A Tho1 Points Light:
P a w J. Tread0 and Michael D. hlo& Department of Chemisby
The University of Michigan Ann Arbor. MI 48109-1055
Most chemists appreciate the impact of the Fourier transform on chemical analysis. IR spectroscopy and NMR spectroscopy were revolutionized by the introduction of multiplexing instrumentation that produced the Fourier transform of the desired spectra. Somewhat less familiar, but no less important, are image- and signal-processing techniques that are based on the Fourier transform. Not all chemists are aware that the Fourier transform is just one of several related linear transforms. The Fourier transform is based on sines and cosines; the other transforms are based on other periodic functions. The Hadamard transform is the most common of these alternative transforms. Based on square waves rather than on trigonometric functions, this transform can be applied in three different areas in analytical chemistry. First, it is used to multiplex or encode analytical signals so that many signals .can he measured simultaneously. In Hadamard techniques the multiplexer turns elements on and off or makes them switch between two values. The same formalism describes multiplexing schemes in which pseudorandom sequences are employed to excite a sys0003-2700/89/0361-723A/$01.5010
@ 1989 Amerlcan Chemical Society
,tern. In general, Fellgett's advantage, the signalhoise ( S / N ) ratio improvement from multiplexing many signals, is sought. Fellgett's advantage can he realized if the detector is noisy, because one unit of detector noise is distributed .among all the measured signals. Second, the transform is used in multiplexed imaging. Here, the goal is usually to obtain spatially resolved signals without focusing a high-intensity source on a sample. Fellgett's advantaee mav he absent. A Dower distribu~"~ tion advantage is avhahle; source powers can he 2 or 3 orders of magnitude higher than for tightly focused radiation. Finally, the Hadamard transform is used in data transformations. Information is redistributed so that signal compression (i,e., truncation of the trans-
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ists are most familiar with the use of Hadamard multiplexing schemes for IR spectroscopy (I).Hadamard encoding of the output of dispersive spectrometers proved less attractive than multiplexing with a Michelson interferometer, which requires a Fourier transform to recover a spectrum. Investigation of other applications of the Hadamard transform has continued in many laboratories worldwide. Elegant technical advances in multiplexing technology have suddenly revived interest in spectroscopic applications. Increased emphasis on spectroscopic microscopy with laser and other intense sources has spurred interest in Hadamard inaging. Early developments in Hadamard techniques were reviewed hy Decker (2) and by Marshall and Comisarow
lNS~RUMEN7A7lON
form) can occur without loss of essential features. The transform is also used before transmission of signals so that the effects of corruption of one or more data words are distributed over the entire signal, rather than causina severe localized loss of information. Here, workers often emphasize the computational efficiency of the Hadamard transform compared with that of the Fourier transform. These areas of application were defined by about 1970. Analytical chem-
(3).There is extensive coverage L. --a. damard transform spectroscopy, imaging, and signal processing in Griffith's 1978 review volume (4). The 1979 Harwit and Sloane monograph (I)on spectroscopic and imaging of .applications .. Hadamard transforms remains the standard comprehensive introduction to both theory and practice in this field. In this article we will emphasize recent developments. Hadamard multiplexing can be understood from the simple three-ele-
ANALYTICAL CHEMISTRY. VOL. 61. NO. 11. JUNE 1, 1989
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INSTRLJMEN7A7ION ment case shown in Figure 1. We can illuminate the three resolution elements xl, x2, and x 3 one a t a time, as in Figure la. Three measurements are required to define the signals at XI, x ~ , and x3. However, we can also illuminate the elements two at a time, as shown in Figure lb. In this case, we can completely define the system by a set of
I" 1
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three equations. These equations can be written in matrix form, as shown. The matrix inversion that recovers xl, xz,andxafrom themeasuredyl,yp,and y3 is called an inverse Hadamard transform. The operation connecting the xi with they; is the Hadamard transform itself. In a practical multiplexing system, hundreds or even thousands of elements are multiplexed. The principles remain the same whatever the number of multiplexed elements. What is important is that the multiplexing involves switching some elements on (i.e., multiplication by +1) while leaving others off (i.e., multiplication by 0). The multiplexing sequence is a systematic set of combinations of odoff elements. Physically, this combination can be realized with mechanical masks, electronic or electrwptical switches, or any appropriate kind of onloff switching system. In practice, cyclic systems are used. Each Combination is generated from the previous one by shifting its elements one position to the left (or right) and placing the overflow in the vacant position. In Figure IC,shifting the fiveelement mask to the left one unit at a time generates the three combinations of Figure lb. We can generalize the system of Figure 1. Any Hadamard code generates the sum of signals from each element weighted by 0 or 1.
