Lecture #7 - Case Western Reserve Universityengr.case.edu/merat_francis/eecs490f07/Lectures/Lect… · · 2012-02-16EECS490: Digital Image Processing Lecture #7 • Image Processing

EECS490: Digital Image Processing

Lecture #7

• Image Processing Example

• Fuzzy logic

– Basics

– Image processing examples

• Fourier Transform

– Inner product, basis functions

– Fourier series


© 2002 R. C. Gonzalez & R. E. Woods

Image Processing Example

Laplacian of image

(c) Sharpened byadding Laplacian

Sobel gradientimage

original image



Image Processing Example

(e) Blurred Sobelgradient image Mask = (c) x (e)

Add Mask to original Power Law IntensityTransform



Basic Fuzzy Logic

“crisp” membershipfunction

“fuzzy” membershipfunction

Probability: there is a 50% chance that a particular person is youngFuzzy logic: a person’s membership with the set of young people is 0.5

Basically everyone is “young” to some degree. A membership functionrepresents that degree.



Basic Fuzzy Logic

=max[ A(z), B(z)] =min[ A(z), B(z)]

OR AND



Basic Fuzzy Logic

Commonly used membership functionsused to describe inputs and outputs.



Fuzzy Input Variables

We will use asingle colorto describe afruit with acolor thatchangesfrom greento yellow tored as itripens.

A particular color zo has amembership value green(zo),

yellow(zo), and red(zo) in all threeinput membership functions.



Fuzzy Output Variables

The fruit can beverdant(unfit toeat), half-mature(ripening), andmature (ripe) The output variable is

maturity which is hard toquantify



Fuzzy System

We nowneed torelate theinputmembershipfunctions tothe outputmembershipfunctions —this iscalledimplication

input membership output membership

This is a simple plot ofthe relationshipbetween the twomembership functions

This is simply themembership function for redAND mature or red mature

red ature(z,v)=min[ red(z), mature(v)]



A fuzzy output for a given input

We nowneed toevaluateeach outputmembershipfunction forthe giveninput value

Q3(v)= red(zo) AND red ature(zo,v)=min[[ red(zo), red ature(zo,v)]

red(zo) is a constant c which clips the outputmembership function as shown above

Q3 is still a membership function!



All fuzzy outputs for a given input

There are 3 differentinput membershipfunctions each ofwhich can be ANDedwith mature.

This gives threedifferent outputmembership functions.

The system output isthe maximum value ateach point orQ=Q1 OR Q2 OR Q3.

This is the maturemembership functionfor a specific color zo



Defuzzification

The output is still aset. The actualmembership value isthe center of gravityof the output set.

v0 =vQ v( )

v=1

K

Q v( )v=1

K



The Entire Process



IF a pixel is dark THEN make it darkerIF a pixel is gray THEN make it grayIF a pixel is bright THEN make itbrighter

The output memberships are only threevalues.

Fuzzy Contrast Enhancement


1. Compute the input membershipfunction AND the output membershipfunction

2. For a specific value of input graylevel we map onto a single output plane.The membership is 1 for deep blacksand gradually decreases to zero. Dothis for each output.

Contrast Enhancement

3. Determine the total membershipfunction and compute the centerof gravity of the output

v0 =μdark z0( ) vdark + μgray z0( ) vgray + μbright z0( ) vbright

μdark z0( ) + μgray z0( ) + μbright z0( )






Original histogram Equalized histogram

Fuzzy membershipfunctions

Fuzzy contrastenhanced histogram




Fuzzy Boundary Extraction

IF a pixel belongs to a uniform region THEN make it white ELSE make it black

IF d2 is zero AND d6 is zero THEN z5 is whiteIF d6 is zero AND d8 is zero THEN z5 is whiteIF d8 is zero AND d4 is zero THEN z5 is whiteIF d4 is zero AND d2 is zero THEN z5 is white ELSE z5 is black



Fuzzy Boundary Extraction Rules




Input membership functionfor ZERO intensitydifferences

Output membership function for blackand white


Fuzzy System

input membership output membership

This is a plot of therelationship betweenthe two membershipfunctions

This is the membershipfunction for difference ANDwhite


A fuzzy output for a given input

We nowneed toevaluateeach outputmembershipfunction forthe giveninput value

This would be the output membership for aspecific intensity difference input ANDwhite





Sum of Functions


The terms shown (blue)

sum to the rippling

square wave (black).

