Target recognition under nonuniform illumination conditions

Target recognition under nonuniformillumination conditions

Victor H. Diaz-Ramirez1,* and Vitaly Kober2

1Centro de Investigacion y Desarrollo de Tecnologia Digital, Instituto Politecnico Nacional, Avenida del Parque No. 1310, Mesade Otay, Tijuana, B.C., 22510, Mexico

2Computer Science Department, Centro de Investigacíon Científica y de Educación Superior de Ensenada,Km 107 Carretera Tijuana-Ensenada, Ensenada B. C. 22860, Mexico

*Corresponding author: [email protected]

Received 9 July 2008; accepted 19 January 2009;posted 23 January 2009 (Doc. ID 98458); published 26 February 2009

A two-step algorithm for reliable recognition of a target imbedded into a two-dimensional nonuniformlyilluminated and noisy scene is presented. The input scene is preprocessed with a space-domain pointwiseprocedure followed by an optimum correlation. The preprocessing is based on an estimate of the sourceillumination function, whereas the correlation filter is optimized with respect to the mean-squared-errorcriterion for detecting a target in the preprocessed scene. Computer simulations are provided and com-pared with those of various common techniques in terms of recognition performance and tolerance tononuniform illumination as well as to additive noise. Experimental optodigital results are also providedand discussed. © 2009 Optical Society of America

OCIS codes: 070.4550, 070.5010, 100.3008.

1. Introduction

Pattern recognition using the correlation operationhas been vastly used in the last decades because itcan be easily implemented in real-time optodigitalprocessors [1–3].However, theperformance of thepro-cessors degrades rapidly when input signals are dis-torted owing to imperfection of imaging sensors,quantization errors, and nonuniform illuminationconditions. Various correlation filters were proposedfor reliable detection of a target in the presence of ad-ditive and disjoint noise [4–7]. On the other hand, afew techniques were proposed for pattern recognitionunderdifferent illumination conditions [8–11]; that is,input scenes are affected by multiplicative nonsta-tionary interference signals. Arsenault and Lefebvre[8] suggested homomorphic cameo filtering (HCF). Itconsists of a logarithmic prepossessing of an inputscene followed by a synthetic discriminant function(SDF) filter [12,13]. The filtering yields good results

in the absence of additive sensor noise. Another fruit-ful approach isaweighted sliced orthogonalnonlineargeneralized (WSONG) correlation [10]. In thismethodthe correlation output is a sum of weighted correla-tions between binary images obtained by thresholddecomposition of the input scene. A drawback of themethod is its high computational complexity. An opti-mum correlation filter for detecting a target em-bedded into a spatially uniform multiplicative andadditive interference signals was proposed [5]. How-ever, for spatially inhomogeneous disturbances thisfilter may yield a poor performance. Recently, Valleset al. [14] presented an illumination-invariant techni-que for 3D object recognition. The method takes intoaccount reflectance properties of surfaces and physi-cal relation between a distant light-source and the ob-ject surface.Actually, reflectancemodels describehowinput scene signals are related to the light-source di-rection [15]. If illuminant direction parameters areknown, an estimation of the illumination functioncan be carried out [16].

In this paper, we propose a two-step algorithmfor reliable recognition of a target under nonuniform

0003-6935/09/071408-11$15.00/0© 2009 Optical Society of America

1408 APPLIED OPTICS / Vol. 48, No. 7 / 1 March 2009

illumination conditions. First, a pointwise preproces-sing of an input scene using an estimate of the illu-mination function is carried out. Next, a minimummean squared-error filter for pattern recognition isdesigned [6,17]. The paper is organized as follows.In Section 2 we present a brief review of common cor-relation techniques for pattern recognition undervariant illumination conditions. In Section 3 reflec-tance models are recalled. In Section 4 we describethe proposed method. In Section 5 computer simula-tions and experimental optodigital results areprovided and discussed. Finally, Section 6 sum-marizes our conclusions.

2. Related Works

Here basic strategies for illumination-invariant pat-tern recognition are described. We consider the HCF[8] and the WSONG correlation [10]. Since any arbi-trary illumination function can be modeled as amultiplicative noise function, an optimum filter de-sign (OF) [5] is also reviewed. For simplicity, one-dimensional notation is used. Let tðxÞ and f ðxÞ bea target signal and an input scene, respectively. Sup-pose that the scene is nonuniformly illuminated.

