Design of correlation filters for recognition of linearly distorted objects in linearly degraded scenes

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E. M. Ramos-Michel and V. Kober Vol. 24, No. 11 /November 2007 /J. Opt. Soc. Am. A 3403

Design of correlation filters for recognition oflinearly distorted objects in linearly

degraded scenes

Erika M. Ramos-Michel and Vitaly Kober*

Department of Computer Science, Centro de Investigación Científica y de Educacíon Superior de Ensenada,Km. 107 Carretera Tijuana-Ensenada, Ensenada 22860, B.C., Mexico

*Corresponding author: [email protected]

Received April 9, 2007; revised August 6, 2007; accepted August 14, 2007;posted August 24, 2007 (Doc. ID 81929); published October 1, 2007

Generalized correlation filters are proposed to improve recognition of a linearly distorted object embedded in anonoverlapping background when the input scene is degraded with a linear system and additive noise. Severalperformance criteria defined for the nonoverlapping signal model are used for the design of filters. The derivedfilters take into account information about an object to be recognized, disjoint background, noise, and lineardegradations of the target and the input scene. Computer simulation results obtained with the proposed filtersare discussed and compared with those of various correlation filters in terms of discrimination capability, lo-cation errors, and tolerance to input noise. © 2007 Optical Society of America

OCIS codes: 100.2000, 100.5010.

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. INTRODUCTIONattern recognition based on correlation has been a sub-

ect of interest for many years. Various techniques and al-orithms for the design of correlation filters optimizingifferent criteria have been proposed to improve detectionnd localization of objects in images [1–22]. When corre-ation filters are used, these two tasks can be solved inwo steps: first, the detection is carried out by searchingorrelation peaks at the filter output, and then coordi-ates of these peaks are taken as position estimations ofhe target in the input scene. The detection capabilities oforrelation filters can be quantitatively expressed inerms of probability of detection errors (false alarms),ignal-to-noise ratio (SNR), discrimination capability, andeak-to-output energy (POE) ratio [3]. Optimization ofhese criteria leads to reducing false recognition errors,hich can occur everywhere in the correlation plane.ome of the measures can be essentially improved usingn adaptive approach to the filter design [23–25] After theetection task has been solved, we are still faced withmall errors of target position estimations due to distor-ion of the object. The coordinate estimations lie in the vi-inity of their actual values. Therefore accuracy of the tar-et location can be characterized by the variance ofeasurement errors along coordinates [5–7].Basically two kinds of mathematical models of input

cene are used for the design of optimum correlation fil-ers. In the overlapping model (additive model), a targets added to a background at unknown coordinates. In theonoverlapping model, a background is spatially disjointnonoverlapping) with a target [10]. Besides, the inputcene may contain additive noise. Common correlation fil-ers such as the matched spatial filter (MSF) [1], thehase-only filter (POF) [4], and the optimal filter (OF) [5]ere derived from the overlapping model of the input

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cene. On the other hand, from the nonoverlapping modelf input scene, optimum correlation filters such as theeneralized matched filter (GMF) [12], the generalized op-imum filter (GOF) [12], the generalized phase-only filterGPOF) [18], and other generalized filters [10–18] werelso introduced.The quality of detection and localization of a targetay be limited by: (i) presence of additive and disjointoise in observed scenes, (ii) scene degradations owing to

mage formation process (imperfection of capturing de-ices, relative motion between the camera and the scene,nd propagation environment), (iii) target distortionstarget motion, target is out of optical system focus). Con-entional correlation filters are sensitive to these degra-ations. Particular cases of the degradations were takennto account in the filter design [19–22,25–29] Recently, aorrelation filter optimized with respect to the meanquared error for detection a target embedded in a back-round noise in environmental degradation was proposed28]. In this case, environmental compensation was in-luded in the filter design. However, it appears that theroblem of detection and localization with correlation fil-ers has not been solved when the input scene degradedith a linear system (degradation during image forma-

ion) contains a linearly distorted target embedded on aonoverlapping background and additive noise.The purpose of this paper is to derive optimum correla-

ion filters, which take into account a priori informationbout an object to be recognized, background noise, linearegradations of the input scene and target, and additiveensor’s noise. We assume that degradations of the targetnd the scene are physically independent of each other.he proposed filters are optimized with respect to recog-ition criteria defined for the nonoverlapping signalodel. The POE criterion is defined as the ratio of the

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3404 J. Opt. Soc. Am. A/Vol. 24, No. 11 /November 2007 E. M. Ramos-Michel and V. Kober

quare of the expected value of the correlation peak to thexpected value of the output signal energy [12]:

POE = �E�y�x0,x0��2/E��y�x,x0��2�, �1�

here y�x ,x0� is the filter output when the target is lo-ated at position x0 in the input scene, E�.� denotes sta-istical averaging, and the overbar symbol in the denomi-ator denotes spatial averaging over x. The secondriterion referred to as the peak-to-average output vari-nce (SNR) is defined as the ratio of the square of the ex-ected value of the correlation peak to the average outputariance [12]:

SNR = �E�y�x0,x0��2/Var�y�x,x0��, �2�

here Var{.} denotes the variance. We also consider theriterion of light efficiency [4], which is an important mea-ure in optical pattern recognition. It is defined as the ra-io of the total light intensity in the correlation plane tohe total light intensity in the input.

