Methods for computing weighting tables based on local power expansion for tristimulus values computations

Changjun Li; Claudio Oleari; Manuel Melgosa; Yang Xu

doi:10.1364/JOSAA.28.002243

Journal of the Optical Society of America A
Vol. 28,
Issue 11,
pp. 2243-2252
(2011)
•https://doi.org/10.1364/JOSAA.28.002243

Methods for computing weighting tables based on local power expansion for tristimulus values computations

Changjun Li, Claudio Oleari, Manuel Melgosa, and Yang Xu

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Changjun Li, Claudio Oleari, Manuel Melgosa, and Yang Xu, "Methods for computing weighting tables based on local power expansion for tristimulus values computations," J. Opt. Soc. Am. A 28, 2243-2252 (2011)

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Abstract

In this paper, two types of weighting tables are derived by applying the local power expansion method proposed by Oleari [Color Res. Appl. 25, 176 (2000)]. Both tables at two different levels consider the deconvolution of the spectrophotometric data for monochromator triangular transmittance. The first one, named zero-order weighting table, is similar to weighting table 5 of American Society for Testing and Materials (ASTM) used with the measured spectral reflectance factors (SRFs) corrected by the Stearns and Stearns formula. The second one, named second-order weighting table, is similar to weighting table 6 of ASTM and must be used with the undeconvoluted SRFs. It is hoped that the results of this paper will aid the International Commission on Illumination TC 1-71 on tristimulus integration in focusing on ongoing methods, testing, and recommendations.

