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Optical Engineering 48�6�, 067201 �June 2009�

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onlinear composite filter performance

icardo Enrique Guerrero-Morenoosué Álvarez-Borregoentro de Investigación Científica y de Educación

Superior de Ensenadapplied Physics Division/Optics Departmentm 107 Carretera Tijuana-Ensenadansenada, Baja California 22860exico-mail: [email protected]

Abstract. We use nonlinear composite filters in object recognition,even when they have rotation, scale, noise, and illumination distortions.We generated 936 images of the letters E, F, H, P, and B. The imagesconsisted of these letters scaled from 70% to 130% and rotated 360 deg.The maximum number of images supported by these filters was deter-mined by a numerical experiment. Considering a system confidencelevel of at least 80%, the maximum number of images is around 216. Wefound a “rotation problem” when the filter contained the letter rotated360 deg, since circles were artificially introduced, and this creates com-plications when working with images that also have circles in their spec-trum. Due to this, we propose a segmented filter that breaks the circularsymmetry. Experiments where carried out in order to find the noise tol-erance of each filter, and the use of Spearman’s rank correlation �inconjunction with the nonlinear method, SNM� is proposed in order toincrease that tolerance. We also made an assessment of the impact thatillumination changes had in the correlation output, in the problem image,and we propose the use of SNM to obtain illumination invariance. Wetested these filters with two real-life problems; nonlinear composite filterscan recognize the target in the presence of distortions. © 2009 Society ofPhoto-Optical Instrumentation Engineers. �DOI: 10.1117/1.3152777�

Subject terms: linear correlation; nonlinear filter; discrimination coefficient; pat-tern recognition; Spearman correlation.

Paper 090089PR received Feb. 5, 2009; revised manuscript received Apr. 14,2009; accepted for publication Apr. 16, 2009; published online Jun. 26, 2009. Thispaper is a revision of a paper presented at the SPIE conference on Applicationsin Digital Image Processing XXXI, August 2008, San Diego, California. The paperpresented there appears �unrefereed� in SPIE Proceedings Vol. 7073.

Introduction

igital correlation systems with linear filters are an attrac-ive solution in pattern recognition since they posses thenherent ability not only to establish whether an object isresent, but also to determine its location within the inputcene. Most of these types of filters do not work well withven small distortions of the target to be recognized; if anbject varies in size, rotation, or illumination, they will, ineneral, have a poor performance. In this way, there existany approaches that have tried to solve the problem of

ize, rotation,1–5 and illumination.6,7 However, in recentears, there has been a lot of effort in the study and devel-pment of invariant correlation systems using linear andonlinear filters. The nonlinear filters have advantagesompared with the classical matched filter,8 the phase-onlylter,9 and other linear filters; due to their great capacity toiscriminate objects, the maximum value of the correlationeak is well localized, and the output plane is less noisy.10

o different contributions have been proposed for compos-te filters in order to be used for distortion-invariant sys-ems in the field of pattern recognition and image analysis.ne of the latest techniques used is the adaptive syntheticiscriminant functions �ASDF�,11 also used in an opticalystem.12 This kind of filter had a very good performance todentify the target when different kinds of noise wereresent in the input scene. More composite filters have been

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presented in different invariant digital systems.13,14 In thispaper, we will focus on examining the capabilities of k-lawnonlinear composite filters as a method of achieving rota-tion, scale, illumination, and/or noise invariance. A com-puter was used to run simulations of the performance of thefilter with different amounts of distortions. This was doneusing rotated and scaled images of different letters. First,we obtain the maximum number of images supported bythe filter through a numerical experiment. When we con-sider rotated images around 360 deg, a problem of rotationsymmetry arises. A segmented filter is proposed in order tobreak this circular symmetry. Experiments with noise werealso done in order to find out their tolerance to it, and anonlinear rank correlation is used to improve noise toler-ance. Examples of filter performance, when the input imagehas mixed distortions present, are shown. We also made anevaluation of the filter’s performance when the input imagehad different levels of contrast �illumination�, and we pro-pose the use of the nonlinear Spearman’s rank correlation�in conjunction with the nonlinear method, SNM� in orderto achieve illumination invariance. Last, in order to showthe wide range of applications of these filters, they wereused with two real-life problems, discriminating betweendifferent copepod species �tiny crustaceous� and identifyingand locating, within specifications, the CPU of a computersound card.

