Volume 57, Number 11, 2003 APPLIED SPECTROSCOPY 13530003-7028 / 03 / 5711-1353$2.00 / 0q 2003 Society for Applied Spectroscopy

Band-Target Entropy Minimization (BTEM) Applied toHyperspectral Raman Image Data

EFFENDI WIDJAJA, NICOLE CRANE, TSO-CHING CHEN,MICHAEL D. MORRIS,* MICHAEL A. IGNELZI, JR., andBARBARA R. MCCREADIEDepartment of Chemistry (E.W., N.C.,T.-c.C., M.D.M.), Department of Pediatric Dentistry (M.A.I., Jr.), and Orthopaedics ResearchLaboratories (B.R.McC.), University of Michigan, Ann Arbor, Michigan 48109

Band-target entropy minimization (BTEM) has been applied to ex-traction of component spectra from hyperspectral Raman images.In this method singular value decomposition is used to calculate theeigenvectors of the spectroscopic image data set. Bands in non-noiseeigenvectors that would normally be used for recovery of spectraare examined for localized spectral features. For a targeted (iden-ti� ed) band, information entropy minimization or a closely relatedalgorithm is used to recover the spectrum containing this featurefrom the non-noise eigenvectors, plus the next 5–30 eigenvectors, inwhich noise predominates. Tests for which eigenvectors to includeare described. The method is demonstrated on one synthesized Ra-man image data set and two bone tissue specimens. By inclusion ofsmall amounts of signal that would be unused in other methods,BTEM enables the extraction of a larger number of componentspectra than are otherwise obtainable. An improvement in signal/noise ratio of the recovered spectra is also obtained.

Index Headings: Hyperspectral Raman imaging; Self-modeling curveresolution; SMCR; Band-target entropy minimization; BTEM; Fac-tor analysis.

INTRODUCTION

Self-modeling curve resolution (SMCR) is a major toolfor extraction of information in vibrational spectroscopicimaging because an image data set may contain tens ofthousands of mixture spectra. The goal is to extract chem-ical information and spatial distribution without any apriori information about the composition of the objectbeing imaged. The problem is simplest in Raman spec-troscopy because a mixed component spectrum is just alinear superposition of the pure component spectra andthese each scale with the number of scattering centers.Because the data set is over-determined, most workershave applied multivariate data reduction schemes to theirimage data. One problem is that there are an in� nite num-ber of linear combinations of the underlying spectra, so� nding a unique and physically realistic solution requiresconstraints. Typical constraints include non-negativity,closure, and selectivity (peak shape, usually expressed asa maximum allowable width of a second derivative).1

To date, none of the chemometrics tools that have beenapplied to the problem have been completely satisfactory.While most of them appear to work well on rather simplesystems containing chemical components in similar pro-portions, they may fail to resolve minor components incomplex systems or in systems in which relative amountsvary widely.

The � rst multivariate SMCR method to analyze vibra-

Received 6 May 2003; accepted 7 July 2003.* Author to whom correspondence should be sent.

tional spectroscopic image data was developed byDrumm and Morris.2 They utilized interactive manual ei-genvector rotation following principal component analy-sis (PCA) to transform abstract chemical factors intomeaningful pure component Raman spectra. Non-nega-tivities of pure spectra and associated scores and partic-ular band shapes were used as transformation constraints.This PCA-based method has been successfully applied toRaman imaging data obtained from a wide variety of syn-thetic materials and biological specimens.2–11 The advan-tage of this approach is that it reduces the large data setinto a small number of principal components, and thepure component spectra of observable components arewell resolved even if there are low signal-to-noise ratiosin the spectra at individual pixels.

Other approaches have been applied. Multivariatecurve resolution–alternating least squares (MCR-ALS)has been employed to analyze Raman image data of poly-styrene spheres in water and of emulsions of detergentcomponents.12 Hopke and co-workers have proposed amodi� ed alternating least-squares procedure to reducecomputational time required for curve resolution of sim-ilar image systems.13 In addition to using the same MCR-ALS method, Fulghum and Artyushkova14 have appliedevolving factor analysis (EFA) and Simplisma to a sim-ilar problem: resolution of chemical components from X-ray photoelectron spectroscopy (XPS) images acquiredfrom blends of poly(vinyl chloride) and poly(methylmethacrylate). Recently, Batonneau et al. applied Sim-plisma to analyze crystal powder spectra acquired usingconfocal Raman microspectrometry.15

Recently, a newly developed SMCR method, band-tar-get entropy minimization (BTEM), has been applied toFourier transform infrared (FT-IR) reaction data of or-ganometallic and homogeneous catalytic reactions16–20

and FT-Raman data of environmental lead compounds. 21

It was shown that BTEM could recover minor compo-nents having very weak spectral signals. BTEM also en-hanced the signal-to-noise ratio of the recovered purecomponent spectra. BTEM is an example of a nonlinearconstrained optimization approach, which uses importantconcepts from information-entropy theory.22 In this paperwe investigate the use of BTEM in the Raman imagingcase and demonstrate its ability to resolve or extract com-ponents that are missed by older methods.