A. Homomorphic Cameo Filtering

Assume that the target size is sufficiently small, sothe target is approximately uniformly illuminated ata position x0:

~tðx − x0Þ ¼ btðx − x0Þ; ð1Þwhere b is an unknown illumination constant. TheHCF performs a logarithmic transformation of theinput scene to convert the multiplicative model tothe additive one:

log½f ðxÞ� ¼ log½tðx − x0Þ� þ log½b�: ð2ÞNext, a cameo correlation filter for the target

detection is designed. The impulse response of thecameo filter hcmðxÞ can be expressed as

hcmðxÞ ¼ a1 log½tðxÞ� þ a2 log½wtðxÞ�; ð3Þwhere a1 and a2 are coefficients to be determined,and wtðxÞ is a binary function defined as unity insidethe target area and zero elsewhere. Note that hcmðxÞcorresponds to the impulse response of a conven-tional SDF filter [12]. The impulse response of thefilter is synthesized in such a way to yield prespeci-fied central correlation outputs in the response tolog½wtðxÞ� and log½tðxÞ�, respectively.

B. Weighted Sliced Orthogonal Nonlinear GeneralizedCorrelation

According to the threshold decomposition concept[18], a quantized signal can be expressed as a sumof binary slices:

aðxÞ ¼XN−1

i¼0

iBINi½aðxÞ�; ð4Þ

where N is the number of gray levels of the signal,and “BIN” is defined as

BINi½aðxÞ� ¼�1; aðxÞ ¼ i0; aðxÞ ≠ i

: ð5Þ

The correlation “⊗” between images f ðxÞ and gðxÞcan be computed as

f ðxÞ ⊗ gðxÞ ¼XN−1

i¼0

XN−1

j¼1

ijBINi½f ðxÞ� ⊗ BINj½gðxÞ�: ð6Þ

The WSONG correlation is defined as follows [10]:

Ωf ;gðxÞ ¼XN−1

i¼0

XN−1

j¼1

WijBINi½f ðxÞ� ⊗ BINj½gðxÞ�; ð7Þ

where WijðxÞ is a matrix of weighting coefficients,which are able to take into account nonuniform illu-mination conditions [10].

C. Optimum Filtering

Assume that the target is imbedded into a nonover-lapping background at unknown coordinate x0. Theinput scene is corrupted by a stationary multiplica-tive function uðxÞ and a zero-mean additive noisenðxÞ:

f ðxÞ ¼ uðxÞ½tðx − x0Þ þ wtðx − x0ÞbðxÞ� þ nðxÞ; ð8Þ

where wtðxÞ is an inverse support function defined aszero within the target area and unity elsewhere. It isassumed that the random variable x0 is uniformlydistributed over the scene area, white additive noisenðxÞ and multiplicative interference uðxÞ are statisti-cally independent of each other. The OF is optimizedwith respect to the peak-to-output-energy [19]:

Hof ðωÞ ¼μm½TðωÞ þ μbWtðωÞ��

12πNmðωÞ ⊗

�jTðωÞ þ μbWtðωÞj2 þ 1

2πNb0ðωÞ ⊗ jWtðωÞj2 þNðωÞ� ; ð9Þ

1 March 2009 / Vol. 48, No. 7 / APPLIED OPTICS 1409

where WtðωÞ is the Fourier transform of wtðxÞ,μm ¼ EfuðxÞg, and μb ¼ EfbðxÞg; NmðωÞ, NðωÞ, andNb0ðωÞ are power spectral densities of uðxÞ, nðxÞ,and b0ðxÞ ¼ bðxÞ − μb, respectively.

