E. Marom and H. Inbar Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. A 1325

New interpretations of Wiener filters for image recognition

Emanuel Marom and Hanni Inbar

Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel

Received November 20, 1995; accepted January 29, 1996

Wiener filters for restoration of images and Wiener filters for image recognition are well established. Wepresent the relations between two versions of Wiener restoration filters and develop new conceptual inter-pretations of the Wiener recognition filter, in the process of which interesting features are revealed. In oneinterpretation we show that the correlation signal resulting from Wiener recognition filtering may be regardedas the restoration of a hidden delta function. In another interpretation we regard the Wiener recognition fil-ter as a two-filter scheme whereby an inverse filter and a Wiener restoration filter are operated sequentially.Results of simulations of the latter representation are presented for various input-noise models, providinguseful insight into the implications of such an interpretation. 1996 Optical Society of America

1. BACKGROUNDA restoration filter is designed to extract as well aspossible a desired signal from a mixture of signal andnoise. There are many possibilities for a criterion ofgoodness of performance, that is, for a measure of howwell the restoration task is performed. The criterion ofleast mean square error, used in the development of theWiener theory,1 has the advantage that it leads generallyto a workable analysis. A review of Wiener restorationfilters is given in the following subsections.

For pattern recognition the form of the original imageis not important; solely the presence or absence of the im-age is of concern. Wiener filters for pattern recognitionare discussed in Section 2. In Section 3 we propose newinterpretations of Wiener filtering for image recognition.Computer simulation results illustrating one such inter-pretation are given in Section 4.

A. Restoration of Unblurred Images Embedded in NoiseLet us first consider a restoration process in which arestored image dsx, yd is obtained from an observed imagessx, yd, where

ssx, yd ­ dsx, yd 1 nsx, yd , (1)

dsx, yd and nsx, yd being, respectively, the original im-age and an input-additive, zero-mean colored noise.Dsu, vd, Ssu, vd, Dsu, vd, and N su, vd denote the respec-tive Fourier transforms (FT’s). For a well-defined esti-mation problem, nsx, yd and dsx, yd are assumed to beuncorrelated, stationary random processes, with respec-tive power spectral densities2 Pnsu, vd ­ jN su, vdj2 andPdsu, vd ­ jDsu, vdj2, the overbar representing ensem-ble averaging. The linear restoration filter Hpsu, vd,which minimizes the mean square error between the re-stored image dsx, yd and the original image dsx, yd, isthe Wiener filter,3

H1psu, vd ­

Pdsu, vdfPdsu, vd 1 Pnsu, vdg

, (2)

or, equivalently,

0740-3232/96/071325-06$10.00

H1psu, vd ­

jDsu, vdj2

fjDsu, vdj2 1 Pnsu, vdg, (3)

the asterisk signifying a complex conjugate operation.The inverse FT of H su, vd is hsx, yd, the desired impulseresponse used for synthesizing the filter. It follows fromEqs. (1) and (2) that

Dsu, vd ­ H1psu, vdSsu, vd

­ H1psu, vdfDsu, vd 1 N su, vdg

­ Dsu, vdPdsu, vd

fPdsu, vd 1 Pnsu, vdg

1 N su, vdPdsu, vd

fPdsu, vd 1 Pnsu, vdg. (4)

This expression reveals that the filter transmits most ofthe original-image energy at frequencies su, vd for whichthis energy is much greater than that of the noise, whilegreatly suppressing the original-image energy at frequen-cies su, vd for which this energy is negligible comparedwith that of the noise.3 Also, when noise energy is smallcompared with that of the original image, its transmis-sion will only slightly affect the restored image and thuswill not be detrimental to restoration, whereas at fre-quencies for which noise energy is much greater thanthat of the original image, it will be almost blocked,since Pdsu, vdyPnsu, vd ,, 1. Evidently, the power spec-tral density of the filtered noise is greatly suppressed atfrequency bands for which the original signal-to-noise ra-tio (SNR) was low.

