Postprocessing of correlation for orientation estimation

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1 Postprocessing of correlation for orientation estimation Laurence G. Hassebrook, MEMBER SPIE Michael E. Lhamon, MEMBER SPIE Mao Wang Jyoti P. Chatterjee University of Kentucky Department of Electrical Engineering 453 AH Lexington, Kentucky Abstract. We present a high-snr approach to estimating the orientation of istorte target objects, where the istortion is out-of-plane rotation. The presence an position of the targets are etecte with a bank of istortion-invariant correlation filters. The filter set we use, known as hybri composite filters, yiels complex responses at the target locations. The peak magnitue responses inicate target locations. The correlation phase angles, at the target locations, are linearly combine into complex signatures unique to a particular orientation. A maximumlikelihoo M-ary classification algorithm is use to etermine the most likely orientation from a finite number of orientations. In practice, we o not use all the hybri filters an the ones we o use are optimally selecte an combine for etection an iscrimination. Given the target location, aitional filtering can be accomplishe with an inner-prouct operation rather than correlation. A secon set of filters, optimize for angle estimation, are applie as inner proucts at the target locations to construct the orientation signatures. A rigorous mathematical treatment of the optimum signature etection architecture is presente an numerical simulations emonstrate the capabilities of this approach Society of Photo-Optical Instrumentation Engineers. [S (97) ] Subject terms: correlation pattern recognition; istortion-invariant; correlation; filter banks; parameter estimation. Paper CPR-01 receive Feb. 11, 1997; revise manuscript receive May 17, 1997; accepte for publication May 3, This paper is a revision of a paper presente at the SPIE conference on Optical Pattern Recognition VI, April 1995, Orlano, Floria. The paper presente there appears (unreferee) in SPIE Proceeings Vol Introuction We present a metho of estimating the istortion parameter of an arbitrary input target from the complex correlation responses of a filter bank. The istortion parameter variation is represente by training set images, which in this case are istorte by out-of-plane rotation. It is assume that the target class images resie in the same scene as the nontarget, or clutter, class images. We are not intereste in the clutter class images so it is necessary to etect an locate the target class images while simultaneously iscriminating clutter class objects. This etection an iscrimination is performe in the presence of aitive white Gaussian noise AWGN, which is assume to be sensor noise. Once the target is etecte an locate we elementwise multiply an integrate i.e., the correlation is evaluate at one point by a vector inner-prouct IP result a series of complex filter templates registere to the etecte target image. For each filter IP, we have a single complex value, an these values are groupe in a vector representing a signature of the input image. The signature vector is then matche with possible response signature vectors an the IP match with the largest real component value inicates which istortion parameter, in this case a rotation angle, is most likely. We limit our attention to istortion parameter estimation using moel-base optical correlation filter esigns. 1 Both parameter estimation an istortion-invariant object etection literature inclue hunres of sources. Algorithms combine to form both object etection an orientation are less common an can be groupe into one of two categories, feature-base an correlation-base algorithms. The feature-base techniques reuce the problem imensionality by operating on specific features. Examples of this inclue the etermination an use of image bounaries 4 or special transformations. 5 Feature-base approaches can reuce the numerical buren of etection an estimation but usually require sophisticate algorithms, which are best implemente on general purpose computers. Correlationbase algorithms represent a simple approach 6 to etection an orientation estimation. However, correlation requires many low-level operations, which are best implemente on special purpose igital an optical processors. We limit out attention to istortion parameter estimation using moelbase correlation filter esigns. 1 We also limit our estimation to a finite number of rotations as represente by a training set. One of the most basic correlation approaches 6 to etection an parameter estimation is to correlate the input image with all the training images in the form of matche filters. The problem with the matche filter approach is that it requires all training images to be correlate. By using complex istortion-invariant filters, the number of correlations is successively increase only until a esire performance measure is achieve rather than requiring all filters to be use. 710 Opt. Eng. 36(10) (October 1997) /97/$ Society of Photo-Optical Instrumentation Engineers