This paper is organized as follows: In Sec. 2, the k-lawnonlinear filtering technique is described. Section 3 showsthe basic construction of a composite nonlinear filter. The

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Correlation plane

Non-linear operations


TargetFourierPlane

ProductInverseFouriertransformInput scene

Fig. 1 Diagram of the necessary operations to obtain the output plane of the nonlinear recognitionsystem.

Correlation plane


Synthesized non-linear composite filterNon-linearoperations

Training images Input scene

{}⋅ℑ{}⋅ℑ−1

{}{}⋅ℑ*

Fig. 2 Diagram of the operations necessary to synthesize a nonlinear composite filter and its appli-cation to an input scene.

Fig. 3 Confidence level of different type A filters.


Fig. 4 Confidence level of different type B filters.

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etric used in this work is described in Sec. 4. The com-uter simulations and results are shown in Sec. 5, and last,ur conclusions are given in Sec. 6.

K-Law Nonlinear Correlation Filtershe system presented in this work is based on a nonlinearrocessor, which performs several nonlinear correlationsetween an input scene and different reference targets. Ineneral, we can express a nonlinear filter �NF� as:15

F = �F�u,v��k exp�− i��u,v��, 0 � k � 1, �1�

here F�u ,v� is the Fourier transform of the object weant to recognize, � is the phase of the target, �·� is theodulus of F�u ,v�, and k is the nonlinearity strength. By

hanging the value of k to 1, we get a classical matchedlter �CMF�, 0 for a phase-only filter �POF�, and −1 for an

nverse filter �IF�.When a nonlinear operator modifies the Fourier trans-

orms of both the scene and the target, we consider therocessor to be a nonlinear correlator.16 Intermediate valuesf k permit us to vary the features of the processor, such asts discrimination capabilities or its illumination invariance.

ig. 5 Input scenes �left� and correlation between them and a 216-mage composite filter type B of the letter E �right�.

(a) (b)

Fig. 6 Letter E’s spectrum �a�, composite filterspectrum �d�.


In Fig. 1, we show, in a diagram, the necessary operationsto obtain the output plane of the nonlinear recognition sys-tem.

3 Nonlinear Composite FiltersAs mentioned, matched filters are not distortion tolerant—ifa problem image presents different distortions, as in mostreal-life cases, we must look for a different approach. Analternative is to use nonlinear composite filters that can beoptimized in order to make them less sensible to scale,rotation, and/or illumination variations.17,18 Composite fil-ters are created from representative images of the objectthat are called training images. The proper selection andgrouping of these training images is of the highest impor-tance in composite filter design. In general, the trainingimages considered in the filter design are variations inscale, rotation, or/and illumination of the target to recog-nize. Figure 2 shows a diagram of the necessary operationsto synthesize a nonlinear composite filter and its applicationto an input scene. In particular, since the training imagesconsist of possible scales of the object, the synthesis of ascale-invariant filter for the letter B is shown.

4 Metric Used in Performance EvaluationThere are several metrics for correlation filter performanceevaluation. In this work, the discrimination coefficient �alsoknown as discrimination capability� was used, which is for-mally defined19 as the ability of a filter to distinguish a

(c) (d)

trum �b�, letter B’s spectrum �c�, and letter P’s

Fig. 7 Correlations between the nonlinear composite filter and theletter E at 70% and all of its rotations.

’s spec

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arget among other different objects. If we consider that anbject is embedded in a noise background, the discrimina-ion coefficient would be given by

C = 1 −�CN�0,0��2

�COBJ�0,0��2, �2�

here the correlation peak produced by the object would beOBJ, and the highest peak of just the noise backgroundould be CN.