As the number of terms

in the sum becomes large,

it approaches a square

wave (red).

( ) ( )++

==

tnn

tn

122

sin12

1sqOdd-order harmonics

Fact: Any Real Signal has a Frequency-Domain Representation

1999-2007 by Richard Alan Peters II


( ) ( )++

==

tnn

tn

122

sin12

1sq

The sinusoids are called

“basis functions”.

Any periodic signal can be described by a sum of sinusoids.

The multipliers are called

“Fourier coefficients”.

Frequency-Domain Representation



( ) ( )++

==

tnn

tn

122

sin12

1sq






Basisfunctions




( ) ( )++

==

tnn

tn

122

sin12

1sq






The Fouriercoefficients(of a squarewave).




The similarity between functions f and g on the interval - / 2, / 2( )

can be defined by

f , g = f t( ) g* t( )/2

/2

dt

where g* t( ) is the complex conjugate of g t( ).

. onto of projection theasor in ofamount theas ofthought

be also can , and thecalled number, This

gffg

gf of product inner

common. in nothing have

and then0, , if handother theOn . if maximal

isproduct inner their thenenergy, same thehave and If

gfgfgf

gf

==

The Inner Product: a Measure ofSimilarity



f , g = f t( ) cos 2 t( ) j sin 2 t( )/2

/2

dt

= f t( )ej2

t

/2

/2

dt

= f t( )e j t

/2

/2

dt

( ) ( )dtttfgf =

2/

2/

2sin, ( ) ( )dtttfgf =

2/

2/

2cos,

ej2

t= cos 2 t( ) j sin 2 t( )

=2

3 differentrepresentations

Inner Product of a Periodic Function and aSinusoid



( ) ( )dtttfgf =

2/

2/

2sin,

( ) ( ) ( )[ ]

( )

( ) dtetf

dtetf

dttittfgf

ti

ti

=

=

=

2/

2/

2/

2/

2

2/

2/

22sincos,

( ) ( )dtttfgf =

2/

2/

2cos,

real number resultsyield the amplitudeof that sinusoid inthe function.




( ) ( )dtttfgf =

2/

2/

2sin,

( ) ( ) ( )[ ]

( )

( ) dtetf

dtetf

dttittfgf

ti

ti

=

=

=

2/

2/

2/

2/

2

2/

2/

22sincos,

( ) ( )dtttfgf =

2/

2/

2cos,

Complex number resultyields the amplitude andphase of that sinusoid inthe function.




is the decomposition of a -periodic signal into a sum of sinusoids.

( )=

++=1

0

2sin

2cos

n

nn tn

Btn

AAtf

( )

( ) 02

sin2

02

cos2

2/

2/

2/

2/

=

=

ndttn

tfB

ndttn

tfA

nn

nn

for

for

Fourier coefficients aregenerated by taking theinner product of thefunction with the basis.

The representation of afunction by its FourierSeries is the sum of sinu-soidal “basis functions”multiplied by coefficients.).()(: tfntf =± that such periodic

The Fourier Series



can also be written in terms of complex exponentials

f t( ) = Cn

n=

e+ j 2 n t

= Cn

n=

e+ j 2 n t + n

= Cn cos2 n

t + n + j Cn sin2 n

t + n

n=

Cn = Cn e+ j n =1

f t( ) ej 2 n t

/2

/2

dt

=1

f t( ) cos2 n

t n j sin2 n

t n

/2

/2

dt

j = 1

n

tfntf

intergers all for

)()( =+

Cn = Cn e + j n

e ± j x= cos x ± j sin x

The Fourier Series



f (t) = Cn

n=

e+ j 2 n t

where Cn = Cn e+ j n .

Cn represents the

amplitude, A=|Cn|,

and relative phase, ,

of that part of the

original signal, f (t),

that is a sinusoid of

frequency n = n / .

0

Why are Fourier Coefficients ComplexNumbers?


Documents

Lecture #7 - Case Western Reserve Universityengr.case.edu/merat_francis/eecs490f07/Lectures/Lect… · · 2012-02-16EECS490: Digital Image Processing Lecture #7 • Image Processing