The system can be solved by calculating the inverse of matrix X,according to Equation 4.
x=s-1.y
(4)
The matrix S-I is computationally easy to generate and is given by Equation 5.
Here W is a matrix that has -1's where S T bas 0's and +l's where ST has +l's. Matrix S, called a Sylvester matrix, is derived from a Hadamard matrix, which contains elements +1 and -1 only. Despite this, the operation is called a Hadamard transform. Rules and algorithms for generating suitable combinations of elements sij have been described by Harwit and Sloane ( I ) . The Hadamard transform may be viewed as describing functions using a basis set of square waves. In contrast, Fourier transforms use sine waves for the basis set. The Hadamard variable corresponding to frequency is termed sequency. Many of the familiar properties of Fourier transforms are observed with Hadamard transforms. The general theory of Hadamard transforms and the uses of the transform and sequency in image and signal processing have been discussed by Harmuth (5). As with the Fourier transform, there is a fast Hadamard transform (FHT). In both cases, the matrix defining the transform must be factorable. In the case of the Sylvester matrices, the FHT " can be defined if the number of eleyj = qjxXi (1) ments is equal to Zk - 1,where k is an ;=I integer. For the Hadamard matrix, containing +l's and -I%, the FHT is In Equation 1, yj is the signal from defined if the number of elements is the j-th combination of onloff elesome 2k. Because the FHT requires ments, and x ; is the signal generated at the i-th element. The vector sj = (s~j, only addition and subtraction, it executes about 8-10 times faster than the su, . . ., s,J has values 1 for each onfast Fourier transform (FFT)on a genelement and 0 for each off-element. eral-purpose computer. To recover all n signals, we need n In general, Hadamard transform measurements. These can be defined multiplexing bas the same kinds of adby the n linear independent equations vantages and limitations as multiplexthat completely define the system. ing using Fourier encoding. Fellgett's advantage, for example, can be obtained in a spectroscopic system where detector noise is dominant, but not in a shot noise limited system. Fellgett's " advantage is ahout the same order of Yz = %Xi (2.2) magnitude for both multiplexing i=1 schemes. In each case, the instrument analog-to-digital converter dynamic ... range must be increased by fi,where n is the number of channels or signals multiplexed relative to the requirements of a single-channel measurement. The choice between Fourier and HaIn matrix notation, Equations 2.1-2.n damard encoding depends largely on may be written as Equation 3. the kind of experiment. A Michelson interferometer automatically produces Y=S.X (3)
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ANALYTICAL CHEMISTRY. VOL. 61, NO. 11, JUNE 1, 1989
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a Fourier encoding. For imaging applications, the simplest spatial encoding is a system of open and closed apertures, which automatically produces a Hadamard encoding. Arguments about small differences in multiplex advantage or even about computation time are overshadowed by the requirements of the experiment.
