analytical relationship between the performance of sdf ﬁlt ...€¦ · scenes. in correlation...

Analytical relationship between the performance of SDF filters and

correlation peak constraints

MOHAMED I ALKANHALComputer and Electronics Research Institute

King Abdulaziz City for Science and TechnologyPO Box 6086Riyadh 11442

SAUDI [email protected]

Abstract: The correlation outputs of synthetic discriminant function (SDF) filters are constrained in response totraining images. In this paper, we establish an analytical connection between the constraint values and resultingcorrelation filter performance. An interesting result fromthis analysis is that we show that linear phase correlationpeak constraints are attractive.

Key–Words:Synthetic discrimination function (SDF), correlation filters, image recognition, convex hull.

1 Introduction

Two dimensional correlation filters are methodsproposed to detect, locate and classify objects inscenes. In correlation filters, the input scene is cross-correlated with a carefully designed correlation tem-plate and the resulting output is searched for largepeaks. Correlation filters are attractive for patternrecognition because of their shift invariance and po-tential for distortion. A significant advance in thedesign of correlation filters was the equal correlationpeak (ECP) synthetic discriminant function (SDF) fil-ters introduced by Hester and Casasent [1]. The ECPSDF synthesizes the filter template as a weightedsum of training images where the weights are cho-sen such that the resulting correlation output valuesat the origin (more commonly referred to as correla-tion peaks) take on pre-specified values. From thatsimple start, the SDF filter design has evolved sig-nificantly including the minimum average correlationenergy (MACE) filter [2], optimal trade-off SDF fil-ter [3], maximum average correlation height (MACH)filter [4], extended MACH (EMACH) [6] filter and thepolynomial distance classifier correlation filter (PD-CCF) [7].

In most SDF filter design approaches, we con-strain the correlation output values at the origin in re-

sponse to training images, e.g. the filter is often de-signed so that the correlation output is 1 for all train-ing images from the desired (or true) class and it is0 for all training images from the undesired (false)class. It is hoped that when such a correlation filteris used to test an input image, the resulting correla-tion output at the origin (from now loosely referred toas the correlation peak) will be close to 1 if the testimage was from the true class and will be close to 0if the test image was from the false class. While sig-nificant attention was devoted to optimizing variousperformance metrics relevant to correlation filters, lit-tle attention has been paid to the role of the correlationpeak constraints. Kallman [8] was one of the first touse constraint phases as additional degrees of freedomin composite correlation filter design. Kumaret al [9]showed that clutter rejection capabilities of SDF filtercan be improved by using correlation peak constraintswith magnitude of 1 and linearly increasing phases.

In above efforts, constraint phases and magni-tudes were used as additional degrees of freedom in anoptimization problem. Instead, we attempt in this pa-per to establish an analytical connection between theconstraint values and resulting correlation filter per-formance. In particular, we introduce two new crite-ria, namely the indiscrimination measure and the dis-

Proc. of the 6th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, Elounda, Greece, August 21-23, 2006 (pp39-44)

tortion tolerance measure. As their names suggest, theformer measure characterizes the correlation filter’sinability to discriminate the desired class from otherwhere the latter measure quantifies the filter’s abilityto tolerate distortions as exemplified by the trainingimages. We will establish an analytical relationshipbetween the these performance metrics and correla-tion constraints. An interesting result from this analy-sis is that we show that linear phase correlation peakconstraints are more attractive. Our analysis should beable to provide a framework for more careful choice ofcorrelation peak constraints. The remainder of this pa-per is organised as follows. Next section provide s thenecessary background and sets up notation. Section3 introduces the two performance metrics and derivetheir analytical reltionship with the correlation peakconstraints. These analytical relationships are used inSection 4 to examine some special cases. Finally Sec-tion 5 contains a summary and conclusions.