2 Fig. 1 HC filter esign equation. There are many istortion-invariant correlation filter esigns 1 available for etection an location of input targets. We use hybri composite HC filters 7 because these filters are representative of various other filter esign types such as synthetic iscriminant function 8 SDF, minimum average correlation energy 9 MACE, an linear phase Coefficient composite 10 LPCC. We give backgroun on HC filters in Sec.. The parameter estimation theory is presente in Sec. 3. Optimal IP filter selection criteria are presente in Sec. 4. Results are given in Sec. 5 an conclusions are given in Sec. 6. HC Filter Backgroun We etect, iscriminate, an estimate rotation angle with a istortion-invariant correlation filter family known as HC filters. 7 In general, any metho of etecting an iscriminating the target will function, but to successively estimate the rotation angle we nee an algorithm that uses the presence of the target to extract rotation information irrespective of rotation. We chose HC filter esigns for this problem because their peak magnitue responses are istortion invariant, while the associate phase responses are sensitive to in-plane an/or out-of-plane rotation. Furthermore, since HC filters are a unification of classical SDF, MACE, an LPCC filter esigns, our analysis is vali for these filter types as well. Figure 1 shows a iagram of the HC filter family matrix structure. The filter matrix contains, as its columns, N filter images, in lexicographic vector form. Each filter image contains pixels. The HC matrix structure is centere aroun the target training set matrix, which has as its columns the training images in lexicographic form. Preceing the training set matrix is the preprocessing matrix. The function of the preprocessing matrix is to moify the -D spectrum of the training set an thereby moify the intensity shape of the correlation plane an simultaneously moify the output noise response. For example, classical SDF filter esign has a preprocessing matrix equivalent to an ientity matrix, while MACE filter esign has a preprocessing matrix that is approximately the training set spectrum. The SDF preprocessing matrix has no effect on the output correlation plane, while the MACE preprocessing matrix acts to prewhiten the input images, thereby making the output correlation shape be a sharp spike with suppresse sielobes. The trae-off for the MACE preprocessing is high-frequency noise sensitivity. The HC filter preprocessing matrix allows for a transitional variation between the classical SDF an the MACE filter preprocessing matrix. Immeiately following the training set matrix, in Fig. 1, is the smoothing matrix. The function of this matrix is to linearly weight the training set as to achieve a specific response at the origin of the correlation plane given a specific input training image. Typically the more constraine the origin output response is, the less generalizable the filter is to images not in the training set. On the other han, not enough constraint allows the output response to vary below the sielobe response, thereby loosing both etection an iscrimination ability. For example, the smoothing matrix acts to constrain the output origin responses to constant values for MACE an SDF esigns. The result is loss in generalization an iscrimination in both esigns. On the other han, the smoothing matrix is an ientity matrix for LPCC filter esign an imposes no constraint on the origin response. The result is improve, an more robust, etection, iscrimination, an generalization performance for specific types of istortion, such as in-plane rotation. This improve performance, however, egraes graually as the istortion becomes more generalize as in the case of outof-plane rotation. By constraining the output responses using a smoothing matrix in transition between no constraints, as in LPCC filter esign, an SDF/MACE filter constraints, the etection, iscrimination, an generalization performance improves