SPECTRAL RECONSTRUCTION IN CASE OF DIFFERENT ILLUMINANTS

Transcription

1 AALS of Faulty Engneerng Hunedoara Internatonal Journal of Engneerng Tome XIV [6] Fasule [February] ISS: [prnt; onlne] ISS: [CD-Rom; onlne] a free-aess multdsplnary publaton of the Faulty of Engneerng Hunedoara. Zsolt SÁVOLI,. Balázs KRÁICZ, 3. András HORVÁTH SPECTRAL RECOSTRUCTIO I CASE OF DIFFERET ILLUIATS -. Interdsplnary Dotoral Shool of Engneerng, Széheny István Unversty, Győr, HUGARY 3. Department of Physs and Chemstry, Széheny István Unversty, Győr, HUGARY ABSTACT: Fndng the spetral features of olor sample sets s a man ssue of olormetry. It s wdespread to apply Prnpal Component Analyss n these researhes. Several studes were wrtten out the reonstruton of samples wth known spetra usng prnpal omponents, and some works dealt wth approxmate reonstruton from trstmulus values. Our study examnes how spetral reonstruton done wth genet optmzaton works n ase of dfferent llumnants. The authors have taken only the trstmulus values of the olor samples gven. Also, they examne the reonstruton n ase of llumnants that have unknown spetral power dstrbutons, then they gve the soluton for these ases n bref. Keywords: Spetral reonstruton, trstmulus values, prnpal omponent analyss, genet optmzaton, llumnants. ITRODUCTIO It s a bas thess of olormetry that any olor stmulus an unequvoally be gven by three numbers. These numbers an be the CIE X, Y, Z trstmulus values or the oordnates of any other sutle olor spae ( Y, x, y; L, a, b ; L, u, v et.). However, the desrpton of self-lumnous objets (lght soures) and surfaes (seondary lght soures) s often gven wth spetral features, n other words wth spetra. Ths type of desrpton provdes muh more nformaton out the observed objet, t needs, therefore, more than three parameters. The researhers n olormetry started examnng how they an determne or gve approxmately the refleton spetra of surfaes wth only a few numbers. Prnpal Component Analyss (PCA), whh s based on elements of mathematal statsts and lnear algebra, has appeared to be an espeally strong and nterestng tool. Several studes deal wth the usage of Prnpal Component Analyss n olormetry, therefore ts mathematal presentaton s not the subjet of the present artle. Publatons []-[7] provde a general vew how t works. In order to apply ths method effetvely, t s neessary to have a set wth a large number of known spetra. Prnpal Component Analyss produes the egenvetors belongng to the sample set. The lnear ombnaton of these egenvetors helps to reonstrut the spetrum. The ove mentoned studes present reonstruton of samples that have known and measured spetra. The queston s whether t s possble to say anythng out the spetrum of olor samples that have unknown refletane funtons f we only know ther trstmulus values X,Y, Z. In some of the studes fousng on ths problem [8], [9], the spetral reonstruton s done wth an algorthm usng pseudo nvers matrx operatons. However, Prnpal Component Analyss or weghed Prnpal Component Analyss has been used n other studes. Correspondng to the trstmulus values, the three egenvetors, whose egenvalues are the greatest ones, are enough to get the same X,Y, Z values as the result of reonstruton [], []. ore aurate results an be reahed wth more omponents, but they lead to undetermned 45 Fasule

