I I I. RELATIONSHIP TO POLYNOMIAL CLASSIFIERS

Motivated by our encouraging experimental results, we tried to understand the contrast measure in a more rigorous manner. Sur- prisingly, this contrast measure is strongly related to polynomial- based classification, when both used in the context of image seg- mentation. In fact, this result i s relevant not only to establish the connection between the two methods, but also to simplify the soh- ing methodology of key inspection problems. The contrast mea- sure can be considered a particular case of polynomial classifi- cation, where only two features are used. In particular, this relationship may be used to simplify the solution of the solder ball inspection problem presented in [I].

The binaryimageB(m,n)containstwoclasses, C, and C2,defined as the sets of pixels

C, = {(m, n):G(m, n) > T }

C, = {(m, n):G(m, n) 5 T } .

Since

the above conditions mean that the expression p(m, n) - ~ d m , n) - T E is checked for positive and negative or zero values. This is a linear expression in thetwoimagefeaturesp(m, n)and u(m, n), with parameters T and T E . For the theory of polynomial classifiers, a fea- ture vector ( f,, - * , fJ is mapped into a decision space with the family of polynomial mappings Pl( fl, , fJ, * . . , P,( fl, . . . , fs), whichcisthenumberofexpectedclasses[2],[7l. It has beenshown that this process can be implemented in parallel architectures [I]. A decision on every feature vector is made by assigning the class i such that

P,( f,, * . , fJ = max {P,4 f,, . 1 , fJ}. l S , S C

The training phase for polynomial classification leads to "optimal" choices for thecoefficients of the polynomial mappings. In thetwo- class problem, the polynomial coefficients satisfy P, + Pz = 1 for all feature vectors. This means that the maximum-likelihood deci- sion function max (fl, P2) yields classes C,, C, given by {( f,, . . . , fJ: Pl( f,, * - . , fJ > i} and {( f,, * , 0: fl( f,, . . . , fs) s i}, respectively.

By adjusting the independent term of our decision function, we get P1(p , o) = p - TU - TE + as the linear polynomial associated to the image feature vector ( p , U). Our image segmentation tech- nique can therefore be cast in terms of polynomial-based decision theory. In our experiments, we have chosen E and 7 in a heuristic manner.

An important consequenceof the relationship established in this letter is that the parameters E and 7, which were determined by trial and error,can now becomputed usingsomeerrorcriteriaand deci- sion-theoretic techniques. Therefore, it would be interesting to comparethevaluesobtained foreand zin the previous section with those arising from minimum misclassification or other optimiza- tion functions.

IV. CONCLUSION

In this letter, a simple contrast measure was used for image seg- mentation. The method yields very good results when applied to an important and difficult machine vision problem arising in inte- grated circuit inspection. Motivated by these results, a study was conducted todeterminethe relationshipof this method with other decision-theoretic segmentation techniques. It was found that the contrast measure segmentation is a particular case of polynomial- based pixel classification. This relationship enables the application of nonheuristic estimation methods for computing the parameters E and T . These parameters govern the performance of contrast mea- sure-based image binarization.

REFERENCES

[I] W. E. Blanz, J. Sanz, and E. Hinkle, "Image analysis methods for solder ball visual inspection in integrated circuit manu- facturing," Res. Rep. RJ-5163, IBM Almaden Research Center,

San Jose, CA, 1986, also to appear in IEEE]. Robotics Automat., vol. 4, no. 1, 1988.

[2] W. E. Blanz, "Feature selection and polynomial classifiers for industrial decision analysis," Res. Rep. RJ-5242, IBM Almaden Research Center, San Jose, CA, 1986.

[3] K. Fu, J. Mu, "A survey on image segmentation," Pattern Recogn., vol. 13, pp. 3-16,1981.

[4] E. B. Hinkle and J. L. C. Sanz, "Fast image segmentation for some machinevision applications," presented at Int. Conf. on Acoustic, Speech and Signal Processing, Dallas, TX, Apr. 1987.

[5] W. Pratt, Digitallmage Processing. New York, NY Wiley, 1978. [6] J. Sanz and A. K. Jain, "Machine vision methods for inspection

of printed wiring boards and thick-film circuits,"]. Opt. Soc. America, Sept. 1986.