3. Illumination Models

Reflectance models describe relations between the il-luminant direction and the surface shape. Surfacescan be Lambertian, specular, or hybrid (combinationof both) [15]. Figure 1 shows a geometrical model of asurface being illuminated by a light source. The zaxis is an observation point. In the Lambertian mod-el, surfaces reflect the light in all directions withequal amplitudes [20]: IL ¼ cosðθÞ, where θ is the in-cident angle, i.e., the angle between the surface-normal vector N and the illuminant direction vectorS (see Fig. 1). Specular surfaces reflect the light as amirror. They can be described as follows [20]: IS ¼δðθs − θÞ, where δðxÞ is the Dirac delta function,

θs ¼ θN − θv, θN is the angle between N and the z axis,and θv is the angle between N and the viewer’s direc-tion. Hybrid surfaces are combinations of Lamber-

tian and specular reflection components: IH ¼ k1ILþk2IS, where k1 þ k2 ¼ 1. Different hybrid reflectancemodels were proposed [21–23]. An extended reviewof illumination reflectance models can be found in[20,24,25]. We use the Lambertian reflectance model.It can be seen from Fig. 1 that the illuminant direc-tion is determined by the following parameters: τ(slant angle), α (tilt angle), and r (magnitude of vectorS0). These parameters define the position of the lightsource with respect to the surface origin. For a Lam-bertian surface, the reflected light from a pointðx0; y0Þ is given by

Iðx0; y0Þ ¼ cosðθÞ ¼ cos�π2− β

�; ð10Þ

where β is

β ¼ arctan�

Sz

½ðSx − x0Þ2 þ ðSy − y0Þ2�1=2�¼ arctan

�r

cosðτÞ½ðr × tan τ × cos α − x0Þ2 þ ðr × tan τ × sin α − y0Þ2�1=2�:

ð11Þ

Substituting Eq. (11) into Eq. (10), the reflected light is given by

Iðx0; y0Þ ¼ cos�π2− arctan

�r

cosðτÞ½ðr × tan τ × cos α − x0Þ2 þ ðr × tan τ × sin α − y0Þ2�1=2��

: ð12Þ

Note that Iðx0; y0Þ depends on the illuminant direc-tion parameters τ, α, and r. In real applications theseparameters are either known or can be estimated.Zheng and Chellappa [16] proposed a technique forestimating τ, α, and the surface albedo ρ from anobserved image as follows:

α ¼ arctan

0BBB@E

�~xL=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~x2L þ ~y2L

q �

E

�~yL=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~x2L þ ~y2L

q �1CCCA;

�~xL~yL

�¼ ðBtBÞ−1Bt

2666664

δI1δI2...

δIN

3777775; ð13Þ

where ð~xL; ~yLÞ are the ðx; yÞ components of the tilt’slocal estimate, δIi is the derivative of the image in-tensity in the ðδxi; δyiÞ direction, and B is the N × 2direction matrix. Slant τ is evaluated by solvingthe nonlinear equation:Fig. 1. Illumination model geometry.


EfIgEfI2g ¼ 0:5577þ 0:6240 cos τ þ 0:1882 cos2 τ

− 0:6514 cos3 τ − 0:5350 cos4 τþ 0:9282 cos5 τ þ 0:3476 cos6 τ− 0:4984 cos7 τ: ð14Þ

Finally, ρ can be computed as

ρ ¼ EfIgf 1ðτÞ þ ðEfI2gf 2ðτÞÞ1=2f 21ðτÞ þ f 2ðτÞ

; ð15Þ

where f 1ðτÞ ≈P

7i¼0 aicosiτ, and f 2ðτÞ ≈

P7i¼0 bi cos

i τ[16]. The coefficients are a0 ¼ 0:1615, a1 ¼ 0:3959,a2 ¼ 0:3757, a3 ¼ −0:0392, a4 ¼ −0:3077, a5 ¼0:1174, a6 ¼ 0:1803,a7 ¼ −0:0984, b0 ¼ 0:0834, b1 ¼0:2169, b2 ¼ 0:2487, b3 ¼ 0:1836, b4 ¼ 0:0048,b5 ¼ −0:1086, b6 ¼ −0:0043, and b7 ¼ 0:0424.

