IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. SMC-4, NO. 4, juLY 1974 371

[17] C. K. Chow, "On optimum recognition error and reject trade- expanding subinterval performance-adaptive S.O.C. algorithm,"off," IEEE Trans. Inform. Theory, vol. IT-16, pp. 41-46, Jan. 1970. J. Cybern., vol. 3, Apr.-June 1973.

[18] T. M. Cover and P. E. Hart, "Nearest neighbor pattern classifica- [22] H. P. Whitaker, "Design capabilities of model reference adaptivetion," IEEE Trans. Inform. Theory, vol. IT-13, pp. 21-27, Jan. systems," in Proc. 1962 Nat. Electron. Conf., vol. 18, pp. 241-249.1967. [23] K. S. Fu, "Learning control systems-Review and outlook," IEEE

[19] T. Lissack -and K. S. Fu, "A separability measure for feature Trans. Automat. Contr., vol. AC-15, pp. 210-221, Apr. 1970.selection and error estimation in pattern recognition," Purdue [24] Y. C. Ho and A. K. Agrawala, "On pattern classification al-Univ., Lafayette, Ind., Tech. Rep. TR-EE 72.15, 1972. gorithms-Introduction and survey," IEEE Trans. Automat.

[20] G. N. Saridis, "On a class of performance-adaptive self-organizing Contr., vol. AC-13, pp. 676-690, Dec. 1968.control systems," in Pattern Recognition and Machine Learning, [25] R. F. Hofstadter, "A pattern recognition approach to the clas-K. S. Fu, Ed. New York: Plenum, 1971. sification of stochastic nonlinear systems," Ph.D. dissertation,

[21] G. N. Saridis and P. A. Fensel, "Stability considerations on the School of Elec. Eng., Purdue Univ., Lafayette, Ind., 1974.

Identification of Three-nDimensional ObjectsUsing Fourier Descriptors of the

Boundary CurveCHARLES W. RICHARD, JR., AND HOOSHANG HEMAMI, MEMBER, IEEE

Abstract-The feasibility of a method for the identification of a was first suggested by Cosgriff [2] and developed bythree-dimensional object from information contained in the boundary of Fritzsche [3], Raudseps [4], Borel [5], and Brill [6]. Moreits silhouettes is demonstrated. A silhouette is characterized by parametricrepresentation of its boundary curve in the complex plane. After normal- recent use of Fourier descriptors for the angle versus lengthization and transformation, a set of Fourier descriptors is derived for function is reported by Barrow and Popplestone [7] andevery silhouette. A minimum distance classifier uses the descriptors to Zahn and Roskies [8]. Features derived from the Fourieridentify the three-dimensional object and to estimate its position and coefficients of the boundary curve were used by Granlundattitude with respect to a known reference coordinate system. The method [9] for the recognition of hand printed characters. Co-was tested for identification of four aircraft representing complex and ordinates of the boundary points are directly used with anonconvex objects. Simulation results, quantitative and statistical, arepresented. nonlinear regression algorithm for template matching in

[10]. This technique has been extended to estimate theINTRODUCTION translation and rotation parameters of a three-dimensional

object from the boundary curve of the perspective projectionTHE METHOD of classifying planar shapes by select- on the image plane in [11]. Moment invariants developeding features based on the boundary of the shape has by Hu [12] have been applied to the boundary points to

been investigated by many researchers. Common to these identify a three-dimensional object and estimate its positionefforts is the intuitively compelling belief that much of the and orientation in space [13]. Sklansky and Davison [14]significant information required for recognition is contained use a Fourier expansion of a slope density function of thein the edges and, in particular, in theboundarycurve of an boundary curve in order to classify three-dimensionalisolated shape. A review of methods used for extracting objects. They describe a novel parallel machine for comput-features based on edges and contours is included in a ing the density of slopes of the silhouette boundary and,survey by Levine [1]. as in this paper, apply the method to classification ofThe advantage of using a boundary curve description is aircraft at a very limited number of orientations.

that features may be chosen that are independent of transla- Brill and Zahn use the angle versus length function totion, rotation, and the size of similar shapes. The use of the describe the plane boundary curve. Granlund characterizescoefficients in a Fourier series expansion of the tangent- th e boundarycurved a rametricrresetaion

angl vesusarc-engh dscritio ofa bonday crvethe boundary curve directly by a parametric representationangle versus arc-length description of a boundary curve z(t) in the complex plane. Features that are independent oftranslation, rotation, and dilation are derived from Fourier

