Reflectance spectra recovery from tristimulus values by adaptive estimation with metameric shape correction

Simone Bianco

doi:10.1364/JOSAA.27.001868

Journal of the Optical Society of America A
Vol. 27,
Issue 8,
pp. 1868-1877
(2010)
•https://doi.org/10.1364/JOSAA.27.001868

Reflectance spectra recovery from tristimulus values by adaptive estimation with metameric shape correction

Simone Bianco

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Simone Bianco, "Reflectance spectra recovery from tristimulus values by adaptive estimation with metameric shape correction," J. Opt. Soc. Am. A 27, 1868-1877 (2010)

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About this Article
History
- Original Manuscript: February 16, 2010
- Revised Manuscript: June 23, 2010
- Manuscript Accepted: June 23, 2010
- Published: July 30, 2010
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 5, Iss. 12

Abstract

In this work a local optimization-based method that is able to recover the reflectance spectra with the desired tristimulus values, choosing the metamer with the most similar shape to the reflectances available in the training set, is proposed. Four different datasets of reflectance spectra and three different error metrics have been used in this study. According to all the error metrics considered, the proposed algorithm was able to recover the spectral reflectances with a higher accuracy than all the state of the art methods considered.

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Method		$Δ E_{94}$ under Ill. D65			$Δ E_{94}$ under Ill. A			$Δ E_{94}$ under Ill. F2			PSNR (%)			GFC (%)
Method	Dataset	Mean	95%	std	Mean	95%	std	Mean	95%	std	Poor	Accurate	Good	Poor	Accurate	Good	Excellent
PINV [4]	MUNStr	0.0042	0.0000	0.0848	0.7843	2.3871	0.8457	0.7625	2.2980	0.8460	44.2520	33.8583	21.8898	20.0000	58.1102	21.8898	0.0000
	MUNSte	0.0036	0.0000	0.0523	0.8457	3.0222	0.9255	0.7718	2.3908	0.8035	44.4795	33.4385	22.0820	21.9243	56.9401	21.1356	0.0000
	VHRELte	0.0940	0.0432	0.5234	1.1286	2.5267	1.0055	1.0399	2.5929	1.3368	69.4118	17.0588	13.5294	71.7647	17.0588	11.1765	0.0000
	MCCte	0.0830	0.5974	0.4065	1.3037	4.1796	1.3174	1.4769	5.2472	1.7519	75.0000	12.5000	12.5000	45.8333	41.6667	12.5000	0.0000
	MDCte	0.0428	0.0917	0.2745	1.1370	3.7136	1.1817	1.2244	3.7235	1.4539	62.2093	22.0930	15.6977	37.7907	44.7674	17.4419	0.0000

PINV-PCA [5]	MUNStr	0.0051	0.0000	0.0786	0.8097	2.7608	0.9418	0.7699	2.3452	0.8257	48.3465	36.8504	14.8031	28.0315	58.2677	13.0709	0.6299
	MUNSte	0.0038	0.0000	0.0440	0.8609	3.4918	1.0436	0.7739	2.3586	0.7929	47.4763	36.4353	16.0883	26.4984	60.0946	12.6183	0.7886
	VHRELte	0.0920	0.3329	0.4898	1.1299	2.5514	0.9649	1.0437	2.4659	1.2800	71.1765	15.8824	12.9412	74.7059	17.6471	7.6471	0.0000
	MCCte	0.0750	0.5401	0.3675	1.2713	4.0579	1.3615	1.4458	4.9712	1.6787	66.6667	12.5000	20.8333	50.0000	37.5000	8.3333	4.1667
	MDCte	0.0513	0.1544	0.2688	1.1707	3.8038	1.2388	1.2456	3.5843	1.4073	63.9535	25.5814	10.4651	41.2791	40.6977	18.0233	0.0000