2 AALS of Faulty Engneerng Hunedoara Internatonal Journal of Engneerng equaton systems, as the trstmulus values an be gven n many ways from more than three omponents. Beause of ths rtal statement, three vetors have been onsdered to be satsfatory n most ases. Publaton [] desrbes the authors method that deals wth ths problem. They used the frst fve egenvetors nstead of the frst three vetors and the features of the refleton funtons of real olor samples for gener optmzaton. It was onsdered that the refleton funtons of the real samples are smooth, wthout strong osllaton and they are non-negatve. Ths method provdes a muh more aurate reonstruton than the earler ones. (Fgure ) The set of 83 textle samples wth known refletane funtons were used. Furthermore, egenvetors 48 flower samples, 565 pant samples and human skn samples were appled. The formalsm of optmzaton s to be shown n bref. ore detals an be read n publaton [].. THE FORALIS OF OPTIIZATIO Durng alulaton, the authors work wth spetra (refletane funtons) whose resoluton s gven, therefore, fnte-dmensonal vetors are used nstead of ontnuous funtons. Let denote the dmenson number of these vetors. For example, f a spetrum s wth a range of 4 nm - 7 nm and wth an equdstant-wavelength step of at nm, 3. The egenvalues of the PCA method arranged n dereasng order are denoted by τ τ τ, the egenvetors relatng to the egenvalues are denoted by v, v,, v, and the mean vetor by m. The lnear ombnaton of egenvetors and the prnpal omponents, provdes the followng spetrum. (,,, ) 46 Fasule (,,, ) v + f () m f s also an -dmensonal vetor. Havng fxed, varles, determne the spetrum of the reonstruted f aordng to equaton (). As a next step, a funton s to be reated that measures the dfferene between ths type of spetrum and the deal spetrum. The dfferene s small for smooth and non-negatve metamers, and t s greater and greater, f the trstmulus values devate from the stpulated ones or f the funton osllates strongly or t takes up negatve values. It s easy to alulate the trstmulus values of the spetrum by the applaton of olor-mathng funtons. X f S x, Y f S y, Z f S z () Fgure. Reonstruton wth three and wth fve In Eq., x, y, z denote the dsrete versons of the CIE olor-mathng funtons whh have the same resoluton as that of the spetra, S. s the dsrete spetral power dstrbuton of the llumnant. Obvously, the values of X, Y, Z depend on the oeffents, but ths dependenes not emphaszed for the sake of brefness. It s possble to alulate the squared sum of the dfferenes to show how muh the values X, Y, Z devate from the predefned values X, Y, Z d ( ) ( ) ( ) ( ), X X + Y Y + Z Z (3) Ths d value s non-negatve and t s equal to when a metamer omples wth the defnton. If 3, the equaton system of the metamer has a sngle soluton. A lot of earler studes whh used PCA ended wth gvng ths soluton. If > 3, t has an nfnte number of solutons, and the most realst one an be hosen wth the use of onstrants on negatvty and strong osllaton. It s possble to desrbe the negatvty of the funton bythe ntegral of the negatve and the postve part, or wth the rato of ther sums n the dsrete ase. Denote: + f max( f,); f mn( f,) (4)- (5)

3 + + f ISS: [prnt]; ISS: [onlne] F ; F f (6)- (7) Defntons (4) and (5) resemble the terms of lower and upper overng funtons used n analyss. The penalty term on negatvty s: F Pn W (8) + n ( F + F ) Wn s the weght fator whh s used to set the relatve weght of ths term wthn the optmzaton funton. It s obvous that P n f the funton has only non-negatve values andw n, and the more negatve parts, f ontans the greater postve values P n has got. It s n extreme ases. The osllaton of the funton s defned by the squared sum of the devaton between the neghborng terms: The penalty term on osllaton s ( f f ) V (9) P v + v ( ) V W () The ost funton whose mnmum s assumed to determne the metamer wth the best qualtatve features s the followng. d (, ) d + Pn + Pv () It s possble to get d from (3), Pn from (8) and P v from (). The weghts W n and W v show the mportane of one or the other penalty terms. Aordng to the pre-alulaton we have already made, the useful values are Wn,W. A lttle hange n them wll not nfluene the fnal result. All n all, d s a non-lnear funton wth varles, whose mnmum orresponds to the best funton for the researhers, n other words, vetor (, ) whh gves the loaton of the extreme values, ontans the optmal weght of the egenvetors used n the reonstruton. In order to fnd the mnmum pont of funton d, whh has been gven n equaton () ove, we use our own genet optmzaton program. The genet algorthm was hosen beause d has a lot of loal mnma (manly beause of the osllaton term) and the gradent-based methods generally annot fnd the global mnmum n these ases. Our genet algorthm uses the standard genet operators, e.g. mutaton and rossng, and n order to aelerate the searh for loal maxma, t uses hll-lmbng steps. The authors had already appled ths ode to solve more ndustral optmzaton problems [3]. Values desrbng the auray of the reonstruton It s possble to desrbe the auray of the reonstruton by several measurng numbers n a quanttatve way. Also, t s possble to measure the olor dfferene between the examned sample and the reonstruted sample, the spetral devaton and auray between the orgnal and the reonstruted refleton funton. The olor dfferene an be gven as t follows. As the frst step, the trstmulus values X, Y, Z of the X, Y, Z are the trstmulus values samples have to be transformed nto values L, a, b (), where n n n of the referene whte trstmulus values under a gven llumnant. Y X Y Y Z L 6 f 6, a 5 f f, b f f Yn X n Yn Yn Z n t, t > ( ) 9 () f t t, t In ase of all the used samples, the ondton 6 t > s met. The X n, Yn, Z n values of the referene 9 whte trstmulus under the llumnant wth S( λ) spetral power dstrbuton are gven by equaton (3). 47 Fasule