It follows that best restoration results when Ssu, vd ismost heavily weighted at frequencies for which the ratio ofsignal spectral density to noise spectral density is largest.The best the filter can do to recover the signal from thenoise is to favor the frequency bands that contain largelysignal energy over those frequency bands that containlargely noise energy.1

B. Restoration of Blurred Images Embedded in NoiseIf the observed image contains a blurred version f sx, yd ofthe original image as well as additive noise, then one has

1996 Optical Society of America

1326 J. Opt. Soc. Am. A/Vol. 13, No. 7/July 1996 E. Marom and H. Inbar

ssx, yd ­ f sx, yd 1 nsx, yd , (5)

where the convolution

f sx, yd ­ dsx, yd p rsx, yd , (6)

or, equivalently, Ssu, vd ­ Dsu, vdRsu, vd 1 N su, vd.Here rsx, yd is a blurring function whose FT is Rsu, vd.With Eq. (2), the Wiener filter for generating a restoredf sx, yd from ssx, yd is

H2psu, vd ­

Pf su, vdfPf su, vd 1 Pnsu, vdg

­jRsu, vdj2Pdsu, vd

fjRsu, vdj2Pdsu, vd 1 Pnsu, vdg, (7)

where Pf su, vd is the power spectral density of f sx, yd andthe FT of f sx, yd is F su, vd ­ H2

psu, vdSsu, vd. SinceDsu, vd ­ F su, vdyRsu, vd, the Wiener filter yieldingdsx, yd—the restoration of the unblurred image—is4

H3psu, vd ­

H2psu, vd

Rsu, vd

­Rpsu, vdPdsu, vd

fjRsu, vdj2Pdsu, vd 1 Pnsu, vdg

­Rpsu, vd

fjRsu, vdj2 1 Pnsu, vdyPdsu, vdg. (8)

One should note that knowledge of the blurring functionrsx, yd, or, equivalently, its FT Rsu, vd, is necessary toprovide optimal restoration quality4,5 in the sense of mini-mal mean square error between the exact true solutiondsx, yd and the estimate dsx, yd.5 Inspection of Eq. (8)indicates that in the absence of noise fPnsu, vd ­ 0g thisrelation reduces to the inverse filter,6 i.e., 1yRsu, vd.

In this section we reviewed two versions of Wienerfilters used for obtaining a restored image dsx, yd from anobserved image ssx, yd: H1

psu, vd is used for nonblurredimages, whereas H3

psu, vd is used for blurred images. InSection 2 we analyze Wiener filters for image recognition.

2. RECOGNITION OF IMAGESEMBEDDED IN NOISEAlthough Wiener filters were developed primarily forsignal restoration, they can also serve extremely wellfor the task of pattern recognition. Optical patternrecognition is commonly achieved by utilizing coherentlyilluminated correlation setups. The detection problemconsists mainly in deciding whether an input scene ssx, ydcontains a reference object rsx, yd. For optical patternrecognition utilizing the frequency-plane correlator (FPC)architecture,7 ssx, yd is filtered by a frequency-plane maskHpsu, vd, which is synthesized on the basis of knowledgeof the reference object rsx, yd. As defined in Section 1,the inverse FT of Hsu, vd is hsx, yd, the desired impulseresponse used for synthesizing the frequency-plane filtermask. The light-field distribution behind this mask con-tains the term Csu, vd ­ Hpsu, vdSsu, vd, where Csu, vdis the FT of the correlation csx, yd ­ ssx, yd ? hsx, yd ­RR

ssx0, y 0dhpsx0 2 x, y 0 2 yddx0dy 0, with ? signifying acorrelation operation. Here

ssx, yd ­ f sx, yd 1 nsx, yd , (9)

f sx, yd being an input object. For autocorrelation casesf sx, yd ­ rsx, yd, whereas for cross-correlation casesf sx, yd represents any other object. The Wiener filterfor pattern recognition is8

H4psu, vd ­

Rpsu, vdfjRsu, vdj2 1 Pnsu, vdg

. (10)

The correlation signal resulting from application ofthis filter was shown to exhibit optimality trade-off be-tween the performance criteria noise tolerance and peaksharpness.8

Wiener filtering for image recognition can be opticallyimplemented through the FPC architecture, as describedabove. Recently we showed9 that Wiener recognition fil-tering can also be implemented with the joint transformcorrelator (JTC) architecture,10 thus extending the capa-bilities of conventional JTC systems. Moreover, this newJTC-based approach permits on-line adaptive estimationof the noise power spectral density when noise inten-sity and statistics, necessary for the filtering design, areunknown in advance. We also showed9 that, comparedwith the FPC case, JTC-based Wiener recognition filter-ing exhibits superior correlation performance with respectto light efficiency. This advantage adds to the inherentadvantages of using JTC schemes over using FPC ones:simplicity and ease of implementation, as well as real-time adjustability for substitution of a reference objectfrom a data bank when a search is desired.