With the help of computer simulation we analyze theerformance of the obtained and conventional correlationlters in terms of discrimination capability [5] and loca-ion errors [7]. We assume that information about all deg-adations is available. In other words, parameters of ainear system, target degradation, and input noise can bestimated taking into account the nature of these degra-ations [30–36]. The presentation is organized as follows.n Section 2, new correlation filters for different models ofegradations are derived. Computer simulation resultsre presented and discussed in Section 3. The perfor-ance of the proposed filters is compared with that of

arious correlation filters and two-step cascade techniquerestoration and subsequent correlation filtering). We alsonvestigate the tolerance of the proposed filters to estima-ion errors of degradation parameters and input noise.ection 4 summarizes our conclusions.

. DESIGN OF FILTERSifferent causes produce blurred and partially blurred in-ut scenes during the image formation process. Since theerformance of conventional correlation filters degradesapidly with image distortions, we suggest to take into ac-ount three types of degradations, that is, (i) degradationy means of a linear system with a known real impulseesponse hLD�x� (it can be atmospheric turbulence, rela-ive motion between a recording camera and the scene,ptical defocusing, etc.), (ii) partial image blurring owingo a linear target degradation hTD�x� (for instance, motionf a target across a still background), and (iii) combina-ion of the two preceding degradations. Besides, we as-ume that a target is embedded into a background noise,nd input scenes are always corrupted by additive noisewing to imaging sensors and quantization errors [35,37].n this section one-dimensional notation is used for sim-licity, and integrals are taken between infinite limits.hroughout the paper we use the same notation for a ran-om process and its realization.

. Generalized Correlation Filters for Recognition of anndistorted Object in Noisy Scene Degraded with ainear Systemet us consider the nonoverlapping signal model. The in-ut scene s�x� degraded with a linear system contains aarget t�x� located at unknown coordinate x0, spatiallyisjoint background b�x ,x0�, and additive noise n�x�:

s�x,x0� = �t�x − x0� + b�x,x0�� • hLD�x� + n�x�, �3�

here “•” denotes the convolution operation, andhLD�x�dx=1.

The following notations and assumptions are used:

1. The nonoverlapping background signal b�x ,x0� is re-arded as a product of a realization b�x� from a stationaryandom process (with expected value �b) and an inverseupport function of the target w�x� defined as zero withinhe target area and unity elsewhere:

b�x,x0� = b�x�w�x,x0�. �4�

2. The coordinate x0 is considered as a random vari-ble.3. B0�� is the power spectral density of b0�x�

b�x�−�b.4. n�x� is a realization from a zero-mean, stationary

rocess with the power spectral density N��.5. T��, W��, and HLD�� are the Fourier transforms

f t�x�, w�x�, and hLD�x�, respectively. The filter output�x� is given by y�x ,x0�=s�x ,x0�•h�x�, where h�x� is a realmpulse response of a correlation filter to be designed.

�� is the transfer function of the filter.6. The stationary processes and the random target lo-

ation x0 are statistically independent of each other.

ext, from the model given in Eq. (3) we derive optimumorrelation filters. Actually, the obtained filters are modi-ed versions of the following correlation filters: the GOF12], GMF [12], and GPOF [18]. The transfer functions ofhe designed filters for recognition of an undistorted ob-ect in an input scene degraded with a linear system areeferred to as GOFLD, GMFLD, and GPOFLD, which areptimal with respect to the POE, SNR, and light effi-iency, respectively.

. Generalized Optimum Filter �GOFLD�he proposed GOFLD maximizes the POE given in Eq. (1).he filter maximizes the intensity correlation peak of a

arget and minimizes the average output-signal energy.rom Eq. (3) the expected value of the filter output�y�x ,x0�� can be expressed as

E�y�x,x0�� =1

2� �T�� + �bW��HLD��H��

�exp�j��x − x0��d�, �5�

nd the square of the expected value of the output peakan be written as

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uppose that the output-signal energy is finite (for in-tance, spatial extent of the filter output is L [12]); the de-ominator of the POE can computed as

E��y�x,x0��2� = Var�y�x,x0�� + �E�y�x,x0��2. �7�

ote that the spatial averaging converts a nonstationaryrocess at the filter output to a stationary process. Thisimplifies the derivation of the optimum filter. In Appen-ices A and B we show that the expressions for the aver-ge of the output-signal variance Var�y�x ,x0�� and the av-rage energy of the expected value of the filter outputE�y�x ,x0��2 are respectively given by

Var�y�x,x0�� =1

2� ��

2�B0�� • �W��2 �HLD��2

+ N��H��2d�, �8�

��+�bW��, which defines a new target to be detected.

