EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

44

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2

-D M

ATH

FU

NCTI

ON

S

CO

NTI

NU

OU

S 2

-D F

UN

CTI

ON

S

•Th

e f

un

cti

on

f d

escri

bes a

su

rfa

ce

in s

pa

ce

•Th

e h

eig

ht

of

the s

urf

ace a

t (x

,y)

is

f(x,y

)

•In

ou

r ca

se, f(

x,y

) re

pre

sen

ts a

con

tin

uou

s p

hysic

al va

ria

ble

, su

ch

a

s im

ag

e irr

ad

ian

ce (

wa

tts/m

-2

)

•Sp

ecia

l ca

ses

Sep

ara

ble

fu

ncti

on

s

Ra

dia

l fu

ncti

on

s

fx

y,(

)

x

y

f(x,y

)

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

45

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2

-D M

ATH

FU

NCTI

ON

S

2-D

va

ria

tion

s o

f th

e d

elt

a

fun

cti

on

delt

a

bla

de

δx

x 0y

y 0–

,–

()

y 0x 0

1

y

x

δy

y 0–

() y 0

1

y

x

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

46

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2

-D M

ATH

FU

NCTI

ON

S

gri

ll

gri

d

com

by b---

|b|

y

xb

com

bx a---

y b---,

a

bδ

xm

ay

nb

–,

–(

)m

∞–=∞ ∑

n∞–

=∞ ∑=

y

xb

a

|a ||

b|

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

47

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2

-D M

ATH

FU

NCTI

ON

S

Sep

ara

ble

Fu

ncti

on

s

•Sp

ecia

l ca

se o

f a

ll c

on

tin

uou

s f

un

cti

on

s

•Pro

du

ct

of

two 1

-D f

un

cti

on

s, on

e in

x a

nd

th

e o

ther

in y

•A

llow

s s

om

e im

port

an

t op

era

tion

s t

o b

e d

on

e in

1-D

•Exa

mp

le s

ep

ara

ble

fu

ncti

on

2-D

recta

ng

le f

un

cti

on

Mu

ltip

ly t

wo 1

-D r

ecta

ng

le f

un

cti

on

s

fx

y,(

)f 1

x()f

2y(

)=

rect

x a---y b--- ,

rect

x a---

r

ect

y b---

=

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

48

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2

-D M

ATH

FU

NCTI

ON

S

by

x

y

x

a

rect

x a---

rect

y b---

y

x

a

rect

x a---y b---,

b

X=

y

x

yp

lan

vie

w

x

y

x

y

a

bX

=

pers

pecti

ve v

iew

are

ctx a---

b

rect

y b---

rect

x a---y b---,

ampl

itud

e =

1, |

x|<

a/2,

|y|<

b/2

=

0, o

ther

wis

e

= 1

/2, |

x|=

a/2,

|y|=

b/2

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

49

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2

-D M

ATH

FU

NCTI

ON

S

Som

e s

ep

ara

ble

2-D

fu

ncti

on

s

func

tion

sepa

rabl

e fo

rm

δx

y,(

)δ

x()δ

y()

δx

x 0y

y 0–

,–

()

δx

x 0–

()δ

yy 0

–(

)

rect

x a---y b---,

rect

x a---

r

ect

y b---

sinc

x a---y b---,

sinc

x a---

s

inc

y b---

tri

x a---y b---,

tri

x a---

t

riy b---

ga

us

x a---y b---,

ga

us

x a---

g

au

sy b---

com

bx a---

y b---,

co

mb

x a---

c

om

by b---

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

50

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

Sep

ara

bilit

y im

plies a

lig

nm

en

t to

coord

ina

te a

xes

Ga

ussia

n e

xa

mp

le sep

ara

ble

non

sep

ara

ble

x

y

x

y

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

51

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

Ra

dia

l Fu

ncti

on

s

•Fu

ncti

on

va

lue d

ep

en

ds o

n r

ad

ius

from

ori

gin

Eq

uiv

ale

nt

to a

1-D

fu

ncti

on

of

rad

ius, r

Gen

era

te 2

-D s

urf

ace b

y r

ota

tin

g “

gen

era

tin

g

fun

cti

on

” f

(r)

aro

un

d o

rig

in

fx

y,(

)f

x2y2

+(

)f

r()

==

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

52

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

Exa

mp

les o

f ra

dia

l fu

ncti

on

s

cylin

der

Ga

ussia

n

cyl

r d---

1,

r ≤

d/2

0,

else

whe

re=

y

x

radi

us =

d/2

1

ga

us

r d---

eπ

rd⁄

()2

–=

x

y

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

53

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

Som

bre

ro

• J

1 is a

fir

st-

ord

er

Bessel fu

ncti

on

of

the f

irst

kin

d.