Manuscript received April 4, 1973; revised January 18, 1974. This expansion of z(t). One advantage of using the coordinateswork was supported in part by the Air Force Office of Scientific z()nstaofaagetnle()rpeettin fthResearch, Grant No. AFOSR-71-2048. An early version of this paper zt nta fatnetage0t ersna1no hwas presented at the Computer Image Processing and Recognition boundary curve is that the complex coordinate functionSymposium on August 24-26, 1972, at the University of Missouri, z(t) - x(t) + iy(t) is less sensitive to the noise inherent

C. W. Richard, Jr., is with the Department of Mathematics, Air in a fuzzy boundary. The angle function is related to theForce Institutie of Technology, Wright-Pattersonl AFB, Ohio. derivative of the coordinate function; i.e. z'(t) =exp (16(t)),H.aHemamiiserwith,thC eprmetofEecrcabngnern,Ohio 420 where the curve is parameterized by its arc length. Small

372 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, JULY 1974

variations in the coordinate values of the boundary points induces a norm and metric on S:can result in large variations in the direction of the tangentvector. In this paper the complex representation of the iIZII = /(Z,Z) (3)boundary curve is used. A subset of the Fourier coefficients P(Z1,Z2) = iZ1-Z2 II. (4)is used directly as the set of descriptors in order to retainthe translation, rotation, and size information as well as The metric p(Zw,Z2) is the root-mean-square (RMS) valuethe shape information (identification). A Euclidian metric of oAZ(t)o,where aZ(t) = Zl(t)s- Z2(t).is introduced in the image plane to measure shape similarity. In order to characterize closed contours with the sameA one-dimensional Fourier expansion of the complex valued geometric shape but which may differ in position, orienta-boundary curve Z(t) is made, and the normalized coef- tion, and size, the shape preserving operations are listed inficients are the Fourier descriptors (FD's) of the shape. terms of the parametric representations.The metric is preserved in this transformation, and a a) Translation:minimum distance classifier is used in the space of Fourier Z'(t) = Z(t) + B, B complex. (5)coefficients. The magnitude spectrum of the FD's is in-variant with respect to translation, reflection, and rotation b) Rotation about the origin by real angle o:of the curve Z(t) and independent of the choice of initial Z'(t) = eia Z(t). (6)point Z(O). The rotation, reflection, and shift in initialpoint compared with a known reference can be recovered c) Change of scale:from the FD's in a simple manner and is a by-product of Z'(t) = KZ(t), K > 0. (7)the metric calculation. The method is applied to identifica- d) Reflection about the x-axis:tion of aircraft at any orientation in space from the outlineof its projection on the image plane by an optical system Z'(t) = Z(1 - t). (8)and is used to estimate the three translation and three rota- e) Shift in initial point:tion parameters of the aircraft. Use of a truncated set ofFD's (ideal low-pass filter) is shown to provide good Z'(t) = Z(t + to). (9)discrimination in the presence of additive Gaussian noise The minus sign appears in the reflection to preserve thefor this classification problem with more than 2600 classes. counterclockwise orientation of the contour.

The closed contour Z' will be said to be similar to theCHARACTERIZATION OF CLOSED CURVES closed contour Z in case the image of Z can be mapped

The set S of curves in the plane to be considered is the set into the image of Z' by a sequence of reflection, rotation,of piecewise smooth closed curves in the plane or the set of translation, and change of scale transformations. From

closed contours in the complex plane [15]. A closed contour these parametric representations we have the result that Z'Z is a continuous mapping from the unit interval onto the is similar to Z in case there are real positive parametersset of complex numbers with parametric representation to a, K, and a complex parameter B such that for all t Egiven by [0,1] either

