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E. Fatemizadeh, Sharif University of Technology, 2012
1
Digital Image Processing
Intensity Transformations and Spatial Filtering
1
• Image Enhancement:
– No Explicit definition
• Methods
– Spatial Domain: • Linear
• Nonlinear
– Frequency Domain: • Linear
• Nonlinear
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Spatial Domain Process
, ,g x y T f x y
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• For 1×1 neighborhood: – Contrast Enhancement/Stretching/Point Process
• For w×w neighborhood: – Filtering/Mask/Kernel/Window/Template Processing
s T r
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Intensity Transformation Functions:
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Image Negatives:
1s L r
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Log Transform:
log 1s c r
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Power-Law Transform:
s cr
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Gamma Correction:
11.8 2.5
r
1.8 2.5r
1.8 2.5r
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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Original =0.6
=0.4 =0.3
• Gamma Correction
– Too Dark Image
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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=5
Original =3
=4
• Gamma Correction
– Too Bright Image
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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Original
C. S. THR.
• Contrast Stretching
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Gray Level Slicing
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Example Using Fig 3.11 (a) Using Fig 3.11 (b)
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Bit-Plane Slicing:
– Highlighting effect of a single bit!
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Example:
– LSB -> MSB (Left to Right and Top to Bottom)
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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Bits (7,8) Bits (6,7,8) Bits (5,6,7,8)
• Image Reconstruction from bit-planes
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Histogram Processing:
– Enhancement based on statistical Properties:
• Local
• Global
– Histogram Definition:
, 0, 1 , 0,
1
k k k k
kk k
h r n r L n M N
np r n
n M N
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Histogram Visual Meaning:
– Dark
– Light
– Low Contrast
– High Contrast
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Histogram Equalization:
– Continuous Case.
– Seek for a suitable transform (Except for negative):
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Effect of Point Process on Histogram:
• Special Case (CDF):
1 S r
s T r drP s P r
r T s ds
0
Uniform Distribution on [0, -1]
1 1
1 1, 0 1
1
r
r r
s r r
r
dT rdss T r L p w dw L p r
dr dr
drp s p r p r s L
ds p r
L
L
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Concept Illustration:
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Discrete Case
• Perfect equalization is NOT possible
0 0
1
min2
min
, 0,1,2, , 1
11 , 0,1, , 1
ˆ 0.5
ˆ 1 0.51
k kr k
k k
k k r j k
j J
k k k
k kk
k
n np r k L
MN n
LS T r L p r n k L
MN
S S round S
S SS L
L S
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Numerical Example:
0
7k
k r j
j
S p r
𝑆𝑘 𝑆 𝑘
1.33 1
3.08 3
4.55 5
5.67 6
6.23 6
6.56 7
6.86 7
7.00 7
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Numerical Examples (Cont.)
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Real Experiment:
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Gray-Level Transfer Function
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Histogram Matching and Modification:
– Goal: Specify the shape of the histogram:
– Example: Pages: 133-136
?
0 1 1
0
1
1
r z
r
r
z
z
p r p z
s T r L p w dw
z G T r G s
G z L p t dt
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Example (Mars image and its histogram):
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Histogram Matching:
– Washout Gray
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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Desired
Transform using
Initial CDF
Transform using
Initial Modified CDF
• Histogram Matching:
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Local Histogram Enhancement
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Histogram Statistics For Image Enhancement:
– Use of Global Statistical Measures
– Gross adjustments in overall intensity (m) and contrast (µ2)
1
0 1 1
1
0 1 1
1,
1,
L M Nnn
n i i
i x y
L M N
i i
i x y
r r m p r f x y mMN
m r p r f x yMN
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Histogram Statistics For Image Enhancement:
– Local mean and local variance:
– Local information intensity and contrast (edges)
1
0 ,
1 2 22
0 ,
1, ,
1, , , ,
: Neighborhood centered on ,
xy xy
xy
xy xy xy xy
xy
L
S i S i
i s t Sxy
L
S i S S i S
i s t Sxy
xy
m x y r p r f s tS
x y r m x y p r f s t m x yS
S x y
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• A simple enhancement algorithm for SEM image:
0 1 2
0 1 2
. , , ,,
, .