In the information theory context, entropy, H , is a mea-sure of the disorder of a system or the degree of infor-mation dispersion across the data set, and is de� ned byEq. 1:

1354 Volume 57, Number 11, 2003

H 5 2 p ln p (1)O i i

In Eq. 1 p i is the discrete probability distribution of i andcan take values between zero and one. Therefore, it isexpected that minimizing the entropy of resolved spectrawill localize spectral features.

Band-target entropy minimization is an extension ofthe classical entropy minimization technique � rst de-scribed by Sasaki et al.23,24 for sets of infrared and ultra-violet spectra and further developed by Garland et al.25–27

Classical entropy minimization has some drawbacks. Ex-tensive computation is required and the method has prob-lems with over-subtraction, which results in missingbands.

Garland and co-workers introduced BTEM to solvethese problems. In BTEM, one targets a spectral featureobserved directly in the � rst few eigenvectors of the co-variance matrix of mixture spectra. In conventional PCA-based methods only those eigenvectors that describe mostof the system variance are rotated to extract data. BTEMuses these and many other eigenvectors that contain smallamounts of determinate data, but which would normallybe discarded because they describe little of the total sys-tem variance. The expanded set of eigenvectors issearched to attain the global minimum value of an ap-propriate objective function, which de� nes the criteria fora solution. Forty or � fty eigenvectors may be used toextract information about a system containing six or eightindependent spectroscopic descriptors. As a consequence,BTEM can resolve components having similar spatialdistributions and can recover minor components thatmight otherwise be missed.

In this paper we describe the application of BTEM toRaman image data and brie� y compare the results tothose obtained by interactive manual rotation,2,3 MCR-ALS,12 Simplisma,28 and interactive principal componentanalysis (IPCA).29 The strengths and limitations ofBTEM are also discussed. We use as imaged objects onesimulated data set and two specimens of bone tissue se-lected from groups of specimens employed in studies ofbone diseases.

THEORY

The spectral image data is � rst reorganized from athree-way array to a two-way array. Array unfolding isdone by stacking up each data column of the image ab-scissa into a single column. This de� nes a new two-waydata array A k3 p , where k is number of mixture spectraand p is number of pixels in one spectrum. A k3 p is abilinear data structure and can be described as the productof two submatrices C k3 s and a s3 p. C k3 s is the matrix ofrelative concentration variables (usually called scores),and a s3 p is a matrix of the pure component spectral var-iables (frequently called factors):

A k3 p 5 C k3 s ·a s3 p (2)

The data array A k3 p is subjected to singular value de-composition (SVD)30 to calculate its eigenvectors. SVDgives three independent matrices, which are the scoresmatrix U, the matrix of singular values S, and the load-ings matrix V T according to the relation:

A k3 p 5 U k3 kSk3 pVTp3 p (3)

Since V T contains the abstract factors of pure componentspectra, a linear combination of some of these vectorscan produce physically meaningful pure component spec-tra. In the traditional approach to vector rotation, s ei-genvectors are retained and s pure component spectra,including background spectra, are generated. Selection ofs is made by one of several statistical tests, such as thecommonly employed Scree test.31

In BTEM one pure spectrum at a time is resolved. Thenumber of eigenvectors, z, taken for inclusion in the ro-tation is usually much larger than s and is chosen inter-actively by the user. Typically, z . 2s. A pure spectruma13 p is extracted by targeting a selected band, usually inthe � rst few eigenvectors. The number of eigenvectorsexamined is never less than s, but more may be includedif they have determinate features even though they arerejected by the standard test. This procedure is repeatedfor all unique narrow bands observable in the choseneigenvectors. This process continues until it producesonly noise. In most cases, usually not more than 50 vec-tors z are taken for vector rotation:

a13 p 5 T13 zV Tz3 p (4)

Signi� cant spectral features observed in the V T vectorsare targeted one at a time and are retained during theprocess of rotation. Retention is attained by normalizingthe targeted band peak intensity to 1.0. When the processis complete, relative intensities of the vectors are recov-ered by projecting them on to the original data set.

