Universitat Autònoma de Barcelona
!# $%$ #&'$ #( $
$ # * #
# + *,
- ,
. - -- * #
## * /
## / ,
0 -# *
#-
. # ,
.# 11 *
2*- * *###-
31 #
4
* # 5 #
# . * # #
11 # # +# 11 5 .
# *
# # +# */
* * +# .
6
. +# * # #
# 4 # #
.#,4 *7#2.473 *#
1 #
* *# * -. #
# * * # #
# * #
#
.# ##
*
/ - * . * * -*
# # ## *###
. ##
# * .
+# *# , 8
## #*#
7 9 : & # ; # 1
&# # < 5 % #
5 +)/ +
/ & 9 ;:/1
< :/ & / + = #:
9 9 # # #& :
/+ / # 1 /
: > ##
5 :/ 9 ? ) = #:
< ## & :/ 9 # % :&
7 # :
/ ;/ 2
& 5 3; 1#
& #
! / : 5 :/ 9 + / # # ,
; 5 # #
# ## # 9 > /
# # # +# # ) /
# % 5 < &
#;:)/& # +,
# ## :/%
:+# 7 :#
# & )9:6
<:; :+# # # # / ,
# # > #
# .# 4 7
·
)# 2.473 / # ,# 5 &:#
/ 9 # 7 # 9 :5 9
#; /# # /
: # &# %
< # ; :; ;
# # +
# & :/& /; /
# ## &# #
/ 9 ; #&## & 7
::/# ## : & ;
: 7 +# #
:& ##
# / #
$
$ 4# @
$674$ A
$67 $ 4# B
@ $ 4#
@ 6 C D
$ $
$
$
E > $ F C11,>
B .A @
.0 (4 $ F E
$ . D
@ $ F > #
F # $ F
E $ # $ F @
E F ,11$ ,F
E C11$ ,F B
B 4 G
!"#$#"! %&$!
@ H* $ ,F 7+# @
@ F*!4 C11$ F @B
@ @B
@ H @D
@@ C11$ ,F ! !8
@@
@@ H G
@ ! G
' (
$ ! E
> ## " .@!,IC EB
@ A $ ,F 4 C B
#C C11$ F B
.4 ,IC B
.! ,4 C
@ ..#,4 *$ $C @
E .C$ ,F D
E ..#,4 *7#$C D
E C1147 DB
E@ H DG
B ! GE
!! !!! ))
E $ F $ #( ## GG
E $ F >
E $ 4#.
E $ @
E@ $
E $ ,F $ ,
E@ I 6 $ , !# G
E 6 6 H
E 7+#
EE ! G
* !! # +
B E
B CJ - D
$#"
! '
4 11 K!
@ 4 , +# @E
@ 8# # ,
@@ $ +# * # @
@ $ # +# #
$67<8
@E + # +# B
@B $ $ . #2$$.3$67+ ,
+# D
> I ED
H
M AE f it
2×10 −2
3 IEG
@ $ # # # I 2I3
1 *K
1 B
> 4 ,I BG
E H
M AE f it
2×10 −2
3 4 ,I,
B $ # 8 1
*K 1
> ! ,4
D H
M AE f it
2×10 −2
3 ! ,4 ,E
G H 2I2I3
4 ,I24I3 ! ,4 2!433 E
$ # 8 1
*K 1 E
> .#,4 *$ D
H
M AE f it
2×10 −2
3 .#,4 *$ ,
G
@ H # # # *2
I 2I3 4 ,I 24I3 ! ,4 2!43
.#,4 *$ 2.4$33 G
$ # 8 1
*K 1 D
E > .#,4 *7# 2
+# -# 3 G
B H
M AE f it
2×10 −2
3 .#,4 *7# ,
G
H # # # *2
I2I34 ,I24I3! ,4 2!43
.#,4 *$ 2.4$3 .#,4 *
7#2.4733 G
D $ # 8 1
, +# K4J GE
G 4 # G
E H 8* -
G
E ! +#,
E@ H +# >
E
E > +# #,
B
EE !