Fig. 8 Input scene �a�, correlation between inpuand SBF2 �c�, and binarized and summed corre

Fig. 9 Performance of type A filters in the pr


In this work, we refer to the confidence level of a givenfilter as the probability of confidence to recognize the targetin the input scene.

5 Computer Simulations and ResultsTo determine the maximum number of supported images bya nonlinear composite filter, first, 936 images of the lettersE, F, H, P, and B in Arial font were generated. Theseimages consisted of each of the letters at 13 different scales,

and SBF1 �b�, correlation between input scene�Figs. 8�b� and 8�c�� d�.

of Gaussian noise �a� and S&PP noise �b�.

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hich range from 70 to 130% in increments of 5%, and 72ifferent angles, which range the full 360 deg in incrementsf 5 deg. Then a numerical experiment was done, whichonsisted of generating different composite nonlinear filtersith diverse numbers of images contained within them,

omputing the correlation for each of the images containedn the filters, as well as the equivalent images of the otheretters, and obtaining the confidence level of each of theseomposite nonlinear filters. A nonlinearity value k of 0.1as been considered as optimum, since it offers the besterformance for this type of filters.16 Because we have 936

...

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Σ

Training images

{}{}⋅ℑ−1

Fig. 10 Diagram of operations to perform

Fig. 11 Performance of type A filters in the pres�a�, 26 �b�, and 39 �c� images.


images of each letter in 13 different scales and 72 differentangles, respectively, tests were done with two types of non-linear composite filters: one where all the different scaleswere present, and we added more images of the letter atdifferent angles to increase the number of images, whichwe will call type A; and another where all of the angleswere present, and more scales were added to increase thenumber of images, which we will call type B. In Fig. 3, weshow the result of these experiments for the first case, typeA filter, and in Fig. 4, we show the results for the secondcase, type B filter.

{}{}⋅ℑ

Non-linearoperations

{}⋅ℑ−1

Spearman’s RankCorrelation

Input image

Correlation plane

C with a nonlinear composite filter, SNM.

f additive Gaussian noise, using the SNM of 13

an SR

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According to a study of Culverhouse et al.,20 taxono-ists can vary their characterization performance depend-

ng on several factors, but on average, their accuracy isround 80%. Since the final objective of this work is topply this method to microorganisms, we consider everylter with a performance above 80% to be a “good” filter.n Fig. 3, we can see that the maximum number of imagesupported by the nonlinear composite filter, type A, isround 430, which is the worst case scenario �which in thisase is the letter E�. In Fig. 4 �type B filter�, we can see thathis number is reduced to around 216 images.

Some tests were done using a nonlinear composite filter

Fig. 12 Performance of type A filters in the pres26 �b�, 39 �c�, and 52 �d� images.

(b) (c) (d)

(e) (f) (h)

(a)

ig. 13 Target free of noise �a�; with additive Gaussian noise ofean zero and variance of 0.1 �b�, 0.2 �c�, and 0.3 �d�; and with&PP noise of noise density of 0.1 �e�, 0.2 �f�, and 0.3 �g�.


of the letter E, with the maximum amount of images, de-termined by the worst case scenario of both cases �Fig. 4�,which was 216. The tests consisted of input scenes contain-ing two letters, one true object and one false. Figure 5shows the input scene and the correlation results betweenthem and the filter mentioned earlier.

We can clearly appreciate that the nonlinear compositefilter works perfectly when trying to discriminate betweenthe letter E �contained in the filter� and the letter F �notcontained in the filter, but very similar to E�. However itdoes not perform well for the letter P: As we can see, thereis a lot of noise in the correlation plane where the letter P

f additive S&PP noise, using the SNM of 13 �a�,

Fig. 14 Input scene with multiple true and false objects, as well asadditive Gaussian noise of variance 0.1 and mean 0 �a�, and corre-lation plane using a 13-image type A filter of the letter E �b�.