[q J. Schuermann, Polynomklassifikatoren fuer die Zeichener- kennung. Munich, FRC: R. Oldenbourg, 1977.

[8] J. Serra, Image Analysis and Mathematical Morphology. Lon- don, UK Academic Press, 1982.

A Hybrid E-Pulselleast Squares Technique for Natural Resonance Extraction

E. 1. ROTHWELL AND K. M. CHEN

A new technique to extract the resonant frequencies of a radar target is presented. The scheme is completely automated, with only the number of natural modes expected and the beginning of late- time as inputs. Results using experimental data demonstrate the insensitivity o f the method to random noise, and to estimates of modal content. Further, the technique is computationally efficient, taking only a few minutes to execute on a PC.

I. INTRODUCTION

Recent interest in using natural resonances in thediscrimination of radar targets has prompted the introduction of several new schemes for extracting natural resonance frequencies from a mea- sured transient response [I]-[5]. None of these has proven com- pletely adequate. A truly useful numerical technique should have the following characteristics: I) computational efficiency: can run on a microcomputer or work station in a short amount of time; 2) automatic operation: no need for user intervention during long iterative procedures, and no initial guesses required; 3) insensi- tivity to random noise and to estimates of the number of modes present.

This letter introduces a technique which will address all three requirements. The recently developed E-pulse scheme [I] was cho- sen as a starting point since it has been shown to meet the third provision.

I!. THEORY

radar target can be represented as a sum of natural modes Assume that the late-time measured response of a conducting

N

m(t) = 2 aneon' cos (ant + pn), TL c t c T, (1)

wheres, = U, + j q , is the natural frequencyof thenth target mode, anand 9,aretheamplitudeand phaseof thenth mode, TLdescribes the beginning of the late-time period, T, describes the end of the measurement window, and the number of modes in the response, N, is determined by the finite frequency content of the waveform exciting the target.

An E-pulse, e(t), i s defined as a waveform of finite duration T,

n = l

Manuscript received August 13, 1987. This work was supported by the

The authors are with the Department of Electrical Engineering, Michigan

IEEE Log Number 8717956.

Office of Naval Research under Contract N00014-67-K-0336.

State University, East Lansing, M I 48824, USA.

296

0018-9219/88/0300-0296$01.00 0 1988 IEEE

PROCEEDINGS OF THE IEEE, VOL. 76, NO. 3, MARCH 1988

which satisfies [6]

c(t) = e@)* m(t) = e(t') m(t - t') dt' = 0,

(2)

If this integral equation can be solved for the unknown €-pulse waveform, then the complex natural frequencies contained in m(t) are the solutions {sn} to €(s) = 0, where €(s) i s the Laplace trans- form of e(t) [6].

A solution to (2) can be obtained by using the method of moments. Expanding

Iore t > TL + T,.

(3)

where { f k } i s an appropriate set of basis functions, substituting into(2),and taking inner productswith asetofweightingfunctions {w,,,} gives

m = 1,2,3, , M. (4)

In the standard €-pulse technique, the selection of M = K = 2N, resulting in a "natural" €-pulse, makes (4) a homogeneous matrix equation. Solutions for { (Yk} thus exist only for certain values of T, which cause the matrix to be singular. An alternative approach is to choose M = 2N, K = 2N + 1, resulting in a "forced" €-pulse. Then (4) becomes an inhomogeneous matrix equation, with solu- tions corresponding to any choice of Te which does not cause the matrix to be singular.

Thenatural frequencies in m(t)can be found mosteasily by using subsectional basis functions of width A in (3), since €(s) = 0 then reduces to the polynomial equation [I]

where Z = exp (-SA). To maximize computational efficiency, rectangular basis func-

tionsareused in (3)whileimpulsefunctionsare used forweighting (point matching). The integration on t in (4) is then trivial, while the integration on t' i s done using the trapezoidal rule.