4 AALS of Faulty Engneerng Hunedoara Internatonal Journal of Engneerng 7 ( λ) x ω ( λ) dλ, Yn, Z n k ϕ S( λ) z ( λ) dλ X n k ϕ S ω (3) 4 ( λ) y ( λ) dλ 7 4 k (4) ϕ 7 The normalsaton oeffent s denoted by ϕ 4 S ω k and x ω ( λ), y ( λ), z ω( λ) funtons. The values of the omparle olor samples are L the CIE L olor dfferene an be determned (5). E ω, a, b ( L L ) + ( a a ) + ( b b ) are the CIE olor mathng, and L, a,. Knowng these, b E (5) If E, the test samples are the same. If E, t gves the just pereptble dfferene under a gven llumnant. Two dfferent values are gven for the spetral auray by the publatons. One of them s a numeral value, GFC (goodness of ft oeffent) (6). 3 ϕ( λ ) ϕr ( λ ) GFC (6) 3 3 ϕ( λ ) ϕ ( λ ) ϕ( λ ) denotes the sample refleton funton at wavelength ϕ r λ denotes the reonstruted spetrum at λ. If GFC, the reonstruted funton s perfetly dental wth the orgnal one. Therefore, the loser GFC gets to, the more aurate the spetral reonstruton s. The formula n (6) orresponds to the osne value of the angle of two strat vetors. The other auray spefyng value s RS (root mean square), whh gves the reonstruton error aordng to the dfferenes between the orgnal and the reonstruted spetra (7). r λ. ( ) 3 RS ( ϕ( λ ) ϕr ( λ )) (7) 3 The notatons n equaton (7) are the same as the notatons n equaton (6). The smaller the value s, the smaller the spetral error s and the greater the reonstruton auray s. The llumnants used n the test The authors used only one well-known llumnant, the CIE E,.e. the equenerget llumnant n ther former publaton []. Ths llumnant was used for the reonstruton of textle samples, flower samples, skn samples and,- pant samples. ow the queston s how the spetral reonstruton wll work n ases of dfferent llumnants wth famlar spetral power dstrbutons. oreover, the seond queston s what happens f the spetral omponents of the llumnant are unknown. The llumnants are as t follows. CIE D65 standard llumnant, whh orresponds to the daylght dstrbuton wth 6 54K orrelated olor temperature. (Fgure ) Fgure. The spetal power dstrbuton of CIE Fgure 3. The spetral power dstrbuton of CIE D65 llumnant D5 llumnant CIE D5 standard llumnant represents the daylght wth a orrelated olor temperature of 5 3 K. (Fgure 3) CIE A standard llumnant s ntended to represent tungsten-flament lghtng. Its 48 Fasule

