continuous 1-d functions - university of arizonadial/ece533/notes2.pdfece/opti533 digital image...

EC

E/O

PT

I533 Digital Im

age Processing class notes 10 D

r. Robert A

. Schowengerdt 2003

1

-D M

ATH

REV

IEW

CO

NTIN

UO

US 1

-D F

UN

CTIO

NS

•K

ron

ecker d

elta

fun

ctio

n a

nd

its re

lativ

es

delta

fun

ctio

n

NO

TE: Th

e d

elta

fun

ctio

n’s

am

plitu

de is

infin

ite a

nd

its a

rea

is 1

. The a

mp

litud

e is

sh

ow

n a

s 1

for c

on

ven

ien

ce in

plo

ts.

Write

the e

qu

atio

n th

at d

efin

es th

e a

rea

of a

delta

fu

nctio

n a

s 1

.

Revie

w th

e d

efin

ition

of d

elta

fun

ctio

n in

term

s o

f the lim

it of c

on

ven

tion

al fu

nctio

ns, s

uch

as th

e re

cta

ng

le fu

nctio

n

δx

x0

–(

)

x

1

x0

x

1x

0 = 0

EC

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I533 Digital Im


r. Robert A

. Schowengerdt 2003

1

-D M

ATH

REV

IEW

even

delta

pa

ir

od

d d

elta

pa

ir

δδx

x0

–b-------------

bδ

xx

0–

b+

()

δx

x0

–b

–(

)+

[]

=

x

|b|

bx

- b

|b|

x0 +

bx

0 - b

x0 =

0x

0 ≠

0

δδx

x0

–b-------------

bδ

xx

0–

b+

()

δx

x0

–b

–(

)–

[]

=

x

|b|

bx

- b

x0 +

b

x0 - b

x0 =

0

-|b|

|b|

-|b|

x0 ≠

0

EC

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I533 Digital Im


r. Robert A

. Schowengerdt 2003

1

-D M

ATH

REV

IEW

com

b (s

ha

h) co

mb

xx

0–b

-------------

b

δx

x0

–n

b–

()

n∞–

= ∞∑=

x

|b|

b- b

x0 =

0

02b

- 2bx

|b|

x0 +

bx

0 -b0

x0 +

2bx

0 -2bx

0

. . .. . .

. . .. . .x

0 ≠

0

Even

delta

pa

ir, od

d d

elta

pa

ir a

nd

com

b h

ave

am

plitu

de o

f b

EC

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I533 Digital Im


r. Robert A

. Schowengerdt 2003

1

-D M

ATH

REV

IEW

•U

se o

f the

δ

fun

ctio

n

siftin

g

• N

OTE

: Siftin

g is

a c

on

volu

tion

, eva

lua

ted

for a

pa

rticu

lar s

hift

• F

ind

s th

e v

alu

e o

f a fu

nctio

n a

t a s

pecific

va

lue o

f the in

dep

en

den

t va

riab

le (s

imila

r to a

look-

up

tab

le) f

α()δ

αx

0–

()

αd∞– ∞∫

fx

0(

)constant

==

EC

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1

-D M

ATH

REV

IEW

sa

mp

ling

• N

OTE

: Sa

mp

ling

is a

mu

litplic

atio

n

• O

utp

ut is

a d

elta

fun

ctio

n, w

ith a

rea

dete

rmin

ed

by th

e v

alu

e o

f the fu

nctio

n a

t the s

pecifie

d

va

lue o

f the in

dep

en

den

t va

riab

le.

un

iform

sa

mp

ling

• N

OTE

: Mu

st d

ivid

e c

om

b fu

nctio

n b

y |b

| to re

tain

am

plitu

de o

f f(x).

• N

OTE

: f(x) m

od

ula

tes th

e c

om

b fu

nctio

n.

fx()δ

xx

0–

()

fx

0(

)δx

x0

–(

)=x

x0

xx

0x

x=

f(x)f(x0 )

x0

1b -----f

x()com

bx

x0

–b-------------

fx

0n

b+

()δ

xx

0–

nb

–(

)n

∞–= ∞∑

=

xx

0 +b

x0 -b

0x

0 +2b

x0 -2b

x0

. . .. . .

xx

0 +b

x0 -b

0x

0 +2b

x0 -2b

x0

. . .. . .

