Vf = ,4Re E E n+l (afm + ib m)e-imOP'(cos 0) (9)

SHORT NOTES

V(V' + V") dS = 0 (8)

where dS is a vector element of the bounding surface.In the original solution for V', V", and V, replace (r, 0, X)

by (rlk, 0, 4. Equation (7) will still be satisfied. Equation(8) will also be satisfied on the new boundary since the orien-tation of VV', VV", and dS will be unchanged. Thereforethe solution for the scaled torso is obtained by replacingr by rlk in the original solution. From (1) the potential inan unbounded medium becomes

1 "O n 1Vf =,4Re E E n+l (afm + ib m)e-imOP'(cos 0) (9)421g n=1 m=O

r

where Re designates real part and

anm + lbem = kn+'(anm + ibnm). (10)Hence the new solution for the scaled torso is valid for

an equivalent multipole generator where the original dipoleterm is multiplied by k2, the quadrupole by k3, etc. Corre-spondingly, the transfer coefficients Tnm must be multipliedby 1/kn+ '. For example, ifk = 2, an identical surface potentialwould imply that the dipole was four times larger and thequadrupole eight times as large as for the original torso.Note that the product of transfer coefficient and multipolecomponent does not change.

DISCUSSIONOur laboratory has been studying the multipole ap-

proach to electrocardiography in which the cardiac sourcedistribution is expressed in terms of its multipole expansion.Knowledge of the multipole components can provide in-formation about the source distribution m, as indicated in(2). The multipole expansion will be useful only if a limitednumber of terms, e.g., the dipole and quadrupole, containa substantial fraction of the available information. To testthis idea the dipole-quadrupole components were evalu-ated on a normal subject utilizing voltage data collected atclose to 300 points on the body surface. Isopotential mapsreconstructed from the dipole and quadrupole data werefound to preserve essential features of maps constructedutilizing the original voltage data.To be practical, the dipole and quadrupole components

must be obtainable from a limited number ofelectrode sites.A computer search led to a subset of 16 electrocardiogramwhich gave substantially the same dipole and quadrupoleas were obtained from the entire set [6], [7]. The iso-potential maps reconstructed from these components arenot shown here, but they would be very similar to Figs.l(b), 2(b), and 3(b).The 17 electrode sites for recording the 16 ECGs were

chosen at points which could be conveniently locatedusing a protractor jig with the subject in a supine position.Tests on a series of 50 subjects have led us to believe thatthis lead system is an accurate one for determining dipoleand quadrupole components on adult male subjects [7].We have not compared actual and reconstructed iso-potential maps on other subjects. Therefore we are not ina position to state whether our present scheme for con-

structing isopotential maps from data obtained in ECGs isof general usefulness. On the original subject, however, thepattern of surface isopotential contours can be constructedreasonably well at each instant during the cardiac cyclefrom data obtained at 17 electrode sites.Note that the present analysis treats the surface electro-

cardiogram as a sequence of isopotential maps. Thus fea-ture extraction is on an instant by instant basis withoutregard to waveform. An alternative approach is to look atthe waveforms recorded at various electrode sites withoutregard to the distribution of potential on the body surface[8]. Use of "intrinsic components" or "principal com-ponents" provides an interesting tool for extracting sig-nificant features from a large number of simultaneouslyrecorded electrocardiographic waveforms [9], [10]. Todate, however, relatively little use has been made of thisapproach.

REFERENCES[1] W. Einthoven, G. Fahr, and J. DeWaart, "Uber die Richtung und

die manifeste Grosse der Potentialschwankungen im menslichenHerzen und uber den Einfluss der Herlzlage auf die form des Elektro-kardiogramms," Pfluegers Arch. Ges. Physiol., vol. 150, 1913, p. 275.

[2] D. B. Geselowitz, "Multipole representations for an equivalentcardiac generator," Proc. IRE, vol. 48, Jan. 1960, pp. 75-79.

[3] B. Taccardi, "Distribution of heart potentials on the thoracic surfaceof normal human subjects," Circulation Res., vol. 12, 1963, p. 341.