• z

ero

s o

f som

b(r

/d)

are

at

r/d

= 1

.22

, 2

.23

, 3

.24

, . . .

som

br d---

2J 1

πr d------

πr d------

--------

--------

----=

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

54

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

Bessel Fu

ncti

on

s

For

inte

ger

n, J n

is a

solu

tion

to d

iffe

ren

tia

l eq

ua

tion

s o

f th

e f

orm

,

J n is e

va

lua

ted

as a

n in

fin

ite s

eri

es

n=

0 (

zero

ord

er)

n=

1 (

firs

t ord

er)

Recu

rsio

n r

ela

tion

s e

xis

t b

etw

een

Jn, J n

+1 a

nd

Jn

-1

J 0(x

) a

nd

J1(x

) a

re t

ab

ula

ted

as f

un

cti

on

s o

f x, or

ca

n b

e c

alc

ula

ted

as n

eed

ed

x2

x2

2

ddy

xx

ddyx2

n2–

()y

++

0=

J 0x(

)1

x2

221

!(

)2----

--------

------

–x4

242

!(

)2----

--------

------

x6

263

!(

)2----

--------

------

–…

++

=

J 1x(

)x 2---

x3

231

!2!

--------

--------

–x5

252

!3!

--------

--------

x7

273

!4!

--------

--------

–…

++

=

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

55

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

2-D

CO

NTI

NU

OU

S S

YSTE

MS

•Gen

era

l syste

m

inp

ut

f(x,

y), ou

tpu

t g(

x,y)

, syste

m o

pera

tor

T

•Lin

ea

rity

•Sh

ift

Inva

ria

nce

gx

y,(

)T

fx

y,(

)[

]=

Ta

f 1x

y,(

)b

f 2x

y,(

)+

[]

aT

f 1x

y,(

)[

]b

Tf 2

xy,

()

[]

+=

gx

y,(

)T

fx

y,(

)[

]=

Tf

xα

–y

β–

,(

)[

]g

xα

–y

β–

,(

)=

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

56

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

•Syste

m I

mp

uls

e R

esp

on

se

Con

tin

uou

s in

pu

t sig

na

l,

For

lin

ea

r syste

ms,

defi

ne

Then

, fo

r LSI

syste

ms,

fx

y,(

)f

αβ,

()δ

xα

–y

β–

,(

)αd

βdβ∫

α∫=

gx

y,(

)T

fx

y,(

)[

]=

Tf

αβ,

()δ

xα

–y

β–

,(

)αd

βdβ∫

α∫=

fα

β,(

)Tδ

xα

–y

β–

,(

)[

]αd

βdβ∫

α∫= h

xy,

()

Tδ

xy,

()

[]

=

Tδ

xα

–y

β–

,(

)[

]h

xα

–y

β–

,(

)=

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

57

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

an

d

is c

alled

th

e s

yste

m im

pu

lse r

esp

on

se

The im

pu

lse r

esp

on

se o

f a

n o

pti

ca

l im

ag

ing

syste

m is r

ep

resen

ted

by

gx

y,(

)f

αβ,

()h

xα

–y

β–

,(

)αd

βdβ∫

α∫=

hx

y,(

)

hx

y,(

)

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

58

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

DIS

CR

ETE

FU

NCTI

ON

S

•Th

e f

un

cti

on

f d

escri

bes a

n a

rra

y o

f d

elt

a f

un

cti

on

s in

sp

ace

•Th

e a

mp

litu

de o

f th

e a

rra

y a

t (m

,n)

is f

(m,n

)