1

EC

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I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

sh

ifting

rep

lica

ting

• N

OTE

: Mu

st d

ivid

e b

y |b

| to re

tain

am

plitu

de o

f f(x)

gx()

fx() ❉

δx

x0

–(

)f

α()δ

xx

0–

α–

()

αd∞– ∞∫

fx

x0

–(

)=

==

xx

0x

❉=

f(x)

xx

0

g(x)1

gx()

1b -----f

x() ❉ co

mb

xx

0–b

-------------

=

xx

0 +b

x0 -b

0x

0 -2bx

0

. . .. . .

1

x

❉

f(x)

xx

0 +b

x0 -b

0x

0 -2bx

0

. . .. . .

g(x)

=

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

recta

ng

le (s

qu

are

pu

lse)

trian

gle

rectxb ---

0

xb⁄

12⁄

>

12

x

b⁄⁄

12⁄

=

1

xb⁄

12⁄

<

=

trixb ---

0

xb⁄

1≥

1x

b⁄

xb⁄

1<

–=

Wh

at is

the v

alu

e o

f b in

the

ab

ove g

rap

h?

-0.2 0

0.2

0.4

0.6

0.8 1

1.2-60-40

-200

2040

60

recttri

f(x)

x

For a

giv

en

b,

the tri fu

nctio

n

is tw

ice a

s w

ide

as th

e re

ct

fun

ctio

n

EC

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

sin

c

sin

c-s

qu

are

d

sinc(xb

)⁄

πx

b⁄[

]sinπ

xb⁄

--------------------------=

sinc2

xb⁄

()

Wh

at is

the v

alu

e o

f b in

th

e a

bove g

rap

h?

-0.4

-0.2 0

0.2

0.4

0.6

0.8 1

1.2-60-40

-200

2040

60

sincsinc squared

f(x)

x

EC

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I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

ga

us(s

ian

) ga

us

xb⁄

()

eπ

xb⁄

() 2

–=

Wh

at is

the v

alu

e o

f b in

the

ab

ove g

rap

h?

-0.2 0

0.2

0.4

0.6

0.8 1

1.2-60-40

-200

2040

60

gaus

f(x)

x

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

Wh

at is

the v

alu

e o

f b in

th

e a

bove g

rap

h?

-1.2

-0.8

-0.4 0

0.4

0.8

1.2-60-40

-200

2040

60

cossin

f(x)x

cosin

e

sin

e

2πx

b⁄(

)cos

ej2π

xb⁄

()

ej

–2π

xb⁄

()

+2----------------------------------------------

=

2πx

b⁄(

)sin

ej2π

xb⁄

()

ej

–2π

xb⁄

()

–2j

----------------------------------------------=

EC

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

CO

NV

OLU

TION

(1-D

)

Wh

y is

it imp

orta

nt?

•D

escrib

es th

e e

ffect o

f a L

inea

r Sh

ift Inva

rian

t (LSI) s

yste

m o

n

inp

ut s

ign

als

• L

is th

e s

yste

m o

pera

tor

•D

escrip

tion

of g

en

era

l syste

m

•For a

n L

SI s

yste

m, L

is a

con

volu

tion

system

(operator L)

inputsignal f(x)

outputsignal g(x)

gx()

Lf

x()[

]=

gx()

fx() ❉

hx()

fα(

)hx

α–

()

αd∞– ∞∫

==

EC

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

Exa

mp

le

2

03

x or α

f(x) or f(α)

03

x or α

h(x) or h(α)

-1

1

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

α

h(- α)

1

x = 0

α

f(α)h(0 - α

)

area = g(0)

α

f(α)h(3 - α

)

area = g(3)

α

f(α)h(2 - α

)

area = g(2)

α

f(α)h(1 - α

)

area = g(1)

α

f(α)h(4 - α

)

area = g(4)

α

h(1 - α)

1x = 1

α

h(2 - α)

1

x = 2

α

h(3 - α)

1

x = 3

α

h(4 - α)

