Mark G. Moran and Bruce R. Kowalski Laboratory for Chemometrics, Department of Chemistry University of Washington, Seattle, Wash. 98195
I M A G E A N A L Y S I S
T h e " H a r p e r Encyclopedia of Science" defines " i m a g e " as "an optical replicant of a luminous or illuminated object formed by a mirror or a lens" ( / ) . In this article it will be seen bow the restrictions implicit in this definition may be relaxed or removed. First, an image need not be a real image or a visual representation. By one method or another an image or a virtual image may be represented purely mathematically. T h e object that gives rise to the image may do so by interacting with electromagnetic radiation either by absorption or emission. However, it is not necessary to restrict attention entirely to this situation. Various types of scattering experiments also result in the generation of images. A familiar example of an ordinary two-dimensional image is a photo graph. A photograph is a distribution of light intensity and possibly color over a surface. More formally, the information contained in a photograph is encoded by specifying the value of intensity fix, y) over the points of a surface (x, y). T h e set of intensities that constitute the image of f(x, y) is known as the gray scale. Intuitively, the gray scale ranges from the extremes of totally black to totally white. A reasonable assignment of the gray scale may be to specify a monotonically increasing function of the density of perfectly opaque, reflecting dots in a given area. An example is given in Figure 1. If a photograph is a familiar example of a two-dimensional image, then by similar reasoning, a spectrum can be thought of as a familiar example of a one-dimensional chemical image. A light spectrum is an image t h a t results from encoding the intensity of radiation absorbed or emitted from an object as a function of wavelength. This kind of image is constructed to extract specific information from the object under study. Display of an infared spectrum of a solution of acetone shows a prominent band at about 17:50 c m " 1 . Visual inspection of this image allows interpretation using h u m a n experience and ability in recognizing patterns. T h e location of the band lends insight into the n a t u r e of the species giving rise to it. T h i s signal may also be used to determine the relative a m o u n t of species present as well as other details concerning the local environment surrounding it.
776 A · ANALYTICAL CHEMISTRY, VOL. 51, NO. 7, JUNE 1979
Photographs, molecular and atomic spectra, and other forms of images, real or virtual, are common to many areas of science and engineering. Chemists are now developing and using instruments capable of generating images of not only one but two and higher dimensionality. Fortunately, there is a great deal known about constructing, transmitting, storing, and improving images, as well as extracting information from them. Powerful mathematical tools primarily developed for application to medical and satellite images have been developed, but by and large are not used in chemistry. Many of the existing methods for analyzing one-dimensional images are also useful when extended to higher dimensional spaces. These methods, when used in conjunction with the observer's intrinsic pattern recognition ability and experience, greatly extend the amount of information t h a t can be extracted from images. T h e purpose of this article is to introduce the analytical chemist to the great similarity among images produced by a variety of methods and to describe a few of the available tools for image analysis.
Image Construction T h e construction of an image may be tackled by any one of several direct or indirect methods. T h e image construction process reflects the nature of the information that the experimentalist desires to glean from the image. In carrying out the q u a n t i t a tive analysis of a substance by analyzing the absorption spectrum of a species, one is usually satisfied with knowing the bulk average concentration of the species in the sample. As detection limits are pressed ever lower and sample matrices become increasingly complex, there are situations where this information is insufficient. In the analysis of a catalytic surface, for instance, it is often desirable to know the concentration of a species as a function of location on the surface. Alternatively, the identity, amount, and distribution of pollutants in a body of water or a smokestack plume may be sought. T h e latter problem could be solved by taking a sufficient n u m b e r of samples at discrete locations on the object to be analyzed. T h e s e values could be used in constructing a spatial " i m a g e " of the object. However, a method based on 0003-2700/79/0351 -776A$01.00/0 © 1979 American Chemical Society
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ζ = 0
ζ = 3
ζ = 2
ζ = 1
Figure 1 . E x a m p l e of gray s c a l e quantized at Κ = 2 (4 levels)