HAW [8]	MUNStr	0.0000	0.0000	0.0000	0.7996	2.7808	0.8978	0.9844	3.6244	1.1516	55.2756	22.6772	22.0472	29.9213	52.1260	17.9528	0.0000
	MUNSte	0.0000	0.0000	0.0000	0.8645	3.0196	0.9949	0.9969	3.7305	1.1228	56.6246	20.8202	22.5552	31.8612	49.8423	18.2965	0.0000
	VHRELte	0.0000	0.0000	0.0000	1.2896	2.9000	0.9396	1.1256	3.1554	1.0787	74.1176	15.2941	10.5882	76.4706	19.4118	4.1176	0.0000
	MCCte	0.0000	0.0000	0.0000	1.0593	3.1735	1.1359	1.4547	5.9282	1.9238	62.5000	20.8333	16.6667	58.3333	20.8333	20.8333	0.0000
	MDCte	0.0000	0.0000	0.0000	1.3299	3.6502	1.2720	1.4741	5.7267	1.6407	77.3256	11.6279	11.0465	51.7442	42.4419	4.0698	1.7442

adaPCA [3]	MUNStr	0.0000	0.0000	0.0000	0.4642	1.1604	0.4309	0.4570	1.1321	0.3426	28.0315	37.7953	34.1732	9.7638	51.8110	37.4803	0.9449
	MUNSte	0.0000	0.0000	0.0000	0.5164	1.4516	0.4932	0.4722	1.1692	0.3762	27.7603	38.8013	33.4385	11.6719	51.5773	35.8044	0.9464
	VHRELte	0.0078	0.0000	0.0773	0.9492	2.2537	0.7310	0.7765	1.9004	0.6225	67.6471	17.0588	15.2941	70.0000	19.4118	10.5882	0.0000
	MCCte	0.0000	0.0000	0.0000	0.5218	1.9816	0.5884	0.5576	1.8724	0.5655	37.5000	37.5000	25.0000	16.6667	37.5000	45.8333	0.0000
	MDCte	0.0018	0.0000	0.0242	0.5719	1.7880	0.5890	0.5645	1.5703	0.4561	43.6047	33.7209	22.6744	16.2791	51.1628	31.9767	0.5814

PCAemRe [6]	MUNStr	0.0032	0.0000	0.0234	0.6730	2.0324	0.7731	0.7071	1.8671	0.5755	50.7087	32.4409	16.8504	39.3701	40.7874	19.5276	0.3150
	MUNSte	0.0027	0.0000	0.0193	0.7139	2.5211	0.8421	0.7106	1.7824	0.5479	49.5268	33.7539	16.7192	39.5899	41.6404	18.1388	0.6309
	VHRELte	0.0661	0.2741	0.3463	1.3007	2.7657	0.9474	1.0788	2.7816	0.8968	70.5882	15.2941	14.1176	78.8235	17.6471	3.5294	0.0000
	MCCte	0.0361	0.2516	0.0830	1.3640	5.4708	1.7111	1.3008	4.4822	1.2857	66.6667	16.6667	16.6667	66.6667	20.8333	8.3333	4.1667
	MDCte	0.0272	0.1867	0.1131	0.8984	2.8645	0.8924	0.9995	2.6132	0.8467	71.5116	15.1163	13.3721	58.7209	30.8140	9.3023	1.1628

PCAmuBa [2]	MUNStr	0.0000	0.0000	0.0000	0.3989	1.2129	0.4563	0.3899	1.0863	0.3488	18.5827	40.0000	41.4173	12.4409	40.0000	44.0945	3.4646
	MUNSte	0.0005	0.0000	0.0119	0.4501	1.6673	0.5543	0.4106	1.2480	0.3913	21.6088	35.8044	42.5868	13.8801	39.5899	43.3754	3.1546
	VHRELte	0.0028	0.0000	0.0370	1.2769	2.8849	0.8846	1.0444	2.5242	0.7793	74.1176	14.7059	11.1765	74.1176	21.7647	4.1176	0.0000
	MCCte	0.0000	0.0000	0.0000	0.7286	2.2277	0.7233	0.6732	2.6753	0.7423	58.3333	20.8333	20.8333	29.1667	41.6667	25.0000	4.1667
	MDCte	0.0000	0.0000	0.0001	0.6103	2.4397	0.6835	0.6077	1.7330	0.5487	48.2558	27.9070	23.8372	20.9302	42.4419	36.6279	0.0000

