ANALYSIS OF NON-NEGATIVE TIKHONOV AND TRUNCATED SINGULAR VALUE DECOMPOSITION REGULARIZATION INVERSION IN PCS

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1 1 th February 13. Vol. 48 No JATIT & LLS. All rghts reserved. ISSN: E-ISSN: ANALYSIS OF NON-NEGATIVE TIKHONOV AND TRUNCATED SINGULAR VALUE DECOMPOSITION REGULARIZATION INVERSION IN PCS 1 YAJING WANG, 1 JIN SHEN, GANG ZHENG, 1 ZHENHAI DOU, 1 ZHENMEI LI 1 School of Electrcal and Electronc Engneerng, Shandong Unversty of Technology, Zbo 5591, Shandong, Chna College of Optcs and Electronc Informaton Engneerng, Shangha Unversty of Scence and Technology, Shangha 93, Chna ABSTRACT Consderng non-negatve characterstc of the partcle sze dstrbuton (PSD), based on trust-regonreflectve Newton method, two non-negatve regularzaton methods of truncated sngular value decomposton (TSVD) and Tkhonov (TIK) for photon correlaton spectroscopy (PCS) are proposed n ths paper. Combnng two regularzaton parameter crterons of and, two non-negatve regularzaton methods are studed. The study results show that, compared wth TIK, TSVD has bgger truncaton effect, poorer smoothness and narrower dstrbuton wdth of nverson PSD, n the case of nose, TSVD has smaller relatve error and peak value error of PSD, better capacty to dscrmnate bmodalty and stronger ant-nose, but at nose-free case, TSVD hasn t obvous advantages, TIK and TSVD are respectvely more sutable for usng and crteron to determne the regularzaton parameter. Keywords: Photon Correlaton Spectroscopy, Partcle Sze, Non-negatve Regularzaton, Inverson 1. INTRODUCTION Photon correlaton spectroscopy (PCS, s also called dynamc lght scatterng) technology has become an effectve method for measurng submcron and nano-partcles sze[1], whch obtan the partcle sze dstrbuton (PSD) by measurng and nvertng the autocorrelaton functon (ACF) of the ntensty fluctuatons scattered by the nvestgated sample. However, nvertng PSD from ACF s a hgh ll-posed problem, whch has been the dffculty n PCS technology. Only when ACF measured s no noses and data calculaton s no roundng errors, does the equaton have a unque soluton n theory. In the measurement, because of the presence of the nose and roundng errors, the exstence, unqueness and stablty of solutons are dffcult to guarantee. Therefore, n PCS technology, we can t solve true PSD, and only obtan ts approxmate value. For approxmate soluton, researchers have proposed numerous methods such as Cumulants method [], exponental samplng method [3], the Bayesan strateges method [4], and the neural network method [5]. Consders non-negatvty of PSD, there are some other methods such as CONTIN [6] and NNLS [7], whch have been verfed to mprove the nverson accuracy of PSD. However, these methods have the drawbacks of senstvty to nose, dffculty n choosng a regularzaton parameter, and complcatons n operaton, poor the capacty to dscrmnate bmodalty. For the solvng of ll-posed equaton, regularzaton s one of most powerful approaches. In PCS nverson, consderng together the regularzaton and non-negatvty should be very effectve. Trustregon-reflectve Newton method [8] s a wdely used non-negatve constrant method whch s ncluded n the Matlab toolbox. Therefore, based on Trust-regon-reflectve Newton method, nonnegatve regularzaton for PCS s proposed n ths paper. The regularzaton are many methods such as Tkhonov (TIK) regularzaton, truncated sngular value decomposton (TSVD) regularzaton, landweber regularzaton, conjugate gradent(cg) regularzaton and so on. So far, there s not an optmal regularzaton method sutable for any problem. Therefore, the study of dfferent regularzaton methods s very meanngful n PCS nverson. In ths paper, TIK and TSVD were compared. In regularzaton method, selecton of regularzaton parameter s crtcal. The generalzed 33

