Table I. Mean and Vectors. Linear Regression Formulas for 

 Calculating Scalar Multiples from Log Exposure Values 



" T", = 11.799 log e.\po.*ure 1< + 0.133 ](.« e.vposiire G + 0.,S32 

 log exposure B — 0.679. 



' y« -^ .'I.OIO log e.xposiiie A' + 1.887 log e.xpo.'iure (1 — :'..192 

 log exposure H + 0.596. 



'' Vs - 3.025 log exposure h' — 0.550 log cxpi.suie (1 — 2.6.S3 

 log exjiosuie li + 0.6.50. 



We luive successfully applied the techiikpie 1o the recouslnic- 

 tioii of visible al.sorplion spectra of a group of nine cultured 

 mariiic ak.-ie. We filtered .50 ml of each culture ont.o a plain 

 white gla-s-fiber filter, 25 mm in diameter. The al)sori)tion 

 spectrum of each liilcrcd sample w;is measured in a Beckman 

 1)K-1.\ speciroijlioloiii.ler aci'ording to Yentsch's method.^ 

 Within 10 inin a phoinLjr.-iph was taken (.f each of the filtered 

 algae. The film was liastman .5242, Ektachrome EF, type B; 

 illumination was by elect ronii' Mash with an 85B filter used over 

 the 3.5-min c.amei'a lens X gray .scale .and a blank glas.s-fiber 

 filter were .also photograjjhed for calibration and reference. The 

 Welch Densichron was \ised to measure the tri-color densities. 

 It was calibialeil against a sle)> wedge of known densities. The 

 ern.r in mcasurenienl was d:0.01 o|.lical densily. 



I'roin the gray scale the l> h"Z h curves are consli'Ucled for 

 each of the i liree layers of the film. The.--e ])rovide a check on 

 proce.-ising opi^rations and film sensiiivity f.\S.\ speed). The 

 18"^, gray patch im.'iged with ea('h filler- is u.sed to correct for 

 inevitable variations in exjiosure. The first IS'^j gray patch is 

 used as a stan<l.ard for each roll of film. .'Xny variations in (he 

 tri-color densities of subseiiueni patches are converted to log 

 exposures. The .-am.e log expo.^iue correction is made for the 

 filter image. In this way all fillers are reduced to the equivalent 

 e.xp<js\ire. 



The actual absorption .sjieclra were subjected to characteristic 

 vector iin.alysis using an IBM 1130 com|)Uter. The operation 

 ill outline was as follows: 



(1) Write the r optical densities of the n algal spectra in 

 lows, one sp(^ctruni per row. The /• columns are then the o])tical 

 (U.nsitios at lO-iim incremenls. 



(2) Find I he mean optical densily of each column. 



(3) Subtract the mean optical density from all of the optical 

 densities in the corresponding colunni. This array is called the 

 n-row by 7'-colunm mean corrected data matrix.' 



(4) Prejiare the transpose of the matrix generated in step 3. 

 This new matrix has r rows and n columns. Each clement 

 o„,,. of the stc]) '■'> mat rix becomes the element a^.n of the transpose. 



(5) Tremulliply matrix 3 by m.atrix 4 (its tf.anspose) to 

 yield an r by )• matrix. 



(6) Calculate in eigen-vectors and -vahies of matrix .5. 

 Weighing coefficients w„ are calculated t)y dividing each element 

 of each veclor by the corresponding root to yield an i X r matrix. 

 Scalar multiiiies //,„ „ .are calculated by summing the products of 

 ic„., X (nie.-m o.d.)„.r. The scalar mull ijjliers indicate how much 

 of each vecl.or is needed to add to the mean to reconstitute any 

 particular absorption spectrum. Table I shows the average 

 values and t he three eigenvectors for a grouj) of .algae. 



The three-layer color film can be thought of as a three-channel 

 .synchronous recojding device. To .see if three vectors are enough 

 to account for all the variation among samples it w,as only nec- 

 essary to see what percent of the trace of the determinant of the 

 above matrix w.as accounted for by each root. The first root 

 accounted for 92.28';o f'le first two for 96.91''r. No substantial 

 imijrovemeiit was observed by increasing the mmiber beyond 

 t hree, which gave a t otal of 98. 11 % for the trace. 



The red, green, and blue (Wratten 92, 92, 94 filters) densities 

 were converted lo log exposure of each ])icture and these w'ere 

 regres.sed in turn ui)on each of the three corresponding scalar 

 multiples to give a least -.stpiare fit. Examples of reconstitiOed 

 s]K'ctr,a and their fit witli the original .s]iectra are shown in (Fig. 

 1. The scalar multiples for these fits are calculated from film 

 densities. 



Ciood fits of (he actiial spectra were found for most algae be- 

 cau.se of the dominance of chlorophyll a in determining the mean. 

 It is noted that less perfect agreement is obtained in the region 

 in which the acce.ssory ])igments, carotenoids, and chromopro- 

 teins, absorb. The over-all fit of reconstructed spectra can be 



Fig. 1. .\bsorplioii spectra of foiu' marine algae. 5S9 is \anno- 

 rhloris alomus; .590 is Aphnnizamenon holsgniii iiiii; 592 is Por- 

 pliorirlium sp.; and 605 is Phacodaclylum In'corniiliim. The .solid 

 line = measured spectra, and the dotted line = reconstituted 

 spectra. 



December 1969 / Vol. 8, No. 12 / APPLIED OPTICS 2567 



31-3 



