O n a d a p tiv e n a n I n tr o d u c tio n
D a v id A lle y s s o n &
A s s u m in g th a t th e p h o to re c e p to r re s p o n s e o f th e h u m a n v is u a l s y s te m
is a d a p tiv e a n d n o n -lin e a r, w e c a n d e riv e
m a th e m a tic a l p ro p e rtie s th a t c a n a c c o u n t fo r b o th c o lo r d is c rim in a tio n a n d c h ro m a tic a d a p ta tio n . T h is c o u ld b e d u e to th e p h o to re c e p to rs re s p o n s e to illu m in a tio n ,
0 .4
0 .2
0 .1
W e a n d th e m o
c a n a rg u e n o n -lin e a re la y b e tw d e le d w ith
L
L
W h ic h c a n b e re w ritte n u s in g a n a u to m a tic g a in c o n tro l a d a p ta tio n a s fo llo w :
0 .2
0 .3
0 .4
0 .5
T h is m a trix c a n b e d e c o m p o s e d a s x
0 .7
0 .6
th a t p h o to re c e p to r rity in th e v is u a l s y e e n re tin a a n d c e re a d a p tiv e n o n -lin e a
M
S S
0
l
X P h o to re c e p to r tra n s d u c tio n
0
x
A c h ro m a tic a n d c o lo r o p p o s itio n s c o d in g
c
A
C C
0
D D
0
a
Y
c
d W
120
L
y
R J B
450
450
400
400
100
M
350
S
80
60
250 200
300 250 200
150
150
100
0
M
0
50
100
50
0
L
150
200
250
300
350
0
50
0
L
-50
400
100
150
200
250
300
0
400
40
200 150
20
T h u s
L
120
140
160
M
W h ic h is e q u iv a le n t to v o n K rie s lin e a r a d a p ta tio n m o d e l
100 50
60
50
40
30
20
90
100
L
110
120
130
140
150
-50 15
160
20
25
30
M
35
40
500
500
100
400
400
S
200
40
50
100
150
200
300
350
400
50
100
150
L
-100 200
250
300
350
0
400
50
40
30
160
180
-50 70
120
300 250
90
100
L
110
120
130
140
150
160
M d
0
S
80
is th e
n n e ls . e rro r
T h e in c a n b e o p p o n a d a p ta
ig g e r e rro r b u t c o n s is te n t d e te rm in a tio n . p re s e rv in g + e ig e n v a lu e s .
60
60
40
P
20
20
0
20
40
P
0 60
80
A s p re v io u s ly , w e c a n th b e tte r p re d ic ts c o rre s p o n n o n -lin e a rity b y its e lf. N a n d th a t it is n o t p o s s ib le lin e a r o p tim iz a tio n , s u c h c h o o e u se N a k lm s a
s e n to th e H a -R u s n d l 'm
c o P E h to 's '
100
e re d in o te to a s
120
0
20
40
fo re te s t if th e th re g c o lo r d a ta th a n p th a t th e th re e -la y e d e te rm in e a ll m o d e ig e n v a lu e s d e c o m
m p u te m a trix n la w o c a n b e
th e to n L e x p
c o rre sp o n tra n s fo rm M S . N o w re sse d a s
60
e -la y e r m h o to re c e r m o d e l e l p a ra m p o s itio n
d in g X Y Z th e c a lin e
80
o d p to is n e te .
c o lo r d a ta to L M S . o rre sp o n d a r re la tio n
100
E ig e n V a lu e s H P E L in e a r B ra d fo rd S h a rp 3 -la y e r m o d e l
6 .9 5
1 4 .5 1 9 .5 3 8 .3 0 6 .7 6
-50 15
20
25
30
35
M
40
45
50
55
60
65
10
20
a b ility to u s e in te rp re te d th e n t c o lo r is n o tio n . A ls o a n
W e h a v e sh o p h o to re c e p to d is c rim in a tio lin e a r p h o to r a n d v o n K rie a firs t a p p ro x
S
80
0
30
40
50
60
70
80
90
E i g e n v a l u e d e c o m p o s i t i o n i s n o t p o s s i b l e w i t h L a m 's d a t a a s th e e ig e n v a lu e s b e c o m e c o m p le x . W e u s e th e s in g u la r d e c o m p o s itio n in s te a d , s o w e w rite :
120
e l r
o t lin e a r r b y a in lm s W e th e n a n c e .
T h e v o n re s u lts w o f th e re sp a c e a n p h y s io lo m o d e l. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
13. 14. 15. 16.