[4] M. S. Spach, W. P. Silberberg, J. P. Boineau, R. C. Barr, E. C. Long,T. M. Gallie, M. A. Gabor, and A. G. Wallace, "Body surface iso-potential maps in normal children, ages 4 to 14 years," Amer.Heart J., vol. 22, 1966, p. 640.

[5] D. B. Geselowitz and 0. H. Schmitt, "Electrocardiography," inBiological Engineering, H. P. Schwan, Ed. New York: McGraw-Hill, 1969.

[6] R. M. Arthur, "Evaluation and use of a human dipole plus quad-rupole equivalent cardiac generator," Ph.D. dissertation, Univ.Pennsylvania, Philadelphia, Pa., 1968.

[7] R. F. Trost, "Experimental studies of a human dipole plus quad-rupole lead system for electrocardiography," Ph.D. dissertation,Univ. Pennsylvania, Philadelphia, Pa., 1970.

[8] T. Y. Young and W. H. Huggins, "On the representation of electro-cardiograms," IEEE Trans. Bio-Med. Electron., vol. BME-10, July1963, pp. 86-95.

[9] ,"The intrinsic component theory of electrocardiography,"IRE Trans. Bio-Med. Electron., vol. BME-9, Oct. 1962, pp. 214-221.

[10] L. G. Horan, N. C. Flowers, and D. A. Brody, "Principal factorwaveforms of the thoracic QRS complex," Circulation Res., vol. 15,1964,p. 131.

Automatic Ship Photo Interpretation bythe Method of Moments

FRED W. SMITH, MEMBER, IEEE, AND MARGARET H. WRIGHT

Abstract-The results of a study undertaken to determine thefeasibility of automatic interpretation of ship photographs using thespatial moments of the image as features to characterize the image arereported. The photo interpretation consisted of estimating the loca-tion, orientation, dimensions, and heading of the ship. The study used

Manuscript received November 30, 1970; revised April 26, 1971. Apreliminary version of this note was presented at the IEEE Symposiumon Feature Extraction and Selection in Pattern Recognition, Argonne,Ill., October 5-7, 1970.

The authors are with Sylvania Electronic Systems, Mountain View,Calif. 94040.

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simulated ship images in which the outline of the ship was randomlyfilled with black and white cells to give a low-resolution high-contrastimage of the ship such as might be obtained by a high-resolution radar.The estimates were made using polynomials of invariant momentsformed by transformations of the original spatial moments, e.g.,density-invariant moments, central moments, rotation-invariant mo-ments, etc. The transformations to invariant moments were chosenusing physical reasoning. The best moments for the polynomials werechosen using linear regression. The performance of the method ofmoments on these low-resolution images was found to be comparableto that of a human photointerpreter and to certain heuristic tech-niques that would be more difficult to implement than the method ofmoments.

Index Terms-Automatic photo interpretation, comparison withhuman, invariant moments, linear regression, method of moments,ship images.

INTRODUCTIONThis short note presents the results achieved by the ap-

plication of physical reasoning and linear regression tech-niques to feature selection in the automatic interpretationof ship photographs. An image is described by invariantspatial moments of its intensity function. The functions forestimating image features from these moments are poly-nomials computed by linear regression on a set of typicalimages. This approach differs from the classical method ofmoments, where functions are computed analytically fromthe probability distribution of the moments. Invariantmoments are shown to be very suitable for feature selectionbecause they allow a significant portion of the feature selec-tion to be based on physical reasoning.

Interpretation of ship images is believed to be typical ofmany other practical problems, for example, other photointerpretation tasks, counting of chromosomes and similarobjects, classification of simple line figures such as hand-printed characters, and any other image processing thatdepends on general shapes rather than details of the image.The quality ofresults obtained for ship photo interpretation[1] is therefore probably indicative of possible achieve-ment on other problems.