Z'(t) = K exp (i2x)Z(t + to) + B (lOa)Z(t) = (x(t), y(t)), t E [0,1], Z(O) = Z(l) (1) or

where x and y are continuous real valued functions on the Z'(t) = K exp (ioc)Z(- t - to) + B. (lOb)interval [0,1]. The derivatives x', y' are piecewise continuous Since K = 0, the general linear transformation is invertible;on (0,1). that is if Z' is similar to Z, then Z is similar to Z'. ThisThe parameter t will be chosen to be proportional to arc eati is av'aid equivalence ren (reflexive, s Tic

length; i.e., the speed IdZ/dtl is a constant. The orientation .. ..of the curve will be taken to be counterclockwise; i.e., the and transitive) and may be used to partition S into equival-interior of the region enclosed by the contour will be to the ence classes. Each class would be represented by its proto-left as the curve is traversed for increasing values of the type, and the distance between any two classes would be

measured as the distance between the prototypes. TheparameterconsideredtoTetheomplexivaluedefun Z for t prototype is characterized by zero first moment (centroid atshall be considered to be the periodic extension of Z for t orgi)an.eodmmn fuiy(ntnr) hin the basic interval [O 1]. origin), and second moment of unity (unit norm). The

indthe nasi ite ]. . . angular orientation and starting point on the contour willbe arbitrary but fixed for each prototype. If S, c S is an(complex) scalar are defined in the usual way. An inner

product of two contours Z1, Z2 e S is introduced as equivalence class of similar contours, and Z1 E S, is chosenproduct Of two contours Zl, Z2 E- S iS introduced asQ Pfrn- ti .vuflIN nrml7 t nM *' ~~~~~~~asa reference, then Z1 would be normalized to form Z1*

(Z1,Z2) = Xf Z1(t)Z2(t) dt. (2) Z1*(t) = Z1(t)-1 (11)o ~~~~~~~~~~~~~~~~~~liz1- [til

Since every Z E S is continuous this Riemann integral wherealways exists, and it is clear that S is a subspace of the 1,= r Z,(t) dt (12)complex Hilbert space L2(0,1) [16]. This inner product JO

RICHARD AND HEMAMI: IDENTIFICATION OF THREE-DIMNSIONAL OBJECTS 373

is the centroid of the contour and USE OF FOURIER DESCRIPTORSFor any closed contour in S, the periodic function Z(t)

liZ-8iIill2 = J 1Z1(t) - t1i2 dt. (13) can be represented by its Fourier series00

MEASURE OF DISTANCE BETWEEN CONTOURS Z (t) = E Ck exp (i2rkt) (19)k==-oo

For the purpose of two-dimensional pattern recognition, wherewhere scale, rotation, or translation are not important, 1distance between contours Z1 and Z2 is defined such that Ck = A Z(t) exp (- i27tkt) dt. (19a)the effects of scale, rotation, and translation are eliminated. oTo measure the distance between two contours Z, and Z2, Writing the complex coefficient ck = ak exp (ick), the realthe contours are first normalized, and then one is reflected, numbers ak and 01k are the kth harmonic "amplitude" androtated, and the initial point shifted for a best fit in the mean- "phase angle," respectively. The set {ak} is called thesquare sense. The distance is defined to be amplitude spectrum and {Xk} the phase spectrum. The

d(Z1,Z2) = min (da,db) (14) piecewise smoothness of Z(t) is sufficient to insure that thewhere Fourier series converges absolutely at any point t and

converges uniformly on any closed interval. The amplitudeda = min IIZi*(t) - exp (ifl)Z2*(t + T)II (15a) spectrum is invariant with respect to translations, rotations,

# T reflections, or shift in starting point of the contour. More-db = min IIZi*(t) - exp (i/l)Z2*( t- )1f (15b) over, for continuous piecewise smooth functions, the ak

#,t,r decrease at a rate proportional to l/k2 so that most of theThe minimizations are taken over the ranges 0 . / < 27, shape information is contained in the low-order harmonics.