De� nition of an objective function for optimizing theelements of T13 z is based on the principle of maximizingthe simplicity of the pure component spectral features andlocalization of band signals. Maximizing spectral sim-plicity means that the minimization of the proposed ob-jective functions is performed. Both an information en-tropy function and a spectral derivative plus integratedband areas function have been employed as objectivefunctions for BTEM. In the latter case, a second deriva-tive is usually employed. In addition, in order to limit thespectra l-estimates search space, non-negativity con-straints are also imposed.

The � nal choice of objective function is made afterpreliminary spectral resolution trials. Even though infor-mation entropy is not always the objective function min-imized, self-modeling curve resolution employing eitherof these objective functions and similar constrained iter-ative minimizations are generally called BTEM.

The Shannon entropy function, G, is minimized ac-cording to Eq. 5:

maxˆmin 5 2 h ln h 1 P (a , C , a ) (5)O p p 13p k31 13 pG p

where:

dap) )dph 5 (6)v

dapO ) )dpp

The � rst term on the right-hand side of Eq. 5 is theShannon entropy function based on a � rst derivative ofthe pure component spectrum estimate a13 p. The second

APPLIED SPECTROSCOPY 1355

term, P, is a penalty function that insures (1) non-nega-tivity in each term of a pure component spectral estimatea13 p and its estimated concentration C k31, and (2) a rea-sonable maximum intensity, a .max

13 p

The penalty function is given in detail by Eq. 7:

P(a13 p , C k31, a ) 5 gaF1(ap) 1 gcF2(C k) 1 gmaxmax13 p (7)

where:2F (a ) 5 (a ) " a , 0 and (8)O1 p p p

p

2ˆ ˆ ˆF (C ) 5 (C ) " C , 0 (9)O2 k k kk

In Eq. 7, ga, gc , gmax are penalty coef� cients for con-straints de� ned by Eqs. 10, 11, and 12:

ì0 F (a ) , l1 p 1ïíg 5 10 l # F (a ) , l (10)a 1 1 p 2ï

410 F (a ) $ lî 1 p 2

In Eq. 10, l1 5 1023 and l2 5 1022 are bounds for theintensity constraint:

3 ˆg 5 10 " F (C ) (11)c 2 k

4 max10 a . a13 pg 5 (12)max max50 a # a13 p

In Eq. 12, a is a coef� cient that sets the maximum in-tensity of a resolved pure spectrum, in relation to thetarget band peak intensity a , which is normalized totarget

13 p

unity.It is also possible to use a second type of objective

function, Fobj, in which a spectral derivative plus the areaunder the spectrum are substituted for the Shannon en-tropy, and minimized according to Eqs. 13 and 14:

maxˆmin 5 ds 1 a 1 P (a , C , a ) (13)O Op p 13 p k31 13 pF p pobj

nd apds 5 ; n 5 1, 2, or 4 (14)p ndp

This type of objective function includes the minimizationof the summation of the � rst, second, or fourth derivativeof the intensity estimate a13 p in Eq. 14, and minimizationof the integrated area of the intensity estimate a13 p. Sim-ilar constraints to those used in Shannon entropy mini-mization are imposed.

The choice of derivative depends on the signal/noiseratio in the underlying data set. The higher the S/N, thehigher the order of the derivative that can be employed.In Raman imaging, � rst and second derivatives are usu-ally the most satisfactory.

Band-target entropy minimization requires a goodglobal optimizer in order to solve the highly nonlinearobjective function. We employ Corana’s simulated an-nealing.32 Because the optimization is iterative the com-putational effort is substantial, but it is easily within thecapabilities of modern personal computers.

EXPERIMENTAL

Because the emphasis in this paper is on methodologydevelopment for Raman image spectral analysis, threedata sets with very different characteristics have been

chosen. The � rst data set is a synthesized Raman image,and the last two data sets are from biological specimens.

A simulated image containing 20 3 20 pixels was con-structed by a linear combination of 785 nm excitationpolymer Raman spectra taken from our own laboratorydata archive. The polymers are poly(tetra� uoroethylene)(PTFE), polystyrene, polyethylene, a labware polycar-bonate, and poly(methyl methacrylate) (PMMA). Imagedata was constructed as a set of 20 3 20 simulated ran-dom concentrations of these � ve components. Polycar-bonate and PMMA were assigned as minor componentswith signal intensities purposely limited to approximately0.25% of the other three major components. Then 0.15%distributed random noise was also added to the imagedata. The experimental pure component spectra and theircorresponding simulated component distributions areshown in Fig. 1.