for all three filter types. The last matrix in Fig. 1 is the eigenvector matrix. Its function is to expan what woul be a single istortioninvariant filter into a family of filters. A family of filters allows the filter esigner to isolate filtering properties, such as SNR an iscrimination ability, better than if a single filter is use. This result becomes more pronounce when a filter bank of the most esire filters are use. Typically, filter banks have the avantage of attaining asymptotically improve results over an iniviual filter as the number of filters in the bank are increase. For example, with LPCC an HC filters, it was foun that 3 filters significantly out performe a single filter. However, 10 filters i not significantly improve performance over 3 filters. Ieally, the last matrix is the eigenvector matrix of the training set correlation matrix where the magnitue values of the eigenvector matrix elements are constraine to unity. In HC filter esign, this matrix is the iscrete Fourier transform DFT or Fourier matrix where the elements are complex exponentials, linear in phase with respect to their element row/ column inex. Avantages of complex response values 11,1 as well as filter banks have been note. 10,13 16 The origin response to HC filters is complex, tens to have constant magnitue, an tens to be linear in phase with respect to the training image an filter orer. If the constraints are mae rigi, as in SDF or MACE filter esigns, or if the training set has ieal cyclic properties, as in LPCC filter esign, then the k th orer filter response at the origin to the n th training image is Optical Engineering, Vol. 36 No. 10, October

3 tt,k exp jkn/nx T t,n h k. 1 n T Es n T H tt,0, tt,1 exp jn/n,..., tt,n1 The filter matrix structure, shown in Fig. 1, has a significance that goes beyon HC filter esign. The HC matrix structure originate as a general matrix structure for SDF, minimum variance SDF MVSDF Ref. 17, MACE, lock an tumbler 15 LAT, LPCC, an linear phase response 10 LPR filter esigns. 18 This structure is also vali for optimum trae-off, 19 GMACE Ref. 0, minimum noise an correlation energy 1 MINMACE, fractional power SDF Ref., an many other SDF filter esigns. The iniviual matrix values vary among these filter esigns, but the significance of the structure shown in Fig. 1 is that the function of the iniviual matrices are isolate. This functional isolation gives insight into the effects of the iniviual matrices on the filter performance, thereby enabling the filter esigner to customize these effects to a specific application. 3 Parameter Estimation Theory an Architecture We use correlation filters to etect an iscriminate the target location from clutter images. Given the target location we implement IP filters to etermine a response signature unique to a particular rotation. The IP filters are ientical in esign to HC correlation filters, but since the target location is known, a numerically efficient IP is performe either optically or electronically. An M-ary signature etector is use to etermine the most likely rotation parameter. We evelop the signature etector esign for both aitive colore Gaussian noise as well as for nonorthogonal IP filters. The general solution is simplifie by assuming orthogonal filters an AWGN. The input moel is an aitive Gaussian noise moel such that for the n th 1 target training vector x t,n we have s n x t,n w, where w is assume to be a spatially stationary zero-mean Gaussian noise vector. In general, the noise vector may be colore Gaussian noise. The output response of the k th orer IP filter to an input s n is yks n T h k tt,k exp jkn/nw T h k, where y(k) is a complex ranom variable. By augmenting all the IP filter orers into a N matrix we obtain an N 1 signature vector y such that y T n s T n Hx T t,n Hw T H. We esign an M-ary maximum-likelihoo ML etector that can etermine which target response y n, out of the N possible responses, is present thereby etermining the most likely rotation angle. The ML etector architecture is etermine from the probability ensity functions associate with all possible response vectors. Since we assume a Gaussian process, we nee only etermine the mean vectors an covariance matrices for the N possible rotations. The mean vectors are given by 3 4 exp jnn1/n. The mean vector is use to etermine the NN covariance matrix as C yy Ey n n y n n T EH T ww T H where C ww Eww T. H T C ww H, The covariance matrix C yy is inepenent of the istortion parameter. Given that A t is the conition that a target is present an