0 80
a trix . a c e a n a l c h a a lle r
10
200
50
0
140
100
100
50
120
80
150
20
L
M
400
S
250 200 150
100
60
350
300
100
80
40
450
350
S
60
10 60
20
G F 450
70
M
65
0
0
80
60
100
-100 250
55
200
0
L
0
50
300
100
20
0
S
300
45
ll 3 x 3 m c o n e sp in d iv id u a lu e s. S m
W e fo u n d th a t th e le a s t-s q u a re s e rro r c a lc u la te d in X Y Z is s ig n ific a n t s m a lle r th a n w ith o th e r m e th o d , a s s h o w n in th e fo llo w in g ta b le .
0
80
120
60
120
150
50
W R J B
80
100
250
100
-50 70
180
80
200
0 100
60
300
S
250
L -M
30
80
40
350
300
S
50
60
20
400
350
60
10 40
M
-50
350
G F
70
M
100
50
20
T h e re fo re , m a th e m a tic a lly , p h o to re c e p to r a d a p tiv e n o n -lin e a rity is c o m p a tib le w ith th e c o lo r d is c rim in a tio n m o d e l a n d c h ro m a tic a d a p ta tio n m o d e l.
70
100
W e h a v e sp a c e . W a p p ly th e b e tw e e n
350
S
300
40
W h e n c o n s id e rin g c h ro m a tic a d a p ta tio n , w e w is h to m a p th e a p p e a ra n c e o f a c o lo r u n d e r tw o a d a p tin g c o n d itio n s . U s in g p h o to re c e p to r fu n c tio n a s a p p e a ra n c e m o d e l, th is le a d s to
80
Modele lineaire:Bradford dans XYZ:lam.da.dat
100
0
T h is m o d e l is a b le to a c c o u n t fo r e llip s o id s in e v e ry o rin e ta tio n a s s h o w n in a p re v io u s p a p e r (A lle y s s o n & H e ra u lt C R A ).
m
n
is c r im in a tio n
120
40
G a n g lio n fu n c tio n 0
fu S e v
Modele lineaire:eig dans XYZ:lam.da.dat 120
s m
A
w h e re D is a d ia g o n a l m a trix a n d M a lin e a r tra n s fo rm a tio n fro m X Y Z to L M D is c o m p o s e d o f s c a lin g fa c to rs fo r th - M a n d D a re e ig e n v e c to rs a n d e ig e n b u t n o t m e a n in g fu ll d e c o m p o s itio n . - H u n t, R L A B , M = H P E , D = M i/M j, b w ith v o n K rie s m o d e l. - B ra d fo rd , D = M i/M j, M le a s t s q u a re - F in la y s o n & S u s s tru n k W h ite -p o in t
s a re n o t th e o n ly s ite o f a d a p ta tio n s te m . G a n g lio n c e lls , w h ic h fo rm b ra l c o rte x , s h o u ld a ls o b e rity .
M
0
w ith
l
r d tio
A w a y fo r e s tim a tin g T is to m in im iz e th e le a s t-s q u a re s e rro r b e tw e e n th e e s tim a te g iv e n b y th e lin e a r m o d e l a n d m e a s u re d S .
0 .6
a d a p ta tio n s ta te
p e ra te s a s a s c a lin g fa c to r, c o n d itio n a n d th e p o s itio n in tio n a llo w s c o n v e rtin g sp h e re s.
lo ta
T is a 3 x 3 m a trix
c o lo r d a ta .
n o n -lin e a rity o th e a d a p ta tio n h is tra n s fo rm a e llip s o id s in to
c o a p
W e c a ll P a m a trix o f c o lo r trip le ts o f N c o lo r s u rfa c e s u n d e r illu m in a n t i a n d S a n o th e r s e t o f c o lo r trip le ts c o rre s p o n d in g to th e s a m e p e rc ie v e d c o lo r u n d e r a s e c o n d illu m in a n t j. A c c o rd in g to v o n K rie s c h ro m a tic a d a p ta tio n m o d e l, th e re la tio n s h ip b e tw e e n c o rre s p o n d in g c o lo rs is lin e a r. S v a lu e s c a n th e re fo re b e d e riv e d a s a lin e a r c o m b in a tio n o f P v a lu e s .