4.0, 0.4, 0.02, 0.4
S G G S GE f x y m x y k m and k x y kg x y
f x y OW
E k k k
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Graphical Illustration:
Local Mean Local Var E or one
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Enhanced Image
– Global histogram equalization
– Use of statistical moments
Original Local Histogram Local Statistics
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Fundamentals of Spatial Filtering:
, , ,a b
s a t b
g x y w s t f x s y t
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Spatial Correlation () and Convolution ()
• Reflection/Rotation in convolution:
, , , ,
, , , ,
a b
s a t b
a b
s a t b
w x y f x y w s t f x s y t
w x y f x y w s t f x s y t
4 6 6 45 5
3 7
7 3
2 8
2
1 9
98 1
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Smoothing Spatial Filters:
– Linear Filters (averaging, lowpass)
– General Formulation
, ,
,
,
a b
s a t b
a b
s a t b
w s t f x s y t
g x y
w s t
2 2
22,
x y
G x y e
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Square averaging Filter:
– Denoising/Smoothing
– Blurring (Equal Value Kernel
1 3
5 9
15 35
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Blurring Usage:
– Delete unwanted (small) subjects.
Original Smoothed Thresholded
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Order Statistics Filters:
– Impulsive noise: • Mono Level : Salt, Pepper noises
• Bi Level: Salt-Pepper noises
– Filter • Median
• Max
• Min
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Example:
Salt-Pepper Noise 3×3 Averaging 3×3 Median
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Sharpening Spatial Filter
– Highlights Intensity Transitions
– First and Second order Derivatives
1, ,
, 1,
0.5 1, 1,
f x y f x yf
f x y f x yx
f x y f x y
2
21, 2 , 1,
ff x y f x y f x y
x
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• First Order Derivative:
– Zero in flat region
– Non-zero at start of step/ramp region
– Non-zero along ramp
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Second Order Derivative:
– Zero in flat region
– Non-zero at start/end of step/ramp region
– Zero along ramp
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Comparison:
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• 1st and 2nd Order Derivative Comparison:
– First Derivative: • Thicker Edge;
• Strong Response for step changes;
– Second Derivative: • Strong response for fine details and isolated points;
• Double response at step changes.
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Laplacian as an isotropic Enhancer:
• Discrete Implementation:
2 22
2 2
f ff
x y
2 1, , 1 1, , 1 4 ,f f x y f x y f x y f x y f x y
0 1 0
1 4 1 90
0 1 0
isotropic
1 1 1
1 8 1 45
1 1 1
isotropic
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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Practically use:
• Laplacian Masks
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Intensity Transformations and Spatial Filtering
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• Background Recovering:
2
2
, ,,
, ,
f x y f x y signg x y
f x y f x y sign
0 1 0
1 1 90
0 0
5
1
isotropic
1 1 1
1 1 45
1 1
9
1
isotropic
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Example:
Non-Scaled and scaled Laplacian
Sharpened using 90º and 45º degree isotropic Laplacian
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Unsharp Masking and High-Boost Filtering:
• k≥0
– k=1: Unsharp Masking
– k>1: High Boost
• Another mask:
– Laplacian and any highpass filter
, , , , , : Blurred image
g , , ,
mask
mask
g x y f x y f x y f x y
x y f x y k g x y
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• One Dimensional Illustration
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Two Dimensional Example
Blurred, 5×5, σ=3
Unsharp mask
Unsharp masking , k=1
Highboost, k=4.5
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Intensity Transformations and Spatial Filtering
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• First Derivative - Gradient:
1 222
TT
x y
f ff G G
x y
f f f ff
x y x y
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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Roberts Cross Gradient
9 5 8 6x yG z z G z z
7 8 9 1 2 3
3 6 9 1 4 7
2 2
2 2
x
y
G z z z z z z
G z z z z z z
Sobel Gradient
• Discrete Implementation
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Example (Sobel Mask):
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Image Subtraction:
– Physically static object and background
– Intensity changes in object but not in background
, , ,g x y f x y f x y
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Intensity Transformations and Spatial Filtering
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• Example:
– Original
– Discard Some bits
– Difference
– Histogram Equalized
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Image Averaging: – Consider an additive noise condition:
g(x,y)=f(x,y)+η(x,y)
– Conditions: • Noise, η(x,y):
– Uncorrelated
– i.i.d
– Zero Mean
• Subject, f(x,y): – Physical Stationary
– Repeatable Experiments
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Intensity Transformations and Spatial Filtering
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• Image Averaging:
– For i.i.d RV’s:
2 2
, ,1
, , , , ,
1 1, ,
i i
N
i g x y x yi
g x y f x y x y E g x y f x y
g x y g x yN N
1
Var1, Var
N
i
i
E EN N
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Intensity Transformations and Spatial Filtering
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• Astronomical Application:
– Repeatable Experiments!