2 1 th February 13. Vol. 48 No JATIT & LLS. All rghts reserved. ISSN: E-ISSN: cross-valdaton () crteron [9] and crteron [1] are more wdely used n practce. and crteron are also compared n the PCS. By the comparson, we fnd out that characterstcs and sutable regularzaton parameter crteron of each method, and draw a concluson that TSVD has better capacty to dscrmnate bmodalty and stronger ant-nose.. INVERSION PRINCIPLES OF PCS For the lght feld of the Gaussan dstrbuton, ACF of scattered lght ntensty s gven by a Segert relatonshp. For the polydsperse partcles, the normalzed ACF of scattered lght ntensty s expressed as g ( τ) = G( Γ)exp( Γτ) dγ G ( Γ) dγ = 1 (1) where τ s the samplng tme, Γ s the decay wdth, G (Γ) s normalzed dstrbuton functon of the decay wdth. In Eq.(1), the relatonshp of decay wdth and the partcle sze s as follow, 4πn θ kbτ Γ = Dq, q = sn( ), D = () λ 3 πηd where D s dffuson coeffcent, q s the scatterng wave vector, n s the refractve ndex of the solvent, λ s the wavelength of the ncdent lght n vacuum, θ s the scatterng angle, k s the Β Boltzman constant, T s absolute temperature, η s solvent vscosty, and d s the dameter of equvalent sphercal partcles. Eq.(1) s a hgh llposed equaton. In theory, we can nvert G (Γ) from measured g (τ), G (Γ) s retreval PSD. 3. REGULARIZATION INVERSION METHOD In the practcal soluton, Eq.(1) s dscretzed as Ax = b (3) where elements of matrx b, x and A are b j = g( τ j ), x = G( Γ ) and a, j = exp( Γ τ j ), respectvely. Soluton of Eq.(3) can be expressed as the followng least squares(ls) problem, Ax b = mn = (4) x LS A powerful tool for the solvng of LS problem s the sngular value decomposton (SVD). Then, SVD of A s as follows A n T = U V = = 1 u σ T v (5) where = dag ( σ1, σ, σn ), σ are the sngular values of A, U and V s left and rght sngular values vectors, respectvely. LS soluton of Eq. (3) s also expressed as n < u, b > x = v (6) LS = 1 σ In Eq.(6), f b contans noses, then σ wll be nfntely small, whch wll lead to bg devaton of x LS. For solvng of ths problem, regularzaton s an effectve method. 3.1 Tkhonov Regularzaton (TIK) TIK adds the -norm of soluton as a constrants condton to Eq.(3). Then, LS soluton of Eq.(3) s approxmately equal to followng problem. mn{ Ax b + λ L( x-x ) } (7) where L s unt matrx, x s ntal soluton, λ s the regularzaton parameter. When x =, based on SVD theory, TIK soluton of Eq.(3) s can expressed as n σ < u, b > xtik = ( ) v (8) = 1 σ + λ σ From Eq.(8), we can see that TIK soluton of Eq.(3) actually adds the followng flter factors to the orgnal soluton of Eq.(3). σ f = (9) σ + λ In order to avod solvng of SVD and reduce the computaton, Eq.(8) can also be expressed as A b mn x- (1) λi 3. Truncated Sngular Value Decomposton Regularzaton (TSVD) Prncple of TSVD s to remove small sngular value whch amplfes the dsturbances. Prncple of TSVD s to remove small sngular value whch amplfes the dsturbances. Based on ths prncple, after truncatng small sngular value, matrx A can be expressed as 34

3 1 th February 13. Vol. 48 No JATIT & LLS. All rghts reserved. ISSN: E-ISSN: A k k = = 1 u σ v T (11) where k s regularzaton parameter. Thus, solvng of Eq.(3) s changed nto solvng of followng well-posed equatons A k x = b (1) Accordng to Eq.(5) and Eq.(11), Soluton of Eq.(1) s expressed as x TSVD = k < u, = 1 σ b > v (13) From Eq.(13), we can see that flter factors of TSVD s 1 σ > λ f = (14) σ < λ where λ s threshold, whch meets σ1 > σ > σ k > λ > σk + 1 > σn > Soluton of TSVD can also be expressed as mn A k x- b (15) 3.3 Selecton of the Regularzaton Parameter crteron crteron computes the curvature of the followng curve n log-log scale ( l g( xλ ), lg( b Ax )) λ (wth λ as ts parameter) and seek the pont wth maxmum curvature, whch s defned as the curve's corner. The curve's corner λ corresponds to the optmal regularzaton parameter value. For dscrete llposed problems, because ths curve almost always has a character-stc L-shaped appearance wth a dstnct corner separatng the vertcal and the horzontal parts of the