2 BackgroundLet x1,x2, . . . ,xN denoteN d2-dimensional columnvectors each obtained by the lexicographic scanningof a training image withd × d = d2 pixels. All Ntraining images are from the true class. Leth denotethe synthesized space-domain template from whichthe SDF filter can be constructed. The correlation out-putsci at the origin are nothing but the inner productsof training imagesxi and filter templateh, i.e.,

ci = |ci| ejθi = hTxi , i = 1, 2, . . . ,N (1)

where the subscriptT denotes the transpose and where|ci| andθi denote the magnitude and the phase, respec-tively, of theith constraint value.

2.1 Convex hull problem

Most early SDF filter designs simply assumed thatci = c, a constant for all training images from the de-sired class. The idea is that when a test image yieldsa correlation peak value close toc, that image canbe declared as coming from the desired class. Thisconstraint leads to poor clutter rejection performanceas explained in the following. Suppose we representthe N vectorsx1,x2, . . . ,xN asN points in ad2-dimensional image space. Then the convex hull of thetraining set is defined as the set of all weighted sumsof training vectors where the weights are non-negative

X

X

X

X X

x1

x2 x3

x4

x5

yi

m

zi

Figure 1: Five training vectors indicated by crossesand the associated convex hull in the shaded region.

Figure 2: A41 × 41 pixel image of digit′′3′′.

and add up to unity, i.e.,

ψ = {x =

N∑i=1

λixi, λi ≥ 0,

N∑i=1

λi = 1} (2)

The convex hull of five2-dimensional vectors isshown as the shaded region in Fig. 1. The verticesare the5 training vectors.

It is easy to verify [9] thathT x = c for anyx ∈ψ, i.e., the correlation output will be same for all im-ages in the convex hull of the training set. While suchinvariance is desirable for images similar to trainingimages ( i.e., in the vicinity of the vertices of the con-vex hull in Fig. 1), it is probably not a good attributefor images in the interior of the convex hull. As an ex-ample of this convex hull recognition problem, let usconsiderm = 1

N

∑Ni=1 xi, the average of all training

images. In most cases, this looks more like a “blob”than any of the training images. Yet, the SDF filterwill assign the blob to the desired class.

To illustrate the “blob” like appearance of the av-erage image, we show in Fig. 2 a41× 41 pixel imageof digit ′′3′′. The training set consists ofN = 180 im-ages obtained by rotating this image in plane at incre-ments of2o. Fig. 3 shows digit′′3′′ at different rota-tion angles. The average image, shown in Fig. 4, looksvery different from the training images. Although theaverage image does not look like digit′′3′′, it will berecognized by the correlation filter as an image fromthe desired class. Of course the average might lookmore like a training image if we had used a much


Figure 3: Four images of different in-plane rotatedversions of the digit′′3′′ image of Fig. 2.

Figure 4: The average of the 180 in-plane rotated ver-sions of digit′′3′′. It does not look like′′3′′.

smaller range of distortions.The above illustration suggests that the desired

class images are not well represented by the convexhull of training images. We show in Fig. 5(a) and5(b) the correlation outputs from the MACE filter inresponse to images in Fig. 2 and Fig. 4, respectively.Both correlation outputs display a sharp peak indicat-ing the presence of digit “3”. In fact, the correlationoutput due to the average image looks cleaner thanone due to a training image. This is an indicator ofthe poor discrimination capabilities of the correlationfilter using constant real values for constraints. Thus,these filters cannot suffice. This problem was pointedout by Kumaret al [9]. It was suggested there thatone way to mitigate this average recognition problemwas to constrain the desired correlation peaks to beof linear phase and unit magnitude. This approach isexplained mathematically in the next section.