Original Noisy
N=8 N=16
N=64 N=128
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Intensity Transformations and Spatial Filtering
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• Difference Histogram: N=8
N=16
N=64
N=128
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Combination:
Bone Scan Laplacian
Original+Laplacian Soble of Original
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Intensity Transformations and Spatial Filtering
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Smoothed Sobel (Orig. + L.)*S.Sobel
Orig.+ (Orig.+L.)*S.Sobel Apply Power-Law
• Combination:
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Fuzzy Image Enhancement:
– Brief Introduction to Fuzzy set
– Fuzzy Intensity Transformation
– Fuzzy Spatial Filtering
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Intensity Transformations and Spatial Filtering
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• Introduction
– Logic of real world
– “He is young man” is true of false?
– Fuzzy and Crisp sets:
• Membership function:
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Intensity Transformations and Spatial Filtering
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• Fuzzy Set:
– Definition:
, , :Set of Elements
Empty Set: 0
Equality:
Subset:
Complement: 1
Union (A B, A B): max ,
Intersection (A B, A B): min ,
A
A
A B
A B
AA
C A B
C A B
A z z z Z Z
z
z z
z z
z z
OR z z z
AND z z z
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Intensity Transformations and Spatial Filtering
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• Graphical Illustration
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Intensity Transformations and Spatial Filtering
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• Some Membership Functions:
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Intensity Transformations and Spatial Filtering
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• Using Fuzzy Sets in inferring:
– Example: Using color to categorize fruits
R1: If color is green then the fruit is verdant
OR
R2: If color is yellow then the fruit is half-mature
OR
R3: If color is red then the fruit is mature
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Intensity Transformations and Spatial Filtering
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• We should to do:
– Fuzzify antecedent and consequent of each rule
– Evaluate membership function of each rules (R1, R2, R3),
– Rules aggregation,
– Final crisp value (Defuzzification).
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Intensity Transformations and Spatial Filtering
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• Fuzzification of antecedents and consequents
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Intensity Transformations and Spatial Filtering
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• if-then rule fuzzification:
– Possibility of consequents is less than antecedents • Example: “if x is A then y is D”
– Our Example:
0 01
2 0 0
0 03
, min ,
, min ,
, min ,
green verd
yellow half
red mat
z v z v
z v z v
z v z v
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Intensity Transformations and Spatial Filtering
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• Rule aggregation
R = R1 OR R2 OR … OR Rn
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Intensity Transformations and Spatial Filtering
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• Rule aggregation in our Example
– Q = Q1 OR Q2 OR Q3
1 2 030 0 0, max , , , , ,z v z v z v z v
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Intensity Transformations and Spatial Filtering
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• Defuzzification:
– Max Membership
– Center of Gravity
– ….
1
1
,
N
i i
i
N
i
i
x xx x dx
x xx dx
x
: ,x x x x
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Complex antecedent:
1 2
1 2
1 2 n
1 2 n
R: IF is AND is AND is THEN is B
: IF is THEN is
min , ,
R: IF is OR is OR is THEN is B
: IF is THEN is
, ,
n
n
A A A A
A A A A
x A x A x A y
R x A y B
x x x x
x A x A x A y
R x A y B
x Max x x x
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Fuzzy Intensity Transformation:
R1: If a pixel is Dark then make it Darker
R2: If a pixel is Gray then make it Gray
R3: If a pixel is Bright then make it Brighter
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Results
Original Histogram Equalization Fuzzy Enhancement
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Analysis:
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Digital Image Processing
Intensity Transformations and Spatial Filtering
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• Spatial Filtering Using Fuzzy Sets
– Goal: Boundary Extraction
– General Rule:
– Uniformity: Intensity difference central pixel and its neighbors
– For a 3×3 maks: di = zi - z5
if “a pixel belongs to a uniform region”, then ”make it
white, else make it black”
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Digital Image Processing
Intensity Transformations and Spatial Filtering
84
• Filtering Rule:
IF d2 is zero AND d6 is zero THEN z5 is white
IF d6 is zero AND d8 is zero THEN z5 is white
IF d8 is zero AND d4 is zero THEN z5 is white
IF d4 is zero AND d2 is zero THEN z5 is white
ELSE z5 is black
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
85
Digital Image Processing
Intensity Transformations and Spatial Filtering
85
• Membership Function
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
86
Digital Image Processing
Intensity Transformations and Spatial Filtering
86
• Rules Graphical Interpretation
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
87
Digital Image Processing
Intensity Transformations and Spatial Filtering
87
• Example:
ee.sharif.edu/~dip
E. Fatemizadeh, Sharif University of Technology, 2012
88
Digital Image Processing
Intensity Transformations and Spatial Filtering
88
• MATLAB Command:
– Image Statistics:
• means2, std2, corr2, imhist, regionprops
– Image Intensity Adjustment:
• imadjust, histeq, adapthisteq, imnoise
– Linear Filter:
• imfilter, fspecial, conv2, corr2,
– Nonlinear filter:
• medfilt2, ordfilt2,
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