How Hadamard -y and imaging are dare Both spectroscopic and imaging applications require a sequence of Hadamard masks that encode the information. Either one-dimensional or two-dimensional encodings are possible, as shown in Figure 2. The most common dask technology is a series of apertures that are fabricated on a transparent substrate. Metal films on glass, quartz, or silicon have been used. Slots milled in thin sheets of metal have also been employed. These systems all use cyclic encodings so that the mask is translated or rotated by one aperture width to generate sequential endings. This principle is illustrated in Figure 2a for a one-dimensional encoding. The active aperture area is defmed by a framing mask. After collection of a complete set of data points, an inverse Hadamard transform is calculated to recover the desired spectrum or image. Spectroscopy is performed by using Hadamard masks to encode the dispersed output of a grating spectrometer for presentation to a single detector. Imaging is performed either by encoding the source radiation or by encoding the image itself. The same principle can be used to
generate two-dimensional mask sequences, as shown in Figure 2b. Here, the mask is folded. For the 15-element sequence shown, the f i s t five elements form one row, the next five form the second, and the last five form the third row. The mask is translated one unit aperture width, as with the one-dimensional sequence. The mathematical operations are the same as for the onedimensional sequence. If a plot of the data is required, it must he rearranged from a linear sequence into the appropriate (+, y) coordinates. It is also possible to fold the sequence twice, as shown in Figure 2c. This results in a more compact mask but requires translation in two dimensions. The Fateley group (6-9)has demonstrated that optical shutter arrays (OSAs) can he used as Hadamard mask systems. An OSA, which is also called a spatial light modulator, contains a matrix of electronically addressable, switchable elements that can be made transparent or opaque. A transparent element represents a 1in a Hadamard sequence; an opaque element is a 0. An OSA can generate a complete set of Hadamard sequences but requires no mechanical movement. The OSA can be used to construct multiplexed spectrometers or imaging devices with no moving parts. Such devices are more compact than mechanical mask systems, provide excellent spectral subtraction capability, and may he more rugged. In addition, the OSA allows selective filtering by leaving certain elements always off, that is, opaque (9). Here, important opportunities for S/N ratio improvement exist.
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L Flaure 2. One-
2D array stepped horizontally and venically past the framing mask
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(a) A 15-eiemerd linear Wtem of endlngs. Each e n d i n g is formed by shining the mask sequence one elemsnt to the lm. (b) A twc-dinmnslonal e n d i n g is fwmad ky folding the onedimensional sequence into Wee mws ot five slemsnrs each. The entire m a y 1s shined me element to me len to generate the next enmding In the sequence. (c)Two foldings genarete a more compact mechanical Implementation. The flrst five endings are generaad by hwimntal shIS only. The next five encadlngs are generated by shining a m Urn dawn and tollowinghe same hwizonmi shin sequence.The final five encodinga we genaretd by shiltlng one m e Unit dorm and folbwiq he same horizontal shnt s~qwnce.
OSA technology is currently limited to certain wavelength ranges. For example, liquid crystal devices work in the visible and near-IR hut not in the UV or through most of the fingerprint region of the IR. Hadamard encoding is not limited to optical or beam spectroscopies. Hadamard sequences can he used to define pseudorandom excitation sequences. For example, Hadamard sequences of potential pulses can be used to make electrode impedance measurements. Hadamard sequences of rf pulses can be used for NMR spectroscopy. Of course, any data set can be subjected to Hadamard transformation. Image transmission from the early days of the space program employed this technique. It is used in pattern recognition, image compression, and other postprocessing applications.