2.2 Linear phase constraints

To alleviate the convex hull problem, it was sug-gested [9] to useci = 1 and θi = 2π

N(i − 1) for

i = 1, 2, . . . ,N . Since|ci| = 1, the magnitude ofthe correlation output (namely|hTxi|) will be 1 for

510

1520

2530

3540

10

20

30

400

0.2

0.4

0.6

0.8

1

510

1520

2530

3540

10

20

30

400

0.2

0.4

0.6

0.8

1

(a) (b)

Figure 5: MACE filter correlation output (a) for train-ing image at0o and (b) for the average training imageshown in Fig. 4.

all training images from the desired class. However,the correlation output will be zero when the input isthe averagem of all training images, i.e.,

hTm =1

N

N∑i=1

(hT xi)

=1

N

N∑i=1

ej2π

N(i−1)

= 0 (3)

Thus, using linear phase constraints results in the de-sirable feature that the correlation output is zero whenthe input is the average training imagem.

In fact, the correlation output will be zero forother averages (e.g.,m2, the average of even num-bered training images;m3, the average of training im-agesx1,x4,x7, . . . ). Thus, we see that the correlationoutput will be zero for some interior points of the con-vex hull in Fig. 1. On the other hand, the correlationoutput magnitude will be one for all training images.Thus using linear phase constraints appears to providemore control on the correlation outputs, particularlyfor interior points of the convex hull. Thus, usinglinear phase complex constraints appears to result inimproved discrimination without any degradation inresponse to training images.

Based on the above observation that constraintphases can improve discrimination, two importantquestions arise. Can the constraint magnitudes beused to improve the performance of correlation filters?How can we quantify the discrimination and distortiontolerance of correlation filters? We attempt to answerthese questions in the next two sections.


3 Analytical relationships

In this section, we attempt to quantify the discrimi-nation and distortion capabilities and derive analyticalrelationship with these metrics and constraint values.

3.1 Indiscrimination measure

Let us refer to the convex hull in Fig. 1. Ideally, wewant the correlation output to be large and relativelyconstant for all images represented by the boundarysurface of the convex hull. The premise here is thatimages on the convex hull boundary surface are morelike training images and thus are more likely represen-tative of the desired class. On the other hand, we wantthe correlation output to be small in response to theimages represented by the interior of the convex hull.

One way to develop a simple metric to character-ize the filter’s discrimination capabilities is to considerthe image corresponding toyi(β) defined as follows.See also Fig. 1.

yi(β) , βxi + (1 − β)m (4)

where0 ≤ β ≤ 1. The correlation output in responseto yi(β) is as follows.

hTyi(β) = βci + (1 − β)c̄ (5)

where

c̄ ,1

N

N∑i=1

ci (6)

is the average of all constraint values.A convenient metric for this task is indiscrimina-

tion measure defined below. Larger values of this in-dicates lower discrimination ability.

I(β) ,1

N

N∑i=1

|hT yi(β)|2 (7)

Substituting Eq. (5) in Eq. (7), we obtain the fol-lowing expression forI(β).

I(β) = β2|c|2 + (1 − β2)|c̄|2 (8)

where

|c|2 ,1

N

N∑i=1

|ci|2 (9)

A total indiscrimination measure (TIM ) can be ob-tained by integratingI(β) overβ from 0 to 1, i.e.,

TIM ,

∫ 1

0I(β)dβ

=1

3|c|2 +

2

3|c̄|2 (10)

Eq. (10) captures the effect of constraints on the dis-crimination capabilities of the correlation filter. Wewill consider some special cases in Section 4.

3.2 Distortion tolerance measure

SDF filters aim to improve the distortion tolerance.We need a metric to capture this characteristic. Oneway of stating this desirable distortion tolerance fea-ture is that we want the correlation output to be largenot just at the vertices of the convex hull, but alsoalong the edges connecting these vertices. Accord-ingly, let zi(γ) be a point on the edge connectingxi

to xi+1 as defined below. See also Fig. 1.

zi(γ) = (1−γ)xi+γxi+1 , i = 1, 2, . . . ,N (11)

where0 ≤ γ ≤ 1.

