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Page 1: 0/ &21436573cs.joensuu.fi/spectral/publications/tossavainen2.pdf · !#"%$ &% ')(#*+('%$ ,-*/.1032%.4 $ .45 6 798+:@?ba1:1:

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VminF3;%>%C[R+p#GHGH:<C+CPD�D�G�ST>#NU8+:1S@:zv�R+>�S@:1GH:<S@:1>%nK;�F#C |

� >#Q�;�:18KCP:1G�Sx=H=�C+CL>)>�J�F)8+:<CP:1:<;�:18+D�G R+p#GHGH:<C+CPD�D�GVp

GHD-��>%CP:1:<;�:18+D�NU8+:�R+p#GHGH:<C+CPD�D�NU8+:Vg,p

#

Vg,p = αpVp + βp����| 6#� �

S@:18+8Ppαp

F)G�N#F)S@J�F)GHD�GUCP:1Gp

;�:1yHD�F3;#>#Q�;�:18KCP:1S@D�G �o;%>#Q�;�:18KCP=H8 �@��>#:1GbR+>βp

F)G�;#>#Q�;�:18KCP:zvS@D�GOF#��8+D-C |U��Q�CPD-I�8�D�A1D�N�CPn+F)GH:18Pp%CPD�D�G�;�:1n+nP>#G

ipR+>9;#>#Q�;�:18KCPD-CP=HGxR+p#GHGH:<C+CPD�D�G

Vg,p;#p#A1:1A<A1p9F)G

D�J�p#A1:1GHD >)>#n+:<GHD�G�|

ip =

(Vg,p − Vc,p)γp ;Vc,p < Vg,p

0;Vc,p > Vg,p

����| 6 � �

S@:18+8Ppγp

F)G�D�NU8+J�F)GHD�G�C+CP:�5-RKF#CL>^NU=�CP8+=�CL>)>#Gq��>#S@ST>#NU8+:�| ��A1D�D�GH8Ppw��>#S@ST>�F)Gq>#nK;)F)A<CL>)>#G ;%p�vA1:1A<A1p 63vc��|H��>#S@ST>�n+:1:1JHJH=H=�nP>%CPN%>#:18+D-;%>#8KCP:�G�p3IUCPX)G�8P:<8Lp%:18+D�8KC/p nP>#N#D�GUCPD�D-8+CL>^R+> 8+D RKF)=HyH=�CL>)>#GSTp)p#n+:<CKCLp#S@p)p#G RKF)N�>#:18+D�A1A1DwG�p IUCPX)A1A1DwD�n+:1NU8+D�D�G�|

Vc,pF)G�N#F)S@J�F)GHD�G�CP:1G�J�N�>%CPN%>#:18+=%R+p#GHGH:<CPD#|

�wA1D-CPD-CL>)>#G�5�D-C+CLp��oF)8��oF)n+:1D�S@:18+8+:1F#G�8+=H=Hn+=H=H8�nP>#yH:1F)S@D-CPn+:<8+D�8KCP: S@:<CL>%C+CP=HG�>�nP:<:1JHJH=H={A1:1GHD/v>)>#n+:18+D�8KCP:�8+=H:1QHNU=�;�:1n+nP>#8KCL>�|

Mλ,p = kλ,pip,����| 6 % �

S@:18+8PpMλ,p

F)G{8+J�D�N�CPn+:1GHD�G�8Pp%CPD�:1A<I)D�NU8+:<CL>#GH8+8+:[W/m2µm]

R+>kλ,p

F)G�8+J�D�N�CPn+:1GHD�G{;%>#NU:1FH|��QHyH:18KCLp#STp#A<A�p ����| 6)6 �L5 ����| 6#� ��R+>�����| 6 � � N%>)> ;#>)>#G ����| 6 % �8P>)>#y�>)>#G

Mλ,p = kλ,p

(

αp

[

(Vmax − Vmin) dp(2n − 1)

+ Vmin

]