Recent s p d m m p l c a p p l h t h Sugimoto (10) has proposed a slitless low-resolution near-IR (1-1.8 pm) Hadamard spectrometer using a Hadamard-encoded 16-element array of light-emitting diodes (LEDs) as the source and a 32-element array of germanium diodes as the detector system. The encoded LEDs replace the slits of a conventional spectrometer, thus improving the throughput. Here, the multiplexing provides a Jacquinot or throughput advantage rather than the Fellgett's advantage obtained by multiplexing the dispersed output onto a single detector. The output of the LED array is collimated so that a remote 0.25-m spectrometer with a 1-km path length can be used. With this system, the minimum detectable absorbance is 2 X 10-5 with a 1-s integration time. Sugimoto calculates that an absorbance of 5 X 10-7 could be reached with a 30-min integration time. As configured for environmental measurements, the system is capable of monitoring "3, CHI, CiH4, CsH8, and the HOz radical with parts-per-billion detection limits and 1-stime resolution. The first grating instrument to employ stationary electrooptical encoding masks at the output was an IR spectrometer described by Hammaker et al. (6). They fabricated a 63-element vanadium dioxide thermodiachromatic array, which was operated in the 7501000-cm-' region with 3-cm-' resolution. They demonstrated the expected Fellgett's advantage and described the merits of stationary mask systems. Because of limitations on the size of the mask, the entire fingerprint region could not be sampled at once. It may prove difficult to design a spectrograph with a sufficiently flat field or to find a material for mask fabrication to allow
ANALYTICAL CHEMISTRY, VOL. 61. NO. 11, JUNE 1. 1989
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INSTRUMEN7A7lON
Wavenumbem (cm-1) Flgure 3. The effect of Rayleigh line rejection on Hadamard transform Raman spectra. (a) Spctrum of 2-nitropmpane vim m significant Raylei€& scattered light peMed by the multiplexer. (b) and IC)The same spctrum. but wim increasingly large m n 1 8 of Rayleim a l t e r peMed by the munipiexer. (Adapted vim pnmir6ion horn Rehnence 9.)
acquisition of a complete spectrum. The limited spectral coverage of many electmaptical masks can he an advantage in routine monitoring. Here, only a narrow spectral region is usually needed. In contrast, an interferometer, without prior spectral filtering, records information from the entire spectrum at once, generating both excess noise and some additional computational burden to recover only the region sought. In this area the Hadamard instrument may he both theoretically superior and cost-effective. In visible and near-IR spectroscopies, Fateley and co-workers (6-9,II) have used liquid crystal optical shutter arrays (LC-OSAs) as stationary Hadamard mask systems. Current commercial devices are useful at wavelengths above -530 nm, but future versions may operate further in the blue. With an LC-OSA this group has demonstrated Hadamard Raman and atomic emission spectroscopies. The Fateley group has encoded the output of a 1-m suhtractive double spectrometer equipped with 1180 groove-per-millimeter gratings. Because the LC-OSA used has a 0.6-mm aperture, the available resolution was about 30 cm-1. Raman spectroscopy is a difficult experiment for any multiplex technique, as practitioners of FT-Raman have pointed out (12).The source noise on 728A
the intense Rayleigh line is redistributed across the entire spectrum by either Hadamard or Fourier multiplexing. This problem is exacerbated with the LC-OSA, because the device has appreciable off-state transmission (7,8).Hadamard encoding allows rejection of much of this noise by placement of an extra mask in front of the OSA elementa, which encode the line. The effect can be seen in the 2-nitropropane Raman spectra shown in Figure 3. Multiplexing allows acquisition of Raman spectra using visible or near-IR excitation. Fateley and co-workers have demonstrated that multiplexing generates excellent S I N ratio Raman spectra with uncooled silicon photodiodes as detectors. As with IR spectroscopy, a real advantage of Hadamard multiplexing over Folvier multiplexing is that the entire spectrum need not be encoded at once. Hadamard Raman spectroscopy may prove superior to FT-Raman for routine monitoring over a restricted wavenumber range. Using current technology at least, it is easier to generate a complete multiplexed Raman spectrum interferometrically than with a Hadamard mask system. There is an exact analog of dispersive Hadamard spectrometry available for mass spectrometry (13, 14). The ions dispersed by a conventional magnetic sector instrument are passed through a sequence of Hadamard masks constructed as slits in metal and mechanically translated and then detected with a single-channel detector. Fellgett's advantage is obtained.