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+ �bW��2�HLD��2�H��2d�, �9�

here ��1/L is a normalizing constant [7]. Substitutingqs. (8) and (9) into Eq. (7), we obtain the average outputnergy:

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+1

2�B0�� • �W��2 �HLD��2 + N��

��H��2d�. �10�

sing Eqs. (6) and (10) the POE can be written as

POE =

�2��−1 �T�� + �bW��HLD��H��d�2

��T�� + �bW��2 +1

2�B0�� • �W��2 �HLD��2 + N��H��2d�

. �11�

sing the Schwarz inequality in the last equation leads to the following expression for a filter that maximizes this cri-erion:

GOFLD�� =��T�� + �bW��HLD��*

��T�� + �bW��2 +1

2�B0�� • �W��2 �HLD��2 + N��

, �12�

here the asterisk denotes the complex conjugate. Thebtained filter takes into account information about theinear image degradation and additive noise by means of

LD�� and N��, respectively. It is of interest to note thathe minimum mean-squared-error filter [28] has the samexpression when the correlation filter is designed for de-ection of a target embedded into a background noise ininear environmental degradations.

Note that the transfer function of the GOFLD contains

n other words, the information about the target supportunction and mean value of a background is important asell as a target signal itself.

. Generalized Matched Filter �GMFLD�he filter maximizes the intensity correlation peak of a

arget and minimizes the average output-signal variance.sing Eqs. (2), (6), and (8), the SNR criterion can be writ-

en as

SNR =

�T�� + �bW��HLD��H��d�2

2� �� B0�� • �W��2 �HLD��2 + N��H��2d�

. �13�

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pplying the Schwarz inequality, we obtain an optimumorrelation filter that maximizes the SNR:

GMFLD�� =��T�� + �bW��HLD��*

� �

2�B0�� • �W��2 �HLD��2 + N��

.

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he GMFLD contains information about the linear imageegradation and additive noise. If the input scene is notegraded by a linear system, then the GMFLD is trans-ormed to the conventional GMF [7,12].

. Generalized Phase-Only Filter �GPOFLD�sing the signal model given in Eq. (3) and Eq. (B2), be-

ow, light efficiency can be expressed as

�H = �E�y�x,x0��2dx

�E�s�x,x0��2dx

= ��T�� + �bW��HLD��2�H��2d�

��T�� + �bW��HLD��2d�

. �15�

correlation filter yielding the maximum light efficiencys the GPOFLD, which is given by

GPOFLD�� =�T�� + �bW��*

�T�� + �bW��exp�− j�HLD

��, �16�

here �HLD�� is the phase distribution of the linear deg-

adation. It can be seen that the GPOFLD does not takento account degradation of the input scene by additiveoise. Therefore, it is expected that the filter will be sen-itive to the noise.

. Generalized Correlation Filters for Recognition of ainearly Distorted Object in Noisy Scene Degradedith a Linear Systemhe input scene contains a linearly distorted target lo-ated at unknown coordinate x0 and spatially disjointackground b�x ,x0�. The scene is also degraded with a lin-
ar system and corrupted by additive noise n�x�: g
2�

s�x,x0� = �t�x − x0� • hTD�x� + b�x�wTD�x − x0�� • hLD�x�

+ n�x�, �17�

here hTD�x� is a real impulse response of target degra-ation, �hTD�x�dx=1, wTD�x−x0�=1−wT�x−x0�•hTD�x�,nd wT�x� is a support function of the target defined asnity within the target area and zero elsewhere. We usehe same assumptions and notations as in Subsection 2.A.t is assumed that linear degradations of the target andhe scene do not affect each other. In a similar manner,hree generalized correlation filters for recognition of ainearly distorted object in an input scene degraded withlinear system are derived. The filters are optimized with

espect to the POE, SNR, and light efficiency, and theirransfer functions are referred to as GOFLD�TD,MFLD�TD, and GPOFLD�TD, respectively.

. Generalized Optimum Filter �GOFLD�TD�rom the scene model given in Eq. (17), the expectedalue of the filter output is expressed as

E�y�x,x0�� =1

2� �T��HTD��

+ �bWTD��HLD��H��exp�j��x − x0��d�,

�18�

here HTD�� and WTD�� are the Fourier transforms ofTD�x� and wTD�x�, respectively. The intensity correlationeak can be computed as follows:

�E�y�x0,x0��2 =1

4�2 �T��HTD��

+ �bWTD��HLD��H��d�2

. �19�

sing Appendices A and B, the expressions forar�y�x ,x0�� and �E�y�x ,x0��2 are obtained, respectively,s

Var�y�x,x0�� =1

2� ��

2�B0�� • �WTD��2 �HLD��2

+ N��H��2d�, �20�

�E�y�x,x0��2 =1

2� ��T��HTD��

+ �bWTD��2�HLD��2�H��2d�. �21�

ith the help of Eqs. (1), (7), and (19)–(21), the POE is
iven as follows:
POE =

�2��−1 �T��HTD�� + �bWTD��HLD��H��d�2

��T��HTD�� + �bWTD��2 +1

B0�� • �WTD��2 �HLD��2 + N��H��2d�

. �22�

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pplying the Schwartz inequality to Eq. (22), a generalized optimum filter is derived:

GOFLD�TD�� =��T��HTD�� + �bWTD��HLD��*

��T��HTD�� + �bWTD��2 +1

2�B0�� • �WTD��2 �HLD��2 + N��

. �23�

. Generalized Matched Filter �GMFLD�TD�
rom Eqs. (2), (19), and (20), the SNR can be expressed as
SNR =

�T��HTD�� + �bWTD��HLD��H��d�2

2� ��

2�B0�� • �WTD��2 �HLD��2 + N��H��2d�

. �24�

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se of the Schwartz inequality yields the following opti-um filter:

GMFLD�TD =��T��HTD�� + �bWTD��HLD��*

� �

2�B0�� • �WTD��2 �HLD��2 + N��

.