I l l . DISCUSSION

In theory, these techniques should work for any choice of T,. However, there are practical limitations on the range of T,. It i s boundedon thelowerend byatime Todetermined bythesampling interval used to measure m(t), and on the upper end by T, - TL from the limits of integration in (4). If natural €-pulses are used, then a solution for T, might not exist inside this range of limits. Also, extraneous roots are possible, but there is no provision for describing the "quality" of a solution. Similarly, if forced €-pulses are used, either noise, or a poor estimate of N, or approximations used in calculating the integrals in (4), may result in one choice of T, being "better" than another.

The most computationally efficient approach is to use forced €- pulses along with some method to determine the "best" T,. An easily implemented scheme is to define the best T, as that which minimizes the squared error

E = IImW - h(t)112 = c [m(tJ - h(ti)12 (6) , where h(t) is the reconstructed waveform

N

h(t) = C ine"' cos ( G ~ t + +,) (7)

and the sum is over sampled values between TL and T,. Here {in = 6n + j&} are the solutions to (5), and {&, in} minimize E with T, and {in} fixed.

n = l

IV. EXPERIMENTAL RESULTS

The pulse response of a thin wire circular loop has been pub- lished previously [I, fig. 7. To test the sensitivity of the technique

0 f ./ C .-

-1

a + a+ a+ 0 0

0 THEORY 2 MODES EXTRACTED @

X 4 MODES EXTRACTED + 7 MODES EXTRACTED

LT * @

-4 -3 -2 - 1 Damping coeff icient in G-Np/s

Fig. 1. Resonant frequencies of thin wire circular loop extracted from its transient pulse response, with various numbers of modes assumed present.

to the number of modes assumed present, the natural frequencies of the loop are extracted and compared in Fig. 1 to theory [7l for N = 2'4, and 7. Here TL = 3 and T, = 9 ns have been used. Results are seen to compare quite well even for N as small as 2. Fig. 2 shows c plotted versus T, for N = 7. The global minimum is searched for

0 i 0 1 2 3 4 5 6

E-pulse duration in nonosec

Fig. 2. Squared error versus T, for circular loop, with seven modes assumed present.

measured data best f i t _ _ _ _ _

' 3 Time in nonosec

Fig. 3. Best fit waveform constructed from extracted natural fre- quencies of circular loop, with seven modes assumed present.

PROCEEDINGS OF THE IEEE, VOL. 76, NO. 3, MARCH 1988 297

automatically between To = 0.5 and J, - JL = 6 ns, and located at J, = 3.417s. Notethat reasonable resultsareexpectedoverafairly wide range of J, where E i s small. The reconstructed waveform (7) is shown in Fig. 3 and is seen to faithfully reproduce the measured data.

To test the sensitivity of the scheme to random noise, the pulse response iscontaminated bywhiteGaussian noise, with zero mean and standard deviation 3,5, and 10 percent of the waveform max- imum (15,13, and 10 dB YN). The frequencies extracted from the noisy waveform are shown in Fig. 4. Obviously, the presence of even 10 dB of Gaussian white noise has little effect on the results.

” - W .

0

0 + O X

0% 0. a

@

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 4. Resonant frequencies of thin wire circular loop, extracted with various amounts of white Gaussian noise added, and seven modes assumed present.

Extracting seven modes from the above data took about 5 min on an IBM PC-XT microcomputer. Execution time depends on the number of data points used in (4) and (6) (1024 and 150 points, respectively, for the above data) and the number of iterations needed to locate the optimum T,.