Recent imaging appllcatians
Hadamard multiplexing is well suited to multiplexed imaging. Two types of imaging are possible: signal-encoded imaging and source-encoded imaging. In source encoding, the excitation source is encoded with a sequence of Hadamard masks. In signal encoding, the signal image is multiplexed. The operation of a two-dimensional signalencoded imager is depicted in Figure 4. The image is projected through a series of Hadamard masks, each containing 63 elements folded into 9 X 7 arrays. The encoded signals are then projected onto a single-element detector. Upon reverse Hadamard transform of the 63 independent signals, the spatially resolved image can be recovered. Image (signal) encoding offers advantages with optical spectrwopies involving reflected, scattered, or emitted light. The image flux is usually far less intense than the source flux, so the mask does not have to he rugged. A h , diffraction from the mask apertures of the incoherent image is less severe than that of the coherent laser source. Where imaging detectors do not exist, source-encoded multiplexing, which is generally applicable to any linear spectroscopy, is often the most practical form of imaging. The alternative to source encoding is tight focusing or other generation of a small-area source and raster scanning of the source across the image. Even when there is theoretically little or no classical multiplex advantage, e n d i n g allows use of unfocused beams. In that
Trp Recovered image
ask 1 R
4. Operation of a &element Hadamard imaging sysi
The 63 encodiw of the image of the nullzHai"2" are generated by 63 sB(IuencBs, which are rep868med by t h 63 ~ row of the Sylvester nmtslx, S:The encodingr on average rep8wm abouf haif the imensw in the image. ma Inverse Syiwstw matrix. S-', is multiplied by me vectw of the 63 encOded signaia to IecOvBT he imgs Of the "Um(YBI.
ANALYTICAL CHEMISTRY, VOL. 61, NO. 11. JUNE 1, 1989
case the local source flux or power density can be reduced by at least 2 or 3 orders of magnitude. High signal intensities can be maintained, and the possibility of sample damage is minimized. Our group has called this a power distribution advantage, which perhaps is as commonly sought as classical multiplex advantages in Hadamard imaging. We have investigated the imaging properties of Hadamard systems generally (15-17).We have shown that for a system of uniform mask elements, resolution is ideally determined by convolution with a unit aperture. In an optical system, an encoded source beam can often be condensed to improve resolution or an image can be magnified before encoding. In these ways diffraction-limited Hadamard imaging, in principle, is possible. IR and visible Hadamard imaging was demonstrated by Harwit and coworkers in the early 1970s. All of this work used dispersive spectrometers to achieve spectral resolution. However, Kraenz and Kunath (18) coupled a two-dimensional Hadamard imaging system to an FT-IR micrhscope. By taking a complete interferogram at each step of the Hadamard sequence, they obtained complete spatialhpectral images at spatial resolution up to 64 X 65 square 50-pm pixels. These workers were seeking a spatial multiplex advantage by the use of defocused IR radiation. The photoacoustic experiments of Coufal and co-workers (19) were the first demonstration of source-encoded Hadamard imaging for an important laboratory spectroscopy that bas no imaging detectors. They employed 7and 15-element linear masks placed directly over the sample being imaged to obtain spatially resolved signals. They also obtained Fellgett's advantage, because photoacoustic spectroscopy with an electret microphone is detector noise limited. The depth-resolving capabilities of photoacoustic spectroscopy were maintained in the multiplexed system. Two-dimensional Hadamard masks can also be used. The Nanjing University group (20) has shown that local power density in two-dimensional photoacoustic imaging can be reduced by a factor of 1000. Our own group has used similar systems to generate transverse photothermal deflection images. Like photoacoustic spectroscopy,transverse photothermal deflection has no available imaging detectors. Source-encoded Hadamard multiplexing is necessary. Excimer lasers can be used as sources for UV imaging if the encoding masks are able to withstand high peak power laser intensities (21). Spatial distribution of
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ANALYTICAL CHEMISTRY, VOL. 61. NO. 11, JUNE 1, 1989
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Binki INSTRUM1
INSTRUMEN7A7ION tor. Figure 6 is an image of polycrystalline benzoic acid obtained with this instrument. Multiplexing can be advantageous in combination with multichannel Raman detectors. Two-dimensional (CCD)detectors can provide complete spectra that are spatially resolved in one direction but not in two. Hadamard multiplexing may provide an inexpensive route to two-dimensional spectrdspatial Raman imaging with these devices. Several groups (25-ZS),primarily astronomers, have used signal-encoded Hadamard X-ray telescopes to improve the S/N ratios in observations of nebulae, solar flares, black holes in space, and plasmas in nuclear reactors. The masks are fabricated as holes in metal foils, and the radiation is directed to counting detectors. Figure 5. An image-encoded Hadamard Raman microscope.
