statistical spatial filtering: application to aerial photographs

Statistical spatial filtering: application toaerial photographs

P. Y. Baures and J. Duvernoy

Spatial filtering is shown to apply not only to the identification of deterministic signals but also to the classi-fication of images. The spectral content of images is put into a classifier that extracts the dominant eigen-vectors responsible for statistical features. Principal images that carry most of the information are obtainedby using optical representations of eigenvectors as spatial filters. The statistical stability and the intrinsicdimensionality of Fourier spectra are related to the fast estimation of useful eigenvectors.

1. Introduction

Although optical processing is well suited to thehandling of images, especially a series of high resolutionones, it does not contribute to effective informationprocessing so long as optical statistical operators are notavailable. Because pattern recognition not only meansthe identification of any deterministic signal but alsorequires the capability of separating statistical classesof signals, there is a need for the association of opticalprocessing with computers. The analysis of aerialphotographs gives a typical example of such an ap-proach. This paper addresses the case where it mustbe decided whether a given survey results from the su-perimposition of different (historical) structures. Eachpoint of the photograph is to be assigned to one of theprobable underlying structures. Multispectral data arenot useful, contrary to customary earth resources sur-veys,1 because only structural features are significant(i.e., boundaries of fields rather than varieties of cul-tures). These features are expressed in terms of geo-metrical properties, so that an optical analysis is pref-erable to a digital one in order to preserve maximal ac-curacy. The most popular optical preprocessing con-sists of applying a Fourier transform. There is no usefor such a transform in itself2 if it does not extractcharacteristics or reduce the dimensionality of thephenomenon one has to deal with. Auxiliary opera-tions, such as log3 or Mellin 4 transforms, are then re-quired as the information is not defined (i.e., context-free image improvement). In some cases it is possible

The authors are with Universit6 de Franche Comt6, Laboratoire dePhysique Generale et Optique (associ6 au CNRS 214; Holographieet traitement optique des signaux), 25030 Besancon cedex, France.

Received 13 December 1977.0003-6935/78/1101-3395$0.50/0.©) 1978 Optical Society of America.

to benefit from Fourier spectra when the informationis modeled. Searching for a continuous characteristicfunction that works on spectra permits a survey of thespace-bandwidth product of the image. This has beenapplied to handwriting recognitions by statisticalanalysis of the scriptor's MTFs, extracted from a seriesof pages. Optical modeling (especially of the Fourierspectrum) gives a dimensionally reduced descriptionof the information to be sampled according to the reg-ularity of the characteristic function.

Fourier spectra are to be connected with second-orderstatistical analysis insofar as they represent a spectraldensity. Parameters can be extracted by covarianceanalysis, their number and significance being associatedwith the eigenvectors of spectra. According to themodel that is chosen, they are also related to the in-trinsic dimensionality6 of the phenomenon under study.This paper shows that eigenvectors, which are expressedin terms of spatial frequencies, correspond to statisticalspatial filters to be used in classical optical correlators.They allow the synthesis of optimal images, in the sensethat they display separated classes as a result of filter-ing.

The successive steps of the method are presented:(1) modeling the information by searching for a grat-inglike spectrum; (2) determination of the optimalsampling, with special reference to the statistical sta-bility of spatial frequencies, and estimation of the in-trinsic dimensionality of the spectrum; (3) fast esti-mation of useful eigenvectors and clustering of thespectrum; (4) statistical spatial filtering for the visual-ization of classes.

11. Optical Modeling of the Information

This paper is concerned with the visualization ofremains of Roman cadastral structures in aerial pho-tographs of Southern France. Roman rules of allot-ment were very regular, contrary to those of modern

1 November 1978 / Vol. 17, No. 21 / APPLIED OPTICS 3395

Fig. 1. Typical remains of a Roman cadastration (Yugoslavia).

France. The soil was checkered in squares the side ofwhich was a multiple of 35 m (= 1 actus). Moreover thisstructure was oriented along a particular direction. Theproblem of extracting remains of such a structure canbe reduced to the estimation of a periodic 2-D signalimbedded in 2-D noise, less directional and less periodic(i.e., the modern cadastration).