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as located, and a small peak at the letter E’s location.evertheless, this behavior occurred for any other letterith curves on it �P, J, U, D, G, etc, results not shown�.So why does this happen? It is comprehensible that the

lter could mistake P with E—after all, they are similar—ut letters like J, U, or Q: should not be a problem. Butemember how we built our nonlinear composite filter: 216mages of the letter E at three different scales and 72 dif-erent angles that go around 360 deg. So by including im-ges of the letter rotated 360 deg, we are artificially intro-ucing circles in our nonlinear composite filter, and thisould introduce first-order Bessel functions in its spectrum,

imilar to what we would expect to see in the spectrum ofetters with curves on them. Figure 6 further illustrates this.

In order to analyze the problem, in Fig. 7, we plot theorrelations between the nonlinear composite filter and theetter E at 70% and all of its rotations. As we can see in Fig., every 45 deg, starting from zero, there is a minimum inhe correlation. So in order to break up that “circular sym-

etry” artificially created, a segmented filter is proposed,hich consist of two parts, one containing the target at

very 45 deg�5 deg �72 images� and another containinghe rest of them �144 images�, which we will respectivelyall SFB1 and SFB2.

In Fig. 8, the results of using the segmented filtersSFB1 and SFB2� on the same image that caused problems

ig. 15 Input scene with multiple true and false objects, as well asdditive Gaussian noise of variance 0.025 and mean 0 �a�, andorrelation plane using a 26-image multiobject type A filter of the

etters E and H �b�.

Fig. 16 Illumination invariance performance usiand a 39-image filter �b�.


before are shown. The results were significantly improved.Similar results can be produced with other letters �resultsnot shown�. A threshold to Figs. 8�b� and 8�c� was applied,and the results were binarized and summed in order to ob-tain the final result in Fig. 8�d�.

Experiments where carried out in order to determine thelimitations of the type A filters when the input scene hadadditive Gaussian or salt and pepper �S&PP� noise. Figure9 shows the mean discrimination coefficient for differentfilters, with an upper and lower limit indicating 95% levelof confidence, with 49 statistical samples. The noise addedhad a mean zero and a variance or density as shown in thegraph. In Eq. �2�, we can see that the discrimination coef-ficient �DC� will be above zero only if the correlation peakof the target is bigger than the correlation peak of the noise.This means that with a positive value of DC, the target isrecognized in the input scene. A 13-image filter can supporta noise variance of around 0.14, and a noise density ofaround 0.15, for additive Gaussian and S&PP noise, respec-tively, while a 104-image filter can support a noise varianceand density of only around 0.02 and 0.01, respectively. It isevident that with higher amounts of images in the filter, itsperformance is drastically reduced.

Another way to calculate the correlation between twoimages with nonlinear methods is using rank statistics. Sub-stituting the value of each pixel for its corresponding rank,in the normalized correlation expression, a nonlinear corre-lation expression is obtained �Spearman’s rank correlation�SRC��. Taking this into consideration, the 2-D SRC can beexpressed as:

R�k,l� = 1 −6 · �m,n�W�rt�m,n� − rs�m + k,n + l��2

�W� · ��W�2 − 1�, �3�

where rt is the filter image, rs is the problem or scene im-age, W is the total number of pixels, R is the Spearman’srank correlation, and m, n, k, and l are space coordinates.The SRC was performed in the spatial domain. The opera-tions used in order to create a nonlinear composite filter andcorrelate it with an input scene using the SRC, are shown inFig. 10.

correlation and the SNM of a 13-image filter �a�
ng the
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We carried out experiments, using the type A filters andifferent amounts of additive Gaussian and S&PP noise.he results are shown in Figs. 11 and 12. The performancef a 13-image filter using the SRC is significantly im-roved; the noise tolerance is doubled in both the Gaussiannd S&PP case. As we increase the number of images con-ained in the filter, this improvement gets diminished untilhe use of the SRC is no longer practical, since it offers nomprovement over the regular correlation. Figure 13 showshe input scene, with different amounts of additive Gauss-an and S&PP noise.