+# D
EB H +# >
D
E > +# #,
G
$ , , $67
HI
$ , , $67 G@ ,
LMN E
@ 7+ # +# , - G
.* 8
E 7+ #
4 #
4 - 4# B
$ 1 K
+# 7
@ H < I $67< #*
*
$ 1 C11$ $,
# D
@ 4 +# .+# * #
- #*
# @D
@ > *
+ # 2 # 3
# 5
@@ < +#
* # * -
@ < +# * #
* -
@E 4 # - * ,
2+-*3 B
@B < $67+ #
+# D
$ 1 ##
I EG
$ 1 # <:
7#- EG
@ . #
B
4 . #
N L
+ B
E !# # # B@
B 4 BE
4 ,I # BE
D 7+ # 4 ,I BB
G 4 8 * ,
B
$ 1 ##
# # 4 ,I
.*
1 ! ,4 E
@ $,4
CS(p;t, θ CS )
B.#,4 *$2.4$3
E 1 .#,4 * $
G
B !# # # D
7#,4
ES(p;t, θ ES )
D@D .#,4 *7#2.473 D@
G .47
D
4 8 * ,
DE
0 #
D
# # + # .47
.## G
@ # # + # .47
# # G@
$ # *.47 : 1
K G
E 4J+# ,
# # .47 : 1 G
E # # ## ,
#
E 7+ # ## ##,
, E
E@ 7+ # - B
E 7+ # -
EE 7+ # C -
EB 7+ # +#
E 7+ # # B
ED 7+ # *#
EG 7+ # * 8
#: D
E 7+ # * *
E 7+ # # #*
# 27+ #3
E 7+ # # #*
# 27+ #3 @
, A
(
$ 6 H
$
- $ $ . #
$
. $ F 0
/ $ ,F 4
/ $,4
/ ! ,4
/ ! ,4
%/ 7#,4
./ 0 (4
/ 6,4 $ $
01 < I
01 <IF
(% 7
%/ <:7 4#-
1 I
#(
/% 4/7
/ 4F 4
$
0/( > <4
/1 4 ,I
/1 4 ,I
/2 4J
-/ .#,4 *$
-/ .#,4 *$
-/% .#,4 *7#
-/% .#,4 *7#
. , - #
* - 6#* # #
# * , - J * #
* #
, - C11
# * - * - *#
#
$ ## .
# *,
*@D D # .
$ # ##
-
. - *
- *
0 -
/ * 8 0 *
## # #- **
+#
$ 8# * C
# # # # # ,
# 0 * #
KP Q*# # /
. 8 +
.
. * - *
*###- . 8
* $#
$ # # #,
# 5 -
6 # #
8 # # # #
* ,
# * # ##
*
# *
A ## # * ,
- 6 H .
* * .
# R * . ##
#PBBBQ 7 * +
/* #
.## # * -
5
. # 8## * # # "
, 2$6H3
6$6H +#
+ $6H / #
2 / + #3 -
- # 2 / -3
0 * /
* #
2 :O *:3
0 # ## # J
# +
. * ,
*, - * # . /
# $6H*,
6* $6H PBQPGEQ
# * J
# #
*
. 5
*
. *# # , *
# # K 11 0
+# / ## .
* # 11 #
C * #
##
. #
* $ #
# ##
. # * +# *
6 +# ,*
Q
# # - -
- 5
w 1
w 2
w 3
##
P 1
P 2
P 3
+ #.+#
PEQ2 # 3I
<*PB@Q .I <*"
3! 0 +
Q
# #
P 1
P 2
P 3
"Q = w 1 P 1 + w 2 P 2 + w 3 P 3
*
w 1
w 2
w 3
#
/ 0 + 6*
Q 1
Q 2
+* +"
Q 1 = w 1,1 P 1 + w 1,2 P 2 + w 1,3 P 3 ∧ Q 2 = w 2,1 P 1 + w 2,2 P 2 + w 2,3 P 3
⇓
Q 1 + Q 2 = (w 1,1 + w 2,1 )P 1 + (w 1,2 + w 2,2 )P 2 + (w 1,3 + w 2,3 )P 3
@ -# 0 + $
Q = w 1 P 1 + w 2 P 2 + w 3 P 3
⇓
kQ = (kw 1 )P 1 + (kw 2 )P 2 + (kw 3 )P 3
$ +# *
6 + * $67
,
r(λ)
g(λ)
b(λ)
G@
. * 5 # 2
P 1 = 700.0nm
P 2 = 546.1nm
P 3 = 435.8nm
3 *0
Q(λ)
+# # "Q(λ) = RP 1 + GP 2 + BP 3
23*
R
G
B
# /,232@323"
R =
ω
Q(λ)r(λ)dλ
23G =
ω
Q(λ)g(λ)dλ
2@3B =
ω
Q(λ)b(λ)dλ
23*
ω
* * # @DD
r(λ)
g(λ)
b(λ)
*
400 500 600 700
−0.5 0.0 0.5 1.0 1.5 2.0 2.5