3. NEW OBSERVATIONS ON WIENERRECOGNITION FILTERS

A. Intertwining of Classical Matched and Inverse FiltersThe trade-off between peak sharpness and noise toler-ance exhibited by Wiener recognition filters8 may be intu-itively comprehended by the following observations. TheWiener filter [Eq. (10)] reduces in the noise-free case tothe inverse filter 1yRsu, vd,6 which yields the sharpestpossible correlation peaks11 but is highly vulnerable toadditive noise.12 At the other extreme, in the presenceof very high noise intensity the Wiener recognition fil-ter reduces to the well-known classical matched filterRpsu, vdyPnsu, vd, which is optimal with respect to noisetolerance.13 Wiener filters for image recognition maythus be regarded as combining the properties of boththe classical matched filter and the inverse filter. Oneshould note that the above inherent distinction holdspiecewise over the entire Fourier spectral plane: overcertain portions of the spectral plane the filter behavesas an inverse filter but over others as a classical matchedfilter.

B. Interpretation of Correlation As Restorationof a Hidden Delta FunctionIn this subsection we show that the autocorrelation signalresulting from the use of a Wiener recognition filter is infact the Wiener restoration of a delta function hidden inthe input scene. This hidden function, exposed by theWiener recognition process, indicates both detection andlocalization.

E. Marom and H. Inbar Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. A 1327

The correlation operation is used to provide recognition,i.e., it determines whether the input object equals the ref-erence object: f sx, yd ­ rsx, yd. One can rewrite this re-lation also as f sx, yd ­ dsx, yd p rsx, yd when dsx, yd is adelta function. Regarding the reference object rsx, yd asa blurring function that distorts a hidden delta functiondsx, yd, we hereby propose to reinterpret the correlationprocess conceptually as a restoration process whereby ef-forts are made to restore dsx, yd at the correlator outputplane by placing a suitable mask Hpsu, vd at the correla-tor filter plane when the input plane displays

ssx, yd ­ dsx, yd p rsx, yd 1 nsx, yd . (11)

It follows that a sharp restored dsx, yd at the cor-relator output plane would indicate that dsx, yd ­dsx, yd, i.e., that the input scene ssx, yd indeed con-tains the reference rsx, yd. Inversely, a correlationsignal lacking a well-defined peak would indicate thatdsx, yd fi dsx, yd, i.e., that f sx, yd is different from thereference object, thus representing a cross-correlationcase. The problem now remains to determine thesuitable mask Hpsu, vd. We showed earlier [Eq. (8)]that the Wiener filter used for restoration of dsx, ydis H3

psu, vd ­ Rpsu, vdyfjRsu, vdj2 1 Pnsu, vdyPdsu, vdg.Since dsx, yd ­ dsx, yd, yielding Pdsu, vd ­ 1, this filterreduces to

H5psu, vd ­

Rpsu, vdfjRsu, vdj2 1 Pnsu, vdg

, (12)

which is indeed equivalent to H4psu, vd defined in

Eq. (10), therefore verifying our new interpretation. Itthus follows that the Wiener recognition filter providesa correlation signal that is optimal with respect to theminimal mean square error between the correlation sig-nal and a delta function, i.e., the best possible estimationfor a delta function. For the noise-free case the Wienerrecognition filter reduces to the inverse filter, as indi-cated by Eq. (12) with Pnsu, vd ­ 0. In this special casethe expression for the restored Dsu, vd ­ H5

psu, vdSsu, vdbecomes Dsu, vd ­ Dsu, vd, thus perfectly restoring thecorrelation signal dsx, yd as a delta function.

C. Two-Filter SchemesA common optical correlation architecture for patternrecognition is the FPC, whereby a Wiener filter may beused in the Fourier plane. Here the Wiener recognitionfilter [Eq. (10)] H4

psu, vd ­ Rpsu, vdyfjRsu, vdj2 1 Pnsu, vdgleads to the display of the correlation distribution ssx, yd ?hsx, yd at the FPC output plane.