We wanthTzi(γ) to be large for alli and allγ.Thus, one way to quantify the distortion tolerance isby averaging|hTzi(γ)|

2 for all i, i.e.,

G(γ) = (1 − 2γ + 2γ2)|c|2 + 2γ(1 − γ)τ (12)

where

τ ,1

N

N∑i=1

ℜ{cic∗

i+1} (13)

By integratingG(γ) overγ, we can define the to-


0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.5

0.6

0.7

0.8

0.9

1

1.1

TIM

TDTM

Figure 6: TDTM vs. TIM for different constraintchoices forN = 60 training images.

tal distortion tolerance measure (TDTM ), i.e.,

TDTM =

∫ 1

0G(γ)dγ

=2

3|c|2 +

1

3τ (14)

We want largeTDTM values and smallTIMvalues. In the next section, we consider several specialcases. Thus in aTDTM vs. TIM figure, we wantvalues close toTDTM = 1, TIM = 0.

4 Analysis results

In this section, we will consider several cases to illus-trate how peak constraint values affectTDTM andTIM .

In Fig. 6, we showTDTM vs.TIM for differentconstraint choices forN = 60 training images. Thearrow points at (1, 1) which corresponds toci = 1

for all i. Square corresponds toci = ej2π

N(i−1) for

all i. The line of crosses at upper right is for con-straints with zero phases and magnitude uniformlydistributed in the interval(0.9, 1.1). The filled dotsin the lower left correspond to unit-magnitude con-straints whose phases are uniformly distributed in theinterval(0, 2π).

The four cases shown in Fig. 6 indicate clearlythat the correlation peak constraints phases (but notthe magnitudes) have the potential to offer better dis-

crimination without sacrificing distortion tolerancesignificantly. In particular, linear phase constraintsare significantly more attractive compared to randomphases.

5 ConclusionWe have presented an analysis of the effect of con-straint magnitudes and phases on correlation peak val-ues. This analysis shows the improved discriminationability offered by linear phase constraints.

Acknowledgements: The author acknowledges KingAbdulaziz City for Science and Technology for itssupport.

References:

[1] C. Hester and D. Casasent, Multivariant techniquefor multiclass pattern recognition,Applied Optics,19, 1980, pp. 1758–1761.

[2] A. Mahalanobis, B. V. K. Vijaya Kumar, andD. Casasent, Minimum average correlation en-ergy filters,Applied Optics,26, 1987, pp. 3633–3640.

[3] P. Refregier and V. Laude, Nonlinear joint trans-form correlation: an optimal solution for adaptiveimage discrimination and input noise robustness,Optical Letters,19, 1994, pp. 405–407.

[4] A. Mahalanobis, B. V. K. Vijaya Kumar, S. Song,S. R. F. Sims, and J. F. Epperson, Unconstrainedcorrelation filters,Applied Optics,33, 1994, pp.3751–3759.

[5] A. Mahalanobis, B. V. K. Vijaya Kumar, andS. R. F. Sims, Distance-classifier correlation fil-ters for multiclass target recognition,Applied Op-tics, 35, 1996, pp. 3127–3133.

[6] M. Alkanhal, , B. V. K. Vijaya Kumar, A. Ma-halanobis, Improving the false alarm capabilitiesof the maximum average correlation height corre-lation filter, Optical Engineering, 39, 2000, pp.1133–1141.

[7] M. Alkanhal, , B. V. K. Vijaya Kumar, Polyno-mial distance classifier correlation filter for pat-tern recognition ,Applied Optics, 42, 2003, pp.4688–4708.

[8] R. Kallman, An algorithm to choose which pix-els to zero out in a zero and phase filter,AppliedOptics, 26, 1987, pp. 5200–5201.


[9] B. V. K. Vijaya Kumar, J. Brasher, C. Hester,G. Srinivasan, and S. Bollapragada, Synthetic dis-criminant functions for recognition of images onthe boundary of the convex hull of the training set,Pattern Recognition,24, 1994, pp. 543–548.


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