n inicates that the n th target is present, the multivariate conitional probability ensity function pf is 1 P yn,at yn,a t N/ C yy 1/ exp 1 y n T C 1 yy y n. Thus if we can etect which target is present, then we also know the rotation parameter. The maximum pf value for a given input etermines which target is actually present. This ecision process oes not change if we take the log of the pf s since the log is a monotonic function. Since the covariance matrix is common to all rotations we can ignore the coefficient in Eq. 8 an concentrate on the exponent, which is the result of the log operation. By removing constant terms, which are inepenent of rotation, the ecision function for the n th rotation is G A,n Ry T C 1 yy n 1 T n C 1 yy n, where R inicates the real component. The linear architecture, shown in Eq. 9, is optimum in minimum probability of error MPE for both colore Gaussian noise an nonorthogonal filters. The angle selection is base on the maximum G n for n0,1,...,(n1). The architecture in Eq. 9 is simplifie even further by assuming orthogonal filters an white Gaussian noise. If the filters are orthogonal, the covariance matrix is iagonal an if the noise is white Gaussian then the input noise covariance matrix is proportional to an ientity matrix. Making the assumption of orthogonal filters, but allowing for colore noise, we obtain N1 1 G B,n R k0 yk c n k k 1 N1 tt,k, 10 k0 c k where the signature covariance matrix is C yy iagc 0,c 1,...,c N1. On the other han, if we assume nonorthogonal filters an AWGN, the signature measure is Optical Engineering, Vol. 36 No. 10, October 1997

4 erive the signal energy. In Sec. 4. we erive the noise energy an combine it with the signal energy for a SNR measure. The IP filters with the highest iniviual SNR are selecte first an yiel the highest signature etection SNR. However, SNR cannot be the only criterion for filter selection because certain IP filter orers woul yiel ambiguous results. In Sec. 4.3, IP filter ambiguity is consiere as well as SNR to obtain an optimize filter selection criterion. 4.1 Signature Signal Energy The target image input is assume to be eterministic so that the maximum signature energy is efine as the square of Eq. 13 such that G n Ry T n n Rx T t,n H n, 14 Fig. Target etection, iscrimination, an angle estimation architecture. The HC architecture is use to locate the target using correlation. The signature matching is accomplishe using IP operations. G C,n Ry T H T H 1 n 1 n T H T H 1 n, 11 where the constant coefficient from the noise covariance will not affect the etection so it is iscare. If both the filters are orthogonal an the noise is AWGN the signature covariance matrix becomes C yy HT H N tt, 1 where tt iag tt,0, tt,1,..., tt,n1. Ignoring the constant coefficient of the signature covariance matrix an the sum of all eigenvalues, which are the same for all signatures, we have the ML signature measure efine as N1 G n R k0 yk exp jkn/n N1 R k0 yk n kry T n, 13 where k is an N1 Fourier vector. The entire HC filter bank an IP architecture, given in Eq. 13, is shown in Fig.. 4 Optimum IP Filter Selection In practice, not all the IP filters woul be selecte. Which IP filters to be use in the angle estimation is etermine by their associate SNR as well as their ambiguity. Since the target is alreay locate, clutter iscrimination is not a criterion for the IP filter selection. However, SNR is an important selection criterion. We evelop a signature etection criterion for optimum SNR as a function of IP filter orers. The IP filters with the largest magnitue response yiel the largest SNR an are selecte first for maximum signature etection SNR. To simplify the erivation, we assume that the istortion is ieally cyclic, resulting in orthogonal IP filters, an the noise is AWGN. In Sec. 4.1, we where we assume cyclic istortion so the IP filter matrix is equivalent to the LPCC filter matrix such that HX t, an the response to a single input is y n T n T n T tt. Thus, the signature energy is G n R n T tt n N1 k0 tt,k N For cyclic istortion an N o we know tt,n tt,nn.if we select a subset of all possible IP filters, the signature energy is N G n b 1 tt,0 tt,k N, 18 where N b (N1)/. 4. Signature Noise Energy an SNR The signature noise energy is efine as the signature variance when the IP filter bank has AWGN as its input. The noise is assume to be zero mean AWGN in a vector form. The noise energy, that is the variance, of the signature etector is given by N1 EG n ERy T n E R k0 yk n k, 19 where y T w T H. The same assumptions are mae of cyclic istortion with the IP filter subset, as for the signal energy. The noise variance in each signature measure is given by N EG n,nb E b 1 y0 Ryk n k. 0 By substituting the ientity Optical Engineering, Vol. 36 No. 10, October