Two data sets of Raman images of bone tissue speci-mens were also analyzed. These are Raman images ofbone tissue specimens used in other ongoing studies,whose physiological and clinical results will be reportedelsewhere.

The � rst bone specimen is calvarial (skull) tissue takenfrom fetal-day-18.5 mice harvested according to a Uni-versity of Michigan Institutional Committee on Use andCare of Animals protocol (8287). This tissue was em-bedded in glycol methacrylate and stained with eosin.The second bone specimen is trabecular bone tissue fromthe proximal femur taken from an elderly human femalein the course of hip arthroplasty surgery. The tissue was� rst stained with basic fuchsin and later embedded inpoly(methyl methacrylate) (PMMA).

The hyperspectral line Raman imaging design and in-strumentation have been previously described.8,33,34 Brief-ly, the specimens were placed on a motorized X–Y stage(New England Af� liated Technologies) mounted on amodi� ed epi-� uorescence microscope (Olympus BH-2).In the current application, a line-focused 785 nm near-infrared (NIR) laser (Kaiser Optical Systems Inc., 400mW output) was utilized as the excitation source, and a203/0.75 numerical aperture (NA) objective (Zeiss) wasused to illuminate the specimens. Raman scattered lightwas collected through the objective and focused into anaxial-transmissive spectrograph (Kaiser Optical SystemsInc., Holospec f /1.81). Signal was detected using a back-th inned, deep depletion , thermoelectrically-cooledcharge-coupled device (CCD) camera (Andor Technolo-gy). In both specimens, 126 point transects were ac-quired. For the calvarial tissue, each transect was inte-grated for 3 min with 1.4 mm spatial resolution (0.7 mmsteps) to generate an image containing 175 3 126 pixelsand a � eld of view of approximately 125 3 125 mm 2.For the proximal femur tissue, 2 mm spatial resolution (1mm steps) were employed to generate a 150 3 126 pixelimage and � eld of view of approximately 150 3 150mm 2.

The BTEM method was applied to all three data sets.For comparison purposes, Simplisma was applied to thepolymer data set and the calvarial tissue data set. Simi-larly, interactive manual principal component rotation,MCR-ALS, and IPCA were applied to the calvarial tissuedata set. Full BTEM and Simplisma recovered spectra arepresented for the polymer image data. Full BTEM recov-

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FIG. 1. Experimental pure component Raman spectra and associated simulated component distributions.

FIG. 2. The � rst nine eigenvectors of the V T matrix for the synthesizedpolymer image data: (a) � rst eigenvector, (b) second eigenvector, . . .(i) ninth eigenvector.

ered spectrum and image results are presented for thebone tissue data sets. For calvarial tissue, bone mineraland matrix spectra (but not embedding medium spectra)recovered by all of the comparison techniques are pre-sented.

All data transformations were performed in MATLAB5.3 (The Mathworks Inc., Natick, MA) using locally writ-

ten scripts and vendor-supplied functions. Prior to imageanalysis, median � ltering was used to remove cosmicspikes and the dark current was subtracted from eachspectrum. A Dell Dimension 4500 computer with a 2GHz Pentium IV processor and 768 MB of RAM wasused for all computations.

RESULTS AND DISCUSSION

Simulated Raman Image. Resolving minor compo-nents for which signals are very weak and often in therange of noise signals has been one of the major chal-lenges for most self-modeling curve resolution algo-rithms. Other techniques than BTEM normally use thenumber of non-noise eigenvectors as the number of com-ponents to describe the determinate variance of mixturespectral data sets. However, this assumption can onlywork well for a system having components with relativeamounts that do not vary widely from each other. Prob-lems will occur when the user attempts to extract spectraof minor components with very weak signals. The goalof calculations on this simulated data set is to test theability of BTEM to extract minor components and toshow that the use of a larger number of eigenvectorsindeed improves the spectrum and image data recovery.

The � rst nine V T vectors found by subjecting the un-folded polymer Raman image data to singular value de-composition are shown in Fig. 2. From the � gure, it isclearly seen that although only � ve polymer componentsare present in this system, there are still some signi� cant

APPLIED SPECTROSCOPY 1357

FIG. 3. (a) Pure component spectra estimates recovered via BTEM using only 5 V T vectors, (b) Pure component spectra estimates recovered viaBTEM using 10 VT vectors for major components and 30 V T vectors for minor components. (c) Pure component spectra estimates recovered viaSimplisma.