We also conducted experiments in which there were dif-erent distortions present in the input scene at the sameime. In Fig. 14�a�, we show a scene with multiple trueargets of different size �letter E�, additive Gaussian noisef variance 0.1 and mean 0, as well as multiple false ob-ects. The result of using a 13-image type A filter �of theetter E at 13 different scales� is shown in Fig. 14�b�. Threearge peaks in the correlation plane at the position of therue targets are seen; therefore, the 13-image filter has pro-ided scale invariance. Another test was done, similar tohe previous, except that this time we considered a multi-bject filter of the letters E and H at 13 different scales �26mages�. The input scene Fig. 15�a� is the same as Fig.3�a� except that the noise has a variance of 0.025. Theesults are shown in Fig. 15�b� and show four peaks in theorrelation plane at the position of the true targets.

During the image acquisition process, there can be illu-ination fluctuation from image to image due to sun posi-

ion, artificial light fluctuations, cloud coverage, etc. Be-ause of these, the incorporation of some type ofllumination invariance to a filter can be highly desirable. Inhis work, we simulated illumination fluctuations in an im-ge by changing the contrast in it. For example, if our im-ges where composed of 0s and 1s �0 being black and 1hite�, we would say that we have a 100%contrast, while ife would have 0s and 0.5s �again, 0 being black and 1hite�, we would say we have a 50% contrast. Tests wereone in order to see the filter’s performance with variousypes of contrasts. As would be expected, the energy quan-ity in the correlation plane depends on its contrast �theore contrast, the more energy�, but since just the contrast

f the test image gets changed, the energy distribution inhe correlation plane is the same for all the different con-rasts. The use of the DC is unpractical because it wouldever change; hence, for the evaluation of the filter’s per-ormance, we instead normalized the correlation values tohe value of the correlation peak at 100% contrast. Figure6�a� shows the performance of a 13-image filter with testmages �Fig. 17� that have different contrasts using the cor-elation �line and asterisks� and using the SNM �line andircles�. Figure 16�b� shows the same information as Fig.

(a) (b) (c) (d)

ig. 17 Test image at 100% �a�, 75% �b�, 50% �c�, and 25% �d�ontrast.


16�a�, except that for a 39-image filter, we can clearly seethat the graphics are the same—they differ from each otheronly in the values of DC. Since the SRC converts the valueof the image pixels to their corresponding rank, the contrastin the image really does not matter, since the ranks wouldbe the same, and therefore, the use of the SNM achievesillumination invariance.

In order to validate the functionality of these types offilters, we conducted experiments with two real-life ex-amples. The first real-life experiment was conducted with aset of 420 test images of copepods �tiny crustaceans�,which consisted of 7 species, each with 30 male and 30female samples �see Fig. 18�. A nonlinear composite filterof each sex and species was generated �14 in total� andcompared with the whole set of images. Since the back-ground noise in all the images is mostly repetitive, the im-ages where cleaned using MATLAB functions found inGonzalez et al.21 Some of the results for this experiment areshown in Fig. 19.

For example, Figs. 19�a� and 19�b� show the proper clas-sification of the whole set of images of the copepod speciesCalanus pacificus female and male, respectively. Figures19�c� and 19�d� are the results for the Pleuromamma gra-cilis species, and Figs. 19�e� and 19�f� are the results for theCentropages hamatus. From the complete set of resultgraphs, we notice that the system not only discriminatesbetween species, but it also distinguishes between sex, andin each case, the confidence level is 100%.