curve. So hence ts name s called as crteron Generalzed cross-valdaton () crteron wavelength s 63.8nm, the refractve ndex of scatterng medum(water) 1.331,scatterng angle 9, absolute temperature 5, Botlzman constant J K -1,the vscosty coeffcent of water N S K -1. Generalzed cross-valdaton () selects the regularzaton parameter by mnmzng the followng functon Axreg b G= (16) I [ trace( I AA )] where G s defned as regularzaton parameters I, A s a matrx whch produces the regularzed soluton xreg when multpled wth b, I.e, x = A b,trace represents the matrx trace. reg 4. ANALYSIS OF SIMULATION DATA The solvng of Eq.(1) and Eq.(15) are acheved by trust-regon-reflectve Newton method. In Eq.(1) and Eq.(15), two crterons of regularzaton parameter are respectvely used, and form four methods such as TIK+, TIK+, TSVD+ and TSVD+. In order to verfy the valdty of non-negatve TIK and TSVD based on trust-regon-reflectve Newton method and study ther characterstc n the PCS nverson, usng four methods, nose-free and nosy ACFs of unmodal and bmodal dstrbutons partcles were nverted. Ther nverson results are shown n Fgs.1~6, Inverson data of PSD are shown n Tables1~4. In the Tables1~4, the relatve error s defned as relatve error = x x theory x (17) theory In the study, nose-free and nosy ACFs wth nose level of.5 and.1 respectvely were acqured by the smulaton. The smulaton ntal PSD s Johnson s SB dstrbuton [11]. In the smulatons, the unmodal dstrbutons partcles share parameters of Johnson s SB ρ =, β =1.1, α max =6nm and α mn =5nm, the bmodal dstrbutons partcles utlzed the sum of two Johnson s SB functons of equal ntensty quotents, sharng parameters ρ 1 =3.8, β 1 =.1, ρ =-.4, β =., α max =7nm and α mn =1nm. Smulaton experment condtons are as follows, the 35

4 1 th February 13. Vol. 48 No JATIT & LLS. All rghts reserved. ISSN: E-ISSN: Tabel.1 Inverted Data Of Unmodal Partcles For TIK In The Dfferent Crteron nose levels peak peak value relatve peak peak value relatve value/nm error % error value/nm error % error Tabel. Inverted Data Of Unmodal Partcles For TSVD In The Dfferent Crteron nose levels peak peak value relatve peak peak value relatve Fgure.1: Inverted PSD Of Unmodal Partcles Wth Free-Nose (A) TIK (B) TSVD Fgure.: Inverted PSD Of Unmodal Partcles At Nose Levels.5 (A) TIK (B) TSVD Fgure.3: Inverted PSD Of Unmodal Partcles At Nose Levels.1 (A) TIK (B) TSVD 36

5 1 th February 13. Vol. 48 No JATIT & LLS. All rghts reserved. ISSN: E-ISSN: Tabel.3 Inverted data of bmodal partcles for TIK n the dfferent crteron nose levels peak peak value relatve peak peak value relatve 16.8, , , , , , Fgure.4: Inverted PSD Of Bmodal Partcles Wth Free-Nose (A) TIK (B) TSVD Fgure.5: Inverted PSD of bmodal Partcles at nose levels.5 TIK TSVD Fgure.6: Inverted PSD Of Bmodal Partcles At Nose Levels.1 (A) TIK (B) TSVD Tabel.4 Inverted Data Of Bmodal Partcles For TSVD In The Dfferent Crteron nose levels peak peak value relatve peak peak value relatve value/nm error % error 167.4, 553.6, ,533.6, , , ,546.9, , 533.6, , ,

6 1 th February 13. Vol. 48 No JATIT & LLS. All rghts reserved. ISSN: E-ISSN: From thers nverson results of the Fgs 1~6 and Tables1~4, when non-negatve constrants of TIK and TSVD were acheved by Trust-regon-reflectve Newton method, we can fnd out the followng phenomenon and draw some conclusons. 1) For the unmodal dstrbutons partcles, nverson results of TIK and TSVD are the unmodal PSD. For the bmodal dstrbutons partcles, nverson results of TSVD are the bmodal PSD, at nose level ~.5, TIK + can obtan the bmodal PSD, at nose level.1, PSD of TIK + hasn t apparent bmodal characterstcs. Therefore, at nose level ~.1, TSVD can use for nverson of the unmodal and bmodal dstrbutons partcles, whle TIK can use for nverson of the unmodal dstrbutons partcles and bmodal dstrbutons partcles wth nose level of less than.5. ) The nverson results of TIK and TSVD may emerge fluctuatons. Inverson PSD of