5 RA AA, 1 into Eq. 0, multiplying out the square, an bringing the expecte value function aroun only the ranom variables yiels N b 1 EG n,nb Ey 0 Ey0yk n k Ey0yk n k N b 1 N b 1 u1 Eykyu n k n u Eykyu n k n u N b 1 N b 1 u1 Eykyu n k n u Eykyu n k n u. The expectation in the first term of Eq. yiels Ey 0Ew T h 0 w T h 0 h 0 H Eww T h 0 h 0 H C ww h 0 h 0 H h 0 0 T X T t X t 0 0 T R tt 0 tt,0 T 0 0 N tt,0. 3 The expectations in the secon an thir terms of Eq. yiel Ey0ykEy0ykh 0 H Eww T h k h 0 H C ww h k h 0 H h k 0 T R tt k tt,k T 0 k N tt,0 k. 4 The expectations in the fourth an seventh terms of Eq. yiel EykyuEykyuh H k C ww h u h k H h u k T R tt u tt,u T k u tt,u N tt,u T k Nu knu. 5 The expectations in the fifth an sixth terms of Eq. yiel EykyuEykyuh T k C ww h u h k T h u k H R tt u tt,u H k u N tt,u ku. 6 By incorporating the resulting ientities from Eqs. 3 through 6 into Eq. we obtain the noise variance as EG n,nb N N b 1 tt,0 tt,0 k n k N b 1 tt,0 k n k knu n k n u N b 1 N b 1 u1 N b 1 N b 1 N b1 u1 N b 1 N b 1 u1 tt,k ku n k n u tt,k ku n k n u tt,k N b 1 u1 tt,k knu n k n u. 7 In Eq. 7 the secon an thir terms go to zero because the range of k in the summation never has value of zero, thus the function is always zero. Similarly in the fourth an last term k is never equal to Nu, thus the function is always zero. The only remaining terms are therefore the first, fifth, an sixth. The fifth an sixth terms are ientical so they can be ae together an reuce to a single summation ue to the function. The noise energy in each signature measure is ientical an can be written as a function of the number of IP filters inclue in the IP filter bank as EG n,nb N N b 1 tt,0 N tt,k, 8 where the equality hols for N b (N1)/ an N is o. The SNR ratio, which is Eq. 18 ivie by Eq. 8, is maximize by incluing the IP filters with the largest eigenvalues. The boune signature measure SNR is given by SNR G N b where N tt,0 N N b 1 tt,k N b 1 g k tt,k k0, Optical Engineering, Vol. 36 No. 10, October 1997

6 g k 1 for tt,k tt,0, 30 else an the prime eigenvalues represent the sorte eigenvalues such that tt,0 tt,1 tt,nb Equality is achieve, in Eq. 9 when N b (N1)/. 4.3 IP Filter Ambiguity If SNR is the only criterion use to select IP filters, the resulting signature matching may be ambiguous. This ambiguity is ue to the elements of the signature being complex phasor terms. That is, a single set of phase angles may be associate with several possible rotations. For example, the zero-orer IP filter h 0 has the same response magnitue an phase for all input rotations yet typically has the highest SNR. Thus the zero-orer IP filter yiels completely ambiguous results. On the other han, the first-orer IP filter h 1 has a unique phase associate with all rotation angles. The first-orer filter has no ambiguity but oes not typically have the highest SNR. Higher orer filters have increasing ambiguity an their interaction as a signature can become complicate. For these reasons, we have chosen a practical compromise to the filter selection. We treat the zero-orer filter as the least esirable filter an the firstorer IP filter as the most esirable filter to be selecte. All other filters are selecte by the SNR value. Our results support this selection criterion as requiring less filters than using the either SNR or orering criterions alone. 5 Numerical Results We test the combine as well as the iniviual component performance of the HC/IP-filter-base etection, iscrimination, an angle estimation system. Two signature etectors are use, one given in Eq. 11 is optimize for orthogonal filters an the other, given in Eq. 13, is optimum for orthogonal filters. Both signature etectors are optimize for AWGN. The input ata consists of 36 binary images of numerically simulate aircraft silhouettes. There are four ifferent aircraft types, one aircraft type is the target class an the other three types are the clutter class. Both target an clutter aircraft are uniformly rotate from 0 to 360 eg an viewe 45 eg from the perpenicular. Technically, we refer to this problem as a single parameter, 45-eg out-of-plane rotation istortion. Only the target training set is use in the filter esign, no clutter information is use. The HC correlation filter bank consists of filters