According to its essential features, a Roman cadas-tration is characterized by a periodic 2-D Fourierspectrum, similar to a grating spectrum. On the otherhand the Fourier spectrum of the dominant (modern)structure does not appear so periodic. This fact ac-counts for the use of an optical Fourier transform-forpreprocessing. Then the information is nearly sepa-rable, two kinds of spectral criteria being available:orientation criteria that permit distinction between twocadastrations the directions of which are sufficientlyseparated; and spectral content criteria that take intoaccount the periodicities of any spectral direction inorder to separate the different gratings it may sup-port.

Unfortunately these criteria are not always efficient,because the density of remains is often too weak. As aresult the corresponding Fourier spectrum does notstand out from noise. Moreover, this grating spectrumis more or less sharpened, according to the regularity ofthe remains, and more or less intense, according to thesurface they cover. In this case pure optical techniquesare not efficient as the information to be extracted doesnot appear in a retrievable form in the spectrum and,consequently, cannot easily be spatially filtered.Therefore, the problem is to separate two classes frommeasurements submitted to two constraints: (1) theyare swamped by noise; (2) they represent statisticalvariables that need a suitable analysis. The first ob-stacle can be overcome by maximum entropy restora-tion,7 but this method will not distinguish classes. Thisfact, connected to the second implication, leads to a

more classical approach based on the Karhunen-Loeveexpansion. Lowitz has shown2 that significant eigen-vectors can be selected under the criterion of the normalerror their corresponding eigenvalues exhibit in thepresence of noise. This principle, established formultispectral images, will be translated here for 2-Dmonochromatic Fourier spectra.

Data are recorded in Fourier spectra of aerial pho-tographs by scanning thirty-six directions selected from50 to 5. The spectral content of each direction issampled at forty-four spatial frequencies, the samplinginterval being 1/710 m (i.e., the spectral periodicity ofthe widest Roman square: the century). There are twopossibilities of analyzing these measurements: (1) di-rections Oi being considered as statistical functions ofspatial frequencies Ni; (2) spatial frequencies being,conversely, processed as statistical functions of direc-tions. Two different orthonormal bases are thenavailable: the first one is a set of concentric rings, andthe second one is a fan of directions (Fig. 2). The di-mension of the starting bases is to be reduced as thecomputational cost of the extraction of dominant ei-genvectors increases with the size of the bases' covar-iance matrices. On the other hand, the consistency ofthe description of the information must be preserved.A study of the usefulness of the spectral samples will becarried out.

Ill. Statistical Stability of Spatial Frequencies;Intrinsic Dimensionality

Recognition of Roman cadastral structures is seldomachieved by using only orientation criteria. Both theweakness and spread of these remains make spectralcontent criteria more efficient and lead to first de-scriptions of the spectrum as directions vs spatialfrequencies. Without any knowledge of the regularityof the grating to be extracted, each direction Oi is to besampled at M points, M being the number of degrees offreedom along this direction (M = 10,000 for a 24 X36-mm2 slide Fourier transformed by the usual lens).A 10,000 X 10,000 covariance matrix is difficult tohandle; however, the sampling interval cannot be lessthan the spectral period of the grating. Taking into

direction j

-+ A

/ (sampling point

Fig. 2. Direction basis and spatial frequency basis used for the sta-tistical analysis of Fourier spectra.

3396 APPLIED OPTICS / Vol. 17, No. 21 / 1 November 1978

N

0 "

S

O+sampled frequencies 0 only

Fig. 3. Filtering of a Roman cadastration along the direction with(and without) selection of a characteristic frequency that includes

noise.

account the level of noise in an actual optical Fourierspectrum, the corresponding number of physicallysignificant samples was found to be 44, as specifiedabove. Are all these spatial frequencies equally sig-nificant? Figure 3 shows a counterexample: a pho-tograph that contains a cadastration, the direction ofwhich is known, that is spatially filtered along this di-rection by means of a set of holes the spacing of whichis 1/35 m-1 (no century boundaries were expected).The filtered image does not exhibit any characteristics:some of the selected frequencies carry noise, not struc-tural information. The problem is, then, to selectstructurally significant frequencies; they have to remainstable from direction to direction.