y
0 .8
p ro c e s s in g , a n d s h o w h o w w e c o u ld p re d ic t c o rre s p o n d in g
d a p tiv e d in g o n sp a c e . T in a tio n
r d
C h r o m a tic a d a p ta tio n
w ith
th e d is c u s s io n to a th re e -la y e r m o d e l o f re tin a l c o lo r
T h e a d e p e n c o lo r d ic rim
fo a
w ith T h e n o n -lin e a r m o d e l tra n s fo rm s th e e llip s o id , s u c h th a t it b e c o m e s
d is c rim in a tio n a n d c h ro m a tic a d a p ta tio n d a ta . W e e x te n d
X
r ity a tic
E llip s o id n u m b e r i is d e fin e d b y th e fo llo w in g e q u a tio n
a n d a n a u to m a tic g a in c o n tro l fu n c tio n , w e c a n d e riv e c o lo r
If w e ta k e th e lin e a r a p p ro x im a tio n a rro u n d a c o lo r
e a o m
C o lo r d is c r im in a tio n
s ta te . A s s u m in g th e N a k a -R u s h to n n o n -lin e a r fu n c tio n
T h e p h o to re c e p to r re s p o n s e is g iv e n b y N a k a -R u s th o n la w : lig h t v a lu e s
-lin c h r
S a b in e S ü s s tr u n k L a b o r a to ir e d e la c o m m u n ic a tio n a u d io - v is u e lle L C A V - E P F L S w itz e r la n d { D a v id .A lle y s s o n ,S a b in e .S u s s tr u n k } @ e p f l.c h
w h ic h is n o n -lin e a r a n d v a rie s a c c o rd in g to th e a d a p ta tio n
P h o to r e c e p to r r e sp o n se
o n d
w n th a rs c a n n a n d e c e p to s a d a p im a tio
e ig a t t c in c
e n v th e o n s o m
a lu e d e c o e n c o d in g ta n t a n d v p le te a d a p
m p o s itio n in to lu m in a rie s w ith ta tio n c o u
o n a n c th e ld b
L a m 's d a t a e a n d sta te o f e c o n s id e re d .
C o n c lu s io n t a m o d e l e x p la in b o c h ro m a tic r m o d e l, w ta tio n in L n o f th e re
o f a d a p tiv e n o n -lin e a rity in th v is u a l p h e n o m e n a o f c o lo a d a p ta tio n . A n a d a p tiv e , n o n h ic h is e q u iv a le n t to lin e -e le M S s p a c e , c a n b e c o n s id e re d e l p h e n o m e n o n .
K rie s c h ro m a tic a d a p ta tio n m o d e l d o e h e n a p p lie d in c o n e s p a c e . H o w e v e r, a tin a th a t in c lu d e s th e re la tio n s h ip o f e n d a fte rw a rd s in to o p p o n e n t c o lo r s p a c e g ic a l p ro c e s s w ith th e v o n K rie s c h ro m
r
m e n t a s
s n o t g iv e g o o d th re e -la y e r m o d e l c o d in g in to c o n e c o u ld re la te th e a tic a d a p ta tio n
D. L. MacAdam. “Visual sensitivities to color differences in daylight.” Journal of Optical Society of America, 32, pp. 247-273, 1942. K.M. Lam. “Metamerism and color constancy ” Ph.D. Thesis, University of Bradford, 1985. G.D Finlayson and S. Süsstrunk. “Pe rformance of a chromatic adaptation transform based on th spectral sharpening,”Proc. IS&T/SID 8 Color Imaging Conference , pp. 49-55, 2000. D. Alleysson and J. Herault “Photoreceptor non -linearity can account for the MacAdam ellipses,” Perception Supp., 26, pp. 106, 1997. K.I. Naka and W.A. Rushton “S -potentials from colour units in the retina of fish (Cyprinidae),” Journal of Physiology, 185(3), pp. 536-555, 1966. R.D. Hamer and C.W. Tyler “Phototransduction: Modeling the primate cone flash response,” Visual Neuroscience, 12, pp. 1063-1082, 1995. D.A. Baylor, B.J. Nunn and J.L. Schnapf “Spectral sensitivity of cone of the monkey macaca fascicularis,” Journal of Physiology, 390, pp. 145-160, 1987. S. Süsstrunk, J. Holm and G.D. Finlayson, “Chromatic adaptati on behavior of different RGB sensors,” Proc. SPIE, vol 4300, pp. 172-183, 2001. T. Yeh, V.C. Smith and J. Pokorny, “Chromatic discrimination with variation in chromaticity and luminance: data and theory,”Vision Research, 33, pp. 1835-1845, 1993. M.A. Webster and J.D. Mollon “The influence of contrast adaptation on color appearance,” Vision Research, 34, pp. 1993-2020, 1997. D. Alleysson and J. Hérault “Variability in color discrimination data explained by a generic model with non linear and adaptive processing,” Color Research and Applicatio n, 26, 2001. R. Shapley and C. Enroth -Cugell, “Visual adaptation and retinal gain controls,” In Progress in Retinal Research, Osborne, N. N. and Chader, G. J., editors, volume 3, pp. 263 -346. Oxford: Pergamon Press. G. Wyszecki and M.S. Stile,Color Science, Wiley, New York, pp. 654-676, 1982. Additional figures and author bibliography can be found at: http://lcavwww.epfl.ch/~alleysso/CGIV2002. Corresponding color data sets can be found at: http://colour.derby.ac.uk/colour/info/catweb/ M.D. Fairchild, Color Appearance Models , Addison-Wesley, 1998.
