Intensity Transformations, Spatial Fil i Hi P iFiltering, Histogram Processing

)],([),( yxfTyxg =

Input imageOutput image

Operator

Spatial filtering: Operator isapplied on the neighbors of alocation (origin); then the origin islocation (origin); then the origin ismoved to a new location and theoperation is repeated.

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Intensity Transfer Functions y

Contrast stretching function. Thresholding function.

r: intensity variable of input image. 

2

s: intensity variable of output image. 

Basic Intensity Transfer Functions y

• Linear• Logarithmicoga c• Power‐law

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Negative Transformation gIntensity level: [0, L - 1]

rLs −−= 1

4Original Image Negative Image

Log Transformation g)1log( rcs +=

Compresses the dynamic range of images with large variations in pixel values.

Loss of details in low pixel valuesLoss of details in low pixel values

5Original Image (Fourier Spectrum) Log Transformed Image

Power-Law (Gamma) Transformation ( )γcrs =

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Gamma Correction Process used to correct power-law response phenomena.

Example: CRT monitor.

Tends to produce images darker.

Decrease value of γ

Solution

Decrease value of γ

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Example: storing images in web sites.

Contrast manipulation: Power-Lawp

MRI of a fracturedspine. 6.0=λ

30λ40λ 3.0=λ4.0=λ

Best contrast Washed out

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Best contrast Washed out

Piecewise-Linear Transform FunctionsContrast Stretching

Expands range of intensity level of an image to full intensity range of a device.Expands range of intensity level of an image to full intensity range of a device.

Low-contrast IImage

)0,(),( 11 = msr)1()(

)0,(),( min11 =L

rsr

ThresholdingContrast stretched image

)1,(),( 22 −= Lmsr)1,(),( max22 −= Lrsr

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g

Piecewise-Linear Transform FunctionsIntensity-Level Slicing

Highlighting a specific range of intensities.Highlighting a specific range of intensities.

Binary Image

Brightens (darkens) desired rangedesired range

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Piecewise-Linear Transform FunctionsBit-Plane Slicing

Gray (194) border:1 1 0 0 0 0 1 0 LSB1 1 0 0 0 0 1 0 LSB

MSB

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Bit-Plane Slicing: Image Compressiong g p

The four highest-order bits are sufficient to reconstruct the original image in acceptable detail. 50% less storage.

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Histogram Processingg gHistogram, kk nrh =)(

Normalized Histogram,MNnrp k

k =)(

Usefulness:Usefulness:

• Image EnhancementI C i• Image Compression

• Image Segmentation, etc.

Hi t f hi h t t i

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Histogram of a high-contrast image has a high dynamic range.

Histogram Equalization - Ig q

10)( −≤≤= LrrTs 10 )( ≤≤= LrrTs

(a) T(r) is a monotonically increasing function in

10 −≤≤ Lr

and

(b) 10f1)(0 ≤≤≤≤ LLT(b) 10for 1)(0 −≤≤−≤≤ LrLrT

For inverse:For inverse:

(a) Is changed to

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(a’) T(r) is strictly monotonically increasing function in 10 −≤≤ Lr

Histogram Equalization - IIg qIf pr (r) and T(r) are known, and T(r) is continuous and differentiable, then

dsdrrpsp rs )()( =

A common transformation function in image processing: ∫−==r

r dwwpLrTs0

)()1()(

)()1()()1()(

0

rpLwpdrdL

drrdT

drds

r

r

r −=⎥⎦

⎤⎢⎣

⎡−== ∫

Uniform probability density function

11

)()1(1)()()(

−=

−==

LrpLrp

dsdrrpsp rrs 10 −≤≤ Ls

Uniform probability density function

1)()1( −− LrpLds r

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Histogram Equalization - IIIg qLet,

⎪

⎪⎨⎧ −≤≤

−=10for

)1(2

)( 2 LrL

rrpr

⎪⎩ otherwise 0)(

rr 2 Di t f∫∫ −

=−

=−==rr

r Lrwdw

LdwwpLrTs

0

2

0 112)()1()(

Discrete form:

∑−==k

jjrkk rpLrTs

0)()1()(

1

2)1(2)()(

−

⎥⎦⎤

⎢⎣⎡

−==

drds

Lr

dsdrrpsp rs

=j 0

∑ −=−

=k

jj Lkn

MNL

01,...1,0 )1(

)(

11

2)1(

)1(2

1)1(2

2

12

2 =−

=⎥⎦

⎤⎢⎣

⎡=

−

LrL

Lr

Lr

drd

Lr

=jMN 0

12)1(1)1( −−⎦⎣ −− LrLLdrL

Uniform distribution

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Uniform distribution

Histogram Equalization - Exampleg q p

3 bit image (L=8) of size3-bit image (L=8) of size 64X64 (MN=4096) pixels.