5 ISS: [prnt]; ISS: [onlne] spetral power dstrbuton orresponds to that of a Plankan radator wth a temperature of 856 K(Fgure 4). Fgure 4. The spetral power dstrbuton of thecie Fgure 5. The spetral power dstrbuton of the A llumnant CIE E llumnant CIE E equenerget llumnant. Ths llumnant has onstant spetral power dstrbuton. Ths theoretal radator renders equvalent weght to eah wavelength. (Fgure 5) CIE F standard llumnant. Its spetral power dstrbutons orrespond to the power dstrbutons of narrow-band fluoresent lamp wth 4 K orrelated olor temperature (Fgure 6). Fgure 6. The spetral power dstrbuton of the Fgure 7. The spetral power dstrbuton of the CIE F llumnant whte LEDs wth phosphor Whte LEDs wth phosphor and three-band whte LEDs wth 5 K and 6 54K orrelated olor temperature arealso appled. The whte LED wth phosphor wth 5 K s named as LED, the three-band whte LED wth 5 K s named as LED, the whte LED wth phosphor wth 6 55K s named as LED3 and the three-band LED wth 6 54K s named as LED4. Fgure 7 shows the spetral power dstrbutons of the whte LEDs wth phosphor, Fgure 8 shows the spetral power dstrbutons of the three-band, whte LEDs. Fgure 9 shows the spetral power dstrbuton of the Plank radator whose olor temperature s 6 54 K and whh s also used n ths researh. Fgure 8. The spetral power dstrbuton of the Fgure 9. The spetral power dstrbuton of the three-band LEDs Plank radator at the olor temperature of 654 K The reonstruton of olor samples under llumnants wth known spetral power dstrbutons The reonstruton of the refletane funtons of the depted textle samples has been done by the help of genet optmzaton on the bass of the trstmulus values X,Y, Z. The frst fve egenvetors gven by the PCA and omplemented wth restrtve ondtons on the refletane funtons of real samples were used. Eah sample has been reonstruted under the ove mentoned llumnants. Fgure shows the reonstruton for some samples. 49 Fasule

6 AALS of Faulty Engneerng Hunedoara Internatonal Journal of Engneerng Fgure. The reonstruton of olor samples supposng dfferent llumnants The values E,RS, GFC gvng the Tle. The mean values of reonstruton auray have been determned for eah sample n ase of eah resoure. The numer values are gven wth the use of the orgnal egenvetors. refletane funton whh s presumed unknown and the refletane funton ganed after RS GFC optmzaton under the gven llumnants. Tle CIE E shows the average of the values E,RS, GFC for CIE D CIE D the tested samples. CIE A As for omparson, Tle shows the mean values CIE F RS for the reonstrutons of other textle samples LED ndated n publaton []. The frst three LED egenvetors are used by the help of PCA and wpca. Tle. The mean values of RS wth methods of PCA and wpca and usng three egenvetors. PCA: wpca: Fasule E, RS, GFC ase of dfferent llumnants wth genet optmzaton and wth the use of fve E n

7 ISS: [prnt]; ISS: [onlne] The tles show that our method of reonstruton whh uses fve egenvetors and onsders the restrtons on the forms of real samples provdes muh better results than the lassal PCA and wpca wth the three egenvetors. Fgure and Tle prove that our method does not depend on the type of the known llumnant. Ths method provdes the refletane funton of one of the metamers of the olor sample wth great auray n ase of any llumnants wth known spetral power dstrbuton. Illumnant wth unknown spetral power dstrbuton Is s natural demand to know what the reonstruton wll be lke f the spetral power dstrbutons of the llumnants were not known. Frst, we supposed that we dd not have any nformaton X n,yn, Zn of the llumnants are out the llumnants. Then t s supposed that the whte ponts known (see next hapter). A olor sample s depted and the reonstruton s done as t follows. It s supposed that the spetral power dstrbuton of the real llumnant s unknown. Certanly, t has to be taken nto aount for the sake of the numeral formulaton and the omparson but not for reonstruton. Then the authors use the seven ove mentoned llumnants as the real, unknown llumnants of the samples, and the reonstrutons are done wth all the seven llumnants. That means 7 7 ases altogether. Certanly, the llumnant that s used n realty s among the seven llumnants. The reonstruton s as aurate as n the former ases. When the llumnant s dfferent from the real one, the reonstruton s generally qute weak. Fgure shows the reonstruton of one sample as t s desrbed ove. The ttle of the graph refers to the real but surpassngly unknown llumnant. The sold blak lne ndates the orgnal sample n eah ase. The olorful lnes ndate the reonstruted spetra under the supposed llumnants. Only few of the ases are shown beause of mmensty. Fgure. The reonstruton n ase of an unknown llumnant Tle 3 shows the averages of the numeral values desrbng the reonstruton onsderng the ombnaton of all llumnants, the ombnaton wthout the real llumnant and the orgnal, atual llumnant. Tle 3. The mean values of RS, GFC for dfferent ombnatons of llumnants E, The ombnaton of all llumnants The ombnaton wthout the atual real llumnant RS GFC The orgnal, atual real llumnant E All the graphs show what the tles show: the reonstruton s not aurate f the spetral power dstrbuton of the llumnant s unknown. 5 Fasule