4. Pattern Recognition Under Nonuniform IlluminationConditions

Here we use notations from Section 2. The inputscene f ðxÞ is degraded by a nonuniform illuminationfunction uðxÞ and corrupted by a zero-mean additivenoise nðxÞ. Therefore, the observed signal can be alsodescribed by Eq. (8). Let us rewrite the signal modelas follows:

f ðxÞ ¼ ~tðx; x0ÞuðxÞ þ ~nðx; x0Þ; ð16Þ

where ~tðx; x0Þ is a new target signal defined as

~tðx; x0Þ ¼ tðx − x0Þ þ μbwtðx − x0Þ; ð17Þ

and ~nðx; x0Þ is a new noise function given by

~nðx; x0Þ ¼ nðxÞ þ b0ðxÞwtðx − x0ÞuðxÞ: ð18Þ

In the Fourier domain, Eq. (16) can be written as

FðωÞ ¼ 12π

~TðωÞ ⊗ UðωÞ þ ~NðωÞ; ð19Þ

where ~TðωÞ, UðωÞ, and ~NðωÞ are the Fourier trans-forms of ~tðx; x0Þ, uðxÞ, and ~nðx; x0Þ, respectively. Thelatter expression can be interpreted as follows: theobserved signal FðωÞ contains the target ~TðωÞ de-graded by a linear system with impulse responseUðωÞ and additive noise ~NðωÞ. First, an optimum lin-ear filter for restoring ~TðωÞ in Eq. (19) is designed. Itcan be done with the help of the Wiener filtering [26]in the spatial domain. Let hrðxÞ be a restoration filterin the spatial domain. Therefore, the restored signal~f ðxÞ ¼ f ðxÞhrðxÞ can be expressed as

~f ðxÞ ≈ ~tðx; x0Þ þ ~nðx; x0ÞhrðxÞ; ð20Þ

with

hrðxÞ ¼u�ðxÞP~tðxÞ

P~tðxÞjuðxÞj2 þ P~nðxÞ¼ 1

uðxÞjuðxÞj2

juðxÞj2 þΛðxÞ ;

ð21Þ

where ΛðxÞ ¼ P~nðxÞ=P~tðxÞ is the noise-to-signal ratioat the coordinate x, P~tðxÞ and P~nðxÞ are estimates ofthe power spectra in the spatial domain of ~TðωÞ and~NðωÞ, respectively. Since the signals ~tðx; x0Þ and~nðx; x0Þ are nonstationary, we assume that x0 is ran-dom and uniformly distributed over the scene area.So, P~tðxÞ and P~nðxÞ are evaluated as averages overthe random variable x0. It is interesting to note thatthe signals ~tðx; x0Þ and ~nðx; x0Þ are converted to sta-tionary signals by applying a statistical averagingwith respect to x0. Since we use the Lambertian sur-face model, the illumination function uðxÞ can be es-timated with the help of Eq. (12) for any τ, γ, and r.The restored image [see Eq. (20)] is still degradedwith additive nonstationary noise. Next, an optimumcorrelation filter for detection of a new target is de-signed. We are looking for a filter that yields a de-sired delta-function output at the position of thetarget, i.e., yd ¼ δðx − x0Þ. This filter can be obtainedby minimizing the mean-squared-error (MSE) [6] asfollows:

MSE ¼Z

∞

−∞

EfjydðxÞ − yaðxÞj2gdx

¼Z

∞

−∞

Z∞

−∞

hcðμÞhcðνÞC~tðν − μÞdμdν

− 2Z

∞

−∞

hcðμÞ~tðμÞdμ

þZ

∞

−∞

Z∞

−∞

hcðμÞhcðνÞChrðν − μÞC~nðν − μÞdμdν

þZ

∞

−∞

δ2ðx − x0Þdx; ð22Þ

where ya ¼ ~f ðxÞ ⊗ hcðxÞ is the filer output, hcðxÞ isthe impulse response of the optimum correlation fil-ter, and C~tðxÞ, Chr

ðxÞ, and C~nðxÞ are autocorrelationfunctions of~tðxÞ, hrðxÞ, and ~nðxÞ, respectively. The lat-ter equation can be written in the Fourier domain as

MSE ¼ 12π

Z∞

−∞

jHcðωÞj2StðωÞdω

þ 12π

Z∞

−∞

jHcðωÞj2SnðωÞdω

−1π

Z∞

−∞

HcðωÞTðωÞdωþ c; ð23Þ

where StðωÞ, and SnðωÞ are the power spectral den-sities of ~tðxÞ and ~nðxÞ, respectively, and c is aconstant. Finally, after some manipulations the fre-quency response of the optimum filter is given by


HcðωÞ ¼½TðωÞ þ μbWðωÞ��

jTðωÞ þ μbWðωÞj2 þ 12πB0ðωÞ ⊗ jWðωÞj2 þ 1

2πNðωÞ ⊗ jHrðωÞj2; ð24Þ

where B0ðωÞ and NðωÞ are the power spectral densi-ties of b0ðxÞ and nðxÞ, respectively. The obtained filtertakes into account information about nonuniform il-lumination byHrðωÞ. A block diagram of the proposedmethod is shown in Fig. 2.