FIG. 4. (a) Recovered BTEM polycarbonate spectrum using 10–50eigenvectors, as labeled. (b) Recovered PMMA spectrum via BTEMusing 10–50 eigenvectors, as labeled.

spectral features visible in the sixth eigenvector. Thesefeatures are mainly contributed by the minor componentPMMA Raman spectrum. Therefore, the assumption thatthe number of signi� cant eigenvectors is equal to thenumber of actual components is erroneous for some cas-es, especially if components are present at a very lowmole fraction. However, it is true in this data set that fromthe seventh eigenvector onwards, the eigenvectors aremostly distributed random noise.

Exhaustive band targeting was performed. In the � rstcurve resolution attempt, only the � rst � ve V T vectorswere rotated using BTEM. Because there is no back-

ground spectrum in this system, summation of the spec-tral second derivatives plus summation of the band areasobjective function was interactively chosen. The recov-ered spectra using � ve V T vectors are presented in Fig.3a. Good pure component spectra that are close copiesof the original pure component spectra were resolved,except for that of PMMA. This result implies that rotationwith more eigenvectors is needed. In a second attempt,we used ten eigenvectors to recover major componentsand thirty eigenvectors to recover minor components. Thesuccessfully recovered pure component spectra of these� ve components are presented in Fig. 3b. As expected,the minor component spectra have lower signal/noise ra-tios than the major component spectra.

To justify the use of a higher number of eigenvectorsin resolving minor components, 10–50 eigenvectors wererotated to recover polycarbonate and PMMA. The recov-ered spectra are shown in Fig. 4. Visually, it can be seenthat spectrum quality does not suffer because of the useof a large number of eigenvectors. Inclusion of eigen-vectors containing distributed random noise even im-proves the signal-to-noise ratio of the recovered spectra.However, if too many eigenvectors are taken, the opti-mization burden will increase, and there is danger ofover-� tting. Therefore, a simple test for selecting thenumber of eigenvectors using visual inspection of thenormalized T 13 z elements is applied. The elements shouldremain approximately zero if additional eigenvectors donot contribute signal. An example of this test is presentedfor the calvarial tissue (see below) data analysis. Cross-validation can also be a good technique to justify the useof a particular number of eigenvectors. However, in mostreal cases, including the tissue specimens that we arestudying, there is no training set available for cross-val-idation.

The Simplisma technique was also employed to extract

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TABLE I. Cross-correlation between estimated spectrum and ex-perimental Raman spectrum.

ComponentBTEM with 10 and

30 eigenvectors Simplisma

PTFEPolystyrenePolyethylenePolycarbonatePMMA

0.99970.99930.99990.84670.8034

0.97310.98150.99620.61850.7281

FIG. 5. The � rst twelve eigenvectors of the V T matrix: (a) � rst eigen-vector, (b) second eigenvector, . . . (l ) twelfth eigenvector.

TABLE II. BTEM spectral recovery parameters and species iden-tities.

Band-targetregion(cm21) z

BTEMobjective function

Componentidenti� cation

1–10 20 Summation of 2nd order derivativeand integrated bands

Background 1

450–460700–750560–580

202020

Summation of 2nd order derivativeSummation of 2nd order derivativeSummation of 2nd order derivative

Background 2Background 3Background 4

952–962 50 Summation of 2nd order derivativeand integrated bands

Bone mineral

874–884 30 Summation of 2nd order derivativeand integrated bands

Bone matrix

885–900 30 Summation of 2nd order derivativeand integrated bands

GMA

1268–1274 30 Summation of 2nd order derivativeand integrated bands

Eosin

pure component spectra of the simulated image data. Theresult is presented in Fig. 3c. Only the three major com-ponents, PTFE, polystyrene, and polyethylene, are recov-ered well. The recovered PTFE spectrum has some neg-ative spectral inter ference from other components .PMMA is quite poorly recovered. Only some majorbands are identi� able. In addition, one band from thepolycarbonate spectrum is also found in the PMMA spec-trum. The worst recovery is found for polycarbonate.Only two bands of polycarbonate are recovered. The re-covery results using Simplisma are expected, becausethere are two minor components with signal intensitiesjust above the noise level.