IMAGE DESCRIPTION BY MOMENTSAn image is represented by the spatial moments of its

intensity function; such a representation is termed the"method of moments." The ijth generalized moment is de-fined by

mij= {ij (x, y)I(x, y) dx dy (1)

where I(x, y) is the intensity function representing the image,integration is over the entire picture, and Lj,{x, y) is somefunction of x and y. The jj(x, y) can be a variety of func-tions, for example, xiy j, which then gives the usual definitionof a moment; the product of Chebyshev polynomials in x

and y; or sine and cosine functions of ix andjy, in which case

the mij are the coefficients of the two-dimensional spatialFourier series expansion ofthe image intensity function. Theterm "moment" is also used to refer to normalized momentsthat are in reality functions of the moments defined by (1).The method of moments has occasionally been suggested

for character recognition [2], [3] and for terminal guidance

[5]. It has also been used successfully to classify the two-dimensional histograms ofthe instantaneous amplitude andfrequency of different modulations of HF signals [6]. Mo-ments have also been used in character recognition tonormalize the characters before recognition is attempted byother techniques [7], [8].

Describing images with moments instead of the morecommonly used image features means that global propertiesof the image are used rather than local properties. Neverthe-less, other previously proposed methods of image pro-cessing are related to the method of moments. Templatematching [9]-[11] and the peep-hole technique [12] canbe considered special cases of the method of moments.

Invariant MomentsA simple transformation of the image moments can be

used to make the moments invariant to confusing variablesthat are not of interest, such as the image's location withinthe picture, or orientation, when length is being estimated.This invariance is achieved by normalizing the higher ordermoments with respect to the lower order moments, whichindicate the position, orientation, etc. A common exampleof an invariant moment is the variance of a probability dis-tribution, which is independent of the mean, or translation,of the distribution. When invariant moments are used foralphanumeric character recognition [2], [3] they are madeindependent of height, width, and slant, as well as positionand orientation.Many of the earlier methods for processing two-dimen-

sional images which did not use invariant moments weremade invariant to shifts and rotation by crosscorrelatingthe image with the template. No cross correlation is neces-sary with the use of invariant moments.

Ship-Image MomentsThe successive estimates of the ship location, orientation,

and dimensions must be made using moments that are in-variant to the previously estimated quantity. Initially theship location is computed as the center of gravity of theimage (x, j), where

- fxI(x, y) dx dy

TI(x, y) dx dy

yJlj(x, y) dx dy

JjI(x, y) dx dy

These moments are invariant to the image intensity

M =o I(x Y) dx dy.

The orientation of the ship can be most readily com-puted as

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SHORT NOTES

onm =tan-' (2)nm

where

Cnm = JJ r" cos mOI(r, O)r dr dO

Snm = rn sin mOI(r, O)r dr dO

n =0,1,2,12 , 0 < m < n (3)

and r and 0 are polar coordinates measured relative to (x, y).Moments independent of intensity, location, and rotationcan be computed by measuring 0 with respect to coordinateaxes along the principal axes of the image to give

{Jrn cos m(0 - O)I(r, 0)r dr dOCn =mnm

S3nm sin mO + Cnm cos mO

Jr sin m(O - O)I(r, O)r dr dOSnm =

3Snm cos mO - Cnm sin mOMoo

where O is (2) with any acceptable m and n.

Moments of up to fifth order were used for this study.Some moments given by (4) are not meaningful and are

therefore not considered, for example, all sine moments withm= 0 vanish. Furthermore, because the heading of the shipwas not used in estimating the dimensions, the terms forestimating ship dimensions should be invariant to an end-to-mnd reflection of the ship image. The sine moments of(4) are not useful with the images used in this study becauseof the symmetric ship model used to simulate the data. Theimages had only small random deviations from symmetry;thus the sine moments, which indicate the asymmetry of theimage, also had small and random values. The sine momentsmight still be found useful for images that have significantasymmetries, such as images containing shadows, etc.

Functions ofMomentsLinear, quadratic, and cubic polynomials ofthe moments

were used as estimators of ship descriptors. These nonlinearfunctions can be designed using a standard linear regressionprogram by treating the powers of the moments as new

variables in the linear combination.Physical reasoning can also be used to restrict the terms

used in the polynomials just as in restricting the momentschosen. Because the estimated dimensions of the shipshould not be affected by an end-to-end reflection of theship image, functions ofcosine moments, the Cnm in (4), wereaccordingly restricted, for m odd, to be even functions of 0.Therefore, the squares ofthe variant moments, the Cnm withm odd, were used in the polynomial functions.

