Digital Image Processing & Pattern Analysis (CSCE 563) Intensity Transformations Prof. Amr Goneid...

Digital Image Processing&

Pattern Analysis(CSCE 563)

Intensity Transformations

Digital Image Processing&

Pattern Analysis(CSCE 563)

Intensity Transformations

Prof. Amr GoneidDepartment of Computer Science & Engineering

The American University in Cairo

Prof. Amr Goneid, AUC 2

Intensity TransformationsIntensity Transformations

Pixel Intensity Processing

Histogram Processing

Combining Images


Pixel Intensity Processing

Point Processing

Gray Scale to Binary

Negative

Power Law Transformation

Contrast Stretching

Compression of Dynamic Range

Gray Level Slicing


Point Processing

Local Processing:g(x,y) = T[f(x,y)]

T = Operator on Locality (neighborhood) of (x,y)

e.g. 3 x 3 T [ ] (8-neighborhood)

Point Processing:For 1 x 1 we have Point Processing

S = T[r] = T[ ]S = New Intensity, r = Old Intensity

p


Gray scale to Binary

s = T(r,a) = 0 for r < a, 1 otherwise


Negative (Image Complement)

s =T(r) = 1-r

The MATLAB function:

g = imcomplement(f)

10 r

1

s

Power Law Transformation

Has the form:

The MATLAB function

g = imadjust (f, [low-in high-in], [low-out high-out], gamma)


crs

Power Law TransformationExample:

g = imadjust (f, [0 1], [1 0]);

equivalent to: g = imcomplement (f);

g = imadjust (f, [0.3 0.7], [0.2 1.0]);



Contrast Stretching

Low contrast images do not make full use of the dynamic range of grey levels. In this case the intensities are confined to a narrow range rmin < r < rmax.

We can use a point transformation s = T(r) to stretch the range to smin < s < smax


Contrast Stretching

r = old intensity, s = new intensity

Slope = s/ r > 1 produces

contrast stretching. = [(smax– smin)/(rmax– rmin)]

Hence, s =T(r)

= (r – rmin) + smin

r

s

r

s


Contrast Stretching Example

rmin = 0.0 , rmax = 0.3 smin = 0.0 , smax = 1.0

Contrast Stretching Function


mrforandmrfors

rmrTs

E

E

10lim that Notice

)/(1

1)(

•The function compresses the input

levels lower than m into a narrow

range of dark levels in the output image.

• Similarly, it compresses intensities

above m into a narrow band of light

levels in the output.

•The result is an image of higher contrast.


Summer 2009 Prof. Amr Goneid, AUC 13

The lena image on the right has been obtained by:

load lena;

m = 0.5; E = 0.5;

g = 1./(1+(m./(double(f)+eps)).^E);

gs = im2uint8(mat2gray(g));

imshow(gs);

13Prof. Amr Goneid, AUC



In the limiting case it produces a binary image.



Compression of Dynamic Range

Map r = {0,D} to s = {0,L-1}

where D >> L-1

s = T(r) = c log(1 + r )

c = (L-1)/log(1+D)

e.g. compression from

256 gray levels

to 16 gray levels.



Gray Level Slicing

Examples:S = T(r)

= {1 for A<r<B ;

c otherwise}

S = T(r)

= {1 for r>A; r otherwise}

s

rs

r

1

0

0

1

1

1A

A B

c


Gray Level Slicing Example

S = T(r) = {1 for r>0.6; r otherwise}



A bad contrast image (F) can be enhanced to an image G by transforming every pixel intensity fij to another intensity gij using a method called Histogram Equalization



PDF and CDF

Histogram Equalization (Theory)

Histogram Equalization (Practice)

Histogram Specification


Probability Density Function (PDF)

Suppose the gray levels r are normalized such that

0 < r < 1.0

pr (r) dr = probability of intensity taking a value between r and r+dr.

A plot of pr (r) against r is called the PDF of the image.

The PDF is normalized: 1

0

1)( drrpr


Cumulative Distribution Function (CDF)

The CDF = the probability of having values less than r

For a uniform PDF, ps (s) = 1 and CDF = s

(i.e. CDF is Linear)

r

rr drrprpCDF0

)()(

Uniform

cdf


Histogram Equalization (Theory)

TooDark

BadContrast

TooBright

P(r)

r

UniformHigh Contrast

P(s)

s

Mapping r into s such that ps (s) is uniform


How?