5. Results

A. Computer Simulations

Here we analyze the performance of the proposedmethod in terms of discrimination capability (DC)and tolerance to both nonuniform illumination andinput additive noise. The DC is formally defined asthe ability of a filter to distinguish a target fromother objects. If a target is embedded into a back-ground that contains false objects, then the DCcan be expressed as follows:

DC ¼ 1 −jCBð0Þj2jCTð0Þj2 ; ð25Þ

where jCBj2 is the intensity maximum in the correla-tion plane over the background area to be rejected,and jCT j2 is the intensity maximum in the correla-tion plane over the target area. The target area is de-termined in the close vicinity of the actual targetlocation. The background area is complementary tothe target area. Negative values of the DC indicatethat a tested filter fails to recognize the target.

The results obtained with the proposed method arecompared with those obtained with the HCF [8],the WSONG correlation [10], the OF [5], and thehomomorphic MSE filtering (HMSE). The lattermethod consists of a logarithmic preprocessing ofan input scene followed by a linear correlation opti-mized with respect to the MSE criterion. If a target isembedded into a background, the input signal can bewritten as

f ðxÞ ¼ ½tðx − x0Þ þ wtðx − x0ÞbðxÞ�uðxÞ: ð26Þ

The model ignores additive noise. The logarithmicpreprocessed scene f logðxÞ is written as

f logðxÞ ¼ log½tðx − x0Þ þ bðxÞwtðx − x0Þ� þ log uðxÞ:ð27Þ

Since target and background signals are mutuallyexclusive, then Eq. (27) can be rewritten as

f logðxÞ ¼ log½tðx − x0Þ� þ log½bðxÞwtðx − x0ÞuðxÞ�¼ tlogðx; x0Þ þ nlogðxÞ; ð28Þ

where tlogðx; x0Þ ¼ log½tðx − x0Þ� and nlogðxÞ ¼ log½bðxÞwtðx − x0ÞuðxÞ� are new target and noise functions, re-spectively. Finally, following the procedure describedin Section 4, the frequency response of the optimumcorrelation filter is given by

HhmseðωÞ ¼T�

logðωÞjTlogðωÞj2 þ SNlog

ðωÞ ; ð29Þ

where TlogðωÞ and SNlogðωÞ are the Fourier transform

of tlogðx; x0Þ and the power spectral density of nlogðnÞ,

Fig. 2. Block diagram of the proposed method.

Fig. 3. (a) Input scene. (b) Illumination function. (c) Observed scene.


respectively. The size of all gray-scale images used inour experiments is 256 × 256pixels. The signal rangeis ½0–255�. Figure 3(a) shows a test scene. The meanvalue and standard deviation of the background are75 and 52, respectively. The target is a fish placed atunknown coordinates. The target area is about 76×34pixels. The mean value and standard deviationof the target are 63 and 49, respectively. Nonuniformillumination is modeled by means of the Lambertianfunction given in Eq. (12) with τ ¼ 800, γ ¼ 450,and r ¼ 0:2. This illumination function is shown inFig. 3(b). The observed scene is obtained from thetest scene by applying the nonuniform illuminationand adding a zero-mean noise with a standard devia-tion σn ¼ 16. Figure 3(c) shows a nonuniformly

Fig. 4. (Color online) Correlation intensity planes obtained with: (a) HCF, (b) WSONG, (c) HMSE, (d) OF.

Fig. 5. Preprocessed scene obtained with the proposed method.Fig. 6. (Color online) Correlation intensity plane obtained withthe proposed method.


Fig. 7. Observed scenes with different levels of nonuniform illumination degradation: (a) low severity, (b) medium severity, (c) hardseverity. Illumination functions used in observed scenes with: (d) low severity, (e) medium severity, (f) hard severity.