ZSS [9]	MUNStr	0.0000	0.0000	0.0000	0.8285	2.6205	0.8743	0.7638	2.1633	0.7533	49.4488	30.0787	20.4724	23.3071	56.6929	20.0000	0.0000
	MUNSte	0.0000	0.0000	0.0000	0.8867	3.2234	0.9655	0.7729	2.1966	0.7288	48.5804	29.9685	21.4511	24.6057	56.6246	18.4543	0.3155
	VHRELte	0.0000	0.0000	0.0000	1.0188	2.3545	0.8136	0.8967	2.2590	0.8356	68.8235	17.0588	14.1176	71.1765	18.2353	10.5882	0.0000
	MCCte	0.0000	0.0000	0.0000	1.2886	3.6157	1.2405	1.3732	4.1437	1.3500	75.0000	12.5000	12.5000	45.8333	41.6667	12.5000	0.0000
	MDCte	0.0000	0.0000	0.0000	1.1577	3.5928	1.1852	1.1663	3.3409	1.1905	67.4419	19.1860	13.3721	39.5349	42.4419	18.0233	0.0000

Proposed	MUNStr	0.0000	0.0000	0.0000	0.3299	1.3016	0.4944	0.3710	1.1639	0.3762	18.2677	24.0945	57.6378	10.8661	33.7008	40.1575	15.2756
	MUNSte	0.0000	0.0000	0.0000	0.3769	1.6527	0.5587	0.3881	1.2903	0.4076	15.4574	27.2871	57.2555	11.5142	35.4890	38.0126	14.9842
	VHRELte	0.0000	0.0000	0.0000	0.8304	2.2835	0.7957	0.7264	1.8522	0.5404	58.2353	22.9412	18.8235	60.0000	20.0000	20.0000	0.0000
	MCCte	0.0000	0.0000	0.0000	0.2892	1.6748	0.4690	0.4535	2.0849	0.6328	20.8333	20.8333	58.3333	12.5000	20.8333	50.0000	16.6667
	MDCte	0.0000	0.0000	0.0000	0.3782	1.9669	0.6606	0.4540	1.4844	0.4889	25.0000	23.8372	51.1628	15.1163	22.6744	45.3488	16.8605

Method	Dataset	$Δ E_{2000}$			$Δ E_{CMC 2 : 1}$
Method	Dataset	D65	A	F2	D65	A	F2
PINV [4]	MUNStr	0.0042	0.9079	0.7812	0.0046	1.0869	0.8074
	MUNSte	0.0035	0.9840	0.7964	0.0038	1.1630	0.8165
	VHRELte	0.0949	1.3619	1.0366	0.1090	1.7741	0.9908
	MCCte	0.0840	1.5216	1.4794	0.0904	1.8327	1.4885
	MDCte	0.0429	1.2897	1.2418	0.0462	1.5269	1.2485

PINV-PCA [5]	MUNStr	0.0050	0.9289	0.7794	0.0055	1.0928	0.8021
	MUNSte	0.0038	0.9921	0.7900	0.0042	1.1562	0.8077
	VHRELte	0.0923	1.3656	1.0314	0.1066	1.7888	0.9785
	MCCte	0.0756	1.4631	1.4412	0.0815	1.7590	1.4429
	MDCte	0.0507	1.3141	1.2526	0.0557	1.5470	1.2556