Gray level r {0,1} transformed to

s = T(r) {0,1}

T(r) is single valued and monotonically increasing

To find T(r), force the two CDF’s to be the same:

CDF(r) = CDF(s) = s

Then solve to find s in terms of r


Example

2

0 1

10 also that Notice

)1(1)()(

)1(2)(givesion Normalizat

10)1()(Consider

2

0

s

rdrrprCDFs

rrp

rrrp

r

r

r

r


Will s have a uniform distribution?

on distributi uniform a has)( Hence,

1)]12/(12[)/)(1(2)(

12

1/

11)( mapping inverse The

/)()(

theorylstatistica From

1

)(1

sp

ssdsdrrspNow

sdsdr

ssTr

dsdrrpsp

s

s

sTrrs



In practice, gray levels are discrete (k = 0 .. L-1)

A histogram is an array whose element nk = no. of pixels with gray level k (k = 0 is black, k = L-1 is white).

A histogram is a plot of nk vs rk e.g. for L = 256

In MATLAB

n = imhist(I,256)



Algorithm to build histogram for an image with N rows and M columns (Ntot = NM pixels):

tottothist

kkij

kmax

max

NNM)N(T

operationsarithmeticof.noComplexity

nn;fk

)Mj(for

)Ni(for

;n)rk(for

LrLet

1

1

1

00

1


Histogram EqualizationHistogram EqualizationAlgorithm (1)Algorithm (1)

The histogram is used in the following algorithm:

tottotequ

r

tottotequmax

tottotequ

totmax

r

kkij

N}/)L{()N(T:gives

)L/pyprobabilitequalwith(CaseAveragethethatShow

N)L()N(T),rr(CaseWorst

NNM)N(T),r(CaseBest

operationsarithmeticof.noComplexity

N/r*)n(s;fr

)Mj(for

)Ni(for

25

1

2

330

1

1

0


Histogram EqualizationHistogram EqualizationAlgorithm (2)Algorithm (2)

A better algorithm first builds a Cumulative array:

)L(N)N(T

N/r*Cg;fr

)Mj(for

)Ni(for

:ituseweThen

Lroperationsarithmeticof.no

sumC;nsumsum

)rk(for

;sumC;nsum

tottotequ

totmaxrijij

max

kk

max

12

1

1

1

100


Histogram EqualizationHistogram EqualizationSummary of Total ComplexitySummary of Total Complexity

Algorithm (1):

Algorithm (2):

tottot

tottotmax

tottot

N}/)L{()N(TCaseAverage

N)L()N(T),rr(CaseWorst

N)N(T),r(CaseBest

27

3

40

)L(N)N(TCasesAll tottot 13


Histogram EqualizationHistogram EqualizationNumerical ExampleNumerical Example

L = 256 gray levels

Algorithm (1):

Algorithm (2):

tottot

tottotmax

tottot

N.)N(TCaseAverage

N)N(T),rr(CaseWorst

N)N(T),r(CaseBest

5131

259

40

2553 tottot N)N(TCasesAll



If N = total no. of pixels = numel (I), then the PDF of input image is Pr(rk) = n(k)/N

L)histeq(f, g

MATLAB, In

into level transform will nqualizatio Histogram

)(

)()(1

kk

kk

j

k

jrk

rCDFs

sr

rprCDF



Practical ExamplesFrom: http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip.html

Original Contrast

Stretched

Histogram

Equalized


Negative Ramp


Liver Tissue Image


Girl Image



Histogram Specification

How to Transform an image (A) with Pr(r) to an image (B) with specified Pz(z).

For the input image (A), the uniform variable s = T(r) = CDF(r) For the output image (B), s = H(z) = CDF(z) Hence z = H-1(s) = H-1(T(r))In MATLAB: g = histeq (f , HistB)


Histogram Specification (Example)

Image A has Pr(r) = 2(1-r)

(-ve ramp over r {0,1})

Image B has Pz(z) = 2z

(+ve ramp over z {0,1})

T(r) = CDF(r)= 1 – (1-r)2

H(z) = CDF(z) = z2 H-1(s) = sqrt(s)

z = H-1(T(r)) = sqrt(T(r)) = sqrt {1 – (1-r)2}38Prof. Amr Goneid, AUC


A Practical Example(-)ve To (+)ve Ramp


Combining Images

Arithmetic Operations

Logical Operations


Combining Images

By Arithmetic Operations:e.g. Addition & Subtraction:C = min (A + B , 1);C = max (A – B , 0);

By Logical operations(Binary Images):C = A & B Logical ANDC = A | B Logical ORC = A xor B Logical Exclusive ORC = ~ A Logical NOT


Example: Arithmetic

I=Image; RN = Random Noise; B=boundary

I + RN + B RN - I


Example: Logical

Mask AND Image1 = Image2


Other ExamplesFrom: http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip.html

Image (A) Image (B) ~ B

A | B A & B A xor B A & (~ B)

Digital Image Processing & Pattern Analysis (CSCE 563) Intensity Transformations Prof. Amr Goneid...

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