To compare resolution quality, the cross-correlation co-ef� cient between the recovered spectrum and the corre-sponding experimental Raman spectrum was calculated.The cross-correlation results for the ten and thirty eigen-vector BTEM and for Simplisma are presented in TableI. From this table, it is very clear that BTEM can producebetter spectral recovery.

Calvarial Tissue. A specimen of fetal calvarial tissuerepresents a major challenge to any multivariate imageanalysis procedure. The tissue is delicate and cannotwithstand high laser power. The suture region in partic-ular contains little matrix (protein) and there is little min-eral even in the most heavily mineralized regions. Thus,we expect the spectra at each pixel to have a low signal/noise ratio.

The � rst twelve eigenvectors of the data matrix areshown in Fig. 5. From the � gure it can be seen that thereare many signi� cant spectral features that can be used astargets for BTEM.

Exhaustive band targeting was implemented, and wefound that eight band features could be used to resolvefour background spectra and four pure component Ramanspectra. Two different objective functions were utilized.For the Raman component spectra, which containedmany narrow features, summation of the spectral secondderivatives plus summation of band areas (Eq. 13) wasused as the objective function. For the broad � uorescencebackground spectra, summation of the second derivativeswas used without inclusion of the sum of band areas. Theband targets, number of eigenvectors taken for resolution,BTEM objective function, and the identity of the resolvedpure component, both broad � uorescence background andRaman spectra, are shown in Table II. These spectral es-timates and their score images are presented in Figs. 6Aand 6B. Using BTEM we are able to separate a bonematrix spectrum from a bone mineral spectrum. The min-eral/matrix ratio does not vary greatly across the � eld ofview and we were unable to obtain separate mineral andmatrix factors using interactive manual rotation, MCR-

ALS, Simplisma, or IPCA. The combined mineral andmatrix factors obtained using these four approaches arepresented in Fig. 7.

The ability to resolve spectra that have only slight spa-tial variation is one of the strengths of BTEM. The prop-erty is a direct consequence of inclusion in BTEM ofeigenvectors that contain predominantly noise and wouldbe discarded in other methods. We included the � rst � ftyeigenvectors, as shown in Table II. The spatial variationbetween mineral and matrix is distributed through mostof these eigenvectors. Rotation of only the � rst 15 eigen-vectors with a band target 952–962 cm21 generated amixed spectrum of bone mineral and matrix, as shown inFig. 8.

We note that BTEM is able to recover separate mineraland matrix factors and to increase the signal/noise ratio,despite inclusion of signal from noise-dominated eigen-vectors. In fact, BTEM provides mineral and matrix fac-tors with a signal/noise ratio of about 50, compared toabout 15 for the mineral 1 matrix factor recovered byinteractive manual factor rotation.

A plot of the normalized optimum elements of the T13 z

APPLIED SPECTROSCOPY 1359

FIG. 6. (A) Pure component spectra estimates of � uorescence backgrounds and associated score images obtained via BTEM from calvarial tissue,(a–d ) � uorescence backgrounds. (B) Pure component Raman spectra estimates and associated score images obtained via BTEM from calvarialtissue. (a) Bone mineral, (b) Bone matrix, (c) GMA, (d ) Eosin.

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FIG. 7. Mineral and matrix bone factors obtained via four SMCR tech-niques: (a) Interactive manual rotation, (b) MCR-ALS, (c) Simplisma,and (d ) IPCA.

FIG. 9. Weight distribution of normalized optimum T 13 z: (a) z 5 15,and (b) z 5 50.

FIG. 8. A mixed spectrum of bone mineral and matrix recovered byBTEM with 15 eigenvectors.

TABLE III. Percentage of reconstructed signal intensities of eachcomponent (via BTEM) compared to the total original experimentalsignals.

Component

Integrated signal intensities ofeach component compared to

total original signals, %

Background 1Background 2Background 3Background 4Bone mineralBone matrixGMAEosin

0.51229.02139.32028.2530.2420.7450.8641.036

vector versus pixel number that shows the weight distri-bution for the linear combination of the z eigenvectors isshown in Fig. 9. For 15 eigenvector rotation, it can beseen (Fig. 9a) that the T13 z elements are scattered and donot converge to zero. This result shows that more z vec-tors should be included in the rotation because spectralinformation on the component corresponding to the tar-geted band is still available in later eigenvectors. When50 vectors are taken for rotation (Fig. 9b), the weightdistribution of T 13 z elements is seen to be convergingtowards zero, particularly from the 35th eigenvector on-wards. This means that little spectral information existsin the 35th eigenvector and beyond, and most spectralvariance information has been recovered. Even this com-

putationally expensive procedure requires only three min-utes on our computer.