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$X = K \sum_{i = 0}^{N} [W_{X, i, 0} + W_{X, i, 2} \frac{(Δ λ)^{2}}{6} + \dots] Δ λ$	$Y = K \sum_{i = 0}^{N} [W_{Y, i, 0} + W_{Y, i, 2} \frac{(Δ λ)^{2}}{6} + \dots] Δ λ$	$Z = K \sum_{i = 0}^{N} [W_{Z, i, 0} + W_{Z, i, 2} \frac{(Δ λ)^{2}}{6} + \dots] Δ λ$
$W_{X, i, 0} = R_{i, 0} x_{S, i, 0}$	$W_{Y, i, 0} = R_{i, 0} y_{S, i, 0}$	$W_{Z, i, 0} = R_{i, 0} z_{S, i, 0}$
$W_{X, i, 1} = R_{i, 0} x_{S, i, 1} + R_{i, 1} x_{S, i, 0}$	$W_{Y, i, 1} = R_{i, 0} y_{S, i, 1} + R_{i, 1} y_{S, i, 0}$	$W_{Z, i, 1} = R_{i, 0} z_{S, i, 1} + R_{i, 1} z_{S, i, 0}$
$W_{X, i, 2} = R_{i, 0} x_{S, i, 2} + R_{i, 1} x_{S, i, 1} + R_{i, 2} x_{S, i, 0}$	$W_{Y, i, 2} = R_{i, 0} y_{S, i, 2} + R_{i, 1} y_{S, i, 1} + R_{i, 2} y_{S, i, 0}$	$W_{Z, i, 2} = R_{i, 0} z_{S, i, 2} + R_{i, 1} z_{S, i, 1} + R_{i, 2} z_{S, i, 0}$
$x_{S, i, 0} = \frac{1}{A} [B_{x, i, 0} \sum_{j = - Δ λ}^{Δ λ} j^{4} - B_{x, i, 2} \sum_{j = - Δ λ}^{Δ λ} j^{2}]$	$y_{S, i, 0} = \frac{1}{A} [B_{y, i, 0} \sum_{j = - Δ λ}^{Δ λ} j^{4} - B_{y, i, 2} \sum_{j = - Δ λ}^{Δ λ} j^{2}]$	$z_{S, i, 0} = \frac{1}{A} [B_{z, i, 0} \sum_{j = - Δ λ}^{Δ λ} j^{4} - B_{z, i, 2} \sum_{j = - Δ λ}^{Δ λ} j^{2}]$
$x_{S, i, 1} = B_{x, i, 1} / \sum_{j = - Δ λ}^{Δ λ} j^{2}$	$y_{S, i, 1} = B_{y, i, 1} / \sum_{j = - Δ λ}^{Δ λ} j^{2}$	$z_{S, i, 1} = B_{z, i, 1} / \sum_{j = - Δ λ}^{Δ λ} j^{2}$
$x_{S, i, 2} = \frac{1}{A} [- B_{x, i, 0} \sum_{j = - Δ λ}^{Δ λ} j^{2} + B_{x, i, 2} (2 Δ λ + 1)]$	$y_{S, i, 2} = \frac{1}{A} [- B_{y, i, 0} \sum_{j = - Δ λ}^{Δ λ} j^{2} + B_{y, i, 2} (2 Δ λ + 1)]$	$z_{S, i, 2} = \frac{1}{A} [- B_{z, i, 0} \sum_{j = - Δ λ}^{Δ λ} j^{2} + B_{z, i, 2} (2 Δ λ + 1)]$
$B_{x, i, 0} = \sum_{j = - Δ λ}^{Δ λ} [S_{λ_{i} + j} \bar{x} (λ_{i} + j)]$	$B_{y, i, 0} = \sum_{j = - Δ λ}^{Δ λ} [S_{λ_{i} + j} \bar{y} (λ_{i} + j)]$	$B_{z, i, 0} = \sum_{j = - Δ λ}^{Δ λ} [S_{λ_{i} + j} \bar{z} (λ_{i} + j)]$
$B_{x, i, 1} = \sum_{j = - Δ λ}^{Δ λ} {[S_{λ_{i} + j} \bar{x} (λ_{i} + j)] j}$	$B_{y, i, 1} = \sum_{j = - Δ λ}^{Δ λ} {[S_{λ_{i} + j} \bar{y} (λ_{i} + j)] j}$	$B_{z, i, 1} = \sum_{j = - Δ λ}^{Δ λ} {[S_{λ_{i} + j} \bar{z} (λ_{i} + j)] j}$
$B_{x, i, 2} = \sum_{j = - Δ λ}^{Δ λ} {[S_{λ_{i} + j} \bar{x} (λ_{i} + j)] j^{2}}$	$B_{y, i, 2} = \sum_{j = - Δ λ}^{Δ λ} {[S_{λ_{i} + j} \bar{y} (λ_{i} + j)] j^{2}}$	$B_{z, i, 2} = \sum_{j = - Δ λ}^{Δ λ} {[S_{λ_{i} + j} \bar{z} (λ_{i} + j)] j^{2}}$
$K = 100 / {\sum_{i = 0}^{N} [y_{S, i, 0} + y_{S, i, 2} \frac{(Δ λ)^{2}}{6} + \dots] Δ λ}$		$R_{i, 0} = \frac{1}{12} [- R_{s} (λ_{i - 1}) + 14 R_{s} (λ_{i}) - R_{s} (λ_{i + 1})]$
		$R_{i, 1} = \frac{1}{2 Δ λ} [- R_{s} (λ_{i - 1}) + R_{s} (λ_{i + 1})]$
		$R_{i, 2} = \frac{1}{2 (Δ λ)^{2}} [R_{s} (λ_{i - 1}) - 2 R_{s} (λ_{i}) + R_{s} (λ_{i + 1})]$

λ (nm)	$W_{X, i}^{(1)} (λ)$	$W_{Y, i}^{(1)} (λ)$	$W_{Z, i}^{(1)} (λ)$
380	0.00003436	0.00000373	0.00015139
381	0.00004686	0.00000508	0.00020659
382	0.00006383	0.00000692	0.00028159
383	0.00008673	0.00000939	0.00038280
384	0.00011735	0.00001269	0.00051830
385	0.00015792	0.00001706	0.00069798
500	0.00320380	0.38685446	0.18344768
501	0.00428471	0.39890290	0.17311330
502	0.00583981	0.41104305	0.16299936
503	0.00783577	0.42329480	0.15313912
504	0.01024154	0.43568054	0.14356953
505	0.01302727	0.44821109	0.13432921
775	0.00003276	0.00001296	0.00000000
776	0.00003046	0.00001206	0.00000000
777	0.00002834	0.00001122	0.00000000
778	0.00002638	0.00001045	0.00000000
779	0.00002458	0.00000974	0.00000000
780	0.00002294	0.00000910	0.00000000
Sum	96.71446216	100.00000000	81.42940223