Fig. 8. Recognition performance of tested filters: (a) HMSE, (b) OF, (c) proposed method. Corresponding standard deviations obtainedwith: (d) HMSE, (e) OF, (f) proposed method.


illuminated and noisy input scene. Intensity correla-tion planes obtained with the HCM, theWSONG, theHMSE, and the OFare shown in Fig. 4. Note that theOF is able to recognize the target, whereas the othermethods fail to detect the target. Next, we evaluatethe recognition performance of the proposed method.The illuminant direction parameters are assumed to

be known. A preprocessed scene obtained from theobserved scene by applying the pointwise restorationfiltering given in Eq. (21) is shown in Fig. 5. The in-tensity correlation output of the optimum MSE filterin Eq. (24) is presented in Fig. 6. One may observethat the target can be easily detected. Note that re-cognition of a target placed in an input scene at un-known coordinates under nonuniform illuminationconditions is equivalent to detection of a set of tar-gets with different mean values and standard devia-tions. In other words, the signal parameters dependon the target position. Therefore, the DC of a correla-tion filter is a function of target coordinates. Next, weanalyze the recognition performance of the HMSE,the OF, and the suggested method in the test scenewhen the parameters of nonuniform illuminationand the standard deviation of additive noise σn arevaried. The HCFand the WSONG are not consideredbecause they yield a poor performance in noisy inputscenes. Figure 7(a) shows an input scene degradedwith a low severity of distortion; that is, illuminantparameters: τ ¼ α ¼ 0, r ¼ 1:1 and σn ¼ 1:0.Figure 7(b) shows the scene degraded with a mediumseverity, i.e., r ¼ 0:4 and σn ¼ 9:0. Figure 7(c) shows ahardly degraded scene with parameters of distortion:r ¼ 0:1 and σn ¼ 18:0. Figures 7(d)–7(f) show nonuni-form illumination functions used in Figs. 7(a)–7(c).We vary r and σn while τ and α are fixed. 30 statisticaltrials for each varying parameter (4320 trials in to-tal) were carried out. Figure 8 show results obtainedwith the tested methods. The DC values vary fromblack color (DC ≤ 0) to white color (DC ¼ 1). From

Fig. 9. Nonuniformly illuminated scene containing: (a) small target, (c) enlarged target. (b) Correlation plane profile obtained from (a).(d) Correlation plane profile obtained from (c).

Table 1. DC Obtained with Different Illuminant Parameters Versus Real Parameters “uorg”a

Illuminant parameters σn ¼ 2 σn ¼ 4 σn ¼ 8 σn ¼ 16

uorg: τ ¼ 70°, α ¼ 35°, γ ¼ 0:4 0:89� 0:004 0:87� 0:004 0:85� 0:004 0:84� 0:006τ ¼ 67°, α ¼ 35°, γ ¼ 0:4 0:87� 0:004 0:85� 0:004 0:84� 0:005 0:81� 0:007τ ¼ 73°, α ¼ 35°, γ ¼ 0:4 0:89� 0:004 0:86� 0:007 0:85� 0:006 0:84� 0:006τ ¼ 70°, α ¼ 32°, γ ¼ 0:4 0:89� 0:003 0:86� 0:005 0:84� 0:006 0:83� 0:007τ ¼ 70°, α ¼ 38°, γ ¼ 0:4 0:89� 0:004 0:88� 0:003 0:86� 0:005 0:85� 0:006τ ¼ 70°, α ¼ 35°, γ ¼ 0:6 0:85� 0:010 0:84� 0:011 0:81� 0:014 0:80� 0:012τ ¼ 70°, α ¼ 35°, γ ¼ 0:2 0:80� 0:025 0:75� 0:033 0:76� 0:029 0:75� 0:025τ ¼ 73°, α ¼ 32°, γ ¼ 0:4 0:89� 0:004 0:86� 0:006 0:85� 0:007 0:83� 0:009τ ¼ 67°, α ¼ 38°, γ ¼ 0:4 0:88� 0:005 0:86� 0:005 0:85� 0:006 0:83� 0:007τ ¼ 73°, α ¼ 38°, γ ¼ 0:6 0:82� 0:015 0:80� 0:017 0:80� 0:019 0:77� 0:026

aWith 95% confidence.

