Application of the Second-Order Statistics for Estimation of the Pure Spectra of Individual Components from the Visible

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1 Applcaton of the Second-Order Stattc for Etmaton of the Pure Spectra of Indvdual Component from the Vble Hyperpectral Image of her Mture Sung-Ho Jong Yong-U R Kye-Ryong Sn * (Department of Chemtry, Km Il Sung Unverty, Daeong dtrct, Pyongyang, DPR Korea) *E-mal: ryongnam9@yahoo.com Abtract: he econd-order tattc (SOS) can be appled n etmaton of the pure pectra of chemcal component from the pectrum of ther mture, when SOS eem to be good at etmaton of pectral pattern, but ther peak drecton are oppote n ome cae. In th paper, one method for judgment of the peak drecton of the pure pectra wa propoed, where the bae lne of the pure pectra wa drawn by ung ther htogram and the peak drecton were choen o a to make all of the pure pectra located upward over the bae lne. Reult of the SOS analy on the vble hyperpectral mage of the mture compoed of two or three chemcal component howed that the preent method offered the reaonable hape and drecton of the pure pectra of t component. Key-word: Second-order tattc, Pure pectra of chemcal component, Htogram, Mng matr, Hyperpectral mage 1. Introducton Durng the pat decade the hyperpectral mage analy ha emerged a a powerful tool n qualty control and dentfcaton of varou product ncludng foodtuff and n non-detructve tudy of bomateral, becaue t ha ome uperorte over the tradtonal method of chemcal analy n nvetgaton of the compoton, patal dtrbuton and dynamc feature of the chemcal component n the ample. Hyperpectral mage analy get the nformaton of the ample chemcal conttuton and t patal dtrbuton by analyzng pectral data et(hyperpectral cube) conted of meaured pectral data of every pel n 2D meaurng area. he analy of the hyperpectral cube effectve when every pel have dfferent pectral nformaton. Meanwhle pectral pattern and ntente correpondng to every pel n hyperpectral cube depend on the conttuton and concentraton of the chemcal component at the pel. Conequently, pectral mage analy applcable to the analy of the ample contanng

2 heterogeneou dtrbuton of the chemcal component 2D concentraton dtrbuton of the ndvdual component can be readly determned by the hyperpectral mage reoluton when F-IR or Raman pectra of the pure component were ued a prmary data becaue thee pectra can be clearly dtnguhed n the meaured wavelength nterval. But t not applcable n the vble - hort wave nfrared pectra becaue n th regon (λ=400~1000nm) the pectra of the component can not be ealy eparated, and conequently the more advanced method of pectral mage analy are requred. he key n hyperpectral mage(n th work we call the pectra of the mture) analy the correct etmaton of the number and pattern of the gnal from the ndvdual component. 1-5 Of the earler known method uch a PCA (Prncple Component Analy), 14 the end-member 15, MCR-ALS (Multvarate Curve Reoluton - Alternatng Leat Square), 16 ICA (Independent Component Analy) 10, and SOS (Second-Order Stattc) 6-9 for etmaton of ource gnal ICA and SOS are regarded to be ft up to date. ICA the bet for hgh accuracy of ource gnal detecton, but the ource gnal are not uppoed to have Gauan dtrbuton and furthermore dffculty n optmal electon of the objectve functon lmt t applcaton. SOS not a accurate a ICA but t relatvely mple becaue optmal electon of the objectve functon not requred for the ource gnal eparaton. Specfcally, t a accurate a ICA when the ource gnal have tme ere tructure. Conderng the fact that pectral ntente of one chemcal component n the two dfferent wavelength vary n a certan relaton, SOS uppoed to be more applcable for the pectra mage analy of the mture. here are ome report on applcaton of SOS for obtanng the pure gnal of the ndvdual ource from ther med one n mechancal vbraton, acoutc and electrc communcaton, 12,13 but t unknown on SOS applcaton for the pectral mage analy of the chemcal mture. In th paper preented wa one method of choong the correct peak drecton of the pure pectra of the chemcal component obtaned by SOS analy of the vble pectra of ther mture. 2. Judgment of the correct peak drecton of pure pectra of the chemcal component Source gnal detecton by SOS baed on the ngular value decompoton of the tme delayed covarance matr to eparate unknown ource gnal from the pace-tmely med gnal. 17 he equaton between the meaured gnal and the ource gnal can be wrtten a (t) A (t) N(t) (1) where t dcrete meaurng varable, A m n mng matr, and )) ( t) ( ( t), ( t),, ( t 1 2 m : vector of m output obervaton