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. Generalized Phase Only Filter �GPOFLD�TD�rom the definition of light efficiency given in Eq. (15),

he scene model given in Eq. (17), and Eq. (B2) the GPOFs determined as

GPOFLD�TD�� =�T��HTD�� + �bWTD��*

�T��HTD�� + �bWTD��

�exp�− j�HLD��. �26�

ote that the filter does not take into account the corrup-ion of the input scene by additive noise.

. COMPUTER SIMULATIONSn this section we analyze the performance of the MSF1], POF [4], OF [5], GMF [12], GOF [12], GPOF [18], andhe proposed generalized filters in terms of discriminationapability (DC) and location accuracy for recognition of ei-her a target or a moving object embedded into test inputcenes degraded with a linear system. The DC is formallyefined [5] as the ability of a filter to distinguish a targetrom other different objects. If a target is embedded into aackground that contains false objects, then the DC cane expressed as follows:

DC = 1 −�CB�0��2

�CT�0��2, �27�

here CB is the maximum in the correlation plane overhe background area to be rejected, and CT is the maxi-um in the correlation plane over the area of target po-

ition. The area of target position is determined in thelose vicinity of the actual target location. The back-round area is complementary to the area of target posi-ion. The further accurate estimation of the target loca-

ion with correlation filters can be carried out [7]. In ouromputer simulations the area of target position is chosens the target area. Negative values of the DC indicatehat a tested filter fails to recognize the target. The loca-ion accuracy can be characterized by means of the loca-ion error (LE) defined as

LE = ��xT − xT�2 + �yT − yT�2, �28�

here �xT ,yT� and �xT , yT� are the coordinates of the tar-et exact position and the coordinates of the correlationeak taken as target position estimation, respectively. Allorrelation filters were implemented with the fast Fourierransform. To guarantee statistically correct results, 30tatistical trials of each experiment for different positionsf a target and realizations of random processes were per-ormed.

The size of images used in our experiments is 256256 pixels. The signal range is [0–1]. Figure 1(a) shows

ig. 1. (a) Test real input scene. (b) Test stochastic input scene.c) Objects used in experiments.

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test deterministic input scene. The scene contains twobjects with a similar shape and size (approximately 4428 pixels) but with different gray-level contents. The

arget is the upper butterfly. The mean value and thetandard deviation over the target area are 0.42 and 0.2,espectively. We use a real spatially inhomogeneous back-round. The mean value and the standard deviation ofhe background are 0.37 and 0.27, respectively. Figure(b) shows a test stochastic input scene with the samearget and false object. The disjoint background is a real-zation of a stationary colored random process with the

ean value of 0.37, the standard deviation of 0.27, andhe horizontal and vertical correlation coefficients �H=�V0.5. The target and the false object are shown in Fig.(c). It is of interest to note that parameters of the non-verlapping signal model are very important for a fairomparison of the performance of correlation filters.

Let us analyze the performance of correlation filters forecognition of a target in the test scene in Fig. 1(a). First,o degradations are used. Figure 2(a) shows the perfor-ance of common correlation filters in terms of the DChen the standard deviation of nonoverlapping back-round is fixed, while the mean value of the backgrounds varied. It can be seen that the correlation filters derivedrom the overlapping model (MSF, POF, and OF) have dif-erent intervals of the mean value where they fail to rec-gnize the target. In contrast, the generalized correlationlters defined for the nonoverlapping model (GMF, GOF,nd GPOF) are always able to recognize the target. Theeneralized filters begin to fail to recognize the target inhe given scene when the standard deviation of the back-round is higher than 0.8. Similar conclusions are ob-ained when the same target is embedded in a random

ig. 2. Performance of correlation filters in terms of DC whenhe standard deviation of nonoverlapping background is 0.27 andhe mean value is varied for: (a) deterministic scene in Fig. 1(a),b) stochastic scene in Fig. 1(b).

ackground (realization of either white or colored randomtationary process). Figure 2(b) presents the performancef correlation filters versus the mean value of a disjointandom background with a standard deviation of 0.27.ince the correlation filters MSF, POF, and OF are veryopular in pattern recognition, in our computer simula-ion we choose such parameters of the background whenhese filters do not fail to recognize the target. Therefore,he performance of all tested correlation filters for recog-ition of objects in degraded input scenes may be ana-

yzed and compared.