•In

ou

r ca

se, f(

m,n

) re

pre

sen

ts a

sa

mp

led

va

ria

ble

, su

ch

as a

d

igit

al im

ag

e

•Sp

ecia

l ca

ses

Sep

ara

ble

fu

ncti

on

s

fm

n,(

)

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

59

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

Sa

mp

led

Fu

ncti

on

s in

2-D

•To

sa

mp

le a

con

tin

uou

s f

un

cti

on

, m

ult

iply

by a

com

b

fm

n,(

)1 a

b----

--------

fx

y,(

)co

mb

x a---y b--- ,

=

fm

an

b,

()δ

xm

ay

nb

–,

–(

)m

∞–=∞ ∑

n∞–

=∞ ∑=

n

mb

a

f(m

,n)

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

60

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

Sep

ara

ble

Fu

ncti

on

s

•Exa

mp

les

2-D

ste

p f

un

cti

on

Mu

ltip

ly t

wo 1

-D s

tep

fu

ncti

on

s:

fm

n,(

)f 1

m()f

2n(

)=

fm

n,(

)1

mn

0≥

,,

0el

sew

her

e,

=

fm

n,(

)u

m()u

n()

=

n

m

n

m

um(

)n

m

un(

)

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

61

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

Nu

meri

ca

l exa

mp

le

n

m

63 3

22

mn

11

3

11

2

f 1f 2

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

62

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

•D

eg

rees-O

f-Fre

ed

om

(D

OF)

gen

era

l ,

ha

s N

M D

OF

sep

ara

ble

,

ha

s o

nly

N+M

-1 D

OF

sep

ara

ble

2-D

fu

ncti

on

s a

re c

on

str

ain

ed

com

pa

red

to g

en

era

l 2

-D f

un

cti

on

s

Wh

y N

+M

-1 D

OF?

fm

n,(

)0

mM

1–

≤≤

0n

N1

–≤

≤,

fm

n,(

)f 1

m()f

2n(

)=

0m

M1

–≤

≤0

nN

1–

≤≤

,

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

63

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

2-D

DIS

CR

ETE

SYSTE

MS

•Gen

era

l syste

m

inp

ut

f(m

,n),

ou

tpu

t g(

m,n

), s

yste

m o

pera

tor

T

•Lin

ea

rity

•Sh

ift

Inva

ria

nce

gm

n,(

)T

fm

n,(

)[

]=

Ta

f 1m

n,(

)b

f 2m

n,(

)+

[]

aT

f 1m

n,(

)[

]b

Tf 2

mn,

()

[]

+=

gm

n,(

)T

fm

n,(

)[

]=

Tf

mm

1–

nn 1

–,

()

[]

gm

m1

–n

n 1–

,(

)=

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

64

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

•Syste

m I

mp

uls

e R

esp

on

se

Dis

cre

te in

pu

t sig

na

l,

For

lin

ea

r syste

ms,

defi

ne

Then

, fo

r LSI

syste

ms,

fm

n,(

)f

kl,

()δ

mk

–n

l–

,(

)l∑

k∑=

gm

n,(

)T

fm

n,(

)[

]=

Tf

kl,

()δ

mk

–n

l–

,(

)l∑

k∑=

fk

l,(

)Tδ

mk

–n

l–

,(

)[

]l∑

k∑=

hm

n,(

)T

δm

n,(

)[

]=

Tδ

mk

–n

l–

,(

)[

]h

mk

–n

l–

,(

)=

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

65

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

an

d

is c

alled

th

e s

yste

m im

pu

lse r

esp

on

se

•A

LSI

con

volu

tion

filte

r fo

r d

igit

al im

ag

e p

rocessin

g is r

ep

resen

ted

by

gm

n,(

)f

kl,

()h

mk

–n

l–

,(

)l∑

k∑=

hm

n,(

)

hm

n,(

)

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

66

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

DIG

ITA

L I

MA

GE R

EPR

ESEN

TATI

ON

•re

pre

sen

t a

Kro

necker

delt

a f

un

cti

on in

2-D

wit

h “

am

plitu

de” a

:

aδ(

x,y

) in

con

tin

uou

s-s

pa

ce

aδ(

m,n

) in

dis

cre

te-s

pa

ce

•re

pre

sen

t a

dig

ita

l im

ag

e a

s a

su

m o

f a

mp

litu

de-m

od

ula

ted

delt

a

fun

cti

on

s:

