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Pro
bability
Models
for
Hig
hD
ynam
icR
ange
Imagin
gC
hri
sPal1
,2,R
ick
Sze
lisk
i1,M
att
Uytt
endaele
1and
Nebojs
aJojic
1
1M
icro
soft
Res
earc
h,Red
mond,W
A,U
SA
2U
niv
ersity
ofW
ate
rloo
,Sch
oolofCom
pute
rSci
ence
,Canada
� �� �
Overv
iew
•E
xpan
ddyn
amic
range
byco
mbin
ing
imag
es
take
nw
ith
diff
eren
tca
mer
ase
ttin
gs
•C
urr
ent
tech
niq
ues
assu
me
cam
era
imag
ing
funct
ion
isth
esa
me
mod
ulo
anex
pos
ure
chan
ge
•N
eed
todea
lw
ith
mor
eco
mple
xnon
-lin
ear
tran
sfor
mat
ions
Fig
.1.
Thre
eim
agesofan
HD
Rsc
ene
taken
with
diff
ere
nt
exposu
re
� �� �
Our
Appro
ach
•C
onst
ruct
apro
bab
ility
mod
elw
ith
wea
kpri
ors
for
funct
ions
•E
stim
ate
adiff
eren
tfu
nct
ion
for
each
imag
eus-
ing
only
pix
elin
tensi
tyva
lues
Fig
.2.
Left
tori
ght.
Tw
om
idexposu
reim
ages
and
an
HD
Rco
mposi
teim
age.
� �
� �Im
agin
gD
evic
es
Fac
tors
impac
ting
irra
dia
nce
-pix
elre
lati
onsh
ip:
•ap
ertu
resi
ze(o
rf-st
op)
•sh
utt
ersp
eed
(or
expos
ure
tim
e)
•w
hit
ebal
ance
sett
ings
•IS
Ose
ttin
gs(e
lect
ronic
gain
/bia
s)
•co
lor
satu
rati
onse
ttin
gsan
dge
ner
alD
SP
� �
� �Im
agin
gFunct
ions
� �� �
Para
metr
icForm
s
•G
ross
ber
gan
dN
ayar
(200
3)des
crib
ea
dat
abas
e
ofov
er20
0diff
eren
tre
spon
sefu
nct
ions
•M
ann
(200
0)en
um
erat
espar
amet
ric
form
s,tw
o
mos
tco
mm
onar
e:
f(r
)=
α+
βrγ,
(1)
wher
er
isth
eir
radia
nce
,α,β
and
γar
em
odel
par
amet
ers.
f(r
)=
(
ebra
ebra
+1
)
c ,(2
)
wher
ea,b
,can
de
are
mod
elpar
amet
ers.
� �� �
Sem
i-Para
metr
icForm
s
•D
ebev
ecan
dM
alik
(199
7)
f(r
)=
f(a
r),
(3)
Known
pre
-non
linea
rity
mult
iplic
ativ
ega
ina
•E
stim
ate
h=
lnf−
1 ,usi
ng
smoo
thnes
sre
gu-
lari
zele
ast
squar
es
F=
N∑
i=1
P∑ k=
1
(
h(x
k,i)−
ln(r
i)−
ln(a
k))
2
+λ
xm
ax−
1∑
x=
xm
in+
1
h′′(x
)2 ,
(4)
•M
itsu
nag
aan
dN
ayar
(199
9):
hig
h-o
rder
pol
y-
nom
ialfo
rg
=f−
1
r=
g(x
)=
N∑ n=
0
c nx
n,
(5)
•T
sin
etal
.(2
001)
:w
hit
ebal
anci
ng
ispre
-
non
linea
rity
scal
ing
aan
doff
setb
f(r
)=
f(a
r+
b),
(6)
� �� �
AG
enera
tive
Model
f 1(x
) x 1
1 x 2
1
r 1
r 2
r N
f 2(x
)
f K(x
)
x 21
x KN
Irra
dian
ce (
At a
giv
en s
pati
al lo
cati
on)
Imag
e 1
Imag
e 2
Imag
e K
x 22
Fig
.3.
Apro
bability
modelfo
rim
age
pix
elval-
ues,
irra
dia
nce
sand
imagin
gfu
nct
ions.
•G
ener
ate
anir
radia
nce
r ifr
omunifor
mp(r
i)fo
r
pix
ello
cati
oni
inim
age
k
•f
k=
fk(r
)is
dis
cret
ized
imag
ing
funct
ion
•vk(r
)is
leve
l-dep
enden
tva
rian
ce.
p(x
1...K
,1...N
,r1.