. Recognition of a Target in Linearly Degraded Scenesirst, test input scenes are homogeneously degraded withlinear system. An example of such linear degradation is

wing to a uniform image defocusing by a camera. As-ume that the impulse response of the blurring is an im-ulse disk with a diameter, say D. The values of D used inur experiments are 3, 7, 11, 15, 19, and 23 pixels. Sincedditive sensor’s noise is always present in input scenes,e corrupt the degraded test scenes by additive zero-ean white Gaussian noise with the standard deviation

n. The values of n are equal to 0.02, 0.04, 0.08, 0.12,.16, and 0.17. Figures 3(a) and 3(b) show examples of theest scene linearly degraded with D=7 and corrupted ad-itionally by overlapping noise with n=0.12. Next, we in-roduce and analyze three different strategies of patternecognition based on correlation operation in degradednd noisy scenes, that is, (i) direct object recognition withcorrelation filter, (ii) design of a correlation filter from a

inearly degraded version of a target and subsequent ob-ect recognition with the obtained filter, (iii) restoration ofhe input scene and subsequent object recognition with aorrelation filter in the restored scene.

. Direct Object Recognitionigures 4(a) and 4(b) show the performance of tested cor-elation filters with respect to the DC and the LE whenhe input real scene in Fig. 1(a) is defocused with differ-nt values of D. It can be seen that the proposed filtersOFLD and GMFLD are always able to detect and to local-

ze exactly (no location errors) the target, whereas the LEor other filters increases as a function of D. One can ob-erve that the GOFLD yields the best performance inerms of the used criteria. The proposed GPOFLD is sen-itive to the linear degradation. Although the conven-ional GOF yields a good performance for D19 with re-pect to the DC, it always has large location errors.esides, the MSF fails to recognize the object. Similaronclusions are obtained for the stochastic test scene de-ocused with the same values of D. Table 1 shows the bestnd worst performance of the filters with respect to theC and the LE when the input stochastic scene in Fig.(b) is used. Since the nonoverlapping background is spa-ially homogeneous the three proposed filters are able toecognize the object for all used values of D. In contrast,he other tested filters have much worse performance inerms of the DC and LE. Now we investigate robustnessf filters to additive noise. Figure 5 shows the tolerance ofhe filters for pattern recognition in blurred (with D=7nd 11) and noisy real test scenes when the standard de-iation of additive noise is varied. One can observe that

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he proposed filters GOFLD and GMFLD are always able toetect and localize the object with small location errors,hereas the performance of the other filters worsen rap-

dly as a function of D and n. Table 2 shows the best andorst performance of the filters with respect to the DCnd the LE. It can be seen that the robustness of allested filters is better when the scene background is a re-lization of a stationary (spatially homogeneous) randomrocess. The performance of the GOF and GPOFLD dete-iorates rapidly when signal noise fluctuation increases.he proposed GOFLD and GMFLD provide a good toler-nce to the noise. As it is expected, the GMFLD yieldsinimal location errors for all values of the standard de-

iation [7].Finally, we investigate the tolerance of the proposed fil-

ers to incorrect estimation of the degradation parameter. Assume that the focus blur is space invariant within

he image. It is known that point-spread functions for mo-ion and focus blurs do have zeros in the frequency do-ain, and they can be uniquely identified by the location

f these zero crossings. Therefore, the parameter D can bevaluated by means of the cepstrum analysis of the powerpectrum of the observed signal [30]. We assume also thatnput noise is a zero-mean, white Gaussian process that isncorrelated to the image signal. In this case, the noiseeld is completely characterized by its variance, which isommonly estimated by the sample variance computed

ig. 3. Test scenes defocused with D=7 and corrupted addition-lly by white noise with n=0.12: (a) scene in Fig. 1(a), (b) scenen Fig. 1(b).

ig. 4. Performance of correlation filters when the input real sceb) LE versus D.

ver a low-contrast local region of the observed image36]. The objective of our test is to respond to the followinguestion: “Is the recognition process sensitive to mis-atching of values of D used in the filter design and in

he degradation of the input scene?” We carried out manyxperiments by varying the parameter D in the designrocess and scene degradations. One can conclude thatmall errors (up to 4 pixels) in the parameter estimationoes not effect in a qualitative sense to the performance ofhe proposed filters. Figures 6(a) and 6(b) show the per-ormance of the filters designed with D=7 for pattern rec-gnition in the deterministic scene blurred with differentalues of D. Figures 6(c) and 6(d) illustrate the perfor-ance of the filters designed with D=7 for pattern recog-ition in blurred with D=3 and noisy deterministic testcene when the standard deviation of additive noise isaried. Numerous computer simulation experiments werearried out. We can conclude that the GOFLD filter is veryolerant to estimation errors of the parameter D. On thether hand, if the blurring parameter is small �D4�,ommon generalized filters may be used for reliable pat-ern recognition in blurred and noisy scenes.