•N

ote

, m

corr

esp

on

ds t

o t

he “

row

” in

dex a

nd

n c

orr

esp

on

ds t

o t

he

“colu

mn

” in

dex o

f a

ma

trix

.

fm

n,(

)f

kl,

()δ

mk

–n

l–

,(

)l

∞–=∞ ∑

k∞–

=∞ ∑=

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

67

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

Exa

mp

le

fm

n,(

)

3 m

,n =

0;

1 0

< |m

|,|n|

≤ 1

;

0 e

lsew

here

;

=

fm

n,(

)3

δm

n,(

)δ

mn

1–

,(

)δ

mn

1+

,(

)…

++

+=

3n

m

= z

ero

= a

mpl

itude

1

a = a

mpl

itude

a

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

68

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

Exa

mp

le 2

-D S

eq

uen

ces

•exp

on

en

tia

l

eig

en

fun

cti

on

of

lin

ea

r, s

hif

t-in

va

ria

nt

(LSI)

syste

ms

•sep

ara

ble

imp

ort

an

t sp

ecia

l ca

se o

f 2

-D s

eq

uen

ces

imp

lica

tion

s in

ma

them

ati

ca

l d

escri

pti

on

s a

nd

com

pu

tati

on

exa

mp

le: 2

-D d

elt

a f

un

cti

on

•p

eri

od

ic

ba

se s

eq

uen

ce r

ep

ea

ts e

very

(M

,N)

poin

ts

fm

n,(

)A

αm

βn

=

fm

n,(

)f 1

m()f

2n(

)=

δm

n,(

)δ

m()δ

n()

=

fm

n,(

)f

mM

+n,

()

fm

nN

+,

()

==

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

69

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

VECTO

R-M

ATR

IX

REPR

ESEN

TATI

ON

•Com

pa

ct

nota

tion

•R

ew

rite

MxN

arr

ay a

s v

ecto

r w

ith

le

ng

th M

N

•Lexic

og

rap

hic

ord

eri

ng

, b

y c

olu

mn

s

or

row

s

den

ote

lexic

og

rap

hic

seq

uen

ce a

s

•U

sefu

l fo

r exp

ressin

g lin

ea

r op

era

tion

s s

uch

as c

on

volu

tion

an

d

Fou

rier

an

d o

ther

tra

nsfo

rms

More

la

ter

. . .

f

Exa

mp

le

f2

1

54

36

=

f co

l

2 5 3 1 4 6

=f r

ow

2 1 5 4 3 6

=

ma

trix

rep

resen

tati

on

m

n

2

5

3

1

4

6

arr

ay r

ep

resen

tati

on

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

70

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

IMA

GE F

OR

MA

TS

Alp

ha

bet

sou

p!

•PB

M (

port

ab

le b

itm

ap

), P

GM

(p

ort

ab

le g

raym

ap

), P

NM

(p

ort

ab

le a

nym

ap

)

•GIF

* (g

rap

hic

in

terc

ha

ng

e

form

at)

•TI

FF*

(ta

gg

ed

im

ag

e f

ile

form

at)

•JP

EG (

lossy c

om

pre

ssed

)

* in

clu

des lossle

ss L

em

pel-

Ziv

-Welc

h

(LW

Z)

cod

ing

JPEG

qu

ality

=1

un

com

pre

ssed

EC

E/O

PT

I533

Dig

ital I

mag

e P

roce

ssin

g cl

ass

note

s

71

Dr.

Rob

ert A

. Sch

owen

gerd

t

2003

2-D

MA

TH F

UN

CTI

ON

S

WA

RN

ING!

•D

o n

ot

use J

PEG f

orm

at

for

testi

ng

a

lgori

thm

s

•TI

FF m

ay s

ca

le im

ag

e v

alu

es t

o

[0..2

55

], d

ep

en

din

g o

n d

isp

lay L

UT

•“ra

w” f

orm

at,

IEEE f

loa

tin

g-p

oin

t,

32

bit

s/p

ixel is

best

for

alg

ori

thm

te

sti

ng

!