x 2-D MATH FUNCTIONS
Transcript of x 2-D MATH FUNCTIONS
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
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PT
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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
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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
!