Table 1. Filter Performance with Respect to DCand LE for Recognition of Target in Degraded

Scene

BestPerformance

WorstPerformance

Values of DWhen a Filter IsAble to Recognize

the TargetaDC LE DC LE

F 0.354 1 0.057 2.84 {3,7}SF 0.033 0.88 0.033 0.88 {3}OF 0.413 0.15 0.413 0.15 {3}OF 0.869 1 0.080 5.76 {3,7,11,15}MF 0.009 1 0.002 1 {3,7}POF 0.691 0 0.127 1.03 {3,7,11}OFLD 0.958 0 0.935 0 {3,7,11,15,19,23}MFLD 0.043 0 0.043 0 {3,7,11,15,19,23}POFLD 0.762 0 0.165 0 {3,7,11,15,19,23}

aThe values of D are 3, 7, 11, 15, 19, and 23.

ig. 1(a) is defocused with different values of D: (a) DC versus D,
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GGGGG

F D=1


. Recognition of Degraded Objectirst, we design modified correlation filters based on theSF, POF, OF, GMF, GOF, and GPOF, which are matched

o a linearly degraded version of the target. The targetegradation is the same as in the test input scene. Figureshows the performance of correlation filters with respect

o the DC and the LE when the deterministic input scenen Fig. 1(a) is defocused and corrupted by additive noise.t may be seen that the OF and the POF have a slightlyetter recognition performance compared with that of theame filters in the preceding approach (see Fig. 5). TheF is able to recognize the object when D7 and n0.03. The MSF fails to detect the object. Other filters

Table 2. Filter Performance with Respect to DCand LE for Recognition of Target in Noisy and

Degraded Scene with D=11

BestPerformance

WorstPerformance

Values of n

Where the FilterCan Recognize


OF 0.130 8.03 0.130 8.03 {0.02}POF 0.111 2.72 0.075 3.31 {0.02,0.04}OFLD 0.784 0 0.260 2.36 {0.02,0.04,…0.17}MFLD 0.006 0 0.001 2.34 {0.02,0.04,…0.17}POFLD 0.360 0.05 0.090 3.78 {0.02,0.04,0.08}

aThe values of n are 0.02, 0.04, 0.08, 0.12, 0.16, and 0.17.

ig. 5. Tolerance of correlation filters for pattern recognition in bn with D=7, (c) DC versus n with D=11, (d) LE versus n with

re able to recognize the object when the degradation isow �D3�. However, we can see that the recognition re-ults obtained with this approach are essentially worseompared with those of the proposed filters in the firsttrategy.

. Object Recognition in Restored Scenen this case image restoration is incorporated in recogni-ion process. Assume that degradation parameters are ei-her known or can be estimated using a priori information30–36]. One of the most popular restoration techniques is

linear minimum mean-square error (LMMSE) method.t finds the linear estimate of the ideal image for whichhe mean-square error between the estimate and thedeal image is minimum. The linear operator acting onhe observed image to determine the estimate is obtainedn the basis of a priori second-order statistical informa-ion about the image and noise processes. In the case oftationary processes and space-invariant blurs, theMMSE estimator takes the form of the Wiener filter

35]. The frequency response of the Wiener filter HW��an be written as

HW�� =Ss�� − N��

Ss��HLD��, �29�

here Ss�� is the power spectral density of the observedegraded input scene. First, the ideal Wiener restorationwith known degradation parameters) is performed, and

and noisy real scenes: (a) DC versus n with D=7, (b) LE versus1.

lurred

Fvb

F=


ig. 6. Performance of proposed filters designed with D=7 for pattern recognition in the deterministic scene blurred with differentalues of D: (a) DC versus D, (b) LE versus D, (c) DC versus n when the scene blurred with D=3, (d) LE versus n when the scene
lurred with D=3.
ig. 7. Recognition of blurred object in defocused and noisy deterministic scene: (a) DC versus n with D=3, (b) LE versus n with D3, (c) DC versus with D=7, (d) LE versus with D=7.
n n

ttmr

cflforttphpma

FcD

Fm

Fcm


hen common tested filters are used for pattern recogni-ion in the restored scene. Figure 8 shows the perfor-ance of correlation filters with respect to the used crite-

ia when the input scene in Fig. 1(a) is blurred and

ig. 8. Performance of correlation filters for pattern recognitioorrupted by additive noise: (a) DC versus n with D=7, (b) LE v=11.

ig. 9. Illustration of linear degradation by a uniform targetotion in 3 pixels from left to right.

orrupted by additive noise. It can be seen that if noiseuctuation is low then the restoration improves the per-ormance of the conventional correlation filters in termsf the both criteria compared with that of the direct objectecognition strategy (see Fig. 5). It is remarkable that thiswo-step procedure exploiting even ideal Wiener restora-ion and conventional filters yields worse results com-ared with those of the proposed filters. Thus, with theelp of a simple example, we demonstrate that for betterattern recognition in degraded scenes the restorationust be automatically included in the transfer function offilter by optimizing recognition criteria.

stored deterministic scene, which was originally defocused andn with D=7, (c) DC versus n with D=11, (d) LE versus n with

ig. 10. Test scenes in Fig. 1 degraded with M=9 and D=7 andorrupted additionally by white noise with n=0.12: (a) deter-inistic scene, (b) stochastic scene.