1

x = 4

shiftm

ultiplyintegrate

EC

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

• P

lot g

(x)

2

0

31

g(x)

x

The s

hifts

in

this

exa

mp

le

are

by in

teg

er

ste

ps, fo

r illu

stra

tion

con

ven

ien

ce

EC

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

Recip

e

Con

volu

tion

• 1

. write

both

as a

fun

ctio

n o

f αf(α

) an

d h

(α)

• 2

. flip h

(or f) a

bou

t α =

0h

(-α)

• 3

. sh

ift h (o

r f) by a

n a

mou

nt x

h(x

- α)

• 4

. mu

ltiply

the tw

o fu

nctio

ns

f(α)h

(x -α

)

• 5

. inte

gra

te th

e p

rod

uct fu

nctio

n o

ver a

ll αg

(x)

• 6

. rep

ea

t ste

ps 3

thro

ug

h 5

un

til don

e

EC

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

Con

volu

tion

Pro

pertie

s

1-D C

ON

VO

LU

TIO

N P

RO

PE

RT

IES

property

comm

utative

distributive

associative

fx() ❉

hx()

hx() ❉

fx()

=

fx() ❉

h

1x()

h2

x()+

[]

fx() ❉

h1

x()f

x() ❉ h

2x()

+=

fx() ❉

h

1x() ❉

h2

x()[

]

fx() ❉

h1

x()[

] ❉ h

2x()

=

EC

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

Con

volu

tion

Exa

mp

les

1-D C

ON

VO

LU

TIO

N E

XA

MP

LE

S

f(x)h(x)

g(x)

f(x)δ(x)

f(x)

f(x-x0 )

h(x)g(x-x

0 )

f(x)h(x-x

0 )g(x-x

0 )

rect(x)rect(x)

tri(x)

sinc(x)sinc(x)

sinc(x)

gaus(x)gaus(x)

12-------g

au

sx2

-------

EC

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

FO

UR

IER

TRA

NSFO

RM

S (1

-D)

Wh

y is

it imp

orta

nt?

•For a

n L

SI s

yste

m, th

e c

on

volu

tion

op

era

tor b

ecom

es a

mu

ltiplic

atio

n

op

era

tor in

the F

ou

rier d

om

ain

•Ta

kin

g th

e F

ou

rier tra

nsfo

m o

f the s

yste

m e

qu

atio

n,

wh

ere

G(u

) is th

e s

pectru

m o

f the o

utp

ut s

ign

al, F

(u) is

the s

pectru

m o

f th

e in

pu

t sig

na

l, an

d H

(u) is

the s

yste

m tra

nsfe

r fun

ctio

n

Gu(

)F

u()H

u()

=

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

• In

ma

ny c

ases, it is

ea

sie

r to a

na

lyze a

n L

SI s

yste

m in

the F

ou

rier

dom

ain

• F

orw

ard

tran

sfo

rm

• In

vers

e tra

nsfo

rm F

u()

fx()e

j2πxu

–xd

∞– ∞∫=

fx()

Fu(

)ej2π

xuud

∞– ∞∫=

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

Fou

rier Tra

nsfo

rm P

rop

ertie

s

•f(x

) an

d F

(u) a

re, in

gen

era

l, com

ple

x fu

nctio

ns

•f(x

) rea

l ➞ F

(u) =

F*(-u

)

• F

is H

erm

itian

: Re[F

(u)] e

ven

, Im[F

(u)] o

dd

•f(x

) rea

l an

d e

ven ➞

Im[F

(u)] =

0 , i.e

. F(u

) is re

al

•Forw

ard

tran

sfo

rm is

the a

na

lysis

of f(x

) into

its s

pectru

m F

(u)

•In

vers

e tra

nsfo

rm is

the s

yn

thesis

of f(x

) from

F(u

)

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

Fou

rier Tra

nsfo

rm P

airs

1-D F

OU

RIE

R T

RA

NSF

OR

M PA

IRS

f(x)F

(u)

1δ(u)

δ(x)1

rect(x)sinc(u)

sinc(x)rect(u)

comb(x)

comb(u)

gaus(x)gaus(u)

tri(x)sinc 2(u)