08.3)(7)(7)(7)( 10

1

011 =+=== ∑

=

rprprprTs rrj

jr

We get:We get:

765.6 308.3623.6 133.1

51

40

→=→=→=→=

ssss

700.7 667.5786.6 555.4765.6308.3

73

62

51

→=→=→=→=→→

ssssss

Equalized histogram for r = 7: 11.04096

81122245=

++

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4096

Histogram Matchingg gSpecified Histogram

⎧ 2Let,

⎪⎩

⎪⎨⎧ −≤≤

−=otherwise0

10for )1(

2)( 2 Lr

Lr

rpr⎪⎩ otherwise 0

Desired image whose intensity PDF: 0. otherwise ,10for )1(

3)( 3

2

−≤≤−

= LzL

zzpz )1(L

∫∫ −=

−=−==

rr

r Lrwdw

LdwwpLrTs

2

112)()1()(

LL 00 11

sL

zdwwL

dwwpLzGzz

z ===−= ∫∫ 2

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2 )1()1(3)()1()(

LLz −− ∫∫ 20

20 )1()1(

[ ] [ ] 3/123/12

23/12 )1()1()1( rLrLsLz =⎥⎤

⎢⎡

==

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[ ] [ ])1()1(

)1()1( rLL

LsLz −=⎥⎦

⎢⎣ −

−=−=

Histogram Matching: Example (1)g g p ( )

From previousFrom previous example:

31 == ss

766 53 1

54

32

10

======

ssssss

7 77 6

76

54

== ssss

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Histogram Matching: Example (2)g g p ( )

∑=i

zpzG )(7)( ∑=

=j

jzi zpzG0

)(7)(

245.2)G(z 000.0)( 40 →=→=zG

7007)G(z1051)(695.5)G(z 000.0)(555.4)G(z 000.0)(

62

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→=→=→=→=→=→=

zGzGzG

700.7)G(z 105.1)( 73 →=→=zG

Find the smallest value of z so that G(z ) is the closet to sFind the smallest value of zq so that G(zq) is the closet to sk.

790 19.04096790)( 3 ==zpz

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Histogram Equalization vs Hist M t hi IHistogram Matching - I

Histogram Equalization:g

O i i l i d it hi t

Histogram-equalized image

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Original image and its histogram image

Histogram Equalization vs Hist M t hi IIHistogram Matching - II

Histogram Matching:g g

Specified histogram:

Transformation:

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Histogram StatisticsgMean (average intensity): ∑

−

=1

0)(

L

iii rprm

=0i

Intensity variance (second moment): ∑−

=

−=1

0

22 )()()(

L

iii rpmrrµ

=0i

Drill: Example 3.11

Without histogram:1 11 1 M NM N

[ ]∑∑∑∑−

=

−

=

−

=

−

=

−==1

0

1

0

221

0

1

0

),(1 and ),(1 M

x

N

y

M

x

N

y

myxfMN

yxfMN

m σ

Learn about local statistics.

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Spatial Filteringp gConsists of: (1) A neighborhood.

(2) A d fi d ti th t i hb h d(2) A predefined operation on that neighborhood.

Filter response:Filter response:

)11()11()()00(...),1()0,1()1,1()1,1(),(

++++++−−+−−−−=

ffyxfwyxfwyxg

)1,1()1,1(...),()0,0( +++++ yxfwyxfw

For a mask of size m x n m = 2a+1 n = 2b+1:For a mask of size m x n, m 2a+1, n 2b+1:

∑ ∑ ++=a b

bttysxftswyxg ),(),(),(

−= −=as bt

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Spatial Correlation & ConvolutionpCheck

CheckCheck

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Spatial Correlation & Convolutionp2-D

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Filter Mask

∑=

==+++=9

1992211 ...

k

Tkk zwzwzwzwR zw

Smoothing Filter (low pass) mask:

a b

∑ ∑

∑ ∑−= −=

++= a b

as bt

tsw

tysxftswyxg

),(

),(),(),(

∑ ∑−= −=as bt

),(

Normalization factor

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Equal weight Weighted averageNormalization factor

Smoothing Effectg

Original image 3 X 3 mask

Noise is less pronounced.

5 X 5 mask

35 X 35 mask

C l t l bl d!

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Completely blurred!

Median FilteringgFind the median in the neighborhood, then assign the center pixel value to that median.p

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Sharpening Spatial Filtersp g pFirst-order derivative: )()1( xfxf

xf

−+=∂∂

Second-order derivative: )(2)1()1(2

2

xfxfxfxf

−−++=∂∂

First derivative:

1. Zero at constant area.2. Nonzero at onset.3. Nonzero along ramp.

Second derivative:

1. Zero at constant area.2. Nonzero at onset & end.3. Zero along constant

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gslope ramp.

Laplacian Mask - IpSecond derivative is more useful in edge detection. x, y, and two

2

2

2

22

yf

xff

∂∂

+∂∂

=∇

diagonal directions

),(2),1(),1(

2

2

2

f

yxfyxfyxfxf

y

∂

−−++=∂∂

),1(),1(),(

),(2)1,()1,(

2

2

2

yxfyxfyxf

yxfyxfyxfyf

−++=∇∴

−−++=∂∂

),(4)1,()1,( ),(),(),(

yxfyxfyxfyfyfyf

−−+++

New image:

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[ ]),(),(),( 2 yxfcyxfyxg ∇+=

Laplacian Mask - IIp

Original blurred image

Using Laplacian mask along x, y

Using Laplacian mask along x, y g , y

axesg , y

axes, and two diagonal

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Unsharp Masking & Hi hb st Filt iHighboost Filtering

To sharpen an image1. Blur the original image.2. Subtract the blurred image from the original (result is mask).3 Add the mask to the original

g

3. Add the mask to the original.

)()()(),(),(),(

kfyxfyxfyxgmask −=

),(),(),( yxgkyxfyxg mask×+=

If k = 1, unsharp mask.If k 1, unsharp mask.If k > 1, highboost filtering.

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