The sample 18 illuminated with an unlocuwrd laser beam. me Raman sulnw is coiieclec a r w @ a mnb'mtkml canpaund mlomamps and presentedto a Hademard mask 88qyBnce before passage though a sP63rmWw. Spanal mlution depsnda on Um relation between mtal magnlflultion In the system and the size of a unn aperture in h e mask. ( m a d with parrnlsslon frm Reference 16.)
the h e r light is an attractive alternative to attenuation of expensive laser power by 2 orders of magnitude. Masks containing hundreds of elements are easily generated and employed in these multiplexing systems (15,22).There is little degradation of SlN ratio beyond that caused by the fador of two attenuation of average power that occurs when masking is used. Most source-encoded imaging bas been near-field imaging. The mask is placed as close to the sample as physic a y possible, and the sampled area is equal to the aperture area This approach offers limited resolution. For the most part, minimum aperture dimensions are limited by fabrication technologies and have been 50 pm on a side, or larger. With any optical spectroscopy, it is possible to use coarse masks to e n d e the beams and then to condense the e n d e d beams, similar to the way photolithography is done. This approach has been applied to photothermal detection (16) and Raman imaging (23).Masks with 150-400-pm unit aperture dimensions have been used to generate images with 13-pm resolution (16).There is no reason that diffraction-limited images with resolution of 0.5jun or less cannot be generated by this technique, so long as properly designed imaging optics are used. Source-encoded Hadamard Raman imaging (23)andsignal-encodedHadamard Raman microscopy (24) have been demonstrated. With a signal-en-
coded instrument. Hadamard images have been obtained with nearly diffraction-limited resolution. Figure 5 is a diagram of a Hadamard transform Raman miqroscope constructed in our laboratories. Unfocused laser light illuminates the sample. The scattered signal is collected, magnified, and projected to the encoding mask by the microscope optics. Spectral resolution is provided by the monochroma-
Hadatnardexcitation seqwnce applications Hadamard sequences are pseudorandom pulse sequences. Optical and Xray spectroscopy and imaging use masks based on these sequences. How; ever, they can be used to define pseudorandom excitations of a system. An efficient form of ac polarography (29-31) uses Hadamard sequences of small-amplitude sine-wave harmonics to exercise a dropping mercury electrode. The dc response and the inphase and quadrature components can be obtained simultaneously, allowing complete specification of the cell impedance. The technique is equivalent to conventional ac polarography, as
1-
-,
-
Figure 6. The image of a benzoic acid crystal (992 cm-') obtained with the instrument of Figure 5. (~daptedw h permissionhorn R B ~ W ~ C B 17 )
ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1. I989
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7
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(bottom)ac polarcgrams of 0.1 mM
Excltatlon w w form mplltllde, V, = 50 mV. (Adapted from Reference 30.)
