No Slide Titlesbisc.sharif.edu/~dip/Files... · Intensity Transformations and Spatial Filtering 1 ... Intensity Transformations and Spatial Filtering 4 •Intensity Transformation

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E. Fatemizadeh, Sharif University of Technology, 2012

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Digital Image Processing

Intensity Transformations and Spatial Filtering

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• Image Enhancement:

– No Explicit definition

• Methods

– Spatial Domain: • Linear

• Nonlinear

– Frequency Domain: • Linear

• Nonlinear

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• Spatial Domain Process

, ,g x y T f x y

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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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• Intensity Transformation Functions:

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• Image Negatives:

1s L r

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• Log Transform:

log 1s c r

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• Power-Law Transform:

s cr

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• Gamma Correction:

11.8 2.5

r

1.8 2.5r

1.8 2.5r

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Original =0.6

=0.4 =0.3

• Gamma Correction

– Too Dark Image

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=5

Original =3

=4

• Gamma Correction

– Too Bright Image

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Original

C. S. THR.

• Contrast Stretching

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• Gray Level Slicing

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• Example Using Fig 3.11 (a) Using Fig 3.11 (b)

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• Bit-Plane Slicing:

– Highlighting effect of a single bit!

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• Example:

– LSB -> MSB (Left to Right and Top to Bottom)

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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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• 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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• Histogram Visual Meaning:

– Dark

– Light

– Low Contrast

– High Contrast

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• Histogram Equalization:

– Continuous Case.

– Seek for a suitable transform (Except for negative):

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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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• Concept Illustration:

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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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• 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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• Numerical Examples (Cont.)

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• Real Experiment:

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• Gray-Level Transfer Function

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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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• Example (Mars image and its histogram):

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• Histogram Matching:

– Washout Gray

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Desired

Transform using

Initial CDF

Transform using

Initial Modified CDF

• Histogram Matching:

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• Local Histogram Enhancement

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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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• 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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• 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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• Graphical Illustration:

Local Mean Local Var E or one

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• Enhanced Image

– Global histogram equalization

– Use of statistical moments

Original Local Histogram Local Statistics

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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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• 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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• 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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• Square averaging Filter:

– Denoising/Smoothing

– Blurring (Equal Value Kernel

1 3

5 9

15 35

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• Blurring Usage:

– Delete unwanted (small) subjects.

Original Smoothed Thresholded

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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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• Example:

Salt-Pepper Noise 3×3 Averaging 3×3 Median

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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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• First Order Derivative:

– Zero in flat region

– Non-zero at start of step/ramp region

– Non-zero along ramp

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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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• Comparison:

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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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• 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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Practically use:

• Laplacian Masks

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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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• Example:

Non-Scaled and scaled Laplacian

Sharpened using 90º and 45º degree isotropic Laplacian

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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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• One Dimensional Illustration

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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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• 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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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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• Example (Sobel Mask):

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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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• Example:

– Original

– Discard Some bits

– Difference

– Histogram Equalized

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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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• 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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• Astronomical Application:

– Repeatable Experiments!

Original Noisy

N=8 N=16

N=64 N=128

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• Difference Histogram: N=8

N=16

N=64

N=128

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• Combination:

Bone Scan Laplacian

Original+Laplacian Soble of Original

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Smoothed Sobel (Orig. + L.)*S.Sobel

Orig.+ (Orig.+L.)*S.Sobel Apply Power-Law

• Combination:

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• Fuzzy Image Enhancement:

– Brief Introduction to Fuzzy set

– Fuzzy Intensity Transformation

– Fuzzy 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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• 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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• Graphical Illustration

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• Some Membership Functions:

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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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• 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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• Fuzzification of antecedents and consequents

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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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• Rule aggregation

R = R1 OR R2 OR … OR Rn

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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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• 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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• 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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• 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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• Results

Original Histogram Equalization Fuzzy Enhancement

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• Analysis:

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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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• Filtering Rule:

IF d2 is zero AND d6 is zero THEN z5 is white




ELSE z5 is black

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• Membership Function

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• Rules Graphical Interpretation

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• Example:

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• 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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No Slide Titlesbisc.sharif.edu/~dip/Files... · Intensity Transformations and Spatial Filtering 1 ... Intensity Transformations and Spatial Filtering 4 •Intensity Transformation