3.0 r( λ )
g( λ ) b( λ )
Wavelength (nm)
Tristimulus values
•
.r
2λ
3 * *#
•
. + #
•
.HI *##
•
7 # * ##6 * # $67
#LMN* HI# .*
# #
λ
6 * #
Q(λ)
/ 2E32B32 3"
X =
ω Q(λ)x(λ)dλ
Y =
ω Q(λ)y(λ)dλ
!Z =
ω Q(λ)z(λ)dλ
"*
ω
* #x(λ)
y(λ)
z(λ)
, LMN#*$674$ A2C3
400 500 600 700
0.0 0.5 1.0 1.5 2.0
x( λ ) y( λ )
z( λ )
Wavelength (nm)
Tristimulus values
#$
6 * # ## #
*+# ## + #*M,
F * #
#
. # LMN # # $67
## # $67< $67< *
* ## !# *
## PBQ $67< # *
## * * ## 6
$67< #* * - #* ,
$67< # LMN#
*/ PE Q"
L ∗ = 116
Y Y n
1 3
− 16
2D3a ∗ = 500
X X n
1 3
−
Y Y n
1 3
2G3
b ∗ = 200
Y Y n
1 3
−
Z Z n
1 3
23
*
X/X n , Y /Y n , Z/Z n > 0.01
X n
Y n
Z n
LMN*
C
X/X n , Y /Y n , Z/Z n < 0.01
L ∗
a ∗
b ∗
### / / "
L ∗ m = 903.3
Y Y n
f or Y
Y n ≤ 0.008856
23a ∗ m = 500
f
X X n
− f
Y Y n
23
b ∗ m = 200
f
Y Y n
− f
Z Z n
2@3
f
X X n
=
⎧ ⎪
⎪ ⎪
⎨
⎪ ⎪
⎪ ⎩ X
X n
1 3
X
X n > 0.008856 7.787
X X n
+ 116 16 X X
n ≤ 0.008856
23
f
Y Y n
=
⎧ ⎪
⎪ ⎪
⎨
⎪ ⎪
⎪ ⎩ Y
Y n
1 3
Y
Y n > 0.008856 7.787
Y Y n
+ 116 16 Y Y
n ≤ 0.008856
2E3
f
Z Z n
=
⎧ ⎪
⎪ ⎪
⎨
⎪ ⎪
⎪ ⎩ Z
Z n
1 3
Z
Z n > 0.008856 7.787
Z Z n
+ 116 16 Z Z
n ≤ 0.008856
2B3
. PE Q## #
* *
$67<# *
# # * - #
* "
0($
0 * # #,
# C##
2H2H3 M *2M3 I 2I3 23 ># 2>33 ,*
2M *,H 2MH3 I,M * 2IM3 ,I 2I3 >#, 2>3
H,>#2H>33 7
(# /# 2-3
2*3 (# +
$ #
$ ###,
+ $
$S + + # *#,
0 + # 8 8
$ #
# *
6 @ #
# 0 # EE E (
,# # *
2 / 3
. * , - # **
# 4PEQ 6 -
* <
. # #
s
* #
s i =
ω
E(λ)S(λ)R i (λ)dλ
2 3*
s i
#i
#*
ω
* # . # #* # "
.# # #
R i (λ)
# HI ,# # CE(λ)
## *S(λ)
O#
A ** ,
-* # * #
. #- * # PBQ * #
# . *
# ## #*
# *
PE Q * # .
# * # *
#$67<# 0 ,
** - * / **
# # !# **- *
/
R i (λ)
S(λ)
E(λ)
* ## *8
. # HI # HI
# # P Q .# HI#P@ Q
# ## #*HI
$67G@LMN #*
. 0
# HI# $67#
* A* # * 6
# HI # 1
# # #
- - # . *#
* #
.
*
8 # 0 * # ,
,# - ##
-
6 @ + # * 6### 2
@23 @233 #8HI
0 * ** #
# * ***/
, . * 2@23
@233# ## #
* HI*# 8
*/ 8
#
23 23
23 23
() ( ) * +
, ) ))
+
-
.
8 ###
# 6 * * +# 8
* ## # # #
*
6 * * #
, ##8
-
6* #
## ,
## # ## .
# ## ##
# # * +#$#E
## HI
# # . #
8##8 8
C23 * + #
8* 0 *
* * - ###
* .
# * #
.##
+, - +
/,) - ,
* . # # ( K PEBQ
R * #
#* 8
# 4
# #
4
# HI,
. #
# 8 .
* */
. # P@EDBGDEB @ EEQ
0## #
# # 6,
#*
PEB Q
0 ## + +
4 8
# # +# PQ .
8#
## 4E + #
23 23
()
$ * ##
* .
/ 6 #
8 PG Q
$ ##* *
.8##**
* *# / 6 #
# # PG Q
A # +# *
*## # *
#,# *
* # *- ##
!!"