shown in Figure 7, but is more efficient. As with any Hadamard technique, the required computations involve no multiplication or division. A laboratory microcomputer can perform them in real time or near real time. Hadamard pseudorandom excitation has been proposed to improve the efficiency of multichannel tandem Fourier transform mass spectrometry (FTMSAWMS). McLafferty and coworkers (32,33)suggest the use of Hadamard weighted excitation functions to dissociate differing combinations of precursor ions. In each step, a different set of half of the precursor ions is dissociated, while product ion masses are monitored. The FTMSAWMS spectra for the product ions are recovered by inverse Hadamard transform of the multiplexed spectra. Hadamard transform FTMSAWMS should provide a Fellgett's advantage over single precursor ion measurements. Other tandem MS experiments can be similarly enhanced. The Hadamard transform is a linear transform. The usual inverse Hadamard transformation fails if the system does not respond linearly. B l h i c h and Ziessow describe techniques for treating nonlinear saturation effecta in Hadamard NMR spectrometry (34).Hadamard pulse sequences are useful in NMR because they allow distribution of the excitation power over the entire data acquisition time. Both Hadamard phase change and amplitude change se732A
ANALYTICAL CHEMISTRY, VOL. 61,
quences can be used. However, if high pulse energies are used, it is necessary to solve the Blocb equations numerically and introduce saturation correction terms into the data treatment for recovery of the spectral line shapes.
Data lranrfarmations Hadamard encoding of data sets for image processing and image compression was one of the earliest applications of the transform (35).The computational efficiency of Hadamard transforms was a major force behind much of this work. Computational considerations have driven most applications to chemical data transformations. It was recognized early that the Hadamard and Fourier transforma allowed about the same amount of data compression by truncation of high-frequency or sequency terms. Recently Zupan and eo-workers (36) demonstrated that Hadamard and Fourier transformation are about equally effective in compression of IR spectra. Of course, calculation of the Hadamard transform is about eight times faster than calculation of the Fourier transform. Figure 8 shows that truncation of the high-sequency terms in the benzenesulfonamide spectra leads to broadened recovered spectra with loss of detail. Compressed spectra recovered a f k r Hadamard transformation have the high-intensity narrow portions removed from the top of peaks before broadening is noticeable. NO. 11.
JUNE 1, 1989
conclusions Hadamard transformation remains a field with proven areas of application and many awaiting exploration. Technologies for fabrication of moving or stationary masks are now available for most of the electromagnetic spectrum. Computer addressing of stationary mask sequences (OSAs) is straightforward. Computer control of stepping motors and piezoelectric translators is routine. These advances make oncedifficult imaging and spectroscopic applications quite practical. Hybrid multichannel/multiplex experiments may become increasingly important as ways to use multichannel detectors to obtain complete spectral/ spatial data on samples. Fourier transform spectral encoding and Hadamard spatial encoding have been demonstrated in IR spectral microscopy. A similar approach could prove useful with Raman spectroscopy. Much Hadamard imaging has been near-field imaging. It is now known that near-field visible imaging can be used to resolve features 2 orders of magnitude smaller than the Rayleigh diffraction limit (37,38).Yet sub-Rayleigh Hadamard imaging has not yet been explored in any region of the electromagnetic spectrum. With such exciting prospects we can expect future progress in Hadamard transform applications to be rapid and far from random.
Flgure 8. Effect of truncation of the Fourier transform (FFT) and Hadamard transform (FHT) on the recovered 512point IR spectrum of benzenesulfonamide. Trumllon of the FFT bmadens the opclrum; buncallon of me FHT amnuales hlglntensitv featues. (Adapled vim permiasion hom Refer*me 36.)
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ANALYTICAL CHEMISTRY. VOL. 61, NO. 11, JUNE 1, 1989
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Appl. Opt. 1982.21,116-20.
(1) Harwit, M.