Hadamard modulation spectrometry has been proposed to deal with this class of problems. Figure 2 shows a block diagram of an imaging spectro photometer based on this principle (2). Note that except for the two-di mensional mask in the entrance focal plane, this arrangement is the same as the Hadamard multiplex spectrom eter described by Decker (3, 4). The singly multiplexed dispersing spec trometer encodes spectral information from all wavelengths at once. This is done by expanding the spectral re sponse in an η-dimensional basis set where the coefficients of expansion are zero or one. The zero coefficient corre sponds to the situation where incident radiation falls on the opaque section of the mask, and the unity coefficient corresponds to radiation passing through the mask slots to the detector. The spectrum is reconstructed by solving the resulting set of linear equations generated by the η different mask settings. The purpose of the m Χ ρ mask in the entrance focal plane is to encode the spectral information in the spatial domain. If this were the only encoding mask in the spectrometer, it would be possible to examine the object with the various mp settings of the mask, store each reading, and subsequently reconstruct a picture of the object. Now, with both the m Χ ρ and the η slotted masks, it is possible to gather spatial and spectral information about an object. Using this scheme, a total of mpn different intensity levels has been collected with one detector. De pending on the algorithm used to de code these measurements, η pictures of the object corresponding to η dif ferent bandwidths could be displayed,
each having mp spatial resolution ele ments. Alternatively, mp spectra could be represented, one correspond ing to each spatial resolution element. An image (in the mathematical sense now) can be constructed if the object under scrutiny gives rise to more than one type of response when subjected to a perturbation. An exam ple is the coupled absorption/emission process undergone when a molecule capable of fluorescence is excited by a beam of light at the appropriate wavelength. It is possible to collect si multaneously the excitation and emis sion spectra of a sample which may consist of a pure species or a mixture. In practice, many more measurements are made than the minimum number dictated by the number of compo nents of the mixture. This provides a solution to the problem of overlap ping spectra and also gives a statistical estimate of the validity of the mea surements. The excitation and emis sion spectra can be considered as vec tors |x> and |y>, where T
| x > = ia'i-V], «2X2, • • · otnxn\ | y > = \a1'y1,a2'y2,
• • • oiny,t\
r
(1) (2)
x, and _y, are scalars proportional to the number of photons absorbed and emitted at wavelength λ, and λ ; , re spectively. The mathematical image is constructed by taking the exterior product of these vectors to form the so-called emission/excitation matrix. |x>'2 • · ·
aia'nx\y„
(6)
The shah function picks out or "sifts" through the function f(x, y) retaining only those values at f(m, n) and de stroying all others. Thus, it converts an analog signal to a two-dimensional digital signal. Another interesting and useful property of this function is that it is its own Fourier transform. This type of sampling is also known as digi tal sampling. Care must be taken so that the collection of sampled points is adequate to reconstruct the picture without introduction of serious error. The image function is reconstructed by taking suitably weighted linear combinations of the sample values. In one dimension, this is written as f(t)=
Σ
fikT)g(t-kT)
(7)
where Τ is the sampling interval, and >f>(t — kT) is the weighting or interpo lation function. It can be shown that if f(t) is a periodic function, the sam pling must be conducted in such a fashion that Τ < i/(2/ c ) where fc is the spatial frequency. This condition ensures that the picture function does not overlap with its copies produced in the Fourier domain. Failure to adhere to this condition is the chief cause of the phenomenon known as aliasing. In two-dimensional and higher spaces, the consequences of overlap can be more severe. If the picture function is expanded in terms of orthonormal unit basis vectors a and b, the sam pling criteria can be stated as follows: A function 7(c) whose Fourier trans form vanishes over all but a bounded region of frequency space can be ev erywhere reproduced from its values taken over a lattice space generated by ma + n b m , n = 0, ± 1 , ± 2 , . ... This is so provided the vectors are sufficiently small to ensure nonoverlapping of the spectrum I(w) with its image of a peri odic lattice generated by rp + sq. This is the so-called reciprocal lattice which is the basis for the function I(w). Image Analysis Methods
In this section, a brief review of image analysis methods is given. No attempt is made to be all inclusive due to the depth and breadth of existing techniques (10). A certain philosophy has been borne in mind during the preparation of this section. The tech niques herein are basically distribu tion-free techniques. That is, no as sumptions are generally made about the probability distribution of the gray scale. Considerable ingenuity is often utilized to model, estimate, or
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guess this most useful property. How ever, on m a n y occasions, images may be sampled, smoothed, enhanced, in tensified, and utilized without specific knowledge of the probability distribu tion of gray scale. S m o o t h i n g , E n h a n c e m e n t , and S h a r p e n i n g . Given t h a t an image has somehow been converted to a digital representation, there exists a group of mathematical operations t h a t can pos sibly enhance the quality of the image. It is i m p o r t a n t to bear in mind t h a t these operations do not increase the information content of t h e picture. However, if the image is badly repre sented, say by blurring or obscured by high-frequency noise, then application of image e n h a n c e m e n t techniques may be desirable. Operations per formed on picture functions may be linear or nonlinear operations. Only linear operations will be discussed here. T h e true undistorted image func tion of an object will be henceforth re ferred to as fix, y). Collection of the image function by one technique or another results in a perceived image function denoted by g(x, y). In the ab sence of distortion (i.e., perfect collec tion technique) f(x, y) = g(x, y). T h e mapping of f(x, y) into g (x, y) may be modeled by the following equation (11):