The total integrated signal intensities of each compo-nent resolved via BTEM are presented in Table III. Thedata indicate that bone mineral, bone matrix, GMA, andeosin contribute less than 3% of the total intensity. Morethan 97% of the signal is background. This result is notunexpected because there is some iron � uorescence ex-cited at 785 nm and no attempt was made to remove itprior to image analysis, in part to demonstrate the abilityof BTEM to recover signals obscured by high back-grounds.

The original data was reconstructed by linear combi-nation of the spectral estimates obtained via BTEM andtheir score images. The difference between the originaland the reconstructed data was then calculated. It is only0.0093. This small value indicates that almost all signalshave been reconstructed, both � uorescence backgroundand Raman.

APPLIED SPECTROSCOPY 1361

FIG. 10. Pure component spectra estimates and associated score images recovered via BTEM from proximal femur tissue. (a) Bone mineral 1matrix. (b) A component that consists of the non-carbonated-phosphate band of bone mineral together with a slightly shifted phenylalanine spectrum.(c) PMMA. (d ) An unidenti� ed component.

Proximal Femur Tissue. Proximal femur tissue wasused in this study because it represents a different set ofchallenges. At the same measurement time used for cal-varial tissue, stronger spectra are observed, raising thepossibility of � nding minor components that might havebeen undetected by earlier methods.

For the proximal femur specimen, visual inspectionwas performed on the few � rst VT vectors, and four nar-row spectral features were targeted for spectral resolu-tion. The BTEM objective function was again a sum-mation of the second-order spectral derivative and inte-grated bands. We were able to recover four pure com-ponent spectra, corresponding to bone mineral 1 matrix,a component that consists of the non-carbonated-phos-phate band of bone mineral together with a slightly shift-ed phenylalanine spectrum, PMMA, and an unidenti� edcomponent possibly containing calcium carbonate. Thebest rotation for the bone mineral 1 matrix includes 20eigenvectors. The four pure-component Raman spectralestimates and their score images are presented in Fig. 10.

It is curious that we cannot obtain separate mineral andmatrix factors for this specimen. From band area mea-surements we know that the mineral/matrix ratio varieswidely around the specimen. The mineral/matrix (phos-phate n1 /amide I) ratio varies from 0.4 to 3 over 80% ofthe pixels, although the distribution is not Gaussian. Ourtentative explanation is that uncarbonated mineral isfound mostly near phenylalanine-rich regions of the col-lagen, causing the nominally independent moieties tovary together.

The second unusual spectrum is of factor d (Fig. 10d).The intense band at 1079 cm21 is close to the CO n1 of22

3

calcite, which is observed at 1083 cm21.35 We have ex-amined several other proximal femur tissue images fromsimilar sources and always � nd this factor at 1079–1080cm21. It should be noted that while calcite does have aweak 1434 cm21 band, the mineral has no other Raman-active vibrations at frequencies higher than 1083 cm21. Itis not certain what other component of the specimen isincluded in this Raman spectrum, but the exercise doesshow that no multivariate procedure can completely re-solve every independent component present at low sig-nal/noise ratios.

The six background components that we obtain are al-most certainly incorrectly rotated. A characteristic ofBTEM as we have implemented it is that the objectivefunctions are designed to extract localized features thatare spectra in which the derivatives have large slopes.This selectivity is gained at the expense of poor extrac-tion of � uorescence or other signals that vary slowly withwavenumber. However, in the present application, inac-curate descriptions of such backgrounds are innocuous.

CONCLUSION

Band-target entropy minimization is one of a possiblylarge and useful family of non-linear constrained opti-mizations. Although the computational burden is alwaysgreater than in linear methods, the bene� ts outweigh thiscost. Compared to earlier methods, the bene� ts include

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extraction of spectra from minor components, separationof otherwise unresolved components, and increased sig-nal/noise ratios in the recovered spectra. In any event,extraction of each spectrum requires no more than a fewminutes computation time on a consumer-level personalcomputer. An important limitation of BTEM is that itdoes require that some spectroscopic feature of each re-covered component be localized and visible in one ormore of the eigenvectors that are predominantly signal.

ACKNOWLEDGMENTS

Financial support provided by NIH grant R01 AR47969 (M.D.M.),R29 DE11530 (M.A.I., Jr.), and by the University of Michigan CoreCenter for Musculo-Skeletal Research (NIH grant P30 AR46024,MDM).