New light is shed on comprehending the Wienerrecognition filter if we regard it as a decomposi-tion into a two-filter scheme, whose components are

Hipsu, vd ­

1Rsu, vd

, (13)

Hrpsu, vd ­

jRsu, vdj2

fjRsu, vdj2 1 Pnsu, vdg. (14)

It is readily observed that Hipsu, vd is the inverse filter

and that Hrpsu, vd is the Wiener filter for restoration

of rsx, yd from ssx, yd ­ rsx, yd 1 nsx, yd, as defined in

Eq. (3). A two-filter scheme was reported in the past6

in conjunction with frequency-plane processors, in a to-tally different context, for ease of implementation onlyand not for analyzing the effects of each filter. The twofilters may be placed sequentially: one behind the otherin registration. Alternatively, one can envisage a systemof two consecutive frequency-plane processors, in whichone of the filter components is placed at the Fourier planeof the first processor while the other filter is placed inregistration at the Fourier plane of the second processor.Hence the first filter is imaged on the other filter. Itshould be well understood that such decomposition is notproposed for practical realizations but rather for allow-ing illustration of features demonstrated by each filter insuch a conceptual experiment.

In the first setup of such a representation, denotedsetup 1 (Fig. 1), the first frequency-plane processor ap-plies inverse filtering [Hi

psu, vd of Eq. (13)] to the in-put scene ssx, yd placed at plane Pin. It is well knownthat even when the processor input [ssx, yd at plane Pin]contains very low noise levels, the resulting correlationsignal at the processor output (plane P1) is very noisy.Nevertheless, this noisy distribution of the correlationsignal is not useless and contains decodable informa-tion, as will be demonstrated in Section 4. The secondfrequency-plane processor filters the noisy correlation dis-tribution by applying restoration according to the refer-ence function rsx, yd [Hr

psu, vd of Eq. (14)], displaying theresulting Wiener correlation ssx, yd ? hsx, yd at its outputplane Pout. One should not be concerned here with thevalue assigned to the inverse filter at frequencies su, vd forwhich Rsu, vd ­ 0, since the second frequency-plane pro-cessor blocks any energy that its input may have at thesefrequencies, thus cleaning up the first correlation distri-bution. Therefore, although the response of the inversefilter is extremely noisy, the second filter can still extractthe information hidden within the noisy features.14

A second scheme, denoted setup 2, in which the lo-cations of these two component filters are exchanged(Fig. 2), leads to an additional interesting observation.In this case the first frequency-plane processor performs

Fig. 1. Wiener recognition filtering interpreted as a two-filterscheme: setup 1.

Fig. 2. Wiener recognition filtering interpreted as a two-filterscheme: setup 2.

1328 J. Opt. Soc. Am. A/Vol. 13, No. 7/July 1996 E. Marom and H. Inbar

restoration of rsx, yd by filtering the input scene [ssx, ydat plane Pin] with the use of Hr

psu, vd [Eq. (14)], thus dis-playing the restored rsx, yd at its output plane P2. Thesecond frequency-plane processor in this representationapplies inverse filtering [Hi

psu, vd of Eq. (13)] over the re-stored rsx, yd, finally generating the same correlation asobtained previously ssx, yd ? hsx, yd at its output planePout. Should we have applied inverse filtering directlyon a noisy input scene ssx, yd, it would have resulted inmisrecognition even for low noise levels, as a result ofthe extreme sensitivity of inverse filtering to input noise.It is thus encouraging to note that recognition may beachieved if inverse filtering is applied on the restoredimage rather than on the observed input-scene image.Here, too, the value assigned to the inverse filter at fre-quencies su, vd for which Rsu, vd ­ 0 is unimportant, sincethe first frequency-plane processor blocks any energy thatssx, yd may have at these frequencies.

An illustration of the operation of the two-filter schemesis given in Section 4 by means of computer-simulationresults. The resulting signals at planes P1, P2 and Pout

are presented for the two setups just described.Before proceeding with an additional interesting fea-

ture of Wiener recognition filtering, we would like topropose another interpretation. The Wiener restorationfilter Hr

psu, vd of Eq. (14) can be regarded as the multipli-cation of H4

psu, vd of Eq. (10) and Rsu, vd. Application ofthis conceptual process leads first to Ssu, vdH4

psu, vd, i.e.,to the restoration of a hidden delta function (Section 3.B).Successive multiplication by Rsu, vd represents a convo-lution with rsx, yd. Therefore the restored rsx, yd can beinterpreted as the convolution of the restoration of a hid-den delta function with rsx, yd. Evidently, the quality ofthe restored rsx, yd depends on the quality of the restora-tion of the hidden delta function.