TIK s relatvely smoother, whle that of TSVD fluctuates more serous. Ths s because dfferent flter wndow for truncatng sngular value s respectvely used n TIK and TSVD. Dfferent flter wndow has dfferent truncaton effects, whch leads to fluctuatons of nverson PSD. Steeper the wndow functon s, more outstandng truncaton effect s. Takng nverson of unmodal dstrbuton partcles wth nose level.5 as an example, flter wndows of TIK and TSVD are shown n Fg.7. The sngular value of TSVD was truncated by the rectangular wndow functon n flterng process. Rectangular wndow functon s steeper, ts truncaton effect s outstandng and fluctuatons phenomenon of nverson PSD s more severe. Compared wth TSVD, flterng wndow of TIK s smoother, ts truncaton effect s weaker, and ts nverson PSD s relatvely smoother TIK TSVD Fgure.7: Flter Wndow Functon Of TIK And TSVD 3) For a unmodal dstrbutons partcles nverson, peak value error of two methods s the same. However, from nverson relatve error pont of vew, at free-nose case, relatve error of TIK s smaller, at the nose case, compared wth TIK, relatve error of TSVD s sgnfcantly lower and s lower than that of TIK up to.445; For the bmodal dstrbuton partcles nverson, compared wth TIK, at all noses cases, relatve errors of TSVD were lower and s lower than that of TIK up to From the peak value error the pont of vew, at nose-free case, peak value error of TSVD + s 6.97%, t was sgnfcantly hgher than that of TIK. However, at all nose cases, compared wth TIK, peak value error of TSVD s smaller and most reduce the peak error 3.94%. From the capacty to dscrmnate bmodalty the pont of vew, capacty to dscrmnate bmodalty of TSVD s stronger than TIK, at nose level.1, nverson results of TSVD stll has apparent bmodal characterstcs, whle nverson results of TIK hasn t apparent bmodal characterstcs. From the peak wdth the pont of vew, n all nose cases, nverted PSD of TIK s wder than that of TSVD. In summary, we can draw the followng conclusons: n the nose cases, TSVD has smaller relatve error and peak value error, better capacty to dscrmnate bmodalty, a stronger ant-nose, narrower peak wdth, but at nose-free cases, TSVD haven t obvous advantage than TIK. 4) By comparng and regularzaton parameter crteron n the nverson of TIK and TSVD, we can be seen that, for the unmodal dstrbuton partcles, peak value error of TIK + and TIK + s same, but at the nose level.1, relatve error of crteron s sgnfcantly lower than that of, for the bmodal dstrbuton partcles, when the nose level s greater than or equal to.5, nverson result of crteron can not dscrmnate two peaks, whle crteron can clearly dstngush two peaks at the nose level lower than.1, at the nose level.1, nverson PSD of crteron stll has weak characterstc of two peaks. Therefore, the crteron s more sutable to determne the regularzaton parameter for TIK n PCS nverson; when nvertng PSD by TSVD, and crteron can get more accurate nverson results, however, compared wth crteron, peak value error of crteron s smaller, maxmum peak value error of TSVD+ s less than 6.97%, whle that of TSVD + s 7.94%, moreover, relatve error of TSVD + s generally hgher, at nose level.1, ts capacty to dscrmnate two peaks s relatvely poorer. Therefore, crteron s more sutable to determne the regularzaton parameter for TSVD n PCS nverson. 38

7 1 th February 13. Vol. 48 No JATIT & LLS. All rghts reserved. ISSN: E-ISSN: EXPERIMENTAL RESULTS ACF of scattered lght ntensty was obtaned through PCS experment setup of our research group[1].the materals tested were standard polystyrene latex spheres suspended n purfed water. They are unmodal partcle wth average dameters 1nm and bmodal dstrbutons partcle wth average dameters 6 nm and nm, For the latter, the relatve proporton of two samples was approxmately 1:1. All measurements were made at scatterng angle of 9 and temperture 98 K. Usng TIK+, TIK+, TSVD+ and TSVD+, respectvely, the above measured ACF