HAW [8]	MUNStr	0.0000	0.9416	1.0066	0.0000	1.1301	1.1330
	MUNSte	0.0000	1.0209	1.0239	0.0000	1.2089	1.1419
	VHRELte	0.0000	1.5671	1.1125	0.0000	2.0318	1.1144
	MCCte	0.0000	1.2435	1.4628	0.0000	1.5516	1.4825
	MDCte	0.0000	1.5560	1.4970	0.0000	1.8372	1.5819

adaPCA [3]	MUNStr	0.0000	0.5603	0.4747	0.0000	0.6721	0.4905
	MUNSte	0.0000	0.6297	0.4921	0.0000	0.7347	0.5022
	VHRELte	0.0074	1.1619	0.7609	0.0087	1.5146	0.6827
	MCCte	0.0000	0.6445	0.5608	0.0000	0.7934	0.5791
	MDCte	0.0018	0.6830	0.5777	0.0020	0.7780	0.5935

PCAemRe [6]	MUNStr	0.0032	0.7832	0.7315	0.0039	0.9446	0.7630
	MUNSte	0.0027	0.8309	0.7341	0.0034	0.9889	0.7648
	VHRELte	0.0681	1.5940	1.0528	0.0835	2.1094	1.0079
	MCCte	0.0355	1.5688	1.2551	0.0415	1.8940	1.2932
	MDCte	0.0270	1.0369	1.0170	0.0315	1.1895	1.0493

PCAmuBa [2]	MUNStr	0.0000	0.4805	0.4114	0.0000	0.5656	0.4315
	MUNSte	0.0005	0.5473	0.4341	0.0006	0.6272	0.4466
	VHRELte	0.0025	1.5628	1.0112	0.0029	2.0598	0.9189
	MCCte	0.0000	0.8983	0.6699	0.0000	1.0868	0.6670
	MDCte	0.0000	0.7239	0.6278	0.0000	0.8278	0.6436

ZSS [9]	MUNStr	0.0000	0.9547	0.7763	0.0000	1.1416	0.7924
	MUNSte	0.0000	1.0264	0.7915	0.0000	1.2142	0.8022
	VHRELte	0.0000	1.2261	0.8907	0.0000	1.6023	0.8360
	MCCte	0.0000	1.4855	1.3606	0.0000	1.7823	1.3557
	MDCte	0.0000	1.2944	1.1735	0.0000	1.5244	1.1730

Proposed	MUNStr	0.0000	0.4014	0.3891	0.0000	0.4810	0.4142
	MUNSte	0.0000	0.4644	0.4098	0.0000	0.5382	0.4299
	VHRELte	0.0000	1.0096	0.7278	0.0000	1.2754	0.6767
	MCCte	0.0000	0.3471	0.4645	0.0000	0.4196	0.5141
	MDCte	0.0000	0.4527	0.4733	0.0000	0.5051	0.5017

Method	Scores
	MUNtr			MUNte			VHRELte			MCCte			MDCte
	D65	A	F2	D65	A	F2	D65	A	F2	D65	A	F2	D65	A	F2
PINV	1	0	2	1	0	1	1	3	3	0	0	0	1	0	0
PINV-PCA	2	0	2	2	0	1	2	3	4	1	0	0	0	0	0
HAW	3	3	0	3	3	0	2	0	0	3	0	0	3	0	0
adaPCA	6	5	5	3	4	5	2	3	6	3	6	6	6	6	5
PCAemRe	0	4	0	0	4	1	0	0	0	2	0	0	1	4	0
PCAmuBa	3	6	6	3	6	6	6	0	0	3	0	0	3	6	5
ZSS	3	0	2	3	0	1	2	3	4	3	0	0	3	0	0
Proposed	6	7	7	3	7	7	6	7	7	3	7	7	6	7	7