REFERENCES

r11

r21

[31

[41

r51

[61

r7

E. J. Rothwell, K. M. Chen, and D. P. Nyquist, “Extraction of the natural frequencies of a radar target from a measured response using E-pulse techniques,” /FE€ Trans. Antennas fropagat., vol. AP-35, pp. 715-720, June 1987. C. Chuang and D. Moffatt, “Natural resonances of radar tar- getsvia Prony’s method and target discrimination,” /€€€ Trans. Aerosp. Electron. Syst., vol. AES-12, pp. 583-589, Sept. 1976. A. G. Ramm, “Extraction of resonances from transient fields,” /€E€ Trans. Antennas fropagat., vol. AP-33, pp. 223-226, Feb. 1985. A. J. Mackay and A. McCowen, “An improved pencil-of-func- tions method and comparison with traditional methods of pole extraction,” /€FE Trans. Antennas fropagat., vol. AP-35, pp. 435- 441, Apr. 1987. B. Drachman and E. Rothwell, “A continuation method for identification of the natural frequencies of an object using a measured response,” I€€€ Trans. Antennas fropagat., vol. AP- 33, pp. 445-450, Apr. 1985. E. J. Rothwell, K. M. Chen, D. P. Nyquist, and W. Sun, “Fre- quency domain E-pulse synthesis and target discrimination,” IEEE Trans. Antennas fropagat., vol. AP-35, pp. 426-434, Apr.

On the Joint Statistics of Amplitude and Phase of a Signal With Co-Channel Interference

B. C . SARKAR

The joint pdf of the resultant amplitude and phase of a signal in presence of co-channel interference has been given. The marginal pdf of the amplitude variable has also been evaluated.

INTRODUCTION

A common experience of a communication receiver designer is that the received signal contains, besides the wanted signal and theadditivewhitechannel noise, signalswhich aremeant for other users of the same part of the electromagnetic spectrum [I]. These signals, called interfering signals, may be of equal frequencyto that of the wanted signal, representing the worst case of interference [2], and the relative phase part of these signals randomly fluctuates about thewanted signal phase. In such a situation the receiver per- formance degrades which can be measured in terms of the relative strength of the interfering and the wanted signals. In this letter we have presented a joint probability density function (pdf) of the amplitude and phase of the signal which is the resultant of the wanted signal and a tone co-channel (having same frequency as that of the signal) interferer. This joint pdf can be used to evaluate the receiver performance in presence of interfering signals.

ANALYSIS

Consider that a single co-channel interfering signal is present with the wanted signal, A sin wt, and the ratio of the power of the interfering signal to that of the wanted signal is a’. Then in the noise-free condition one gets received signal r ( t ) as

r ( t ) = A sin wt + aA sin (wt + O ( t ) ) (1 )

where O(t) i s the relative phase of the interference to that of the wanted signal. 8 ( t ) isa random variable uniformlydistributedwithin the range -u to u and so the pdf of O ( t ) i s

1 2u p(0) = -, -u 5 0 5 a.

The resultant received signal can be written as

r ( t ) = a(t) sin (wt + 9(t)) (3)

where

a(t) = A[(I + a cos 0)’ + (a sin O)’]”’ (4a)

and

a sin 0 1 + acoso’ p(t) = arc tan (4b)

An examination of (4a) and (4b) reveals that the amplitude variable a(t) and the phase variable p(t) lie between A(1 - a) toA(1 + a) and arc tan (-a(l - a’)-“) to arc tan (a(1 - CY’)’/’), respectively. Now to the receiver, r ( t ) i s a signal whose amplitude and phase are ran- domly modulated. In this situation the average receiver perfor- mance can be obtained if the joint pdf of a( t ) and (p(t) variables is known. To get the required joint pdf p(a, (p), we start with two variables x‘ and y’, defined as x’ = cos 0 and y’ = sin 0; here 0 is a random variable uniformly distributed within the range -u to a, having the pdf given in (2). Hence random variables x’ and y’ are uncorrelated but not statistically independent. Now observe that, for a particular 0, x‘ and y’ variables lie on the circumference of a circle on the x‘y’-plane whose radius is unity and the center i s the point having coordinates (0,O). So we can take the joint pdf of x’

.~ . .

Manuscript received June 19,1987; revised August 11,1987. The author i s with the Radionics Laboratory, Physics Department, Burd-

1987. E. J. Rothwell and N. Gharsallah, “Determination of the natural frequencies of a thin wire elliptical loop,”lEEE Trans. Antennas wan University, Burdwan-713 104, India. frobagat., vol. AP-35, pp. 1319-1324, Nov. 1987. IEEE Log Number 8717322.

0018-9219/88/0300-0298$01.00 0 1988 IEEE

298 PROCEEDINGS OF THE IEEE, VOL. 76, NO. 3, MARCH 1988