. * -
K P Q * + /
8 6* ,
# K ! P DQ # # *
# #* ,# ,
.
11 PDQ* ,
* # .
, #
11 ## -
#
6*# # , - 11,
11 # 6
*
C k
11 * #μ C k
* #s
## #
μ C k (s)
*[0, 1]
. #*
s
C k
* #
t k
. *# # # #
# 11
11
n
#"
n k =1
μ C k (s) = 1
*μ C k (s) ∈ [0, 1], k = 1, . . . , n
2D30 #
s
* # ##
CD
CD(s) = (μ C 1 (s), ..., μ C n (s))
2G3* #
n
#s
#. #
N (s)
s
. *+
CD(s)
"N (s) = t k max | k max = arg max
k=1,...,n {μ C k (s)}
23*
t k
C k
6 # #
K
n = 11
"C k ∈ {Red, Orange, Brown, Y ellow, Green, Blue, P urple, P ink, Black, Grey, W hite}
C #
/ 23 -*
# H
>-
-
# $% &
. 1 + # 6 $# * #
* - * *
#
6$# #* # #
- - .
8 # *
,
$#@ # #+# 6#*
8 # *- /
# +# 11 6 #
# * # # # #
+# * # # # .
1 # # * - * ##
$# * #
# , - 6
# # ## , # #
# .#,4 *7#,
2.473 * # #
#
$# E ## # # .47
. #
8 - #
# # , # # # 6#
# #
* ## #
C$#B
1 .
+#
$ 1 8#
C # # #
* - # # 8# * # ,
6 # * 1 #
8
$% ' (
$ ## *
0 * # *
.* #
**@D D A /
. * *
20(43# #
+ *
6 D # . M *
## * 8
PBQ 6 * # * #
4 0 1PEQ# # + 8
#
# * *
DE@# # * # PB@Q 4
* # #
.
## # # ##
## #
*8#
# *- * * - # "
J * # ,
# 0(4 #
$ ## # 8 # ,
# !# # # # 7
#8 # *,
. # 2 3 2 3 23
#* * # + $
# + ,
* + #
. # # *
KT! PDQDDB 4 #
8 *8/ PE Q 6
* PDDQ . *
4 - 4#P@@@Q C *
400 500 600 700
0.0 0.2 0.4 0.6 0.8
1.0 S M L
Wavelength (nm)
Relative Absorption
0 )
0 * 0)1$$2$3
0 # #
# 0(4. #
* ## * *,
. # *
#
* *
4 ,
* 0(4 # #
.# #PBGQ# PQ,
# 8 .
** # + *
+ * * # +
# +
## * # # 0PBGQ 0:
## * ## # #
+ 0 *
0 U P @Q ## # #
## # .
# ## # #+# .
## * / 7T 0
PEQ
P D GQ ## - #
<I F 2<IF3.*
* # ! (
PQ 8# # ## <IF
/ - A # +
* # ## .#
“RG”
.* # ##
“RG”
* * . #“Y B”
. #* # #*
“W h − Bl”
7 ## * 8#
4# ##
# # PDDQ
K- # PD@Q #
# # +# .
* * , ##
. * - *8 ,
, *+ 0 !( ! ( P@Q # #
, ## +# #
* K- #
# # 0: ## + #
.# # # +.- *
##
+## PD Q 6!( !( P@QV
## 2 3
# W
! ( ! ( PQ # # +
* J * ,# #
* #,
* + .+ #
+/ V$ $W+
# # N-
PBQ # / - + (
,## # 0 *4
PQ ,# ( X
# # 6 PQ
( * 8 - 05-
PB Q * -
* 8 -( (D N-PB@Q
* ( 0 * * -
+ V W + * -
# + PEDQ
$ - ##
# .**
## ,
* # - *
“W h − Bl”
#
“RG”
“Y B”
# * 0 *
# -# ##
PQ # # +# PQ
# ** - # #
* ,
** * - *
* # #
*
PBEGGQ
) $
$
* * # * A
## +#
* /
A +
.*
- # $
#+ # 8
* - 7 PG Q
# K P Q 8
*# * - #/
K #- ,
82+# D *3 J
* - K* ## #
+ 8
. # V W
# # . # #"
•
6 +#
•
6 *•
6## # 5•
6 # ###- 8
#-#-
6 # #
# .# #"
•
. # #,•
5•
C * # #•
6 + R # #+
.* -* 8* * #
6# :
# * .#*
# 6
#5* - # * 8- C
#*
* 4
# + #2 3
! * #* #,
* - D K
+ * -"
. *
.
8* # .
.
7 * - * * #-##
8
*
# * *+
"
•
46". *-•
466"•
4666" *2 3•
4 6(" *2 *# 3