D. Linear Preprocessing Operator EffectsA linear preprocessing operator tsx, yd applied to bothreference- and input-scene images is sometimes used inrecognition problems to improve the correlation signalquality. For instance, since similarities between imagesoften exist, it is sometimes better to recognize an in-put object by its contour, i.e., by correlating the gradi-ents or the Laplacians of the reference- and input-sceneimages rather than the images themselves.15 We willnow analyze the effects of linear preprocessing operatorson Wiener recognition filtering. In a FPC setup with aWiener recognition filter, application of a linear operatortsx, yd [whose FT is T su, vd] on both rsx, yd and ssx, yd ­rsx, yd 1 nsx, yd results in the following representationsin the Fourier domain: Rsu, vdTsu, vd for the reference,Ssu, vdTsu, vd for the input scene, and Pnsu, vdjT su, vdj2

for the noise. Thus the light field distribution behind themodified Wiener filter is

Ssu, vdTsu, vdRpsu, vdTpsu, vd

fjRsu, vdj2jT su, vdj2 1 Pnsu, vdjT su, vdj2g

­ Ssu, vdRpsu, vd

fjRsu, vdj2 1 Pnsu, vdg. (15)

It is therefore obvious that the correlation signal dis-played at the output plane is indifferent to any linearpreprocessing operator tsx, yd. Alternatively, we can in-

terpret that restoration of the hidden delta function, asdescribed in Section 3.B, is unaffected by application of alinear operator to both the reference- and the input-sceneimages.

4. COMPUTER SIMULATIONSSeveral computer simulations were performed to illus-trate and provide insight into the concepts derived inSection 3.C, whereby pattern recognition through Wienerfiltering is conceived as two-filter schemes: Wienerrestoration and inverse filtering in an arbitrary order.The signals at the output of each frequency-plane proces-sor were computed for the setups of Figs. 1 and 2. Allautocorrelation tests were performed with the referenceobject depicted in Fig. 3(a), and for cross-correlation testsa “false” object shown in Fig. 3(b) was used. The input-scene images in our simulations, limited to 32 3 46 pixels,were assumed to contain additive zero-mean Gaussiannoise, with either low-frequency or high-frequency char-acteristics, and variable intensity.

Figures 4(a)–4(c) present the simulation results for aninput-scene image [Fig. 4(d)] containing low-frequencynoise with a standard deviation sn ­ 0.2. Since thereference object occupies 185 pixels of unit value,this noise level corresponds to a SNR power ratio of185ys0.22ds32ds46d ø 3.1. Figures 4(a) and 4(b) presentthe results obtainable with setup 1. Figure 4(a) showsthat the correlation signal displayed at plane P1, resulting

Fig. 3. Objects used for computer-simulation tests: (a) F18aircraft model, (b) open-wing Tornado aircraft model.

Fig. 4. Autocorrelation case; low-intensity, low-frequency addi-tive noise. Signals exhibited at planes (a) P1 (setup 1), (b) Pout(setups 1 and 2), and (c) P2 (setup 2), for (d) the input scene atplane Pin: the F18 aircraft model embedded in low-frequencynoise with sn ­ 0.2.

E. Marom and H. Inbar Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. A 1329

Fig. 5. Autocorrelation case; low-intensity, high-frequency addi-tive noise. Signals exhibited at planes (a) P1 (setup 1), (b) Pout(setups 1 and 2), and (c) P2 (setup 2), for (d) the input scene atplane Pin: the F18 aircraft model embedded in high-frequencynoise with sn ­ 0.2.

Fig. 6. Autocorrelation case; moderate-intensity, low-frequencyadditive noise. Signals exhibited at planes (a) P1 (setup 1),(b) Pout (setups 1 and 2), and (c) P2 (setup 2), for (d) the in-put scene at plane Pin: the F18 aircraft model embedded inlow-frequency noise with sn ­ 0.75.

from application of inverse filtering directly on the input-scene image of plane Pin [Fig 4(d)], is sharp but somewhatnoisy. Figure 4(b) shows that the restoration applied bythe second frequency-plane processor of setup 1 leads toa well-defined correlation peak at plane Pout. Turningnow to the results for setup 2, we see in Fig. 4(c) therestoration of the reference object from the input-sceneimage-plane Pin, performed by the first frequency-planeprocessor and displayed at plane P2. One can see thatthe Wiener restored image is less noisy than the input-scene image, as reflected in a SNR improvement by afactor of ,2.5. Consecutive application of inverse filter-ing by the second frequency-plane processor of setup 2 onthe restored image at plane P2 results finally in the samesharp and noiseless Wiener correlation signal at planePout, as depicted in Fig. 4(b).