+ βp − Vc,p

)γp

.����| 6D& �

�[yHD�A1A1:18+D�8KCLp^N%>)> ;#>#8KCL>wG�p#QHy�p)p#G�5)D-C+CLpwG�p IUCPX)Gq8+J�D�N�CPnP>)>#A1:18+D�G�8Lp�CLD-:1A<I�D�NU8+:<CL>#GH8+8+:1G�STp#p#nPpn+:1:1JHJH=H= yH:1�):<CL>)>#A<:<A<=�;�=H:18KCL> ��r�# :18+8P> �

dpR+>

n�/5j;�:1yHD�FY�)D�GHD�nP>)>%C+CPF)n+:18KCL> �

Vmin

R+>Vmax

�/5;%>#QU;�:18KCP:1S@D�8KCL> �

αpR+>

βp�/5[rtsuV # 8KCLp �

Vc,pR+>

γ�98+D�N%p�NU=�;#>#JH:1GHG�>#G�F)S@:1G�>#:<8+=H=HNU8P:<8+CL>�R+>

A1F):18KCPD >#:<GHD�D�G�>�N�p IUCLD/CPCLp ;%p#8KCLp@ST>%CPD�n+:�>)>#A<D�:<8KCL>��kλ,p

�/|

Page 13: 0/ &21436573cs.joensuu.fi/spectral/publications/tossavainen2.pdf · !#"%$ &% ')(#*+('%$ ,-*/.1032%.4 $ .45 6 798+:@?ba1:1:

}�J�D�N�CPn+:18+D�G�8Pp%CPD�:1A<I�D�NU8+:<CL>#GH8+8+:1G�ST>#NU8+:1S@:�8P>)>#y�>)>#G�N%>)> ;#>#8KCL>

Mλ,p = kλ,p(αpVmax + βp − Vc,p)γp

����| 6 ' �

R+>�GHF)n+S@D�D�nP>)>#ST>#A<A1>x8+J�D�N�CPn+:1GHD�GY8Pp%CPD�:1A<I)D�NU8+:<CL>#GH8+8+:jST>#NU8+:1S@:�>%nK;)F�F)GH8P>x8P>)>#y�>)>#G

Mλ,p

Mλ,p,max

=

[

αp(Vmax − Vmin)

αpVmax + βp − Vc,p

dp2n − 1

+αpVmin + βp − Vc,pαpVmax + βp − Vc,p

]γp

.����| 6DC �

ZYp)p#n+:<CPD-CLp)p#GOS@F)GH:<CPF)n+:<GO��>#:1G R+>�F#��8+D-C�CPD�n+S@:<Ckg,p

R+>ko,p

8+D�=HnP>)> ;%>#8KCP: #

kg,p =αp (Vmax − Vmin)

αpVmax + βp − Vc,p

ko,p =αpVmin + βp − Vc,pαpVmax + βp − Vc,p

����| 6#� �

kg+ko = 15%N%>)> ;#>#A1A�>�����| 6 ' ��8+=HF)n+:<CPD-CP=HG�GHF)n+S@D�D-nL>%=HNU8PD-G�;�=HF)NU8+:�|#Ve:1D-C I�A1A1DA1:1:zCLp#G�CLp#N#F)nKCP:<A<A1D

R+>�rtsuV # A1A1DVmin

5Vmax

R+>Vc,p

F3;#>�CxJ�p)p)>#8+:�>#A1A<:<8+D�8KCP:[;#>#NU:1F):<CL>�|kg,p

R+>ko,p

n+:1:1JHJH=�;%>%C9;#>#:<G;%>#QU;�:18KCP:1S@D�GO��>#:1GxR+>�F#��8+D-C�CPD�n+S@D�:18KCLp�|

Mλ,p

Mλ,p,max

=

(

kg,pdp2n − 1

+ ko,p

)γp

.����| � � �

}�J�D�N�CPn+:1GHD�GYnP>#yH:�>#GH8+8+:Lλ

;�F):1y�>)>#G�I�QHyH:18KCLp)p ���r >#nK;)F):1QH:1GO8+D�=HnP>)> ;#>#8KCP: #

Lλ =Mλ

π.

����| ��7 �

��>)> ;#>�����| ��6 � I�QHyH:18KCLp)pdp# G�8+J�D�N�CPn+:18+D�D�GYnP>#yH:�>#GH8+8+:1:1G

Lλ,p =

Lλ,p,max

(

kg,pdp

2n−1+ ko,p

)γp NU=HG (

kg,pdp

2n−1+ ko,p

)

≥ 0

0NU=HG (

kg,pdp

2n−1+ ko,p

)

< 0.