The statistical stability of each part of the Fourierspectrum is investigated assuming the sample is sur-rounded by a small area of continuity. The successionof forty-four samples associated with a direction isconsidered as the superposition of two intercalatedseries of twenty-two samples, shifted from one samplinginterval. The two series are processed as two inde-pendent realizations of the same spectrum. They un-dergo statistical variations as the direction moves from01 to 036- Comparing the statistical parameters of thetwo realizations, it is possible to decide which sampledfrequencies are significant: Let xj = the variance ofNkj (kth sampled spatial frequency, jth realization):

Nkj = k/710 m- 1, with k = 1, . . . 22 and j = 1,2,

a' = ( Nkj - (Nkj ) 2),

(1)

(2)

where the brackets denote an average over the thirty-sixdirections. What is a structurally significant spatialfrequency? Contrary to the noise, which is assumed tohave the same statistical properties, spatial frequenciescarrying information about the structures have statis-tical properties that vary from direction to direction.Typically, such a spatial frequency exhibits a variancesignificantly different from zero. How to decide the

significance? The normal error: here two estimationsof the variance of each spatial frequency can be obtainedfrom Eq. (2); their mean value is

2rk ak (3)

for the frequency Nk, the upper bar denoting an averageover the realizations (i.e., j); the variance of the twoestimations is

-hi Ik, (4)

and the normal error on the estimation of the varianceis given by

Pk = ak k (5)

It must be decided whether the variance is significantlydifferent from zero. This is not the case if its standarddeviation is of the same order as its mean value; then theprobability that zero constitutes the true value of themean is very high. More precisely, a normal error equalto 2 means that zero is not the true mean with a proba-bility of 0.96. The twenty-two spatial frequencies Nkare examined under this criterion.

This gauging has been applied to the initial forty-foursampled frequencies considered as 2, 3, and 4 realiza-tions of, respectively, 22, 13, and 11 samples. Resultsare presented in Fig. 4. A highly significant spectralperiodicity, the spacing of which equals five sampleintervals, appears as the manifestation of a 2-D gratingstructure, present in the aerial survey (a high and stablevariance denotes strong variations of the geometricalstructure in the photograph as the direction changes;the spectral periodicity indicates that a grating appearssuddenly during the angular scanning). The corre-sponding ground period is 142 m. Another significantspectral periodicity (fourteen intervals) seems to be apart of the previous one. (The converted ground dis-tance is 52 m.)

It can be concluded that among the forty-four sam-pled spatial frequencies, only six are to be retained assufficiently different from isotropic noise. The intrinsicdimensionality of the Fourier spectrum does not exceedsix; this means a smaller number of really interesting

STABILITY___' |t SAMPLING RATE:

4sfu

3 sf u

10 20 30 40 sfu

Fig. 4. Statistical stability of the Fourier spectrum considered asthe superimposition of different realizations of the same sampling (1

sfu = 1/710 m-1 ).


samples are to be taken out, but they are not statisticallydecorrelated. Then a closer determination of the in-trinsic dimensionality is obtained from a study of theorthogonal expansion of the spectrum on the Karhu-nen-Loeve basis.

It has been shown8 that eigenvalues of the data co-variance matrix carry the entropy of data under test.Ranking the set of eigenvalues leads to the selection ofthe first few that account for most of the infomation.According to a given percentage of the total amount ofinformation, the corresponding eigenvectors representintrinsic parameters, and their number is the intrinsicdimensionality of the data. But it is also establishedthat dominant eigenvectors do not always mean clas-sifying parameters. Especially, the influence of noiseis to be investigated, as it has been for spatial frequen-cies.

Again the forty-four spatial frequencies are consid-ered as two realizations of the same set of twenty-twomeasurements. Two (22 X 22) covariance matrices arethen available giving two different estimates, (kl) and(h2) of the same set of eigenvalues (k), ( = 1, . . . ,22).Their stability can be checked by the normal error:

Pk = (Xk)/(AXk), (6)

where

X = kj, j = 1,2, (7)

= (kj - xk)2 . (8)

Note that in this case the averaging is a vectorial oper-ation, because the orientation of eigenvectors sup-porting eigenvalues changes from one realization to theother. The stability of the twenty-two eigenvalues ispresented in Fig. 5. Only the first one seems to be sig-nificant under the criterion of a normal error equal to2. Two conclusions can be derived from this result: (1)the dominant eigenvalues are also stable; (2) the in-trinsic dimensionality of this Fourier spectrum is equalto 1.

However, a direct estimation of the intrinsic dimen-sionality from the stability of spatial frequencies is anefficient procedure because it involves only the com-putation of twenty-two normal errors instead of a 22 X22 covariance matrix.

