance in military systems
Image Enhancement in Spatial Domain
Sharpening Spatial Filters
Image Enhancement in Spatial Domain
Second Derivative
Intensity profile
p(x)
1
Edge
0.5
0
20
40
60
80
100
120
140
160
180
200
0.2
1st derivative
dp
dx
2nd derivative
d2p
dx 2
0.1
0
0
50
100
150
200
0
50
100
150
200
0.05
0
-0.05
Image Enhancement in Spatial Domain
Second Derivative
1.5
1
0.5
p(x)
0
-0.5
0
50
100
150
200
0
50
100
150
200
1.5
1
d2p
p( x ) − 10 2
dx
0.5
0
-0.5
Image Enhancement in Spatial Domain
Second Derivative
Before sharpening
p(x)
After sharpening
d2p
p ( x ) − 10 2
dx
Image Enhancement in Spatial Domain
Second Derivative – the Laplacian
The Laplacian of f(x,y) is defined as
∂ f ∂ f
∇ f = 2 + 2
∂x
∂y
2
2
2
Image Enhancement in Spatial Domain
Second Derivative – the Laplacian
∂ f
= f (x +1, y) + f (x −1, y) − 2 f (x, y)
2
∂x
2
∂ f
= f (x, y +1) + f (x, y −1) − 2 f (x, y)
2
∂y
2
∇2 f = f (x +1, y) + f (x −1, y) + f (x, y +1) + f (x, y −1)
− 4 f (x, y)
Isotropic filters are rotational invariant.
Image Enhancement in Spatial Domain
Second Derivative – the Laplacian
Image Enhancement in Spatial Domain
Second Derivative – the Laplacian
f
∇2 f
scaled
∇ f
2
f − ∇2 f
Image Enhancement in Spatial Domain
Second Derivative – the Laplacian
 f ( x, y ) − ∇ 2 f ( x, y )

g ( x, y ) = 

2
f
(
x
,
y
)
+
∇
f ( x, y )

if the center coefficient of the
Laplacian mask is negative
if the center coefficient of the
Laplacian mask is positive
Image Enhancement in Spatial Domain
Second Derivative – the Laplacian
Image Enhancement in Spatial Domain
Unsharp Masking
fs (x, y) = f (x, y) − fb (x, y)
fs(x,y) is the sharped image.
fb(x,y) is a blurred version of f(x,y).
Image Enhancement in Spatial Domain
High-boost Filtering
fhb (x, y) = Af (x, y) − fb (x, y)
A≥1
fs(x,y) is the high-boost image.
fb(x,y) is a blurred version of f(x,y).
Image Enhancement in Spatial Domain
High-boost Filtering
f hb ( x, y ) = Af ( x, y ) − f b ( x, y )
= ( A − 1) f ( x, y ) + f ( x, y ) − fb ( x, y )
= ( A − 1) f ( x, y ) + f s ( x, y )
 Af ( x, y ) − ∇ 2 f ( x, y )

f hb ( x, y ) = 

2
Af
(
x
,
y
)
+
∇
f ( x, y )

if the center coefficient of the
Laplacian mask is negative
if the center coefficient of the
Laplacian mask is positive
Image Enhancement in Spatial Domain
High-boost Filtering
Image Enhancement in Spatial Domain
High-boost Filtering
Image Enhancement in Spatial Domain
First Derivative – The Gradient
Gradient
 ∂f 
 ∂x 
∇f =  
 ∂f 
 ∂y 
 ∂f  2  ∂f  2 
∇f =    +   
 ∂x   ∂y  
∂f
∂f
∇f ≈
+
∂x ∂y
1
2
Image Enhancement in Spatial Domain
First Derivative – The Gradient
Robert crossgradient operators
Sobel
operators
Image Enhancement in Spatial Domain
Sobel Operators
-1
0
1
-2
0
2
-1
0
1
f
-1 -2 -1
∂f
to compute
∂x
∂f
∂x
0
0
0
1
2
1
∂f
to compute
∂y
∂f
∂y
Image Enhancement in Spatial Domain
First Derivative – The Gradient
f
∂f
∂y
∂f
∂x
∇f
Image Enhancement in Spatial Domain
First Derivative – The Gradient
 ∂f   ∂f 
∇f =   +  
 ∂x   ∂y 
2
2
Image Enhancement in Spatial Domain
Combining Spatial Enhancement Methods
Image Enhancement in Spatial Domain
∇ 2 f Methods
Combining Spatial Enhancement
∇f
B
A
D
smooth
-
+
Sharpening C
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2 nd Edition.
E
Image Enhancement in Spatial Domain
Combining Spatial Enhancement Methods
C
G
E
Power
Law Tr.
Σ
F
Multiplication
A
(Images from Rafael C. Gonzalez and Richard E.
Wood, Digital Image Processing, 2 nd Edition.
H
Image Enhancement in Spatial Domain
Combining Spatial Enhancement Methods
Original image
Result
Image Enhancement in Spatial Domain