Fig. 10. Optical setup used for generation of observed scene withnonuniform illumination.


Fig. 8(a) it can be seen that the black area(correlation filter fails to recognize the target) isabout 39% of the total area. We see that the nonuni-form illumination performs mapping of the statisti-cal parameters from the recognition area to afailure one. Figures 8(d)–8(f) shows the standarddeviations of the corresponding DC values inFigs. 8(a)–8(c), respectively. The HMSE methodyields a high variation of the DC owing to a logarith-mic preprocessing of a noisy input scene. The OFyields a failure area of about 7.6%. However,Fig. 8(e) shows that this method also possesses a highvariability of its performance in terms of the DC. Theproposedmethod yields the best andmost robust per-formance with respect to the DC. In other words, theproposed method is tolerant to nonuniform illumina-tion as well as to input additive noise.One can also observe that after applying nonuni-

form illumination to an input scene, the signal insidethe target area remains approximately homoge-neous. Strictly speaking, this assumption is onlyvalid when the target area is small and the illumina-tion function is smooth. Actually, the performance ofa filter under nonuniform illumination conditions de-teriorates quickly when the target size enlarges.Figures 9(a) and 9(c) show input scenes containingtwo nonuniformly illuminated versions of the target.We see that the point light source distorts the largertarget more. So, it is expected that the recognition

performance of a spatially invariant operation (corre-lation) decreases when the size of a target increases.Figures 9(b) and 9(d) show the profile of maximumcolumn values in the correlation plane of the OFfor recognition of two versions of the target undernonuniform illumination conditions. Finally, robust-ness of the proposedmethod to variation of estimatedilluminant parameters is investigated. We carriedout 30 statistical trials (1200 in total) for differentpositions of a target and additive noise realizationswith σn ¼ 8, while varying the illuminant para-meters: τ, α, and r. The results are presented inTable 1. It can be seen that the method is robustto a variation of the illumination parameters.

B. Optodigital Results

For the test scene shown in Fig. 3(a) under nonuni-form illumination conditions experimental optodigi-tal results were obtained. The observed scene wasgenerated with the help of the optical setup shownin Fig. 10. The test scene was printed in the 2D Lam-bertian surface (mate paper sheet) by means of a la-ser printer. The point light source was locatedaccording to the parameters: τ ¼ 840, α ¼ 550, r ¼20 cmwith respect to the surface origin. The obtainedobserved scene is shown in Fig. 11(a). We estimated[27] that the standard deviation of additive whitenoise is σn ¼ 7. The estimated illumination functionis shown in Fig. 11(b). The preprocessed image is

Fig. 11. (a) Experimental observed scene. (b) Estimated illumination function. (c) Preprocessed image.

Fig. 12. Optical setup of joint transform correlator.


shown in Fig. 11(c). Next, for the target detection weused the joint transform correlator (JTC) [2]. The op-tical setup is shown in Fig. 12. Since the frequencyresponse of the MSE correlation filter given inEq. (24) is a fully complex-valued function, the refer-ence image in the spatial domain is a real-valuedbipolar image. This means that the referenceimage cannot be directly displayed in conventionalamplitude-only spatial light modulators (SLM). Toovercome this problem we used the bipolar decompo-sition method [28]. The method requires two opticalcorrelations and a simple postprocessing. The resul-tant joint input images are shown in Fig. 13. Finally,the intensity correlation plane obtained with theproposed method is shown in Fig. 14. We see thatthe target can be easily detected with a value ofDC ¼ 0:71.

6. Conclusions

A new method for reliable pattern recognition undernonuniform illumination conditions was presented.The method performs an optimum pointwise pre-processing in the spatial domain followed by an

Fig. 13. Joint input images for real experiments: (a) positive part of the reference image, (b) negative part of the reference image.

Fig. 14. Experimental intensity correlation plane obtained withthe proposed method.


optimum correlation filtering in the Fourier domain.Computer simulation results showed that the pro-posed method yields good results for pattern recogni-tion under nonuniform illumination conditions, andit is robust to additive sensor’s noise. A good accor-dance between computer simulation and experimen-tal results was obtained.

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Target recognition under nonuniform illumination conditions