whose weighting an esign are base on to previous work. 7 The IP filters were create an selecte as escribe in this research. There are four experiments. The first experiment emonstrates the etection, iscrimination, an estimation capability without noise. In Fig. 3 we show a sample pixel test image. The aircraft in the lower left is the target SMPR. The clutter class inclues the Phantom lower right, B-747 upper right an the F-104 upper left. We foun the target an clutter peak intensities for the 36 test inputs an forme a histogram of the response istribution. In Fig. 4 we show a peak response istribution of a three-filter HC filter bank. The Fig. 3 Sample 1818 input test image. Target aircraft SMPR (lower left). Clutter images are Phantom (lower right), B-747 (upper right), an F-104 (upper left). graph shows the istribution for 36 clutter an 34 target responses. Two of the target responses are not shown because they are outsie the upper range of the histogram. Note the large thresholing winow between intensities 0.4 an 0.8. Thus, even though the target intensity has a large sprea, there is clearly enough winow region to achieve 0 MPE. All targets were exactly locate an only two IP filters were require to exactly estimate the rotation angle inex. The secon experiment is the same as the first experiment except that the input test images are corrupte by AWGN as shown in Fig. 5. The noise energy is equal to the signal energy for an input SNR110 log (1)0 B. The system functions with 0% probability of error in classifica- Fig. 4 Histogram of peak intensity responses for 36 test images with no noise. Optical Engineering, Vol. 36 No. 10, October

7 Fig. 7 MSE of inex estimation for the IP signature etector. The horizontal axis inicates how many IP filters are use for etection. There are 36 signatures use. Curves are for 0-B SNR, 10-B SNR, an 17-B SNR. For the 17-B SNR case, the correct target locations are known. Fig. 5 Sample 1818 input test image with 0-B SNR. Noise is AWGN an the aircraft types are the same as escribe in Fig. 3. tion, 0% error in location of the target, an requires only two IP filters to achieve 0% error in angle estimation. The target an clutter istribution for a three-filter HC filter bank is shown in Fig. 6. The threshol winow ecreases to between 0.6 an 0.9, which is still aequate for 0-MPE iscrimination. Detection an location of the target are exact an the Gaussian shape, ue to noise, is apparent in Fig. 6. The mean square error MSE in the estimate angle inex 0 to 35 representing 0 to 350 eg is in inclue in Fig. 7. The estimate was performe twice, once by assuming the filters are orthogonal an once using their actual nonorthogonal relationship. A comparison of these two etectors is shown in Fig. 7. For this experiment, the number of filters neee for 0% error were two IP filters. The thir experiment is the same as the secon experiment except that the noise energy is 10 times more than the signal energy such that SNR0.110 log (0.1) 10 B. An example test image is shown in Fig. 8. The target is still locate with 0% error but 15 HC filters are require an the iscrimination ability of the HC filter bank between target an clutter begins to egrae, as shown in Fig. 9 Both IP signature etectors still perform well, as shown by the curves in Fig. 7, but both require four IP filters. The fourth experiment is the same as the thir experiment except that we test only the estimation component of Fig. 6 Histogram of peak intensity responses for 36 test images with 0-B SNR. Fig. 8 Sample 1818 input test image with 10-B SNR. Noise is AWGN an the aircraft types are the same as escribe in Fig Optical Engineering, Vol. 36 No. 10, October 1997

8 Acknowlegments This research was supporte through the National Aeronautics an Space Aministration Cooperative Agreement No. NCCW-60. References Fig. 9 Histogram of peak intensity responses for 36 test images with 10-B SNR. the system. The SNR is 1/5010 log (1/50)17 B. In this case, the etection an iscrimination ability of the correlation component have completely egrae to the point of unreliable target location, but the signature etection continues to function given correct target locations. The curves in Fig. 7 show that the MSE of the angle estimation goes almost to 0 with 1 IP filters, given correct target locations an the orthogonal signature etector. The MSE goes to 0 with 11 IP filters for the nonorthogonal etector an it is apparent from the curves that the nonorthogonal etector has consistently lower MSE than the orthogonal etector esign. 