g(x,y)= ff_"„f and Ν Χ Ν image matrices, the evaluation of the discrete cross correlation is te dious, involving /V4 calculations for each offset value. Fortunately, a gen eralization of the convolution t h e o r u m provides an easier method of evalua tion. For the cross correlation of two functions F(hO[F(h2)]*
(18)
+ 1)|, + D|)
= /·'(/;, • h-z)
(2f)
T h e discrete Fourier transforms of Λ ι and h-i may be evaluated using fast transform procedures (16) that run
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considerably faster than matrix multiplication routines. Thus, searching for a maxima in large arrays is feasible by this procedure. This type of image matching has been used by Morrison in the analysis of NBS-662 steel for niobium and titanium (17). As pointed out, this type of image correlation leads to four-dimensional information. There are two concentration dimensions and two spatial dimensions. As existing scattering techniques improve and new ones are developed, these kinds of data analysis methods will be extremely useful in handling the higher-dimensional data sets generated. Conclusions
(optional)
This article has attempted to introduce analytical chemists to the many ways to create and analyze real and virtual images. Images are nothing more than Ν-dimensional (N > 2) spectra. Therefore, the analysis of im ages should not present a problem for the analytical chemist for two reasons. First, analytical chemists are quite fa CA-02 Moisture Meter with Printer miliar with one-dimensional spectra. Second, many of the concepts and mathematical operations applied to MITSUBISHI CHEMICAL INDUSTRIES LIMITEDone-dimensional spectra extend to Instruments Dept.. Mitsubishi Bldg., 5-2, Marunouchi 2-chome, Chiyoda-ku, higher-dimensional data with the ap Tokyo, 100 Japan Telex: J 2 4 9 0 1 Cable Address: KASEICO TOKYO propriate amount of increased compu tational complexity assumed by the CIRCLE 147 O N READER SERVICE CARD computer. Fortunately, a considerable amount of N-dimensional image anal ysis mathematics and associated com puter software is available. The avail ability is principally due to the large interest in the field of medicine for an alyzing medical images and in remote
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A N A L Y T I C A L CHEMISTRY, VOL. 5 1 , NO. 7, JUNE
1979
Bruce R. Kowalski, professor of chem istry, is the director of the Laboratory [or Chemometrics at the University of Washington. His research interests include the application of novel mathematical and statistical methods to improve the measurement process and to extract useful chemical infor mation from chemical measurements.
s e n s i n g a p p l i c a t i o n s w h e r e images c o m e from s p a c e c r a f t p r o b e s a n d e a r t h s a t e l l i t e s . W e e x p e c t t h a t image a n a l y s i s will b e c o m e a s t a n d a r d tool of t h e a n a l y t i c a l chemist, in t h e f u t u r e as m o r e t y p e s of a n a l y t i c a l i n s t r u m e n t a t i o n a r e d e v e l o p e d or modified t o p r o d u c e h i g h e r - d i m e n s i o n a l images.
References ( 1 ) " H a r p e r Encyclopedia of Science." (2) M. Havvitt, Appl. Opt., 10, 1415 (1971 ). Ci) J . A. Decker, J r . , ibid., 9, 1392(1970). (4) J . A. Decker, -Jr., Anal. Chem.. 44, 127A (1972). (5) 1. M. W a r n e r . E. H. Davidson, a n d G. D. C h r i s t i a n , ibid., 49, 2155 (1977). (β) C. N . H o , G. D. C h r i s t i a n , a n d Ε. Η. Davidson, ibid., 50, 1 108 (1978). (7) G. H. Morrison a n d G. Slodzian, ibid., 47, 932A (1975). (8) -J. D. F a s s e t t , J . K. R o t h , a n d G. H. Morrison, ibid., 49, 2322 (1977). (9) R. Bracewell. " T h e Fourier T r a n s f o r m a n d Its A p p l i c a t i o n s . " 2nd éd., M c G r a w Hill, N e w York, N.Y., 1978. (10) "Digital Image Processing," H a r r y C. A n d r e w s , Ed., I E E E C o m p u t e r Society, 1978. (11) A. Rosenfeld a n d A. K a k , "Digital P i c t u r e Processing," Academic P r e s s , New York. N.Y., 1976. (12) B. R. Kowalski, Anal. Chem., 47, 1152A (1975). (13) "Digital Image P r o c e s s i n g , " H a r r y C. A n d r e w s , Ed., I E E E C o m p u t e r Society, 1978. (14) L. G. R o b e r t s , " M a c h i n e P e r c e p t i o n of T h r e e Dimensional Solids," in " O p t i cal a n d Klectrooptical Information P r o cessing," J . T . T i p p e t et al., E d s . ρ 159, M I T Press, C a m b r i d g e , Mass., 1965. (15) J . D. Fassett a n d G. H . Morrison, Anal. Chem., 50, 1861 (1978). ( 16) J . W . Coolev a n d ,). W . T u k e v , Math. Camp., 19,297 (1965). (17) J . D. Fassett, J . R. Roth, a n d G. H. Morrison, Anal. Chem.. 49, 2322 (1977).
Mark G. Moran is a graduate student in the Department of Chemistry at the University of Washington. His re search interests include newer meth ods of analysis of chemical data, of experimental design, and of mass spectrometry. Mr. Moran is support ed with a research assistantship from the Office of Naval Research.
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