IV. Estimation of Useful Eigenvectors

The intrinsic dimensionality of the Fourier spectrumis expected to be less than the Shannon number;moreover, modeling the spectral content of the infor-mation reduces the number of useful samples. Butstatistically stable spatial frequencies can be larger thanindependent signals (for example, spectral directions).In both cases some samples are superfluous or redun-dant so the size of a useful covariance matrix is reduc-ible. Calculating only dominant eigenvectors is theneffective since their number is small in comparison withthe original dimension of the problem. The two casesare investigated here.

vectorial stability-direct eigenv.-

_____________________ I

1 5 10 20

Fig. 5. Statistical stability of the eigenvalues corresponding to asampling of 2 sfu.

A. Where the Number of Signals is Small

Let n be the number of signals (for example, n = 36spectral directions) and m the number of samples de-scribing each signal; here there are m = 44 spatialfrequencies. Usually the covariance matrix Cmm isobtained from the zero-mean data matrix Dmm (con-sisting of the n signals centered with respect to the av-erage signal):

Cmm = (1/n)Dmn Dnm, (9)

where D m denotes the transposed matrix. ObviouslyCmm is a function of n or less linearly independentvectors; n eigenvalues at most are then expected todiffer from zero, and it is not necessary to compute anddiagonalize a m X m matrix, the last m - n + 1 eigen-values of which are not useful. The procedure proposedby McLaughlin9 deals with the reduced matrixC',, defined by

Cnn = (1/l)DIm Dmn, (10)

its eigenvectors < and eigenvalues A' are related to theeigenvectors Som and eigenvalues Xm of Cmm by

Xm = Xn, m < n,

,pm = Dmn' S'n, m n.

(11)

(12)

Results of the estimation of the first eigenvalues of the44 X 44 covariance matrix by a 36 X 36 reduced matrixare presented in Fig. 6. The error in the eigenvaluesdoes not depend on their magnitude but rather seemsto be fixed around 10-7 by the algorithm and therounding errors of the computer. The error on thecorresponding eigenvectors is also checked by the anglebetween 'Pm and its estimate An; its maximal value is 40°(36th eigenvector).

Such an estimate is readily implemented without anyconditions on the data.

B. When the Intrinsic Dimensionality is Small

Assuming that an upper bound d of the intrinsic di-mensionality of spectra is determined at low computa-tional cost and that d is less than n or m, the extractionof useful eigenvectors is to be restricted to the handlingof some d X d matrix. According to the Fukunagaprocedure,10 the data matrix Din is divided into n/dsmaller matrices D'md (i = 1,... ,n/d). It is shown thateigenvectors *pi and eigenvalues X of the elementarycovariance matrices,


DIRECT

6.74 E-011.15 E-018.85 E-027.41 E-021.90 E-021 .24 E-026.66 E-035.73 E-031.81 E-031.03 E-035.95 E-044.55 E-041.63 E-041.34 E-046.50 E-054.60 E-052.54 E-051 .71 E-05

1,10 E-059.39 E-066.72 E-062.35 E-061 .98 E-06

ESTIM

6.74 E-011.15 E-018.85 E-027.41 E-021.90 E-021 .24 E-026.66 E-035.73 E-031.81 E-031.03 E-035.95 E-044.55 E-041.63 E-041 .34 E-046.50 E-054.60 E-052.54 E-051 71 E-051.10 E-059.38 E-066.77 E-062.35 E-062.04 E-06

ROT.

022

020202020 0202020202020°0°0°02O91 427252

1 0 2

Fig. 6. First twenty-three eigenvalues of the 44 X 44 covariancematrix and their estimates from the 36 X 36 reduced covariance ma-trix. The rotation of the corresponding eigenvectors from their actual

values to their estimated values is indicated.

Qdd = (d)Did*D d 13

yield the estimate of the first d eigenvalues and eigen-vectors of Cmm:

d n/dXm =- j Md, m d, (14)

ni=d n/d

Dm d - d, m d. (15)n i=1

In the previous example, the dimensionality of thespectrum was found to be six. The thirty-six spectraldirections are then divided into six groups, D44.6, theeigenvalues and eigenvectors of which give estimates ofthe first six eigenelements. Only the first eigenvalueis well estimated, with an error less than 5%, the rotationof the corresponding eigenvector being about 4 degrees.Checking the statistical stability of these six estimatesshows (Fig. 7) that only the first eigenvalue can be se-lected. This result is in good agreement with the directextraction of stable elements.