6 Conclusions We present a new technique for estimating the istortion parameter base on IP operations. We use rotation estimation as an example but because of the IP filter esign, the technique can be extene to general istortion parameter estimation. The IP filter response signature matching is optimize for both nonorthogonal filters an colore noise. We evelope IP filter selection criterion, optimal in SNR, given orthogonal IP filters, an an AWGN input moel. The resulting system was robust for noisy images. At 0-B input SNR we coul etect, iscriminate, an estimate the angle with 3 correlation filters an IP filters. At 10-B input SNR, we coul etect, iscriminate, an estimate the angle with 15 correlation filters, an 4 IP filters. At 17-B input SNR, we lost our ability to etect an iscriminate accurately, but if we coul locate the input targets, then the IP filter estimation coul estimate with 0% error with 1 IP filters. We eliminate ambiguity by simply forcing the first IP filter selecte to be the one filter with no ambiguities an the last IP filter to be the one with maximum ambiguity. All filters in between were selecte for maximum SNR. We believe that these results inicate a new an robust metho for istortion parameter estimation. 1. B. V. K. Vijaya Kumar, Tutorial survey of composite filter esigns for optical correlators, Appl. Opt. 313, D. Cyganski an R. F. Vaz, Linear signal ecomposition approach to affine invariant contour ientification, Pattern Recog. 81, M. Wang, L. G. Hassebrook, J. Kirsch, J. Evans, an C. Knapp, Active image registration an recognition, Proc. SPIE 4885, M. Wang, J. Evans, L. G. Hassebrook, an C. Knapp, A multi-stage, optimal active contour, IEEE Trans. Image Process. 511, X. Lou, L. G. Hassebrook, M. Lhamon, an J. Li, Numerically efficient angle, with, offset an iscontinuity estimation of lines using the iscrete Fourier-bilinear transformation, IEEE Trans. Image Process. October H. L. Van Trees, Detection, Estimation an Moulation Theory: Part I, Wiley, New York L. G. Hassebrook, M. Rahmati, an B. V. K. 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Kinser, Fractional-power synthetic iscriminant functions, Pattern Recog. 74, Laurence G. Hassebrook receive his BSEE egree from the University of Nebraska in Lincoln in 1979, his MSEE egree from Syracuse University in 1987, an his PhD in electrical an computer engineering from Carnegie Mellon University in He worke at Lincoln Electric System Corporation from 1980 to 1981 in evelopment an automation of loa analysis systems. From 1981 to 1984 an then uring summer from 1985 through 1987, he worke in non-estructive testing an inspection at IBM Corporation in Enicott, New York. He is presently an associate Optical Engineering, Vol. 36 No. 10, October

9 professor of electrical engineering at the University of Kentucky. His interests are in the area of 3-D ata acquisition, pattern recognition, an opto-electronic signal processing. Michael E. Lhamon receive his BSEE egree from Ohio Northern University in 1990, his MSEE from Georgia Institute of Technology in 1991, an his PhD from the University of Kentucky in He worke for the National Raio Astronomy Observatory at the Very Large Array in Socorro, New Mexico, uring 1989 an 1990 an Multi-Link Incorporate in 199. His present interests inclue automatic object recognition an iscrimination an realtime igital an optical signal processing. Aitional interests inclue real-time istribute processing on parallel igital signal processors an fiel programmable gate arrays. He is a member of SPIE an IEEE. Jyoti P. Chatterjee receive his BE in electronics engineering from Visvesvaraya Regional College of Engineering, Inia, in After completing his BE egree he worke with Gorej & Boyce Manufacturing Company, Inia, from 1990 through 1993 as a service executive. He receive his MS in electrical engineering from the University of Kentucky in Since then he has been employe at Digital Technics Incorporate, Marylan, as a DSP software engineer. Mao Wang receive his BS egree an MS egree in electrical engineering from Nanjing Institute of Technology in 1984 an 1987, respectively. He receive his MS egree in biomeical engineering in 199 an his PhD egree in electrical engineering in 1995 from the University of Kentucky. He is currently a lea engineer at Motorola Cellular Infrastructure Group, Arlington Heights, Illinois. His research interests inclue igital signal an imaging processing an igital communications. 718 Optical Engineering, Vol. 36 No. 10, October 1997