3 )) ( t) ( ( t), ( t),, ( t 1 2 n : vector of n ource gnal N t) ( N ( t), N ( t),, N ( t)) m ( 1 2 : vector of m meaurng noe AMUSE algorthm for ource gnal detecton by SOS a follow: ) to etmate covarance matr R (0) of zero-mean output gnal, where N a number of meaurng varable. ) ˆ N R ˆ (0) 1/ N ( ( (2) k 1 to calculate egenvalue decompoton (EVD) or ngular value decompoton (SVD) of R ˆ (0), Rˆ (0) V V V V V V V (3) U N N N where [ v, v, v ] a m n matr correpondng to acendngly arranged n domnant V 1 2 n egenvalue dag{ } and V N a m (m-n) matr whch contan m-n noe 1 2 n egenvector correpondng to noe egenvalue dag{ } ) to apply prelmnary whtenng converon, 1 2 N n1 n2 m. ( V ( Q( (4) v) to etmate covarance matr of vector ( for pecal tme delay 0 and apply SVD for etmated covarance matr. k1 v) o etmate ource gnal and mng matr, N ˆ R ( ) 1/ N ( ( k ) U V (5) 1 2 Â Q U V U (6) y( ˆ( U ( t) (7) o apply the above SOS analy procedure n the etmaton of the pure pectra of the ndvdual chemcal component from the pectra of ther mture, here, propoed wa one mple model oluton that ha two chemcal component (P1 and P2) a olute (for eample, an aqueou oluton of potaum permanganate KMnO 4 and potaum dchromate K 2 Cr 2 O 7 ) and uppoed that the pectra of both of P 1 oluton and P 2 oluton were already known a hown n Fgure 1.

4 (P 1 ) (P 2 ) Fgure 1. he ntal pure pectra of two component (P 1 and P 2 ) From here, SOS analy on the above model wa carred out by ung MALAB. 18 It wa aumed that the meaurng object conted of 7 pel and the mng matr A compoed a A = [ ; ; ; ; ; 0 1; 1 0 ] and the meaured pectrum of the P 1 -P 2 mture modeled by y() A (), ) ( ( ), ( )) ( 1 2 In th cae, the etmated pectra of the P 1 -P 2 mture and t character were hown n Fgure 2, where 2-a the change of y() accordng to and 2-b the change of column vector of A accordng to pel, howng the concentraton dtrbuton of the component. (a) Fgure 2. he modeled pectra of the P 1 -P 2 mture and the mng matr A (b) he reult of SOS analy by ung y() a the ntal data (the pectra of the P1-P2 oluton)

5 were gven n Fgure 3, where ŝ 1 () and ŝ 2 () are the pure pectra of P 1 and P 2, repectvely. Fgure 3. he etmated pure pectra from the ntal y() By comparng Fgure 1 and 3, t can be een that the pattern of the etmated pure pe ctra of P 1 and P 2 by SOS were qute mlar to the ntal one, but the peak drecton of ŝ 2 () wa the oppote to the ntal pure pectrum of P 2 n Fgure 1. he change profle of column vector of the mng matr A calculated from the etmated pure pectra by the equaton (6) wa hown n Fgure 4, from whch t can be een that the mng matr A from the etmated pectra wa dfferent from the one from the ntal pectra n Fgure 2-b. he dfference n profle change of the column vector between mng matr A from the ntal pectra and that from the etmated one can be attrbuted to the oppote peak drecton of the pure pectra. From the above tral applcaton of SOS n etmaton of the pure pectra from the vble pectrum of ther mture, t can be concluded that one of the key problem here n SOS applcaton for chemcal mture to chooe the correct drecton of the peak n the etmated pure pectra of the chemcal component. Fgure 4. he change profle of column vector of A from the etmated pectra

6 um. o olve th problem, we propoe one method for the correct peak drecton a follow. he frt tep of th method the ratonal defnton of the bae lne for the pure pectr It common that mot of the vble pectra of chemcal ubtance have the pattern of comparatvely long and mooth lne, parallel to the bae lne and the peak located upward over the lne. So t qute reaonable to chooe th mooth lne a the bae lne, where t he htogram value of the pectrum uppoed to be the mamum. herefore, the bae lne b of the th pure pectrum ˆ ( ) choen a follow: [ z y ] ht(ˆ ( )), where k atfe z ma{ z }, j 1, 2,, 10 k j b y k he econd tep to fnd out the peak of pectrum by nterpretaton of the gn of and to locate the mamum peak value p ma and mnmum peak value mn At lat, the correct peak drecton determned by ung the value of follow: - the correct drecton ( p b ) 0}&{( p b ) 0} or {( ma mn {( p ma b ) 0}&{( p mn b ) 0}&{ P ma b P mn - the oppote drecton {( ma mn p b ) 0}&{( p b ) 0} or {( p ma b ) 0}&{( p mn b ) 0}&{ P ma b P mn b } b } p. p ma -b and In cae of the oppote drecton, the gn of the correpondng ˆ ( ) have to be changed. 3. Reult and Dcuon dˆ ( ) / d p mn -b a he preent method for choong the correct peak drecton wa appled n SOS analy for the above model and the reult etmated from the ntal pectra of the P 1 -P 2 mture n Fgure 2 were hown n Fgure 5, 6.