..K
,f1.
..K
,v1.
..K
)=
N∏ i=
1
K∏ k=
1N(x
k,i;f
k(r
i),v
k(r
i))p(r
i)p(f
k)p(v
k).
(7)
� �� �
AG
enera
tive
Modelfo
rFunct
ions
•G
ener
ate
(i−
1)th
der
ivat
ive
byin
tegr
atin
gth
e
ith
wit
hin
tegr
atin
gm
atri
xA
0
•G
ener
ativ
em
odel
for
funct
ions:
random
vari
-
able
sfo
rder
ivat
ives
f′′,f
′an
dsc
alar
sb 1
,b0
•E
nco
de
pri
ors
onsm
ooth
nes
s(s
econ
dder
iva-
tive
)an
dfir
stder
ivat
ive
b 1
b 0
f '' f ' f
b 1
b 0
f ''
f
z f
Fig
.4.
Illu
stra
ting
agenera
tive
modelfo
rfu
nc-
tions
as
aB
ayesi
an
Netw
ork
.
� �� �
Pri
ors
for
Functions
•M
argi
nal
dis
trib
uti
onof
fca
nbe
wri
tten
p(f
(r))
=N
(f;A
µz,Φ
),(8
)
wher
eΦ
=A
ΨA
T.
•C
lose
rela
tion
ship
bet
wee
neq
uat
ion
(8)
and
smoo
thnes
sre
gula
riza
tion
met
hod
s.
� �� �
Optim
ization
ofth
eM
odel
•O
pti
miz
elo
gof
the
mar
ginal
pro
bab
ility
(in-
trac
table
)
•Soln
:C
onst
ruct
vari
atio
nal
bou
nd
onlo
g
mar
ginal
use
Dir
acdel
tas
forQ
.
−E
Q{l
ogP
({x
i,k},{r
i},{
fk},{v
k})}
=
−
[
K∑ k=
1
N∑
i=1
logN
(xk,i;f
k(r
i),v
k(r
i))
+
N∑
i=1
logp(r
i)+
K∑ k=
1
logp(f
k)+
K∑ k=
1
logp(v
k)]
.
(9)
� �� �
Itera
te:
Update
fk(r
)th
en
r i
•V
aria
tion
alpar
amet
ers
are
esti
mat
esof
func-
tion
san
dir
radia
nce
s
•Set
the
der
ivat
ive
of(9
)w
.r.t
.va
riat
ional
pa-
ram
eter
sto
zero
fnew
k=
[
ΛT kΣ−
1k
Λk
+Φ−
1]−
1[
ΛT kΣ−
1k
xk
+Φ−
1 Aµ
z
]
•M
AP
valu
esfo
rrnew
i=
arg
min
r
(
−K∑ k=
1
(
logN
(xk,i;f
k(r
),vk(r
)))
−lo
g(p(r
))
)
•U
pdat
epri
or
para
met
ers
for
funct
ions
usi
ng
Exp
ecta
tion
Max
imiz
atio
n(E
M)
step
s
� �� �
Resu
lts
Fig
.5.
(Upper
left
tolo
wer
right)
hig
hest
,m
id-
dle
,lo
west
gain
image.
HD
Rim
age
with
lu-
min
ance
sre
mapped
usi
ng
the
glo
bal
funct
ion
inR
ein
hard
et
al.
(2002)
010
020
030
040
050
060
0
050100
150
200
250
Est
imat
ed Ir
radi
ance
Pixel Intensity
010
020
030
040
050
060
0
050100
150
200
250
Est
imat
ed Ir
radi
ance
Pixel Intensity
010
020
030
040
050
060
0
050100
150
200
250
Est
imat
ed Ir
radi
ance
Pixel Intensity
100
200
300
400
500
600
050100
150
200
250
Est
imat
ed Ir
radi
ance
Pixel Intensity
Fig
.6.
(Left
toR
ight)
Itera
tion
0,1
,and
6.
(Far
Rig
ht)
Fin
alitera
tion,cu
rves
use
dfo
rFig
.2.
Fig
.7.
Mappin
gth
elo
west
gain
tohig
hest
.(L
eft
toR
ight)
(1)H
ighest
gain
(2)M
ultip
lica
tive
funct
ion
(3)
Multip
lica
tive
and
bia
s.(R
ight)
Our
Alg
ori
thm
.