data were nverted. The nverson PSDs and data are shown n Fgs.8 ~9 and Table 5. As can be noted from Fgs.8~9 and Table5, nverson PSD of TIK and TSVD are both consstent wth ther true PSD. However, compared wth TIK, the peak value of TSVD s more close to the true value, ts peak wdth s also narrower. It shows that TSVD has good nose mmunty. But nverson PSD smoothness of TSVD s worse. In terms of regularzaton parameters crteron, nverson results of TIK + and TSVD + L- curve more close to true dstrbuton. Therefore, from the analyss of the measured data, we can draw the concluson consstent wth smulaton data Fgure.8: Inverson PSD Of 1nm Partcles (A) TIK (B) TSVD Fgure.9: Inverson PSD Of 6nm And nm Partcles (A) TIK (B) TSVD Partcles (nm) Table 5 Inverson Peak Value Of Expermental Unmodal And Bmodal Partcles TIK TSVD , 54,.5 63, ,.5 39

8 1 th February 13. Vol. 48 No JATIT & LLS. All rghts reserved. ISSN: E-ISSN: CONCLUSION For nverson problem of PCS, based on trustregon-reflectve Newton method, at regularzaton parameter crteron of and, two nonnegatve constrants regularzatons wth TIK and TSVD were proposed and compared n ths paper. By the study of smulaton and experment data of unmodal and bmodal dstrbutons partcles, we can be drawn the followng conclusons. Frstly, compared wth TIK, truncaton error of TSVD s bgger, smoothness and peak wdth of ts nverson PSD s poorer and narrower, respectvely. Secondly, n the case of nose, nverson PSD of TSVD has smaller relatve error, smaller peak value error, better capacty to dscrmnate bmodalty and stronger ant-nose, but nose-free case, TSVD hasn t obvous advantages. Thrdly, for PCS nverson, TIK and TSVD are respectvely more sutable for usng and crteron to determne the regularzaton parameter. These conclusons can be treated as a reference for the applcaton of regularzaton method n PCS nverson. ACKNOWLEDGEMENTS Ths work was supported by the Natural Scence Foundaton of Shandong Provnce (ZR1FM5, ZR1EEM8) REFERENCES: [1] K. Takahash, H.Kato and T.Sato, Precse measurement of the sze of nanopartcles by dynamc lght scatterng wth Uncertanty Analyss, Part. Part. Syst. Charact, Vol. 5, No. 1, 8, pp [] D. E. Kopple, Analyss of macromolecular polydspersty n ntensty correlaton spectroscopy:the method of cumulants, J. Chem. Phys, Vol.57, No. 11,197, pp [3] N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pke, Exponental samplng method for lght scatterng polydspersty analyss, J. Mod. Opt, Vol. 8, No. 8, 1981, pp [4] M. Iqbal, On photon correlaton measurements of collodal sze dstrbutons usng Bayesan strateges, J. Comput. Appl. Math, Vol. 16, No. 1-,, pp [5] L. M Guglotta, G. S Stegmayer and L.A Clement, A neural network model for estmatng the partcle sze dstrbuton of dlute latex from multangle dynamc lght scatterng measurements, Part Part Syst Charact, Vol. 6, No.4, 9, pp [6] S.W. Provencher, CONTIN: A general purpose constraned regularzaton program for nvertng nosy lnear algebrac and ntegral equatons, Comput Phy Commun, Vol. 7, 198, pp.9-4. [7] I. D. Morrson and E. F. Grabowsk, Improved Technques for Partcle Sze Determnaton by Quas-Elastc Lght Scatterng, Langmur, No. 1, 1985, pp [8] Coleman, T.F. and Y. L, "An Interor, Trust Regon Approach for Nonlnear Mnmzaton Subject to Bounds," SIAM Journal on Optmzaton, Vol. 6, No., 1996, pp [9] G.H.Golub, M.T.Heath and G. Wahba. Generalzed cross-valdaton as a method for choosng a good rdge parameter, Technometrcs, Vol.1, 1979, pp [1] P. C. Hansen. The Use of the L-Curve n the Regularzaton of Dscrete Ill-posed Problems, SIAM J Sc Comput, Vol. 14, No.6, 1993, pp [11] A. B.Yu, N.Standsh. A study of partcles sze dstrbuton, Powder Technology, Vol. 6, 199, pp [1] W. Lu, J. Shen, and X. Sun, Desgn of multple-tau photon correlaton system mplemented by FPGA. Proceedngs of the Internatonal Conference on Embedded Software and Systems, Inst. of Elec. and Elec. Eng. Computer Socety, July, 9-31, 8, pp