7 Fgure 5. he baelne for the pure pectra of the two component Fgure 6. he pure pectra of P 1 and P 2 etmated by the preent method he change profle of column vector of mng matr A calculated from the pure pectra by the preent method wa hown n Fgure 7. Fgure 7. he change profle of the newly etmated column vector of A

8 In comparon wth thoe n Fgure 4, the change profle n Fgure 7 are n better concdence wth the ntal one n Fgure 2-b. hee reult howed that the propoed method for the correct peak drecton n the SOS etmaton of the pure pectra of the chemcal component from the vble pectra of ther mture can gnfcantly reduce error not only n the drecton of the peak, but alo n the concentraton dtrbuton analy of chemcal component n the ample mture. In addton, poblty of applyng the preent method n mult-component ytem wa teted wth another model oluton wth 3-component olute (P 1, P 2, P 3 ), where the ntal pure pecrtra of P 1, P 2, and P 3 were modeled lke thoe n Fgure 8. (P 1 ) (P 2 ) (P 3 ) Fgure 8. he pectral pattern of the 3 component (P 1, P 2, P 3 ) her mng matr A wa aumed a follow: A = [ ; ; ; ; ; ; ]

9 he pectra of ther mture (P 1 -P 2 -P 3 ) wa modeled by y() A(), ) ( ( ), ( ), ( )) ( (Fgure 9). 10. Fgure 9. he mulated pectra of the P 1 -P 2 -P 3 mture he etmated pure pectra for the P 1 -P 2 -P 3 mture by the preent method were gven n Fgure Fgure 10. he etmated pure pectra of P 1 -P 2 -P 3 mture

10 hee reult mply that the preent method can offer the wder applcaton of SOS analy n the nvetgaton of behavor of the ndvdual component n the mult-component chemcal ytem. 5. Concluon Second-Order Stattc eem to be one of the properou pectral mage analy method n the nvetgaton of the comple chemcal ytem. ral applcaton of SOS n etmaton of the pure pectra of the ndvdual chemcal component from the vble pectra of one model mture compoed of two component howed that t may gve the unreaonable pure pectra of the chemcal component wth the oppote peak drecton, whch can be a reaon of ncreang the error n determnaton of the true pure pectra of the component. Here n th paper preented wa one method for judgment of the correct drecton of peak n the pure pectra of the component obtaned by SOS analy of the pectra of ther mture, where the bae lne defned from the htogram and the mamum- and mnmum-value of the peak were ued for choong the reaonable peak drecton. he preent method wa appled n SOS analy of the pure pectra of the component from the vble pectra of the two- and three-component chemcal mture and gave the better reult n the peak drecton and concentraton dtrbuton, compared wth the common SOS analy. More detaled reearche for SOS applcaton n the pectral mage analy of the complcated mult-component chemcal ytem eem to have good propect n practce of chemtry and SOS telf. Reference [1] Antone Souloumac, A Stable and Effcent Algorthm for Dffcult Non-orthogonal Jont Dagonalzaton Problem, 19th European Sgnal Proceng Conference (EUSIPCO 2011) Barcelona, Span, 2011, Augut 29 - September 2. [2] Adel Belouchran and Andrzej Cchock, A Robut Whtenng Procedure n Blnd Source Separaton Contet, Electronc Letter, 2000, 36, [3] A. Ikhlef, R.; Iferroujene, A.; Boudjellal, K.; Abed-Meram, A.; Adel Belouchran, Contant Modulu Algorthm Ung Hyperbolc Gven Rotaton, 2013, arxv: vl. [4] Meloub Ammar; Karm Abed-Meram; Adel Belouchran, A New Algorthm for Comple Non orthogonal Jont Dagonalzaton Baed on Shear and Gven Rotaton, 2013, arxv: vl. [5] Luca Parra; Paul Sajda, Blnd Source Separaton va Generalzed Egenvalue Decompoton, Journal of Machne Learnng Reearch, 2003, 4, [6] Maart, D.L.; Vandegnte, B.G.M.; Buyden, L.M.C.; Jong, S.; Lew, P.J.; Smeyer-Verbeke, J.,

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