n in reersus

BSLbodsl

Tsa�miwMedttue3psc

fbmwwmGs=rtt

chocsr1

nOtmdiatugteant�mm

GGGGGG

FL


. Recognition of a Moving Target in Linearly Degradedceneset us consider a uniform target motion across a fixedackground. For clarity and simplicity, we assume that anbject moves from left to right with a constant velocity Vuring a time capture interval of �0,T�. The impulse re-ponse of the target degradation can be expressed as fol-ows [30]:

hTM�x� = �1/M, if 0 x M = VT

0, otherwise � . �30�

he target motion leads to a partial blur of the inputcene. Figure 9 illustrates this degradation when a targetnd a background are one-dimensional discrete signalst1 , . . . t6� and �b1 , . . .b9�, respectively, and the targetoves in 3 pixels from left to right. The input signal s�r�

s formed as average of intermediate sequences s��r�. Nowe consider a uniform target motion from left to right onpixels in test scenes. The values of M used in computer

xperiments are 3, 5, 7, 9, 11, and 15 pixels. Moreover, weegraded homogeneously test scenes containing a movingarget with a linear system. Assume that the linear sys-em degradation is the same as in Subsection 3.A. Thus, aniform image defocusing can be characterized by a diam-ter, say D. The values of D used in our experiments are, 7, 9, 15, 19, and 23 pixels. Figures 10(a) and 10(b)resent the test scenes shown in Figures 1(a) and 1(b), re-pectively, degraded with M=9 and D=7 and additionallyorrupted by overlapping noise with n=0.12.

First, let us analyze the performance of tested filtersor recognition of a moving object in still undistortedackground. Figures 11(a) and 11(b) show the perfor-ance of correlation filters with respect to the DC and LEhen the input scene in Fig. 1(a) contains a moving targetith different values of M. In this case generalized opti-um filters referred to as the GMFTD, GOFTD, andPOFTD are obtained from Eqs. (23), (25), and (26), re-

pectively, by substituting into these equations HLD��1. One can observe that the MSF, POF, and OF fail toecognize the moving object. The performance of conven-ional generalized filters deteriorates quickly in terms ofhe DC and the LE when the target displacement in-

ig. 11. Performance of correlation filters for recognition of a mE versus M for the scene.

reases. In contrast, the obtained GOFTD and the GMFTDave no location errors, and they are always able to rec-gnize the target. The GPOFTD fails when M�9. Similaronclusions are obtained for the stochastic scene. Table 3hows the best and worst performance of the filters withespect to the DC and the LE when the input scene in Fig.(b) contains a moving target.Next, recognition of a moving object in defocused and

oisy test scenes is carried out. Since the MSF, POF, andF are not able to recognize a moving object in undis-

orted scenes, they are excluded from the analysis. Nu-erous computer simulations were performed. When the

eterministic scene shown in Fig. 1(a) is used, the follow-ng observations are obtained. The GMFLD�TD is alwaysble to detect and to localize the moving object for anyested values of M, D, and n. However, it yields low val-es of the DC. The proposed GOFLD�TD provides a veryood performance in terms of the DC and the LE when ei-her n0.03 or M5 or D5. The performance of gen-ralized filters for M=5, 9 and D=7 in terms of the DCnd LE as a function of the standard deviation of additiveoise for the deterministic scene is shown in Fig. 12. Inhis case the GOFLD�TD fails to detect the object when M9 and n�0.12 [see Fig. 12(c)]. When the stochastic ho-ogeneous scene shown in Fig. 1(b) is used, the perfor-ance of the filters is slightly better. The performance of

Table 3. Filter Performance with Respect to DCand LE for Recognition of Moving Target in Input

Scene

BestPerformance

WorstPerformance

Values of MWhere the FilterCan Recognize


OF 0.321 1.8 0.321 1.8 {3}MF 0.003 1.8 0.003 1.8 {3}POF 0.417 1.25 0.094 2.75 {3,5}OFLD 0.727 0 0.429 0.15 {3,5,7,9,11,15}MFLD 0.006 0 0.001 0.10 {3,5,7,9,11,15}POFLD 0.521 0 0.001 0.26 {3,5,7,9}

aThe values of M are 3, 5, 7, 9, 11, and 15.

arget in the input scene shown in Fig. 1(a): (a) DC versus M, (b)
oving t

F

Fw


ig. 12. Recognition of a moving object in defocused with D=7 and noisy deterministic scene: (a) DC versus n with M=5, (b) LE versus
n with M=5, (c) DC versus n with M=9, (d) LE versus n with M=9.
ig. 13. Recognition of a moving object in defocused with D=7 and noisy stochastic scene: (a) DC versus n with M=5, (b) LE versus nith M=5, (c) DC versus with M=9, (d) LE versus with M=9.
n n

tLstftgrpm

4Ntogerrtntawbutpsmpdi

avrnp

AIasa=npc

Nwa=dae

wRaf

Ti

wh

xtctm

Tambs

Itdps


he filters for M=5, 9 and D=7 in terms of the DC and theE versus the standard deviation of additive noise for thetochastic scene is shown in Fig. 13. One can observe thathe GOFLD�TD and GMFLD�TD are able to detect the objector any tested values of M, D, and n. Finally, note thathe task of recognition of a moving object in a linearly de-raded and noisy scene is very difficult when even all pa-ameters of degradations are correctly estimated. Theroposed GMFLD�TD and GOFLD�TD yield a good perfor-ance in terms of the used criteria subject.