λ (nm)	$x_{S, i, 0}$	$x_{S, i, 1}$	$x_{S, i, 2}$
380	0.00003101	0.00002114	0.00000277
390	0.00053068	0.00033149	0.00003478
400	0.00814436	0.00193385	0.00013112
410	0.04200163	0.00506032	0.00016218
420	0.10667504	0.00608486	$- 0.00007304$
430	0.16444228	0.00717920	0.00019269
440	0.25011327	0.00649688	$- 0.00030266$
450	0.28177539	$- 0.00065947$	$- 0.00036697$
460	0.24018196	$- 0.00643103$	$- 0.00019704$
470	0.15663699	$- 0.00899410$	$- 0.00003547$
480	0.06672045	$- 0.00733136$	0.00018105
490	0.01326399	$- 0.00311246$	0.00022254
500	0.00356631	0.00091431	0.00019246
510	0.03177856	0.00482783	0.00020077
520	0.10044142	0.00904666	0.00021548
530	0.21046753	0.01177497	0.00004994
540	0.33350309	0.01316623	0.00010687
550	0.47470889	0.01436793	0.00000745
560	0.61854806	0.01399168	$- 0.00004201$
570	0.75470425	0.01317939	$- 0.00004295$
580	0.87641723	0.00833746	$- 0.00044214$
590	0.92398029	0.00426932	0.00002616
600	0.96098717	$- 0.00090373$	$- 0.00056044$
610	0.89576485	$- 0.01115157$	$- 0.00042967$
620	0.74246169	$- 0.01797063$	$- 0.00023055$
630	0.54574012	$- 0.01863040$	0.00014894
640	0.37295911	$- 0.01609793$	0.00010544
650	0.22547488	$- 0.01214197$	0.00027837
660	0.13116016	$- 0.00745118$	0.00018000
670	0.07295322	$- 0.00477592$	0.00010230
680	0.03530725	$- 0.00286383$	0.00008920
690	0.01546068	$- 0.00133205$	0.00006270
700	0.00766929	$- 0.00056331$	0.00001787
710	0.00365665	$- 0.00030696$	0.00000882
720	0.00149686	$- 0.00013754$	0.00000777
730	0.00078986	$- 0.00005126$	0.00000144
740	0.00040596	$- 0.00003074$	0.00000072
750	0.00017083	$- 0.00001696$	0.00000066
760	0.00006739	$- 0.00000571$	0.00000045
770	0.00004571	$- 0.00000211$	$- 0.00000004$
780	0.00002444	$- 0.00000112$	0.00000011

λ (nm)	$y_{S, i, 0}$	$y_{S, i, 1}$	$y_{S, i, 2}$
380	0.00000340	0.00000227	0.00000030
390	0.00005751	0.00003489	0.00000364
400	0.00085404	0.00019992	0.00001343
410	0.00432300	0.00052312	0.00001736
420	0.01118425	0.00076543	0.00000660
430	0.02014246	0.00142327	0.00006250
440	0.04039084	0.00243963	0.00003277
450	0.06774426	0.00300686	0.00003033
460	0.10120362	0.00391913	0.00006711
470	0.14868977	0.00558160	0.00008386
480	0.21184922	0.00630436	$- 0.00000427$
490	0.27652568	0.00851331	0.00025181
500	0.38586002	0.01221308	0.00006343
510	0.51428058	0.01322847	0.00003808
520	0.65034880	0.01360936	$- 0.00000489$
530	0.78077912	0.01014720	$- 0.00033462$
540	0.85121518	0.00528026	$- 0.00013200$
550	0.88838124	0.00122660	$- 0.00026839$
560	0.87448507	$- 0.00355909$	$- 0.00020227$
570	0.82093212	$- 0.00610060$	$- 0.00005178$
580	0.75093431	$- 0.00920146$	$- 0.00025100$
590	0.64228821	$- 0.00942059$	0.00020414
600	0.56294275	$- 0.00891232$	$- 0.00015669$
610	0.45899045	$- 0.01101265$	$- 0.00004318$
620	0.34530401	$- 0.01114203$	0.00003293
630	0.23880865	$- 0.00950113$	0.00012544
640	0.15539119	$- 0.00743816$	0.00008393
650	0.09047376	$- 0.00517190$	0.00013689
660	0.05179826	$- 0.00301511$	0.00007636
670	0.02854808	$- 0.00189451$	0.00004214
680	0.01374585	$- 0.00112171$	0.00003539
690	0.00600698	$- 0.00051883$	0.00002448
700	0.00297720	$- 0.00021885$	0.00000697
710	0.00142040	$- 0.00011909$	0.00000342
720	0.00058238	$- 0.00005337$	0.00000301
730	0.00030801	$- 0.00001990$	0.00000056
740	0.00015873	$- 0.00001197$	0.00000028
750	0.00006700	$- 0.00000662$	0.00000026
760	0.00002652	$- 0.00000223$	0.00000017
770	0.00001805	$- 0.00000083$	$- 0.00000001$
780	0.00000969	$- 0.00000044$	0.00000004