8 AALS of Faulty Engneerng Hunedoara Internatonal Journal of Engneerng Illumnant wth known whte pont but wthout known spetral power dstrbuton As a next step, they examne the reonstruton when the spetral power dstrbuton s unknown but there s some nformaton out the llumnant and ts whte pont s known. In ths ase, we modfy the ost funton () of the genet reonstruton. Instead of the trstmulus values X, Y, Z, we use the values L, a, b for optmzaton (8), (9). ' d (, ) d + Pn + Pv (8) ' Unlke equaton (3), here d does not ontan the trstmulus values of the sample, but the values L, a, b () derved from the trstmulus values. They are used beause the CIE L system onsders the adaptaton of whte. ' d ( ) ( ) ( ) ( ), L L + a a + b b (9) The values L, a, n the equaton are the orgnal values of the sample under an llumnant b wthout known spetral dstrbuton but wth known whte pont. The values L, a, b an be alulated from the reonstruted spetrum. Durng alulaton, the whte pont of the hosen llumnant s used. The prnpal omponents are determned wth the genet optmzaton algorthm. The method of reonstruton s smlar to the former ones. They use a real llumnant wthout known spetral dstrbuton but known whte pont and the reonstruton s done wth seven known llumnants. It means 7 7 ases altogether. Fgure shows the reonstruton of a depted sample. The ttle of the graph ontans the name of the llumnant surpassngly wthout spetral power dstrbuton but wth known whte pont. Only few ases are shown beause of mmensty. Fgure. The reonstruton for an llumnant whose spetral power dstrbuton s unknown but the whte pont s known Tle 4 shows the average value of E,RS, GFC for all possble ombnatons, even f the real, orgnal llumnant s left out when the average values are determned. oreover, the values are examned when the real llumnant s onsdered. Tle 4. The mean values of E,RS, GFC for dfferent ombnatons of llumnants The ombnaton of all llumnants The ombnaton wthout the atual real llumnant RS GFC The orgnal, atual real llumnant E Fasule

9 ISS: [prnt]; ISS: [onlne] Aordng to Fgure and Tle 4, they an state that the reonstruton for an llumnant wth unknown spetral dstrbuton s more aurate f there s some nformaton out t, n other words, the whte pont of t s known. However, the value RS, whh gves the reonstruton error, s stll great. Illumnant wth known whte pont and wth orrelated olor temperature but wthout known spetral power dstrbuton It s supposed that the results of the reonstruton wll be better f they use llumnants wth known spetral power dstrbutons whose orrelated olor temperature values are the same as that of the atual real llumnants whh have unknown spetral power dstrbutons but ther whte ponts are known. They determned the spetrum for samples hosen randomly aordng to the desrpton n the hapter mentoned ove. The reonstruton s done wth llumnants whose spetral power dstrbuton s known nstead of the atual real llumnant whose whte pont, orrelated olor temperature s only known but ts spetral power dstrbuton s unknown. The llumnants have the same orrelated olor temperature as the real one does. The prnpal omponents are provded by the genet optmsaton program. The followng four llumnants are used n all four ases: CIE D65, whte LED wth phosphor (or LED3), three-band, whte LED (or LED4) and the Plankan radator. Fgure 3 shows just a few ases of reonstruton. The ttle of the graph ontans the name of the atual real llumnant that s supposed to be unknown. Fgure 3. The reonstruton for llumnants whose spetral power dstrbutons are unknown, but whh have a known whte pont and a orrelated olour temperature Tle 5. The mean values of E, RS, GFC for dfferent ombnatons of llumnants The ombnaton of The ombnaton wthout the The orgnal, atual real all llumnants atual real llumnant llumnant RS GFC E The authors determne all the ombnatons and the average values of E,RS, GFC, when they do not alulate wth the orgnal llumnant and when they alulated wth only that one. The values are shown by Tle 5. Fgure 3 and Tle 5 show that the reonstruton s effetve for an llumnant wthout known spetral power dstrbuton but wth a whte pont and orrelated olor temperature f the orrelated olor temperature of the llumnant s the same as the orrelated olor temperature of the orgnal llumnant wth unknown spetral power dstrbuton. The Tle 6 shows the summary of former ases. 53 Fasule