From a statistical point of view, features in the Fou-rier spectrum (considered as a set of directions) are tobe related to one intrinsic variable depicted by the firsteigenvector. This parameter is responsible for XI = 67%of the variance that occurs in the spectrum. DespiteKarhunen-Loeve transform decorrelated data, it doesnot always separate signals in classes: signals scatteredby a proper intrinsic variable, which maximizes thevariance of their mapping, are not necessarily wellclustered. Nevertheless the K-L transform constitutesan optimal preprocessing 11 that selects a few variablesto be input to suitable classification algorithms.

V. Statistical Spatial Filtering

The Fourier spectrum of a photograph is studied inorder to separate different geometrical structures: adominant eigenvector is responsible for most of thevariance that occurs as the spectrum is described as aset of directions. Because directions are preserved byFourier transform, results of the statistical analysis alsohold for the photograph itself, considered from a di-rectional point of view. The fact that the intrinsic di-mensionality equals one suggests that the most impor-tant part of the information contained in the photo-graph is packed in the projection of the photograph onthe dominant eigenvector. But it must be decidedwhether such a projection is capable of separatingclasses in its presence. Figure 8 shows the projectionof thirty-six directions in the space of the first threeeigenvectors (following two eigenvectors added to makethe drawing legible). Data are clustered in three classes;in particular, directions 1, 2, 31, and 32 constitute asignificant group, well separated from other direc-tions.

4SQk)

3 1 \ estimates ofthe first 6 eigenv.

2.

- -

l 2 3 4 5 6

Fig. 7. Statistical stability of the estimates of the first six eigenval-ues, obtained from 6 X 6 covariance matrices.

NORMALIZED SPECTRA(36 DIRECTIONS)

2 )32

I11

131

(area Mont~Iimar 033)11 I !

10

Fig. 8. Projection of the Fourier spectrum of an aerial photograph,considered as a set of thirty-six directions in the spatial frequency

basis, in the space of the first three eigenvectors of this basis.


Similar results are obtained when the Fourier spec-trum is described as a set of spatial frequencies in thedirection basis: a cluster of medium frequencies isseparated from two others located in low and highfrequencies, respectively (Fig. 9). In this example thevariance carried by the dominant eigenvector is devotedto the clustering. Such a projection can be translatedin terms of spatial filtering. (The principle of such ananalog approach was first proposed by Frieden12 inorder to determine the coefficients of the expansion ofan image on a prolate spheroidal basis.)

Consider, for example, the spatial frequencies basis:the corresponding first eigenvector is expressed in thisbasis. Its components are given at each sampled spatialfrequency. An optical representation of this eigen-vector consists of a set of concentric rings, the trans-parency of which is proportional to the successivecomponents of the eigenvector in the spatial frequencybasis. The projection of thirty-six directions on thiseigenvector is defined by the scalar product of theirFourier spectrum by the eigenvector. The opticalmeaning of this operation is the spatial filtering of theFourier spectrum of the photograph by an optical rep-resentation of the eigenvector. Conversely, the firsteigenvector of the direction basis is optically describedas a fan of successive circular sectors, the transparencyof which is proportional to the successive componentsof the eigenvector.

Then projections of the photograph on dominanteigenvectors can be visualized by spatial filtering. Sucha filtered image displays what is extracted from theobject considered from the intrinsic dimensionalitypoint of view. This principal image is optimal in thesense of the exhibition of a maximal variance. Twokinds of statistical spatial filters are available here.They cannot be constructed directly as sets of rings orof sectors, because spectral elements transmitted byeach component of the filters are not allowed to inter-fere: the statistical analysis is performed on spectralenergies, and only energies can be recombined. Inter-ference effects are suppressed by incoherent superim-position of elementary images transmitted sector aftersector and recorded during an exposure time propor-tional to the component of the spatial frequency ei-genvector (Fig. 10). In the spatial frequency basis eachring is simulated by a collection of small identical ap-ertures, randomly displayed along the circle corre-sponding to the sampled spatial frequency. Thenumber of apertures is proportional to the componentof the direction's eigenvector on the spatial frequencybeing considered. The spatial filter composed by thefamily of concentric rings is rotated during a fixed ex-posure time in order to remove interferences.