2πu

0 x(

)cos

12

u0

-----------δδuu

0-----

12

x0

-----------δδxx0

-----

2π

ux

0(

)cos

2πu

0 x(

)sin

j2

u0

----------- δδuu

0-----

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

Fou

rier Tra

nsfo

rm P

rop

ertie

s

1-D F

OU

RIE

R T

RA

NSF

OR

M P

RO

PE

RT

IES

name

f(x)F

(u)f(x)

F(u)

f(±x)F

(±u)

F(±x)

scalingf(x/b)

|b|F(bu)

shifting

f(x ± x0 )

derivative

linearity

f1

x() ❉ f

2x()

F1

u()F

2u(

)

fu

+ −()

f1

x()f2

x()F

1u(

) ❉ F

2u(

)

f1

x() ★ f

2x()

F1

u()F

2u–

()

ej2π

x0 u

±F

u()

f1

x()f2

x–(

)F

1u(

) ★ F

2u(

)

ej2π

xu0

±f

x()F

uu

0+ −

()

fk()

x()j2π

u(

)kF

u()

j2πx

–)

kfx()

Fk()

u()

a1 f

1x()

a2 f

2x()

+a

1 F1

u()

a2 F

2u(

)+

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

LIN

EA

R, S

HIF

T-INV

AR

IAN

T (LSI) S

YSTE

MS (1

-D)

•Lin

ea

r: ou

tpu

t of a

su

m o

f inp

uts

is e

qu

al to

the s

um

of th

e in

div

idu

al

ou

tpu

ts

•sh

ift-inva

rian

t: syste

m re

sp

on

se d

oes n

ot c

ha

ng

e o

ver s

pa

ce

Syste

m e

qu

atio

n

• w

here

h(x

) is th

e s

yste

m im

pu

lse re

sp

on

se

g1

x()a

f1

x() ❉ h

x()=

g2

x()b

f2

x() ❉ h

x()=

gx()

af

1x()

bf

2x()

+[

] ❉ h

x()=

af

1x() ❉

hx()

bf

2x() ❉

hx()

+(

)=

g1

x()g

2x()

+=

gx()

fx() ❉

hx()

=

EC

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PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

•Fou

rier tra

nsfo

rm o

f syste

m e

qu

atio

n

• w

here

H(u

) is th

e s

yste

m tra

nsfe

r fun

ctio

n

• H

(u) is

a c

om

ple

x filte

r tha

t mod

ifies th

e s

pectru

m F

(u) o

f f(x)

•For c

om

ple

x fu

nctio

ns G

, F a

nd

H

Gu(

)F

u()H

u()

=

am

pl

Gu(

)[

]a

mp

lF

u()

[]a

mp

lH

u()

[]

=

ph

ase

Gu(

)[

]p

ha

seF

u()

[]

ph

ase

Hu(

)[

]+

=

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

1-D

CA

SCA

DED

SYSTE

MS

N c

asca

ded

LSI s

yste

ms

•Sin

gle

syste

m e

qu

iva

len

t

wh

ere

hn

et is

the n

et s

yste

m im

pu

lse re

sp

on

se

. . .

f(x)g(x)

h1 (x)

hN

(x)

gx()

fx() ❉

h1

x()[

] ❉ h

2x()

{}…

❉ …

hN

x()=

f(x)

hnet (x)

g(x)

hn

etx()

h1

x() ❉ h

2x()…

❉ …

hN

x()=

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

FO

UR

IER

TRA

NSFO

RM

EX

AM

PLES

Ex 1

. sin

c(x

/2) ❉

sin

c(x

/3)

•Con

volu

tion

in th

is c

ase is

very

diffic

ult!

•Ta

ke th

e F

ou

rier tra

nsfo

rm

•Ta

ke th

e in

vers

e F

ou

rier tra

nsfo

rm

2rect

2u

()

3rect

3u

()

⋅6

rect3

u(

)=

21/4-1/4

31/6-1/6

x

uu

61/6-1/6

u

=

63 ---sinc(x/3)2

sinc(x/3)=

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

Ex 2

. Fin

d s

pectru

m o

f sq

ua

re w

ave w

ith a

DC b

ias

•W

rite s

qu

are

wa

ve a

s c

on

volu

tion

•Ta

ke th

e F

ou

rier tra

nsfo

rm

• s

pectru

m is

com

b fu

nctio

n, m

od

ula

ted

by s

inc fu

nctio

n, s

am

ple

d a

t freq

uen

cy in

terv

al ∆

u =

1/5

, i.e

1/p

erio

d. z

ero

s a

t u =

n/2

, n =

± 1

, ±2

, . . .