LINEAR REGRESSIONA linear-regression program attempts to find a set of co-

efficients wi, i= 0, 1, * * , N, of the linear function

Y= Wo + WlXl + W2X2 + + WNXN (5)

that approximates the desired quantity in a least-squaresmanner. When linear regression is used to find estimators ofship dimensions, y is either length, or width, etc. The xi aremoments or powers of moments.The coefficients are determined using K specific observa-

tions of corresponding y and x values,

Xk: Xlk, X2k, XNk k=l, ,K. (6)

Because generally K>> N, the specific values of (6) cannotbe used in (5) to solve for the unknown wi. Instead the wiare chosen to minimize the mean-squared error between theYk and W0+WlXlk+ * + WNXNk, a straightforward matrixinversion problem.The linear regression computer program has two routines

that allow the number of terms in the function (5) to bevaried efficiently. Using physical reasoning, the designerinitially chooses a large set of possibly useful independentvariables (50 or more). These variables are "offered" to theprogram, which selects them in order, using a "term addi-tion" routine. Independent variables are added one at atime to the set already chosen. The variable chosen at eachstep is that least correlated to those already in the set. Nocheck is made to determine whether the chosen independentvariable is helpful in approximating the dependent variable.The selection process is identical to the maximal-pivot-element strategy often used to invert matrices [13]. Theterm addition routine is generally terminated before all ofthe offered terms are used because of limits in computa-tional accuracy.A "term deletion" routine is then used to reduce the set,

one variable at a time. This routine deletes at each step thevariable whose omission causes the smallest increase in thesum of squared error [14]. This deletion strategy givesoptimal one-term deletions, but does not necessarily giveoptimal multiterm deletions.The coefficients of the polynomials for estimation were

designed to minimize the standard deviation of the error inlinear measure. Other error measures could have beenminimized instead, such as the standard deviation of thepercent error, or the maximum-magnitude error. Choice ofthe most suitable error measure to be minimized dependson the use to be made of the estimates. Minimization of thestandard deviation in the units of the quantity being esti-mated was chosen because a computer program for mini-mizing the standard deviation was readily available. It wasfelt that the results using other error measures would be atleast qualitatively the same as those obtained in this study.

SHIP PHOTO-INTERPRETATIONThe coefficients for the estimated ship parameters were

designed and tested using moments from a set of 260simulated ship images, half in a design set and half in a test

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IEEE TRANSACTIONS ON COMPUTERS, SEPTEMBER 1971

MLa ,

*.4I-

MERCHAS H P

kNT DESTROYER SUlOR CRUISER

Fig. 2. Examples of ship outlines.

JBMARINE

r~~~- .

U

Fig. 1. Six examples of simulated ship images.

set. Fig. 1 shows six examples of the images, including thesmallest, least dense image and the largest, most denseimage. These images are believed to be typical of the high-contrast low-resolution images of a ship on the open seathat might be obtained with a synthetic-aperture radar[15]. The study was limited to images containing one andonly one complete ship, and with no return from the wateroutside the ship outline. The study was also limited to top-view images ofships because the top is the projection usuallydisplayed for synthetic-aperture radars.The image was generated by specifying the outline of the

ship and its average density. A grid of square cells was over-layed on the ship, and cells with cell centers on or within theoutline were given intensity 0 or 1 with a probability cor-responding to the average density. The cell size was 83feet square.The ship outline is considered to be inscribed within a

rectangle, and to be symmetric about its long axis. Threeship outlines were used, a merchant ship, a destroyer, and asubmarine. Examples of the outlines are given in Fig. 2.The four basic ship types were short merchant ship, long

merchant ship, destroyer cruiser, and submarine. They

TABLE ISHIP PARAMETER DISTRIBUTION

Ship Type Number Length (ft) Aspect Ratio

Short merchant ship 100 200-350 6.25- 7.25Long merchant ship 100 350-500 7.0 - 8.0Destroyer cruiser 40 200-500 8.0 -10.0Submarine 20 200-500 9.5 -11.5

were generated in quantities approximately proportionalto their occurrence in the actual ship population (see TableI). The length and the aspect ratio (length-to-width ratio)were chosen independently and randomly from uniformdistributions with the ranges given in Table I. The orienta-tion of the outline was also varied randomly over a range ofninety degrees. The average density was randomly chosenwith a uniform distribution from 0.3 to 0.5.