1. R. Tauler, B. Kowalski, and S. Fleming, Anal. Chem. 65, 2040(1993).

2. C. A. Drumm and M. D. Morris, Appl. Spectrosc. 49, 1331 (1995).3. C. A. Drumm and M. D. Morris, Appl. Spectrosc. 50, 708 (1996).4. P. A. Walker III, J. M. Shaver, and M. D. Morris, Appl. Spectrosc.

51, 1394 (1997).5. N. L. Jestel, J. M. Shaver, and M. D. Morris, Appl. Spectrosc. 52,

64 (1998).6. J. M. Shaver, K. A. Christensen, J. A. Pezzuti, and M. D. Morris,

Appl. Spectrosc. 52, 259 (1998).7. J. A. Timlin, A. Carden, and M. D. Morris, Appl. Spectrosc. 53,

1429 (1999).8. J. A. Timlin, A. Carden, M. D. Morris, J. F. Bonadio, C. E. Hof� er

II, K. M. Kozloff, and S. A. Goldstein, J. Biomed. Opt. 4, 28(1999).

9. J. A. Timlin, A. Carden, M. D. Morris, R. M. Rajachar, and D. H.Kohn, Anal. Chem. 72, 2229 (2000).

10. C. P. Tarnowski, M. A. Ignelzi, Jr., and M. D. Morris, J. BoneMineral. Res. 17, 1118 (2002).

11. S. Stewart, D. A. Shea, C. P. Tarnowski, M. D. Morris, D. Wang,

R. Franceschi, D. L. Lin, and E. Keller, J. Raman Spectrosc. 33,536 (2002).

12. J. J. Andrew and T. M. Hancewicz, Appl. Spectrosc. 52, 797 (1998).13. J. H. Wang, P. K. Hopke, T. M. Hancewicz, and S. L. Zhang, Anal.

Chim. Acta 476, 93 (2003).14. K. Artyushkova and J. E. Fulghum, J. Electron Spectrosc. Relat.

Phenom. 121, 33 (2001).15. Y. Batonneau, C. Bremard, J. Laureyns, J. C. Merlin, and W. Win-

dig, J. Phys. Chem. B 107, 1502 (2003).16. W. Chew, E. Widjaja, and M. Garland, Organometallics 21, 1982

(2002).17. E. Widjaja, C. Z. Li, and M. Garland, Organometallics 21, 1991

(2002).18. C. Z. Li, E. Widjaja, W. Chew, and M. Garland, Angew. Chem.

Int. Ed. Engl. 41, 3785 (2002).19. C. Z. Li, E. Widjaja, and M. Garland, J. Am. Chem. Soc. 125, 5540

(2003).20. C. Z. Li, E. Widjaja, and M. Garland, J. Catal. 213, 126 (2003).21. L. R. Ong, E. Widjaja, R. Stanforth, and M. Garland, J. Raman

Spectrosc. 34, 282 (2003).22. C. E. Shannon, Bell Syst. Tech. J. 3, 379 (1948).23. K. Sasaki, S. Kawata, and S. Minami, Appl. Opt. 22, 3599 (1983).24. K. Sasaki, S. Kawata, and S. Minami, Appl. Opt. 23, 1955 (1984).25. Y. Zeng and M. Garland, Anal. Chim. Acta 359, 303 (1998).26. E. Widjaja and M. Garland, J. Comput. Chem. 23, 911 (2002).27. L. Chen, W. Chew, and M. Garland, Appl. Spectrosc. 57, 491

(2003).28. W. Windig and J. Guilment, Anal. Chem. 63, 1425 (1991).29. D. S. Bu and C. W. Brown, Appl. Spectrosc. 54, 1214 (2000).30. G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hop-

kins University Press, Baltimore, 1996).31. E. R. Malinowski, Factor Analysis in Chemistry (John Wiley and

Sons, New York, 2002).32. A. Corana, M. Marchesi, C. Martin, and S. Ridella, ACM. Trans.

Math. Software 13, 262 (1987).33. S. L. Zhang, J. A. Pezzuti, M. D. Morris, A. Appadwedula, C. M.

Hsiung, and M. A. Leugers, Appl. Spectrosc. 52, 1264 (1998).34. K. C. Christensen and M. D. Morris, Appl. Spectrosc. 52, 1145

(1998).35. I. A. Degen and G. A. Newman, Spectrochim. Acta, Part A 49, 859

(1993).