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m r d Tmnsform Optics; Academic: New York, 1919. (2) Decker, J. A,, Jr. Anal. Chem. 1972.44, 121 A-134 A. (3) Marshall, A. G.; Comisarow, M. B. Anal. Chem. 1975,47,491 Ab04 A. (4) Grif?iths, P. R., Ed. Transform Technioues m Chenustrv: _.Plenum: New York. 19is. (5) Harmuth, H. F. Sequency Theory; Academic: New York, 1917. (6) Hammaker, R. M.; Graham, J. A,; Tilotta, D. c.;Fateley, W. G. In Vibmtional Spectra and Structure; Durig, J. R,Ed.; Elsevier: Amsterdam, 1986; Vol. 15; pp. 401-85. (I) Tilotta, D. C.;Hammaker, R.M.; Fateley, W. G. Appl. Spectrosc. 1987,41,721RA
(iYTdotta,D. C.; Hammaker, R.M.; Fateley, W. G. Appl. Opt. 1987,26,4825-92. (9) Tilotta, D. C.;heeman, R.D.; Fateley, W. G. Appl. S ectmc. 1987,41,12t3&81. (10) Sugimotn, Appl. Opt. 1986,25,86365. (11) Tilotta, D.C.;Hammaker, R. M.; Fateley, W. G. Spectmhim. Acta 1987,43A,
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L.;Xu, J.; Zhang, S. Ultrason. Symp. Proc. 1986,1,5014. (21) Fotiou. F. K.; Morris, M. D. Appl. Spectrosc. 1986,40,104-6. (22) Fotiou, F. K.; Morris, M. D. Anal. Chem. 1987,59,18549. (23) Treado, P. J.; Morris, M. D. Appl. Spectrosc. 1988,42,897-901. (24) Treado, P. J.; Moms, M. D. Appl. Spectrosc. 1989,43,19&93. (25) Oda, M. Adu. Space Res. 1983,2,20716. (26) Tsunemi, H.; Kitamotn, S.; Miyamoto, S.; Murai, H.; Shimizu, S. Adu. Space Res. 1983,2,211-20. (21) Miyamoto, S. Space Sci. Instrum. 1977,3,413-81. (28: Skinner, G. K. Sei. Am. 1988,295,84(20) Wei,
(BySeelig, P. F.; de Levie, R Anal. Chem. 1980,52,154€-11. (30)Chang, Chane. C. C.; C.: de Levie, Levie. R.Anal. Chem. 1983,55 isss, 55-35659. 35659. (31) Pmpkil, L.; Fanelli, N. J.Eleetroana1. (31) Chem. 1987,222,361-71. (32) McLafferty, F. W.; Amster, I. J. Int. J. MMSSpectrom. Ion Proceases 1986, 72, 85-91. (33) McLafferty, F. W.; Stauffer, D. B.; Loh, S. Y.; Williams, E. R. Anal. Chem. 1987,59,221213. (34) Bliunieh, B.; Zieaww, D. J. Magn. Reson. 1982,46,3&5405. (35) Pratt, W. K.; Kane, J.; Andrews, H. C. h e . IEEE 1969,57,5&68. (36) Zupan, J.; Bohanec, S.; Razinger, M.; YyiE, M. Anal. Chim.Acta 1988,210,63-
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(38) Betzig, E.; Harootuninn, A,; Lewis, k, Ieaccson, 1900. M. Appl. Opt. 1986, 25, 1890This work was supported in psrt by the National Institutes of Health through grant GM-37W.
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Patrick J . Treado (right) is a third-year graduate student at the University of Michigan pursuing a Ph.D. in analytical chemistry. He received his B.S. degree in chemistry from Georgetown Universit) in 1985. His research interests are in Roman spectroscopy, photothermol spectroscopy, and microscopy. Michael D. Morris (left)is professor of chemistry at the University of Michigan. He received his B.A. degree in chemist0 from Reed College in 1960 and his Ph.D. in chemistry from Harvord University in 1964. After several years on the chemistry faculty at Penn State, he joined the University of Michigan in 1969. His research interests are in analytical laser spectroscopj, with special emphasis on Roman spectroscopy.
ANALYTICAL CHEMISTRY. VOL. El. NO. 11, JUNE 1, 1989