Length EstimationLinear, quadratic, and cubic polynomials were used as

estimators of length. Fig. 3 shows the change in error as thenumber of terms was first increased and then decreasedusing a regression function containing a linear combina-tion of the Cnm of (4) when m was even, or the squares of theC,,, when m was odd. There were 19 such terms; however,only 14 terms were used. The error standard deviation andmaximum-magnitude error in feet are shown in Fig. 3versus the number of terms in the regression function. Theerror is shown as the number of terms was increased from10 to 19, and reduced from 14 to 1. Arrows are used toindicate the different curves for increasing and decreasingthe number of terms.The knee in the standard deviation curve occurs at one

term; the knee in the maximum-magnitude curve occurs ateight terms. The number of terms in the knees of the curves

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0

SHORT NOTES

30

201

X- lMAXIMUM-MAGNITUDE ERRORS15

00

5

ERROR STANDARD DEVIATIONS0

0 5 10 15 20 25 30 35 40

NUMBER OF TERMS

Fig. 3. Error versus numbers of terms for estimated length.

is significant because using more than this number of termsin the regression polynomial yields little or no improve-ment in the errors on the test set. This behavior is typicalof all the regression runs in this study.A six-term cubic polynomial of the moments gave the

best estimate of length with a standard deviation on thetest set of 16.1 ft (approximately two cells). The terms were

C30, C22, C20, C22, C30, and C20. The maximum-magnitudeerror on the test set was 75.8 ft.The higher order regression polynomials significantly re-

duced the errors. The quadratic polynomial performedbetter than the linear, and the cubic better than the qua-dratic. A fourth-order function might be expected to do stillbetter.The results indicate that the reduction option of the linear

regression program does decidedly better than the term-addition option at selecting terms most likely to produce a

minimum standard deviation. It is therefore important forthe regression polynomial to include as many terms as

possible before the reduction is started.The accuracy of the estimated lengths was not limited by

the order (values of m and n in (4)) of the moments used inthis study. Although m and n could take values up to 5, thelargest value of n was 3 in the most useful terms for anylength regression function, and the largest value ofm was 2.Presumably the higher order moments were not found use-

ful because they are more sensitive than the lower ordermoments to the random variations of the image.A meaningful estimate of length can be made using poly-

nomials oforientation-independent moments, i.e., momentsof the form

CnO= J'JrI(r, O)r dr dO.

A second-order polynomial of five such terms yielded a

standard deviation on the test set of 18.9 ft with maximum-magnitude error of 74.0 ft. Estimating functions containingonly orientation-independent moments offer the computa-tional advantage in that no transformation to principalaxes is necessary.

Width EstimationLinear and quadratic polynomials of the moments were

also used as estimators of width. The change of the errorwith the number of terms was similar to that with lengthestimation, except that the knee was not as sharp. A four-term quadratic polynomial of the moments gave the bestestimate of width with a standard deviation of 2.15 ft onthe test set. The four terms were C20, C22, C20, andC22.The maximum-magnitude error was 5.21 ft.The quadratic regression polynomial did not greatly im-

prove the standard deviation on the test set over that of thelinear function. A higher order polynomial was not ex-pected to do significantly better, and none was tried.As with length, the accuracy of the estimates was not

limited by the order of the moments (n and m in (4)). Thehighest value of n was 2 in the terms found most useful inthe quadratic, although in the linear functions C50 and C54were used at the knee; the highest value of m was 2 in themost useful quadratic terms.