Some salient properties of the FD's are summarized in0 . ll < 1. From the triangle inequality for a general norm the following. Let C = {ck}, and C' = {ck'} be the Fourierit follows that 0 . d(Z1,Z2) . 2. Based on these definitions coefficients for the contours Z and Z', respectively.two contours are similar if and only if d(Z1,Z2) = 0. a) The inner product is preserved:

If da = 0, Z2 fits Z1 . If db = 0, Z2 fits the reflection of Z,.For three-dimensional identification, translation, rotation, (Z',Z) = (C'IC) (20)and relative scale of one contour with respect to another where(reference) become important. These parameters are

C

recoverable from the following relations: (C',C)= E Cke'Cka= /* (16a) b) The norm is preserved:

K=IIZ= -p11 (1 6b) I1Z112 = (Z,Z) = (C,C) = IC 112. (21)IIZ2 - 211 c) The average value or centroid of Z is given by the

B = p1 - K exp (io)p2 (16c) zeroth harmonic

where ,B* is the value of ,B in (15a) or (15b) that defines d in (22(14). Alternate expressions for da and db that eliminate the = Z(t) dt = co (22)minimization with respect to the rotation angle ,B may bederived. d) Let C* be the FD's for the normalized contour Z*.

F1 _________ From b) and c) we haveda2=2 [1 - max f Zl*(t)Z2*(t + T)dt] (17a) = 1 c0*=0 (23)

and

db = 2 - max Z1*(t)Z2*(t T ) dt|] (17b) Ck* = Ck/[(l EY C - ICoI2)] for k :A 0. (24)

where e) Let Z' be similar to Z. If there is no reflection

,B* = arg Z1*(t)Z2*(t + T*) dt (18a) ck' = K exp [i(oc + 27rkto)]ck + 6k,oB. (25a)

or If there is reflection

/* = arg [ Z1*(t)Z2*(-t - *) dt (18b) Ck' = Kexp [i(a + 2Jrkto)]ck + 3k.oB (25b)

° ~~~~~~~wherebk,0-= lifk = 0, and 3k,0 = 0,ifk#0 .where arg is the principal value of the angle of the complex f) Amplitude spectra for normalized similar contours arenumber. The equivalent formulas (17a) and (17b) express the same.the fact that our distance measure is just the Euclidian g) Distance measure in the Fourier space: the distancedistance at the point of maximum correlation. between two contours Z1 and Z2 can be measured in the

374 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, JULY 1974

transform space. From the distance relations (17a) and (17b) and the inner product is analogous to the continuous case.Entirely analogous results hold for SN as were developed

da2 221 - max |E CkCk* exp (-i2lrkT) (26a) for the continuous space S with integrals and infinite sumsdk=-L ]replaced by finite sums. The Fourier coefficients are now

given bydb2 = 2 L1 - max | E CkCk* exp (- i2rkT) (26b)db2=2[ t ~k=-oo N

T =-00 ~~~~~~~~~~~~~Ck=-E zj exp (-i27rkjIN). (27)The series in (26a) and (26b) are just the Fourier expansions N J= 1of the cross correlation functions in (17a) and (17b). The only approximation involved here is in representing aRelation (25a), which states how the Fourier coefficients nonpolygonal contour by a polygonal contour. The distancetransform under translation, rotation, shift in initial point, between two contours now isand change of scale, was noted by Granlund [9].

d(Z',Z) = min (da,db) (28)CLASSIFICATION OF CONTOURS AND POLYGONAL CLOSED where

CURVES

In applications to pattern recognition it is assumed that da2 = 2 1- max Y ck ck exp(-i27tkj/N) (29a)there is a finite set of known equivalent classes of contours, j k=-nJand the objective is not only to classify an unknown con- - ]tour in one of the classes but also to recover the orientation d = 2 k= -nleand size of the unknown contour. The identification is with n1 + n N. Here the summations in (29a)performed in the Fourier domain using the minimum aditneciein and (29b) are JUSt the discrete Fourier transforms of Udlstance crlterlon.

Let Zi* be the normalized prototype for class S i and V, respectively, where1 2 ,P, with some arbitrary but fixed initial point and Uk = Ck*'Ck' Vk = Ck*'c (30)angular orientation. Let Ci* ={Cki*} be the set of Fourier k

coefficients for Zi*. Each contour is filtered using an ideal The maximization is performed by taking the fast Fourierlow-pass filter before normalization; that is, a truncated transform of U and V and finding the jth component withseries is used to represent Zi*. The finite set of FD's {cki*, the largest magnitude.k = 0, ±1,+ 2, -.,+ m} is stored for each prototype ZiAi = 1,2,* ,P. An unknown contour Z with Fourier AIRCRAFT IDENTIFICATIONcoefficients {Ck} is classified in class Sj, in case d(Z,Zj*) < This method was applied to the problem of identifyingd(Z,Zi*), i = 1,2,- ,P, i + j. Equations (26a) and (26b) an aircraft and estimating its position and orientation inare used for the distance calculation with finite k. The size, space from its projection on an optical image plane. Air-rotation angle about the centroid, reflection, and centroid craft were selected because they represent complex and non-of the unknown contour Z, with respect to the closest convex solid objects. The body fixed coordinate system is theprototype Zj*, are estimated as before. usual one for aircraft with the origin at the CG, the positive