. CONCLUSIONSew correlation filters were proposed to improve recogni-

ion of a linearly distorted object embedded into a non-verlapping background noise when the input scene is de-raded with a linear system. The obtained filters takexplicitly into account information about an object to beecognized, disjoint background noise, linear system deg-adation, and linear target distortion. We assumed thathis information is available or can be estimated from theature of degradations. Therefore, the proposed filters es-ablish upper bounds of pattern recognition qualitymong correlation filters with respect to the used criteriahen the input scene and the target are degraded. It haseen shown that a two-step procedure (ideal restorationsing known degradation information and subsequent op-imum correlation filtering) yields worse results com-ared with those of the proposed filters. The computerimulation results demonstrated superiority in the perfor-ance of the proposed filters for pattern recognition com-

ared with that of various correlation filters in terms ofiscrimination capability, location errors, and tolerance tonput additive noise.

Finally, note that if a priori information about scenend target degradations is not available, then either con-entional generalized filters (for a low level of linear deg-adations) or a linear combination (synthetic discrimi-ant function filters [8,9]) of the proposed filters forossible representative degradations can be used.

PPENDIX An this appendix we derive an expression for the spatialverage of the output-signal variance when the inputcene is degraded with a linear system and corrupted bydditive noise. Let rb�x ,x0�=b�x ,x0�•hu�x� and rn�x�n�x�•h�x� be random correlation outputs of b�x ,x0� and�x�, respectively, with hu�x�=hLD�x�•h�x�. Since the out-ut random processes are independent of each other, wean write

Var�y�x,x0�� = Var�rb�x,x0�� + Var�rn�x��. �A1�

ote that rn�x� is a zero-mean, stationary random processith the power spectral density Yn��=N��H��2nd with the constant variance Var�rn�x��2��–1�N��Hu��2d�. Therefore, the spatial averageoes not affect to this variance. The output-signal vari-nce of the nonstationary random process rb�x ,x0� may bexpressed as

Var�rb�x,x0�� = E��rb�x,x0� − E�rb�x,x0��2�

= E� b0�x − ��w�x − x0 − ��hu��b0�x

− �w�x − x0

− �hu� ��*d�d � , �A2�

here the asterisk denotes the complex conjugate. Letb�x� be the correlation function of a zero-mean, station-ry random process b0�x�. We can rewrite Eq. (A2) in theollowing way:

Var�rb�x,x0�� = Rb� − ��w�x − x0 − ��hu��w�x − x0

− �hu� ��*d�d = �Rb�� • �hu��w�x − x0

− �� • �hu�− ��w�x − x0 + ��*��=0. �A3�

aking Fourier transforms from the relevant quantitiesn Eq. (A3) we obtain

Var�rb�x,x0��

=1

2� B0�� 1

2�Hu�� • W��exp�j�x − x0��

� � 1

2�Hu�� • W��exp�j�x − x0��*

d�

=1

�2��3 B0��Hu��1� � Hu*��2�W��1 − ��W*��2

− ��exp�j�x − x0��1 − �2��d�d�1d�2, �A4�

here Hu��=HLD��H�� is the Fourier transform ofu�x�.On spatial average over an area L→� with respect to

−x0, exp�j�x−x0��1−�2�� is close to the Dirac delta func-ion, say 2��1−�2��, where ��1/L is a normalizingonstant. Using the sifting property of the delta function,he average power spectral density of rb�x ,x0� can be esti-ated as follows:

Yb�� = � �

2�B0�� • �W��2 �Hu��2. �A5�

herefore, the spatial averaging converts the nonstation-ry process rb�x ,x0� to a stationary process. This alsoeans that the input nonstationary random process

0�x�w�x−x0� becomes a stationary process with the powerpectral density

Yb�� =�

2�B0�� • �W��2. �A6�

f a background noise is a zero-mean, overlapping process,hen W��=2��, and with L→� the power spectralensity given in Eq. (A5) is simplified to the known ex-ression for a stationary random process passed a linearystem, that is, Y ��=B ��H ��2. Finally, from Eqs.
b 0 u

(v

AIatrfc

Ae

Ite

R

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

3

3


A1) and (A5) the spatial average of the output-signalariance can be written as

Var�y�x,x0�� = Var�rb�x,x0�� + Var�rn�x��

=1

2� ��

2�B0�� • �W��2��HLD��2

+ N�� H��2d�. �A7�

PPENDIX Bn this appendix we derive an expression for the spatialverage of the energy of the output expected value whenhe input scene is degraded with a linear system and cor-upted by zero-mean, additive noise. We use the notationsrom Appendix A. The square of the output expected valuean be written as

�E�y�x,x0��2 = ��t�x − x0� + �bw�x − x0�� • hu�x��2. �B1�

pplying the Parseval theorem, we obtain the output en-rgy as

�E�y�x,x0��2dx =1

2� �T�� + �bW��2�Hu��2d�.

�B2�

t is clear that the output energy does not depend on thearget position x0. Finally, the average of the output en-rgy can be written as

�E�y�x,x0��2 =1

L �E�y�x,x0��2dx

=1

2� ��T�� + �bW��2�HLD��2�H��2d�.

�B3�

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Documents

Design of correlation filters for recognition of linearly distorted objects in linearly degraded scenes