����| ��6 �

E#F#C+CL>w8+J�D�N�CPn+:18KCLp�nP>#yH:�>#GH8+8+:�>^;)F):1y�>)>#Gq>#nK;�:1F):1y�> ���r�>#nK;)F):18KCL>�5%CLp3IUC IUILλ,p,max, kg, koR+>

γCP=HG�CPD >9RKF)N%>#:18+D�A1A1D9N�>#G�> ;#>#A1A1D#|!1^p#S@pTJ�>#nP>#S@D-CPn+:<C^8P>)>#y�>#>#GbSTp)p#n+:<CPD-C+C I�p@nP>#yH:1F)S@D-CPn+: v

8+:18KCLp�S@:<C+CL>#=HNU8+:18KCL>�|3��>)> ;#>����| ��6 ��;�F):1y�>)>#G >�R+>%CPD�A1A�>eN%>#G�> ;#>#G 8+J�D�N�CPn+:18+D�G�nP>#yH:�>#GH8+8+:1G9ST>#N)v8+:1S@:1G

Lλ,p,max8+N�>)>#A�>#=HNU8+D�NU8+:�|��[8+:1S@D�n+NU:1NU8+:�8+N%>#A�>)>#n+:

R;�F):1y�>)>#GOD�8+:<C+CLp)p�8+D�=HnP>)>3;%>#8KCP: #

Lλ,r = RLλ,r,max.����| �)� �

7��

Page 14: 0/ &21436573cs.joensuu.fi/spectral/publications/tossavainen2.pdf · !#"%$ &% ')(#*+('%$ ,-*/.1032%.4 $ .45 6 798+:@?ba1:1:

��p IUC+CLp#STp#A1A1p9I�QUCLp#A1X):<CLp ����| �)� ��R+>�����| ��6 � ;)F):1y�>)>#G�NU:1noRKF):<C+CL>)>

R =

(

kg,rdr

2n−1+ ko,r

)γr NU=HG (kg,r

dr

2n−1+ ko,r

)

≥ 0

0NU=HG (

kg,rdr

2n−1+ ko,r

)

< 0.

����| �D� �

��p#n+:1S@:zC+CL>#=HNU8+D�G{N%>#GHG�>#A<CL>R

F)G�CPn+:18KCP:1Sx=HA1=H8P>#nK;)FH|u}�D�G{CL>#n+NUN%>b>#nK;)FTR+>YD�J�p#STp)p#nPp#: v8KIUI�8^n+:1:1JHJH=�;#>%C�N�I�8+D�:18+D�8KCLpOrtsuVv G�p3IUCPX)8KCLp�5�GHF)n+ST>#A1:<8+F):<G�CP:<S@D-CPFUyHD�:18KCL>^R+>�CP:1D-C I�:18KCLpqnP>#yH:1F%vS@D-CPn+:1S@:zCKCL>#=HNU8+:18KCL>�|,1^p#:18KCLp@8KI�:18KCLp

RF)GYA�>#:<C+CPD�D�8KCL>qn+:1:1JHJH=�;#>�|�_j=HS@:1G�>#GH8+8+:�>qS@:<CL>%C+CL>#D�8+8P>

JH=HG�>#:18+D�A1A1D�N%>#G�> ;#>%A1A1D9F3;#>%C^;)F):1ST>#8+8P>@8+D�=HnP>)> ;%>%C9r�c�$# G ��rF)S@Sq:18+8+:1F)G�� G�CPD�n+G�>%CP:1F)G�>#A1D�yHDA�� � ��A�>#:<nP>#�)D��WCPn+:18KCP:1Sx=HA<=H8P>#nK;)F#C�#

Xr = 683

Lλ,rxλdλ

Yr = 683

Lλ,ryλdλ

Zr = 683

Lλ,rzλdλ.����| �% �

� >#8KCL>)> ;%>#8KCL:jSx=H:1A<A<DwN%>#G�> ;�:1A1A1D#|�}�: RKF):<C+CL>#S@>#A<A1> ����| �)� �N�>)>3;%>)>#G ����| �% �L5�8P>)>#y�>)>#G