In this section the subset SN ' S of closed contours, x axis toward the nose, positive y axis along the starboardwhich is used in practical applications, is considered. wing, and positive z axis toward the bottom surface. TheEach Z in SN consists of N line segments continuously image plane is the XY plane of the reference system and isconnected to form a simple closed curve. A polygonal considered to be the ground plane, X north, Y east, andcontour Z in SN may arise as a first approximation to a positive Z down. The Euler angles for the aircraft are thecontour Z(t) formed by selecting N equidistant points on roll angle about the x axis, pitch angle of the xy plane withthe unit interval and connecting the points zj = Z(j/N) respect to the ground plane, and azimuth or yaw angle ofon the contour with line segments. Polygonal curves also the projection of the x axis in the ground plane. Four modelarise naturally by connecting the finite set of boundary aircraft, an F-4 Phantom, a Mirage IIIC, a MIG 21, andpoints of a simply connected digital picture with line an F-105 were considered. The solid objects-in this casesegments. It is assumed that each contour in SN consists aircraft-were represented by a wire frame model con-of exactly N line segments, where N is relatively large but structed by selecting nodes on the surface of a 1:72 scalefixed. In practice, N is chosen to be a power of two for model and connecting the nodes with line segments. For aefficiency in using a Cooley-Tukey algorithm [17] to com- given position and orientation in space the wire frame modelpute the FD's. In the case of digital pictures, points may be is projected onto the image plane, and the boundary pointsadded or deleted from the boundarysettogive Napprox- are traced using a numerical algorithm [11]. A typicalimately equidistant points. projection is shown in Fig. 1 with the boundary shown inA polygonal contour in SN can be represented as an Fig. 2. Points are added to the outline to give N= 512

ordered N-tuple over the field of complex numbers Z= boundary points. The Fourier coefficients are calculated,(z1,z2," ,zN), where the boldface is used to denote avector, and 39 low-order normalized components (m = 19) areThe point zj is the initial point of the jth line segment and stored for each aircraft at 50 increments in roll and pitchthe terminal point of the (j-l)th line segment. Now SN angles. The range of pitch angle is from -90° to 900, andis an N-dimensional vector space over the complex field, the range in roll angle is from 0 to 900. FD's for negative

RICHARD AND HEMAMI: IDENTIFICATION OF THREE-DMENSIONAL OBJECTS 375

PLRNE P B HT YRA PITCH ROLL 01E DIST PLANE R a HT TYA PITCH ROLL NO16E DISTF4 0.00 0.00 6900. 190.0 30.0 -50.0 0.00 F4 0.00 0.00 6900. 190.0 30.0 -50.0 0.00

Fig. 1. Projection of wire frame model of F-4. Fig. 2. Boundary curve of Fig. 1.

roll angles need not be stored since the outlines are the Small variations were chosen for X and Y in comparisonreflections of the corresponding positive roll angles. Because with the variations in height (or range) since it is assumedof the bilateral symmetry of an aircraft, a silhouette for a that the unknown silhouette is visible though not centeredpitch angle 4 + 1800 is identical to the silhouette for 4. within a relatively narrow viewing window. In contrast,This quantization in roll and pitch gives 666 reference the complete range of the attitude angles yaw, pitch, andcurves for each aircraft or a total of 2664 classes. roll were allowed. Roll and pitch angles were not restrictedAlthough 39 Fourier coefficients were stored for each to the multiples of 50 used for the reference set. Boundary