Aspect Ratio EstimationAspect ratio was estimated in two ways: directly, by

linear and quadratic polynomials, and indirectly, as theratio of estimated length to estimated width. Becauseaspect ratio is independent of the size of the ship, eachmoment with indices n andm in (4) is normalized by dividingby C.O. A two-term quadratic polynomial gave the bestestimate of aspect ratio with a standard deviation of 0.54on the test set. The two terms were C24 and C44 (normal-ized). This quadratic polynomial did significantly betterthan any linear combination.When aspect ratio was estimated as the ratio ofestimated

length and estimated width, the results were not as good,presumably because numerator and denominator wereestimated separately.The accuracy in estimating the aspect ratio was again not

limited by the order of the moments, since 4 was the largestvalue of n and m in both the best linear and quadraticpolynomials. The number of patterns in the design setmight have limited the accuracy of the results; the error forthe test set remained essentially constant for increasingnumbers of terms beyond the knee in the standard devia-tion, while that of the design set decreased somewhat.

Heading ClassificationThe orientation of the ship images was computed during

the initial normalization. A remaining task is the binarychoice of the heading, or bow end, of the ship. The sub-marines, which are symmetric end to end, were excludedfrom this design and test. The heading was estimated bycomparing the magnitudes of the half-length of the shipfrom the centroid to the bow and the half-length from thecentroid to the stern. Because the centroid ofa ship image iscloser to the stern than the bow, the sign of the differencebetween the estimated half-length to the bow and the esti-mated half-length to the stern can be used to indicate theship's heading. Only moments that change sign when theimage is rotated 180° can supply information about the

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IEEE TRANSACTIONS ON COMPUTERS, SEPTEMBER 1971

heading; the useful moments are therefore limited to thosein (4) with m odd. A linear polynomial in the three mostuseful suitably invariant terms, C55, C53, and C51, gave

22.1-percent classification errors on the design set and 22.3on the test set. These three best terms include the maximumvalue of n in (4), presumably because the heading dependsonly on the details near the ends of the image. Momentswith larger values of n, or weightings that eliminate themiddle portion of the ship, might very well give better re-

sults.OTHER ESTIMATES

To give a basis for judging the method of moments forship photo interpretation, other heuristic or intuitive tech-niques were also tried on the same tasks. Both heuristic andintuitive estimates of length were made. In the heuristicestimate the length was taken as the distance between thetwo most distant points on the image, plus a suitableconstant. The constant (20.4 ft) was set to give zero mean

error on the design set of 130 images.Intuitive estimates of length were made for the images of

the design set. The photo interpreter knew the ship type,the ship heading (but not its orientation), and the fact thatthe length lay between 200 and 500 ft. His procedure was touse his knowledge of the ship outlines to pick out nonzero

cells which he knew with high confidence would lie on theoutline of the ship, to complete mentally the outline of theship and to measure the distance between the end points ofthe mental ship outline. The estimation errors for bothtechniques are given in the Conclusion.

Because both the heuristic and intuitive estimates dependheavily on the location of certain critical cells of the image,the performance of these estimation techniques will bequite sensitive to random returns from reflections outsidethe outline of the ship. When such noise is present, the

performance given in the conclusion may be unobtainable,unless more complicated logic is used to locate the criticalcells.A combination of heuristic and intuitive techniques was

used to estimate width. The technique is less well definedand less simple to carry out than the heuristic technique forlength because it requires a visual (intuitive) determinationof the orientation of the image. After deciding on theorientation, the measurement taken for width is the maxi-mum distance, measured perpendicular to the axis oforientation, between any two points of the image. As withlength, an additive constant (3.01 ft) was determined fromthe design set to make the mean error zero. The results are

summarized in the Conclusion.The aspect ratio was estimated as the ratio ofthe heuristic

estimate of length and the heuristic-intuitive estimate ofwidth. The heading was estimated heuristically by examina-tion of the last quarter of the image at each end, and a

determination of which end contained more points withnonzero density. The end with fewer such points is con-

sidered the bow, and the ship to be heading in that direction.