λ (nm)	$z_{S, i, 0}$	$z_{S, i, 1}$	$z_{S, i, 2}$
380	0.00013590	0.00009376	0.00001229
390	0.00233858	0.00148823	0.00015682
400	0.03659306	0.00885325	0.00060928
410	0.19305030	0.02404448	0.00082818
420	0.50744637	0.03073314	$- 0.00023721$
430	0.81157285	0.03871681	0.00109232
440	1.28245349	0.03847085	$- 0.00134066$
450	1.51550410	0.00444873	$- 0.00182524$
460	1.38623193	$- 0.02397648$	$- 0.00093524$
470	1.05386885	$- 0.03831152$	$- 0.00041241$
480	0.64261238	$- 0.03677961$	0.00053173
490	0.33723709	$- 0.02259929$	0.00078499
500	0.18362784	$- 0.01174115$	0.00030982
510	0.09581346	$- 0.00651981$	0.00022834
520	0.05154976	$- 0.00328789$	0.00009300
530	0.02698481	$- 0.00194496$	0.00004663
540	0.01195007	$- 0.00115315$	0.00003326
550	0.00347008	$- 0.00060472$	0.00002446
560	0.00002057	$- 0.00014432$	0.00001796
570	0.00000000	0.00000000	0.00000000
580	0.00000000	0.00000000	0.00000000
590	0.00000000	0.00000000	0.00000000
600	0.00000000	0.00000000	0.00000000
610	0.00000000	0.00000000	0.00000000
620	0.00000000	0.00000000	0.00000000
630	0.00000000	0.00000000	0.00000000
640	0.00000000	0.00000000	0.00000000
650	0.00000000	0.00000000	0.00000000
660	0.00000000	0.00000000	0.00000000
670	0.00000000	0.00000000	0.00000000
680	0.00000000	0.00000000	0.00000000
690	0.00000000	0.00000000	0.00000000
700	0.00000000	0.00000000	0.00000000
710	0.00000000	0.00000000	0.00000000
720	0.00000000	0.00000000	0.00000000
730	0.00000000	0.00000000	0.00000000
740	0.00000000	0.00000000	0.00000000
750	0.00000000	0.00000000	0.00000000
760	0.00000000	0.00000000	0.00000000
770	0.00000000	0.00000000	0.00000000
780	0.00000000	0.00000000	0.00000000