class, the simulation results that will be described indicate points were extracted from the projections of the wirethat most of the significant shape information is contained frame model of the input aircraft. Additional points werein less than 21 low-order components. The relatively large added to generate a total of 512 points on each boundary.number of points (N = 512) was chosen to approximate a To simulate the effect of detector noise, Gaussian noisecontinuous boundary curve in the image plane in order to with zero mean and different standard deviations wasreduce the quantization error and increase phase sensitivity. added independently to each of the 512 boundary poinits.The error in aligning the starting points resulting from two The standard deviation a of the noise was a given percentagequantizations of the same contour (or phase-shift error of R, where R is the distance from the centroid to thein the Fourier domain) is proportional to 1/N. This clas- farthest point on the boundary. One hundred silhouettessification problem, with 2664 classes using smaller values were generated for each of three cases, a = 0, a 0.1 R,of N (N = 256, 128), gave variations in the distance and a = 0.2R. These 300 silhouettes were subjected to themeasure that approached the distance between neighboring identification method where the type of the plane as well asclasses in the reference set. its parameters was estimated as will be described.

A minimum distance classifier over the 2664 classes in theRESULTS OF SIMULATION reference set was used for identification and estimation of

In order to test this approach in identification of type and roll and pitch. The other four parameters X, Y, height, andestimation of parameters of the aircraft, two experiments yaw were estimated from the centroid, norm, and rotationwere conducted. In the first experiment, a total of 300 angle of the noisy boundary curve. The simpler distaiicesilhouette boundaries were generated as input data with measure D(Z',Z) given byknown true parameters of position and attitude of the m

aircraft. Noise was added to the boundaries and the noisy D2(Z,Z) = E ICk* -_ Cki*l2 (33)boundary was compared with the reference data to identify k=-m

the aircraft type and estimate its parameters. For comparison, was used, where cki* are the normalized FD's for themean absolute errors in each of the parameters of position reference contour Zi, i = 1,2,- * ,2664, and Ck* are the(X, Y, height) and attitude (yaw, pitch, and roll) were FD's for the unknown contour Z. This distance measurecomputed. In the second experiment, one of the four air- eliminates the two Fourier transforms in (29) required forcraft was deleted from the reference set. The deleted aircraft each class and decreases the computation time by a factorwas then used to generate input data. The identification of 15. The true distance measure d is calculated just onceroutine was repeated for 100 samples of the unknown with the Zi that gave minimum D. A typical best fit isaircraft. The following describes the two experiments in shown in Fig. 3. The scattered points in Fig. 3 were pro-more detail. duced as follows. An input aircraft was produced (Fig. 2).The silhouette of an unknown aircraft was simulated by This input aircraft has the following parameters:

choosing one of the four models at random and selectingrandom values for the six parameters uniformly distributed Type X Y Ht Yaw Pitch Rollover the ranges

F-4 0.00 0.00 6900.0 190.0° 30.0 --50.0°CG: -25.< X, Y< 25 ft, 2500.< height . lOGOO0ft

(31) Points on the boundary of Fig. 2 were selected and randlom0 < yaw <30°-90° < itch < 90°noise (zero mean and standard deviation af = 0.1R) was0~~~~~~~~.ya'6~90 ic 0 added to the coordinates of each point to simulate measurie-

-90° . roll . 90°. (32) ment noise. The result is the set of scattered points in Fig. -B.

376 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, JULY 1974

The latter set of points was treated as input to the systemwhere a best match was made to the stored reference set.The solid contour in Fig. 3 is the closest reference with thefollowing estimated parameters:

Type X Y Ht Yaw Pitch Roll %j: * .

F-4 1.93 0.30 6696.0 189.70 30.00 -50.0°

The results of this experiment are tabulated in Table I.Number of errors in the type of aircraft identification andthe mean absolute error in each of the six parameters of

PLANE A 8 HI YAN PITCH ROLL NOISE 01STposition (in feet) and attitude (in degrees) are tabulated a, f4 0.00 0.00 6900. 190.0 30.0 -50.0 0.10for three cases of zero noise, and a = 0.1R and 0.2R, F4 1.93 0.30 6696. 189-7 30.0 -S0.0 0.0691respectively. As mentioned before, there are 100 samples Fig. 3. Best fit to noisy boundary.randomly selected for all three cases. Also, the averageminimum distance d in each of three cases, based on the 100 TABLE Isamples, is recorded. The small error in X and Y (the NUMBER OF IDENTIFICATION ERRORS, AVERAGE PARAMETER ERROR,distance from the optical axis) compared with the error in AND MINIMuM DISTANCE FOR 100 SAMPLESaheight (or range) merely reflects the difference in the max- 0 Percent 10 Percent 20 Percentimum variation allowed for these parameters. In simulating Noise Noise Noisean unknown silhouette, X and Y were chosen at random ID errorg 0 1 8in the interval (-25, 25 ft), while the height was chosen at X error (ft) 1.9 1.9 4.1