The second real-life experiment carried out consisted ofidentifying and locating the CPU of a computer sound cardto within specification. Considering that a system like thiscould be easily implemented in a production line, the tol-erance specifications for this filter where a �5 degrotation—since the position of the camera could be fixed,no scale invariance would be needed. Therefore, a nonlin-ear composite filter of the CPU at all possible rotations, 11images in total �considering steps of 1 deg�, was created.For this, we first took a picture of the sound card, and fromthis image, we extracted the CPU and made rotated ver-sions of it for the construction of the CPU filter. We thenproceeded to take pictures of the sound card with differentilluminations �overexposed and underexposed� and at dif-ferent angles �in order to simulate the placement of the

(c) (d)

(g) (h) (k) (l)

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(i) (j)

(m) (n)

Fig. 18 Images of copepods. Calanus pacificus female �a� andmale �b�; Rhincalanus nasutus female �c� and male �d�; Cen-tropages furcatus female �e� and male �f�; Pleuromamma gracilisfemale �g� and male �h�; Temora discaudata female �i� and male �j�;Acartia tonsa female �k� and male �l�; and Centropages hamatusfemale �m� and male �n�.

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Correlation of different copepod species (cleaned images)

Mean±SE±2*SEExtremes

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Fig. 19 Some results of the correlation of different copepod species, using a nonlinear composite filtercomposed of the images of a single species’ sex.

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CPU at an angle�. Some of the results in this experiment areshow in the Figs. 20–23. The results show that the filter isable to identify and locate the CPU of the sound card evenwhen there are extreme illumination variations.

6 ConclusionsIn this work, we present a deep analysis of k-law compositenonlinear filter advantages and disadvantages. Knowing theproperties of such types of filter would help us better designsystems for pattern recognition. The system was tested us-ing five alphabet letters thought to be very similar in Arialfont as targets. We found that a maximum of around 216images for our so-called type B filter was possible. It wasshown that the proposed segmented filter improves the fil-ter’s performance when dealing with images rotated aroundthe 360-deg range. We also found that the filter’s toleranceto noise quickly decreases along with the increase of im-ages contained in the filter, and the use of the SNM wasproven as an effective method for better noise tolerancewhen the filter contained a relatively small amount of im-ages. We also show the decrease in the correlation peak asthe contrast decreases in the target, and the use of the SNM�Fig. 10� was proved to be an effective method to achieveillumination invariance. Nonlinear composite filters whereused effectively with real-life images in the discriminationof different copepod species and even genders �with a100% confidence level� and to identify and locate the CPUof a computer sound card even with extreme illuminationvariations and some rotation invariance ��5 deg�.

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ig. 20 Original sound card image �a� and its correlation with thePU filter �b�.

ig. 21 Underexposed sound card image �a� and its correlation withhe CPU filter �b�.

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ig. 23 Rotated �within specifications� sound card image �a� and itsorrelation with the CPU filter �b�.

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1. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Pro-cessing Using MATLAB, Prentice Hall, Upper Saddle River, NJ�2004�.

Ricardo Enrique Guerrero-Moreno re-ceived his BSc in electronic engineeringfrom Universidad Autonoma de Baja Califor-nia �UABC�, Mexico, in 2005, and his MScdegree in optics from Centro de Investi-gación Científica y de Educación Superiorde Ensenada �CICESE�, Mexico, in 2008.His research interests include image pro-cessing and pattern recognition.

ptical Engineering 067201-1

Josué Álvarez-Borrego received his BSdegree in physical oceanography in 1981from the Facultad de Ciencias Marinas,Ensenada, Baja California, Mexico, and hisMSc and PhD degrees in optics in 1983 and1993 from CICESE, Mexico. He is a titularresearcher with the Applied Physics Divi-sion, Optics Department, CICESE, Mexico.His research interests are image processingwith applications to biogenic particles andimage processing of the sea surface. He is

the author of several scientific papers and presentations at scientificconferences. He is a member of the Mexican Academy of Optics,National Research System �SNI�, and the Mexican Science Acad-emy.

June 2009/Vol. 48�6�1

http://dx.doi.org/10.1364/AO.28.002358

http://dx.doi.org/10.1117/1.601319

http://dx.doi.org/10.1364/AO.41.002172

http://dx.doi.org/10.3354/meps247017

nonlinear composite ﬁlter performanceusuario.cicese.mx/~josue/pdf/guerrero-alvarez_2009.pdf ·...

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