λ (nm)	${WT}_{0, 10, X}$	${WT}_{0, 10, Y}$	${WT}_{0, 10, Z}$
380	0.00031013	0.00003396	0.00135900
390	0.00530680	0.00057515	0.02338571
400	0.08144315	0.00854033	0.36592870
410	0.42001415	0.04322975	1.93049326
420	1.06674499	0.11184195	5.07443798
430	1.64441451	0.20142354	8.11568739
440	2.50112002	0.40390635	12.82446991
450	2.81773966	0.67743913	15.15496418
460	2.40180739	1.01203111	13.86224902
470	1.56636197	1.48689020	10.53863506
480	0.66720110	2.11848145	6.42609128
490	0.13263925	2.76524280	3.37235378
500	0.03566294	3.85858063	1.83626906
510	0.31778402	5.14277978	0.95812974
520	1.00440912	6.50345507	0.51549500
530	2.10466466	7.80775166	0.26984669
540	3.33501404	8.51210867	0.11950006
550	4.74706484	8.88376739	0.03470067
560	6.18544924	8.74480634	0.00020568
570	7.54700428	8.20927960	0.00000000
580	8.76412791	7.50930501	0.00000000
590	9.23975604	6.42284958	0.00000000
600	9.60982304	5.62939901	0.00000000
610	8.95760306	4.58988128	0.00000000
620	7.42457925	3.45302265	0.00000000
630	5.45737353	2.38807439	0.00000000
640	3.72957218	1.55390404	0.00000000
650	2.25473734	0.90473302	0.00000000
660	1.31159493	0.51798000	0.00000000
670	0.72952847	0.28547934	0.00000000
680	0.35307069	0.13745775	0.00000000
690	0.15460599	0.06006945	0.00000000
700	0.07669254	0.02977187	0.00000000
710	0.03656636	0.01420391	0.00000000
720	0.01496852	0.00582378	0.00000000
730	0.00789859	0.00308011	0.00000000
740	0.00405956	0.00158730	0.00000000
750	0.00170830	0.00067000	0.00000000
760	0.00067393	0.00026523	0.00000000
770	0.00045711	0.00018054	0.00000000
780	0.00024441	0.00009689	0.00000000
Sum	96.71179804	100.00000000	81.42420215

λ (nm)	${WT}_{2, 10, X}$	${WT}_{2, 10, Y}$	${WT}_{2, 10, Z}$
380	$- 0.00261167$	$- 0.00027257$	$- 0.01178335$
390	$- 0.00572926$	$- 0.00055572$	$- 0.02774898$
400	0.06479630	0.00679110	0.28275070
410	0.41615195	0.04161386	1.90403381
420	1.02996025	0.10451464	4.87938376
430	1.68366986	0.19907881	8.28553634
440	2.51001868	0.39579370	12.85956371
450	2.86106476	0.66962219	15.35209740
460	2.43854511	1.00204133	14.06784377
470	1.56719318	1.48245092	10.57076745
480	0.65079472	2.08856416	6.39338838
490	0.10201015	2.76415129	3.30465785
500	0.00103683	3.82760525	1.74844727
510	0.28338817	5.13774877	0.92649321
520	0.98493478	6.53231483	0.49163727
530	2.07557396	7.81401517	0.25937178
540	3.33339694	8.56917547	0.11381170
550	4.74074564	8.90990622	0.03033907
560	6.18769308	8.77100465	$- 0.00168173$
570	7.59251372	8.25254698	$- 0.00145208$
580	8.75266142	7.48606732	0.00000000
590	9.33579848	6.46581226	0.00000000
600	9.63498305	5.60997671	0.00000000
610	9.02728861	4.60179623	0.00000000
620	7.44599024	3.44569596	0.00000000
630	5.47247949	2.37998210	0.00000000
640	3.69007834	1.53050753	0.00000000
650	2.23285047	0.89226707	0.00000000
660	1.27992693	0.50303078	0.00000000
670	0.70745585	0.27634415	0.00000000
680	0.33942494	0.13195082	0.00000000
690	0.14613901	0.05671757	0.00000000
700	0.07063239	0.02740881	0.00000000
710	0.03437784	0.01335160	0.00000000
720	0.01420575	0.00552820	0.00000000
730	0.00717042	0.00279759	0.00000000
740	0.00388493	0.00151961	0.00000000
750	0.00161266	0.00063282	0.00000000
760	0.00062810	0.00024746	0.00000000
770	0.00040424	0.00015980	0.00000000
780	0.00023845	0.00009457	0.00000000
Sum	96.71337878	100.00000000	81.42745733

Abstract

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Tables (10)

Equations (38)

Journal of the Optical Society of America A