Dataset	[0,100,100,1]		$[10^{- 3}, 100, 100, 1]$		$[10^{- 2}, 100, 100, 1]$		$[10^{- 1}, 100, 100, 1]$		[1,100,100,1]		$[10^{1}, 100, 100, 1]$		$[10^{2}, 100, 100, 1]$		$[10^{3}, 100, 100, 1]$		[1,0,0,0]
Dataset	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC
MUNStr	19.8029	0.9965	0.0181	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9972
MUNSte	20.1850	0.9961	0.0106	0.9967	0.0000	0.9967	0.0000	0.9967	0.0000	0.9967	0.0000	0.9967	0.0000	0.9967	0.0000	0.9967	0.0000	0.9965
VHRELte	24.5944	0.9782	0.5200	0.9832	0.0000	0.9854	0.0000	0.9854	0.0000	0.9854	0.0000	0.9854	0.0000	0.9854	0.0000	0.9854	0.0000	0.9843
MCCte	23.0726	0.9933	0.3170	0.9943	0.0000	0.9947	0.0000	0.9947	0.0000	0.9947	0.0000	0.9947	0.0000	0.9947	0.0000	0.9947	0.0000	0.9954
MDCte	21.7940	0.9957	0.0745	0.9964	0.0073	0.9964	0.0000	0.9964	0.0000	0.9964	0.0000	0.9964	0.0000	0.9964	0.0000	0.9964	0.0000	0.9959

Dataset	[1,0,0,1]		$[1, 10^{- 3}, 10^{- 3}, 1]$		$[1, 10^{- 2}, 10^{- 2}, 1]$		$[1, 10^{- 1}, 10^{- 1}, 1]$		[1,1,1,1]		$[1, 10^{1}, 10^{1}, 1]$		$[1, 10^{2}, 10^{2}, 1]$		$[1, 10^{3}, 10^{3}, 1]$		[0,1,1,0]
Dataset	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC	$Δ E_{94}$	GFC
MUNStr	0.0000	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9974	0.0000	0.9972
MUNSte	0.0000	0.9967	0.0000	0.9967	0.0000	0.9967	0.0000	0.9967	0.0000	0.9967	0.0000	0.9967	0.0000	0.9967	0.0000	0.9967	0.0002	0.9967
VHRELte	0.0000	0.9851	0.0000	0.9851	0.0000	0.9852	0.0001	0.9854	0.0000	0.9854	0.0000	0.9854	0.0000	0.9854	0.0000	0.9854	0.0036	0.9851
MCCte	0.0000	0.9947	0.0000	0.9947	0.0000	0.9947	0.0000	0.9947	0.0000	0.9947	0.0000	0.9947	0.0000	0.9947	0.0000	0.9947	0.0000	0.9946
MDCte	0.0000	0.9964	0.0000	0.9964	0.0000	0.9964	0.0000	0.9964	0.0000	0.9964	0.0000	0.9964	0.0000	0.9964	0.0000	0.9964	0.0008	0.9962

Begin
Derive a basis $V_{0}$ for the whole training set
Estimate the recovered spectra
Calculate the spectral residuals between the recovered
and the training set spectra
Derive a basis B for the spectral residuals
Use the basis $V_{0}$ to give an initial estimate of the recovered
spectra using Eq. (5) for each initial estimate of the
recovered spectra
Do
Calculate the GFC with the spectra in the training set
Derive a basis $V_{1}$ for the spectra with $GFC ∊ [p, 1]$
Minimize Eq. (18)
Compute the final recovered spectra using Eq. (17)
End
End

Begin
Derive a basis $V_{0}$ for the whole training set
Estimate the recovered spectra
Calculate the spectral residuals between the recovered
and the training set spectra
Derive a basis B for the spectral residuals
Use the basis $V_{0}$ to give an initial estimate of the recovered
spectra using Eq. (5) for each initial estimate of the
recovered spectra
Do
Calculate the GFC with the spectra in the training set
Derive a basis $V_{1}$ for the spectra with $GFC ∊ [p, 1]$
Minimize Eq. (18)
Compute the final recovered spectra using Eq. (17)
End
End

Abstract

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Figures (7)

Tables (7)

Equations (25)

Journal of the Optical Society of America A