Yerror 1.3 1.3 2.0random in the interval (2500, 10000 ft). The results of Ht error 75 204 700this first experiment as tabulated demonstrate the feasibility Yaw error (0) 1.7 2.1 3.7

Pitch error 1.7 4.8 10.9of using Fourier coefficients as features for the present Roll error 2.5 3.3 9.6identification and parameter estimation problem. Average minimumTo illustrate the ability to reject an object not in the distance d = 0.031 d = 0.086 d = 0.175

reference set, a second experiment was performed. In this a Sample aircraft in reference set.experiment, the 666 sets of FD's for the F-105 were deletedfrom the reference set. One hundred samples of the F-105 TABLE IIwith a = 0.1R were generated and treated as unknown NUMBER OF IDENTIFICATION ERRORS, AVERAGE PARAMETER ERROR,inputs. The minimum d over the set of 100 was sufficiently AND MINIMUM DISTANCE FOR 100 SAMPLESbgreater than the maximum d in the previous experiment so 10 Percent Noisethat a threshold could be set to signal that the aircraftwas not in the reference set. The six parameters were still ID errors 100

X error (ft) 3.9estimated with an overall average error of 14 percent. Y error 3.9These results are given in Table II. Ht error 1591

Yaw error (0) 11.6Several more experiments were conducted in order to Pitch error 28.0study the effect of bandwidth and angle quantization, Roll error 30.5study theeffecAverage minimumand also the effect of selection of the distance measure D, distance d = 0.203which uses only the magnitude of the FD's instead of the b Sample aircraft not in reference set.true distance measure d where the complex values of theFD's are utilized.

In Table III the effectiveness of D (33) versus d (28), TABLE III(29a), and (29b) as a measure of distance is empirically COMPARISON OF D AND d AS DISTANCE MEASURESCstudied through the average errors in recognition and the 0 Percent Noise 10 Percent Noiseaverage absolute error in parameters for 100 samples. Forthe zero noise case, there is no identification error, while D d D dfor 10 percent noise there are two identification errors. The ID errors 0 0 2 0errors in the parameters can be observed from this table Xerrors (ft) 1.9 1.8 1.9 1.9

Yerror (ft) 1.3 1.3 1.3 1.3to be relatively comparable under zero noise and 10 percent Ht error (ft) 75 71 204 194noise. Yaw error (0) 1.7 0.9 2.1 1.5* ~~~~~~~~~~~~~Pitcherror (0) 1.7 1.6 4.8 2.0Table IV shows the effect of decreasing the number of Roll error (0) 2.5 0.9 3.3 2.1FD's used from 39 to 21. As in Table I, the D measure was Aveiratge minimum .3 002008001used for classification with prototypes stored at 5° in-crements in pitch and roll. The results were essentially the c 50 increment, bandwidth = 39.same with identical identification errors and less than2 percent average difference in the six parameters. Based on

RICHARD AND HEMAMI: IDENTIFICATION OF THREE-DIMENSIONAL OBJECTS

these empirical results, the 21 low-order Fourier coefficientsseem adequate for identification and parameter estimation.Comparison of Table IV with Table V shows the effect

TABLE IV of increasing to 100 the pitch and roll increment at whichEFFECT OF BANDWIDTH ON IDEN IFICATION the prototypes are stored. This of course decreases the

number of prototypes in the reference set by a factor ofa = 0.0 v= O.1R four. The D measure was used with a bandwidth of 21.