10 AALS of Faulty Engneerng Hunedoara Internatonal Journal of Engneerng Tle 6. The summary of former ases Illumnant used n the reonstruton, S Illumnant ove the olor samples, S (spetrum always known) spetrum known S Chapter 5 S ost funton s bass: CIE XYZ spetrum unknown - Chapter 6 ost funton s bass: CIE XYZ spetrum unknown, whte pont known L L, a a, b b Chapter 7 ost funton s bass: CIE L spetrum unknown L L, a a, b b Chapter 8 whte pont known orrelated olor temperature known T p ( S ) Tp ( S ) ost funton s bass: CIE L 3. COCLUSIO Present study shows a genet optmsaton proess that helps to reonstrut the refletane funton of olorful samples on the bass of ther trstmulus values. Durng reonstruton the authors use the frst fve egenvetors provded by the PCA together wth restrtve terms for the shape of the refletane funtons of the real samples. It gves the opportunty to hoose those metamers from ther nfnte set n suh a way that the refletane funton orresponds wth the refletane funtons of the real samples. Ths study examnes how the ove mentoned method works n ase of 7 dfferent llumnants whose spetral power dstrbuton s known. It an be found that the genet optmzaton provdes the refletane funton aurately f the spetral power dstrbuton of the llumnant s known and t does not depend on the type of the llumnant. Fnally, the reonstruton s effetve for an llumnant that has unknown spetral power dstrbuton but known whte pont and a orrelated olor temperature when they use suh an llumnant whose orrelated olor temperature s the same as that of the orgnal one. Referenes [] Tzeng DY, Berns RS: A Revew of Prnpal Component Analyss and Its Applaton to Color Tehnology. Color Researh and Applaton, 5 Aprl, Volume 3, umber, pp [] aloney LT: Evaluaton of lnear models of surfae spetral refletane wth small numbers of parameters. Journal of Optal Soety of Amera, 986, 3, pp [3] Jaaskelanen T, Parkknen J, Toyooka S: Vetor-subspae model for olor representaton. Journal of Optal Soety of Amera, 99, 7, pp [4] Vrhel J, Gershon R, Iwan LS: easurement and analyss of objet refletane spetr. Color Researh and Applaton, 994, 9, pp.4 9. [5] Garía-Beltrán A, eves JL, Hernández-Andrés J, Romero J: Lnear bases for spetral refletane funtons of aryl pants. Color Researh and Applaton, 998, 3, pp [6] Tajma J: A huge spetral haratersts datase and ts applaton to olor magng deve desgn. Proeedngs of the 6th IS&T/SID Color Imagng Conferene, IS&T, Sprngfeld, VA; 998. pp [7] Hardeberg JY, Shmtt F, Brettel H: ultspetral olor mage apture usng a lqud rystal tunle flter. Opt Engneerng, 4, pp [8] Farman HS, Brll H: The prnpal omponents of refletanes. Color Researh and Applaton, 4, Vol. 9, o. 4, pp.4-. [9] Harf T, Amrshah SH, Agahan F: Reovery of refletane spetra from olormetr data usng prnpal omponent analyss embedded regresson tehnq.opt. Rev. 5, 8, pp [] Farnaz A, Seyed AA, Seyed HA: Reonstruton of refletane spetra usng weghted prnpal omponent analyss. Color Researh and Applaton, 8, Vol. 33, o. 5, pp [] Smone B: Refletane spetra reovery from trstmulus values by adaptve estmaton wth metametr shape orreton. Journal of Optal Soety of Amera,, 7, 8, pp [] Sávol Zs, Horváth A, Kránz B: Színmnta-halmazhoz lleszkedő, jó kvaltatív tulajdonságú metamer gyors keresése.képfeldolgozók és Alakfelsmerők Társaságának. országos konferenája, Keskemét, 5, pp [3] Horváth A, Horváth Z: Optmal shape desgn of Desel ntake ports wth evolutonary algorthm. Festauer (szerk.) umeral athemats and Advaned Applatons, EUATH 3 54 Fasule