The principal image obtained by spatial filtering ofthe Fourier spectrum by the optical representation ofthe first eigenvector of spatial frequencies is shown inFig. 11. This image carries the most important part ofthe variance of spatial frequencies considered as sta-tistical functions of direction. The River Rhone dividesthe field in two regions. On the right side two straightroads support packs ofindistinct high periodicities; on

/44 FREQUENCIES) 13 23

31 27 121

32 /.\ 30

2 I~~~~~~~~229~~~~~~~ 2

4t 38

14

Fig. 9. Projection of the Fourier spectrum of an aerial photograph,considered as a set of forty-four spatial frequencies in the direction

basis, in the space of the first three eigenvectors of this basis.

smOBJECT [W OBJECTVP , 37 L1F\ ie e 0tr Naeral-kgAp

/ J ' + | K1L6 +IMKAGE

[777]Hj}Lk tmt*T llt. L-

Fig. 10. Optical setup for statistical pa cfiltering: the object isfiltered by an optical representatiion of the eigenvector being con-sidered. In the direction basis (left) the K-L image is obtained byincoherent superimposition of elementary images transmitted sectorby sector. In the spatial frequency basis the K-L image is given byrings the transmittance of which is proportional to the components

of the eigenvector o

ORGIAL

high freq.class

PRINCIPAL COMPONENT

Fig. 11. The principal K-L image of the previous aerial photographdefined as its projection on the first eigenvector of the direction basis.In this basis two classes of spatial frequencies are separated (Fig. 8).Here these classes appear as high frequencies on the right side of the

river, and as medium frequencies on the left side of the river.


the left, a typical squared feature is visible corre-sponding to the medium frequencies (see Fig. 9,frequencies 22 and 23).

VI. Conclusion

Optical information processing is adapted to thestatistical analysis of classes of geometrical elements inimages. The dimensionality of the problem is reducedby modeling the information and by computing its in-trinsic dimensionality. Estimation procedures avoidthe computational cost of a direct analysis of the spec-tral content of images. Dominant eigenvectors play therole of spatial filters and provide principal images thatcarry most of the variance.

Statistical spatial filtering not only concerns theanalysis of single photographs but also applies to a seriesof images, such as handwritten pages.1 3 In this caseeigenvectors are extracted after learning on a set oftraining pages; they can be widely used to process im-ages to be classified.

References1. B. J. Turner, "Cluster Analysis of Multispectral Scanner Remote

Sensor Data," in Remote Sensing of Earth Resources, Vol. 1, F.Shahrokhi, Ed., University of Tennessee, Tullahoma (1972).

2. G. E. Lowitz, Automatisme 21 (314), 83 (1976).3. J. W. Goodman and H. Kato, Opt. Commun. 8, 378 (1973).4. D. Casasent and D. Psaltis, Appl. Opt. 15, 1795 (1976).5. J. Duvernoy, Appl. Opt. 15, 1584 (1976).6. K. Fukunaga, Introduction to Statistical Pattern Recognition

(Academic, New York, 1972), Chap. 10, pp. 288-301.7. B. R. Frieden, J. Opt. Soc. Am. 62, 511 (1972).8. S. Watanabe, "Karhunen-Loeve Expansion and Factor Analysis,"

in Proceedings 4th Prague Conference on Information Theory(1965), pp. 635-660, in Pattern Recognition: Introduction andFoundations, ed. J. Sklansky (Dowden, Hutchinson, Ross Inc.,Stroudsburg, Pennsylvania, 1973).

9. J. A. McLaughlin and J. Raviv, Inf. Control 12, 121 (1969).10. Ref. 6, Chap. 8, pp. 249-250.11. S. Watanabe, Knowing and Guessing (Wiley, New York, 1969),

Sec. 7.6.12. B. R. Frieden, J. Opt. Soc. Am. 57, 1013 (1967).13. J. Duvernoy, D. Charraut, and P. Y. Baures, Opt. Acta 24, 795

(1977).

IFAORS Short Course No. 420"MULTIPLE LIGHT SCATTERING: IN ATMO-SPHERES, OCEANS, CLOUDS AND SNOW to beheld in the Hospitality House,Williamsburg, VA, December 4-8, 1978.Course Registration Fee: $350.00.Sponsored by: Institute for Atmo-spheric Optics and Remote Sensing(IFAORS), Hampton, Va. In cooperationwith: The College of William & Mary,Williamsburg, Va. Course Instructors:Dr. H. C. van de Hulst (Leiden U., TheNetherlands); Dr. G. K. Kattawar(Texas A & M U.); and Dr. A. Deepak,IFAORS. For information write to:IFAORS Short Courses, P. 0. Box P,Hampton, VA 23666, or call804/838-3715.


statistical spatial filtering: application to aerial photographs

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