Xr = R683

Lλ,r,maxxλdλ = RXr,max

Yr = R683

Lλ,r,maxyλdλ = RYr,max

Zr = R683

Lλ,r,maxzλdλ = RZr,max.����| � & �

V�p#Q�p#G�S@D�GHGHD�8+8Pp~RKF)N�>#:18KCL>tN%>#G�> ;%>)>eF)G�N%p#8+:<CPD�A<C I^D�n+:1NU8+D�D�G�| �^GHGHD-CP=HA1A1D~JH:1NU8+D�A1:1A1A<Du8+J�D�N�CKvn+:1GHD�G�nP>#yH:�>#GH8+8+:WF)G�N)F#A1S@D�G�n+:1:1JHJH=HST>%CKCPF)S@>#GN�>#G�>3;%>#G{A1:1GHD >)>#n+:<N)F)S �H:1G�>)>�CP:<FH5jF)A1D/CPCL>#D-G�5D-C+CLp�yHD-CPD�N�CPF)n+: �oS@:<C+CL>#A1>%:zCPD^CL>#:j8+:1A1STp �WI�QHyH:18KCLp)p�JH:1NU8+D�A1:1G�N�>#:1NUNU:jN#F)A1S@D�D�S@:18+8+:1F#CL>�|

X = Xr +Xg +Xb

Y = Yr + Yg + Yb

Z = Zr + Zg + Zb.����| �' �

}�: RKF):<C+CP>#S@>#A<A1>�����| � & � N%>)> ;#>)>#G ����| �' � 8P>)>#y�>)>#G

X = RXr,max +GXg,max +BXb,max

Y = RYr,max +GYg,max +BYb,max

Z = RZr,max +GZg,max +BZb,max.����| � C �

7)7

Page 15: 0/ &21436573cs.joensuu.fi/spectral/publications/tossavainen2.pdf · !#"%$ &% ')(#*+('%$ ,-*/.1032%.4 $ .45 6 798+:@?ba1:1:

}�D�=HnP>)> ;�:1A1A�>xN�>)> ;�F):1A1A1> ��r >#nK;)F#CeI�QHyH:18KCPD-CLp)p#G�r�c�$# G�CPn+:18KCP:1Sx=HA<=H8e>#nK;�F):<QH:1G�|

R =

(kg,rdr

2n−1+ ko,r)

γrNU=HG

(kg,rdr

2n−1+ ko,r) ≥ 0

0NU=HG

(kg,rdr

2n−1+ ko,r) < 0

G =

(kg,gdg

2n−1+ ko,g)

γgNU=HG

(kg,gdg

2n−1+ ko,g) ≥ 0

0NU=HG

(kg,gdg

2n−1+ ko,g) < 0

B =

(kg,bdb

2n−1+ ko,b)

γbNU=HG

(kg,bdb

2n−1+ ko,b) ≥ 0

0NU=HG

(kg,bdb

2n−1+ ko,b) < 0.

X

Y

Z

=

Xr,max Xg,max Xb,max

Yr,max Yg,max Yb,max

Zr,max Zg,max Zb,max

R

G

B

.

����| �)� �

��>)> ;#>#G ����| �)� �t> ;�=HA1A�>x;)F):1y�>)>#G�A�>#8+N#D >�CPn+:18KCP:1Sx=HA<=H8P>#nK;)F#C�yH:1�):zC/>#>#A<:<8+:18KCL>�>#nK;)F):18KCL>�A�p#Q�CP:1D�G�|V�n+:18KCP:1Sx=HA<=H8P>#nK;�F):<8KCL>x8P>)>#y�>)>#G�A�>#8+N)D-C+CP=�>qse���Wvc>#nK;�F#CeN%p)p%GUCPD�:18+D�A1A�pxN�>)>3;%>#A1A�>x8+D�=HnP>)> ;#>#8KCP: #

R

G

B

=

Xr,max Xg,max Xb,max

Yr,max Yg,max Yb,max

Zr,max Zg,max Zb,max

−1

X

Y

Z

.

����| � � �

dr =2n − 1

kg,r

(

R1/γr − ko,r) NU=HG

0 ≤ R ≤ 1

dg =2n − 1

kg,g

(

G1/γg − ko,g) NU=HG

0 ≤ G ≤ 1

db =2n − 1

kg,b

(

B1/γb − ko,b) NU=HG

0 ≤ B ≤ 1.����| �H7 �

7�6

Page 16: 0/ &21436573cs.joensuu.fi/spectral/publications/tossavainen2.pdf · !#"%$ &% ')(#*+('%$ ,-*/.1032%.4 $ .45 6 798+:@?ba1:1:

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