ESTIMATION ACCURACIES

Table II gives a comparison of the estimation accuracies

TABLE IIPERCENT ERRORS IN SHIP PHOTO INTERPRETATION

HeuristicA Method of Computer Human

Priori Moments Techniques Intuition

4.9Length estimate 28.6 6-term cubic 4.8 3.5

5.6Width estimate 34.1 4-term quadratic 7.7

6.7Aspect ratio estimate 9.7 2-term quadratic 10.1 not tested

23.3Heading' classification 50 3-term linear 22.5 10.8

a Classification error.

obtained by the method of moments, a human photo inter-preter, and heuristic computer techniques. It also gives thea priori estimates of length, width, and aspect ratio, whichwere computed using only a knowledge of the distributionsin Table I, without any knowledge of the image. To be use-ful, estimates obtained by any estimation method must bebetter than the a priori estimates. For easier interpretationof the relative magnitude of the error, Table II gives theerror in percent. The cell size, 81 ft, is 2.4 percent of theaverage length (350 ft), and 18 percent of the average width(46.8 ft).

CONCLUSION

The method of moments appears to work quite well forautomatic ship photo interpretation. The standard devia-tions ofTable II show that the performance ofthe method ofmoments on these low-resolution images is comparable tothat of a human photo interpreter, and it is either roughlyequivalent to or better than that of the heuristic techniques.Failure to surpass the performance of a human is notalways a serious liability, however. In many situations,automation is required, even if the resulting performance isnot as good as that of a human. When automation is re-quired, the method of moments is quite possibly the bestmethod because it is easier to implement than the heuristictechniques, which require point-by-point image analysis.

Several conclusions can be drawn about the types ofmoments and functions which are useful for ship photointerpretation.

1) Simple polynomials were adequate for the method ofmoments; the most complicated function required was a6-term cubic.

2) Nonlinear functions of the moments, i.e., quadraticand cubic terms, gave improved estimates over those oflinear functions, generally significantly improved estimates.

3) During the study it was clearly demonstrated that thereduction option of the linear regression program doesdecidedly better than the term-addition option at selectingterms most likely to produce a minimum standard devia-tion. It was also demonstrated that the regression poly-nomial should include as many terms as possible beforethe reduction is started.

4) The variation of the error measures with the number

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SHORT NOTES

of terms in the polynomial function was of the same formfor all photo interpretation tasks. For terms ordered by thereduction option of the linear regression program, thedecrease of the standard deviation with increasing numbersof terms is quite rapid for small numbers of terms; however,after a small number of terms has been included in thepolynomial, the addition of more terms causes little reduc-tion in the standard deviation, and any reduction made isnot statistically significant. The same phenomenon occurs

for the maximum-magnitude error; however, it did notdecrease as rapidly as the standard deviation, possiblybecause it was not being minimized directly but only as a

consequence of the minimization of the standard deviation.The form of the variation of the error discussed in item 4)

if it is typical, raises two questions about the current workto develop better methods of feature selection.

1) The relatively sharp knee in the curve at the relativelysmall number of terms in many problems may obviate theneed for sophisticated measurements selection techniques.The "brute-force" technique of evaluating all sets of mea-surements to find the best becomes feasible.

2) A standard, readily available linear regression pro-

gram appears to do a creditable job of feature selection,which probably can be made slightly better by repeatingthe term-addition and term-reduction options severaltimes. In the face of this performance, can any special-purpose algorithms be expected to do significantly better?

REFERENCES[1] F. W. Smith and M. H. Wright, "Automatic ship photo interpreta-

tion by the method of moments," Sylvania Electronic Systems,Mountain View, Calif., Tech. Memo. SESW-M1 169, Nov. 1967.

[2] M. K. Hu, "Visual pattern recognition by moment invariants," IRETrans. Inform. Theory, vol. IT-8, Feb. 1962, pp. 179-187.

[31 F. L. Alt, "Digital pattern recognition by moments," in OpticalCharacter Recognition, G. L. Fischer et al., Eds. Washington, D. C.:Spartan, 1962, pp. 153-179.

[4] K. Udagawa, J. Toriwaki, and K. Sugino, "Normalization and recog-

nition of two-dimensional patterns with linear distortion by mo-ments," Electron. Commun. Jap., vol. 47, June 1964, pp. 34-46.

[5]S. Moskowitz, "Terminalguidance by pattern recognition-a new

approach," IEEE Trans. Aeronaut. Navig. Electron., vol. ANE-1 1,Dec. 1964, pp. 254-265.