More-impressive and instructive results are demon-strated in Figs. 5(a)–5(c) for an input-scene image[Fig. 5(d)] containing high-frequency noise with a stan-dard deviation sn ­ 0.2. The correlation signal dis-played after inverse filtering at plane P1 [Fig. 5(a)] seemsto imply misrecognition: it is very noisy and has no pri-mary peak, despite the low input-noise level. This is dueto the high sensitivity of inverse filtering to input noise,especially for noise including intense high-frequency com-ponents. Nevertheless, the distribution at this plane(P1) is not useless: the restoration applied by the sec-ond frequency-plane processor of setup 1 results quiteunexpectedly in a well-defined correlation peak at planePout [Fig. 5(b)], thus clearly indicating recognition. Di-recting our attention now to setup 2, the restored image[Fig. 5(c)] exhibits an improvement in SNR by a factor

Fig. 7. Autocorrelation case; high-intensity, low-frequency addi-tive noise. Signals exhibited at planes (a) P1 (setup 1), (b) Pout(setups 1 and 2), and (c) P2 (setup 2), for (d) the input scene atplane Pin: the F18 aircraft model embedded in low-frequencynoise with sn ­ 1.0.

Fig. 8. Autocorrelation case; very-high-intensity, low-frequencyadditive noise. Signals exhibited at planes (a) P1 (setup 1),(b) Pout (setups 1 and 2), and (c) P2 (setup 2), for (d) the in-put scene at plane Pin: the F18 aircraft model embedded inlow-frequency noise with sn ­ 2.0.

1330 J. Opt. Soc. Am. A/Vol. 13, No. 7/July 1996 E. Marom and H. Inbar

Fig. 9. Cross-correlation case; moderate-intensity, low-fre-quency additive noise. Signals exhibited at planes (a) P1(setup 1), (b) Pout (setups 1 and 2), and (c) P2 (setup 2), for (d)the input scene at plane Pin: the open-wing Tornado aircraftmodel embedded in low-frequency noise with sn ­ 0.75.

of ,11.9 compared with that of the input-scene image[Fig. 5(d)]—however, with fuzzy-edge appearance. It isevident that the level of the noise is greatly reduced andthat its high-frequency components are strongly attenu-ated. Successive inverse filtering leads to the resultshown in Fig. 5(b). Since the restored image is muchmore tolerant than the input-scene image to inverse fil-tering, it is instructive to regard the restoration stageas a preprocessing step. The great usefulness of apply-ing this step before inverse filtering is demonstrated bycomparing Figs. 5(b) and 5(a), these figures showing theresults of inverse filtering applied, respectively, on therestored image and on the input-scene image.

Other interesting results are shown in Figs. 6(a)–6(c)for an input-scene image [Fig. 6(d)] containing low-frequency noise with a higher standard deviation ssn ­0.75d, corresponding to a SNR power ratio of ø 0.22.Here one can see clearly the capabilities of Wienerrestoration when comparing the original input-sceneimage [Fig. 6(d), plane Pin] with the restored image[Fig. 6(c), plane P2]. Evidently, it is possible to iden-tify the reference object in the restored image but not inthe original input image.

Considering input-scene images embedded in evenhigher levels of additive low-frequency noise [Figs. 7(d)and 8(d)] reveals that although restoration quality drops[Figs. 7(c) and 8(c)], recognition is still possible throughperforming correlation using a Wiener recognition filter[Figs. 7(b) and 8(b)].

Finally, we compare the autocorrelation test results de-picted in Fig. 6 with cross-correlation test results (Fig. 9).The noise considered in these cases is low-frequency noisewith standard deviation sn ­ 0.75. Neither restorationnor recognition is achieved when the object embedded in

noise is the false object of Fig. 3(b). Thus the discrimina-tion capabilities of the Wiener recognition filtering processare clearly demonstrated.

5. SUMMARYIn this paper we presented new conceptual interpreta-tions of Wiener filters for recognition of images and dis-cussed some of their interesting and powerful features.We showed that a correlation setup utilizing a Wienerrecognition filter may be interpreted as a restorationsetup whereby attempts are made to restore a hiddendelta function indicating recognition. For this interpre-tation the reference object is regarded as the function thatblurs an original delta function. Further, we proposedan interpretation of the Wiener recognition filter as atwo-filter scheme composed of a Wiener restoration filterand an inverse filter. Computer simulations results il-lustrating the behavior and implications of such schemeswere presented. Implementation of Wiener filtering forpattern recognition through FPC as well as JTC archi-tectures was discussed, and the advantages of the latterwere described.

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