+1

-15

10-5

1 f(x)

x

. . .. . .

fx()

15 ---rectx2 ---

❉

com

bx5 ---

=

Fu(

)15 ---

25

sinc2

u(

)co

mb

5u

()

⋅⋅

⋅⋅

=

2sinc

2u

()co

mb

5u

()

=

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

• if s

qu

are

wa

ve p

erio

d P

= 2

x p

uls

e w

idth

, we h

ave th

e c

lassic

sq

ua

re w

ave s

pectru

m a

t

• u

= 0

, ±1

/P, ±

3/P

, ±5

/P, . . .

? Verify

the a

bove s

tate

men

t for a

n a

rbitra

ry

perio

d P

• s

inc(2

u) a

nd

com

b(5

u)

-0.4

-0.2 0

0.2

0.4

0.6

0.8 1

1.2

01/2

13/2

-1/2-1

-3/2

1/52/5

3/54/5

5/56/5

7/58/5

-8/5-7/5

-6/5-5/5

-4/5-3/5

-2/5-1/5

21/5u

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

• s

inc(2

u) tim

es c

om

b(5

u)

-0.4

-0.2 0

0.2

0.4

0.6

0.8 1

1.2

01/2

13/2

-1/2-1

-3/2

1/52/5

3/54/5

5/56/5

7/58/5

-8/5-7/5

-6/5-5/5

-4/5-3/5

-2/5-1/5

2/5

com

b,

mod

ula

ted

by

sin

c fu

nctio

n

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

•Sca

ling

Pro

perty

•W

idth

of F

(u) is

invers

ely

pro

portio

na

l to w

idth

of f(x

)

rect(x)

x

rect(x/2)

x

uu

sinc(u)

2sinc(2u)

rect(x/3)

x

3sinc(3u)

u

F123

1/2-1/2

1-1

3/2-3/2

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

δδ(2x)

x

δδ(x)

x

δδ(2x/3)

x

uuu

cos(πu)

2cos(2πu)

3cos(3πu)

F

1/2-1/2

1-1

3/2-3/2

1/2

13/2

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

δδ(u)/2

u

δδ(u/2)/4

u

δδ(u/3)/6

u

Fcos(2πx)

xxx

cos(4πx)

cos(6πx)

1/2

1/2

1/2

1

2

3

-1

-2

-3

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

Su

perp

ositio

n P

rop

erty

•Fou

rier tra

nsfo

rm o

f su

m o

f fun

ctio

ns e

qu

als

su

m o

f their in

div

idu

al

Fou

rier tra

nsfo

rms

1 + cos(2πx)

x

δ(u)+δδ(u)/2

u

F1/2

1-1

1

cos(2πx)+cos(4πx)

δδ(u)/2+δδ(u/2)/4

u

1/2

1-1

2-2

x

EC

E/O

PT

I533 Digital Im


r. Robert A

. Schowengerdt 2003

1-D

MA

TH R

EV

IEW

SYSTE

M A

NA

LYSIS

WITH

THE F

OU

RIE

R TR

AN

SFO

RM

• Th

ree a

pp

lica

tion

s

Application

Given

Find

Spatial Dom

ainF

ourier Dom

ain

system output

f(x), h(x)

g(x)

system

identificationf(x), g(x)

h(x)N

Aill-conditioned

inversionh(x), g(x)

f(x)N

Aill-conditioned

gx()

fx() ❉

hx()

=G

u()

Fu(

)Hu(

)=

gx()

F1–

Gu(

)[

]=

Hu(

)G

u()

Fu(

)⁄

=

hx()

F1–

Hu(

)[

]=

Fu(

)G

u()

Hu(

)⁄

=

fx()

F1–

Fu(

)[

]=

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