ID errors 0 2 As expected, the average error in pitch and roll approx-X error 1.9 1.9 imately doubled. The aircraft identification errors increasedYerror 1.3 1.3Ht error 77 208 from two to five at the 10 percent noise level, and theYaw error 1.8 2.1 average errors in estimated yaw angle and height alsoPitch error 1.7 5.0Roll error 2.5 3.5 doubled.Average minimum Comparison of Tables V and VII shows the effect of a

distance 0.029 0.069 further decrease in bandwidth with prototypes stored at

d D measure, 50 increment, bandwidth = 21. 100 increments. Although the average errors in the positionand attitude parameters are still nominal, the increase inidentification errors from 0 to 3 at the zero noise level

EFFECT OF QUANTIZATION ON IDENTIFICATION indicates that 11 FD's are not sufficient for the identificationAND PARAMETER ESTIMATION WITH MEASURE De purpose. Also, comparison of Tables V and VI gives some

a = 0.0 a - 0.1R further empirical evidence of the loss in identification andparameter estimation due to use of D instead of d at a

ID errors 0 5 different bandwidth. In the latter case, five identificationXerror 1.9 2.0Yerror 1.5 1.5 errors are significant in Table V compared with no identifi-Ht error 189 276 cation errors in Table VI for the 10 percent noise case.Yaw error 3.8 4.7Pitch error 8.4 11.2 The errors in parameters are about 20 percent less in TableRoll error 6.6 7.6 VI.Average minimum

distance 0.056 0.090 A minimal test with real data was made with two 35 mm- 0' increment, bandwidth = 21.

slides of an F-4. The optical images were digitized using aflying spot scanner over an 80 by 80 mesh size. The bound-ary was extracted using gradient techniques. Points were

TABLE VI added to give 512 points, and the resultant contour usedEFFECT OF QUANTIZATION ON IDENTIFICATION as the unknown. One of the photographs is shown in Fig. 4AND PARAMETER EsTIMATION WITH MEASURE d'

and the resultant best fit in Fig. 5. The best fit in the Fouriera = 0.0 a =.1R domain for the contours in Fig. 5 is given in Fig. 6. The

ID errors 0 0 circled points give the amplitude spectrum for the unknownXerror 1.9 1.9 contour, and the solid lines are for the best fitting referenceY error 1.4 1.4 contour. A two-step classification procedure was used forHt error 151 230Yaw error 3.1 2.8 the real data. The five closest prototypes for each of thePitch error 8.6 8.9 four aircraft types were first found using the distanceRoll error 5.0 4.0Average minimum measure D. The unknown was then classified using the

distance 0.052 0.083 minimum d measure over the reduced set of 20 references.

10° increment, bandwidth = 21. The actual parameters were unknown, but the estimatedyaw, pitch, and roll angles appear reasonable comparedwith visual inspection of the photographs.

TABLEVII__________________ ____

EFFECT OF FURTHER BANDWIDTH REDUCTION ONIDENTIFICATION AND PARAMETER ESTIMATIONg Type X Y Ht Yaw Pitch Roll

F-4 1.1 1.8 6917 190.00 30.00 -50.0°a =0.0 a=0O.1R -

ID errors 3 6 CONCLUSIONSX error 1.9 2.1Yerror 1.4 1.5 The feasibility of using truncated Fourier descriptors ofHt error 206 302Yawerror 5.9 39 1 image boundaries for identification and estimation of three-Pitch error 10.3 12.0 dimensional solid objects, even under moderate amounts ofRoll error 8.9 11.1 noise, has been established. Specific results show that withAverage minimum

distance 0.053 0.077 a sufficient number of Fourier coefficients this identificationD mesur, 10icreent,banwidh =11.and parameter estimation scheme could be carried out

with a simpler distance measure with no appreciable effecton parameters estimates but with two to five percentidentification error. The true distance measure requires

378 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNEIICS, JULY 1974

two Fourier transforms for each comparison with a storedprototype. It involves a fifteen-fold increase in serialprocessing time.

Further work includes optimization of the number ofFourier descriptors, enlarging the classes of solid objects,using correlated noise instead of independent noise, andfinally real time computational, storage, and speed require-ments. Also, the problem of treating degraded imageswhere part of the image may be missirg due to clouds,etc., is of practical importance and should be furtherinvestigated.

Finally, numerous other applications in traffic flowanalysis, parcel post handling and analysis, cell identifica-tion in medicine, and automated assembly lines in industrycan be cited where this approach can be of potential use.This is particularly true when the number of parameters isless than six, and when the solid objects are less complicated.

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