[6] F. W. Smith, "Automatic HF signal classification by the method ofmoments," Sylvania Electronic Systems, Mountain View, Calif.,Tech. Memo. SESW-M1037, Mar. 1967.

[7] R. Bakis, N. M. Herbst, and G. Nagy, "An experimental study ofmachine recognition of hand-printed numerals," IEEE Trans. Syst.Sci. Cybern., vol. SSC-4, July 1968, pp. 119-132.

[8] R. G. Casey, "Moment normalization of handprinted characters,"IBM J. Res. Develop., vol. 14, Sept. 1970, pp. 548-557.

[9] H. D. Block, N. J. Nilsson, and R. 0. Duda, "Determination anddetection of features in patterns," in Computer and InformationSciences, J. T. Tou and R. H. Wilcox, Eds. Washington, D. C.:1964, pp. 75-1 10.

[10] M. Minsky, "Steps toward artificial intelligence," Proc. IRE, vol. 49,Jan. 1961, pp. 8-30.

[11] E. Rosenblatt, Principles of Neurodynamics: Perceptions and theTheory of Brain Mechanisms. Washington, D. C.: Spartan, 1961.

[12] R. Casey and G. Nagy, "Recognition of printed Chinese characters,"IEEE Trans.Electron. Comput., vol.EC-15, Feb. 1966, pp. 91-101.

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Pattern Recognition Signal Processing forMechanical Diagnostics Signature Analysis

ROY L. HOFFMAN, MEMBER, IEEE, AND

KEINOSUKE FUKUNAGA, MEMBER, IEEE

Abstract-Signature analysis of small, complex, cyclic mecha-nisms is discussed. An envelope preprocessor for isolating signalevents relating to mechanical impacts is developed. A recognitionsystem based on second-order (normal) statistics is used and the use

of statistical procedures for signature intepretation is presented.Experimental evidence is presented to support the validity of thisapproach to mechanical diagnostics signature analysis.

Index Terms-Diagnostic systems, envelope detection, mechan-ical diagnostics, pattern recognition, preprocessing, second-orderstatistics (or normal or Gaussian), signal analysis, signal averaging,signature analysis, time trends.

INTRODUCTION

Today there is widespread interest in diagnostic systemswhich can predict impending mechanical system failures[I ]-[8]. Several studies have demonstrated that "secondaryeffects" such as sound, vibration, temperature, pressure,

and other physical phenomena, exhibit telltale changeslong before catastrophic mechanical failures occur and,therefore, can be used to predict and diagnose mechanicalmalfunctions [3]. The technology for monitoring secondaryeffects is called "signature analysis." The signature is theset of time-varying measurements of secondary effects.The analysis of these data is amenable to pattern recog-

nition [4], [9], [10].Previous investigations have shown how signature

analysis applies to large mechanisms which in some re-

spects are easier to diagnose. Sensors can usually be placedto detect the secondary effects of one component or me-

chanical subsystem, independent of similar emissions fromother components or subsystems.

Small mechanical devices, in contrast, have many

moving parts, in highly interrelated mechanical subsystems,in physical proximity, whose secondary effects cannot beindependently sensed. Most sensor outputs are combina-tions of the secondary effects from several subsystems andsuccessful signature analysis often is possible only when thesecondary effects of the component of interest can be iso-lated through signal processing.

SIGNATURE ANALYSIS

In this experimental study we found that the secondaryeffects of many cyclic components can be isolated by usingtime-signal averaging along with filtering and time gating[11 ], [12]. Time-signal averaging has been shown effectivefor reducing random (additive) noise in cyclic signals[3], [4], [7],[13]. Its applicability to isolating the emissionsof concurrently operating mechanisms hinges on the inter-relationships between signals from those mechanisms. If a

Manuscript received November 2, 1970; revised March 29, 1971. Apreliminary version of this note was presented at the IEEE Symposium on

Feature Extraction and Selection in Pattern Recognition, Argonne, Ill.,October 5-7, 1970.

R. L.Hoffman is with IBM General Systems Division, Rochester,

Minn. 55901.K. Fukunaga is with the Department of Electrical Engineering, Pur-

due University, Lafayette, Ind. 47907.

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