1. INTRODUCTIONThis is a companion paper to a concurrent paper titled‘‘lp-norm optimum filters for image recognition. Part I:algorithms.’’1 In that paper a family of filters, termedlp-norm optimum filters, was designed on the basis ofminimizing the output due to the input signal and theoutput due to the input noise. To measure the size of theoutput, an lp-norm metric was used. Theoretical devel-opment of these filters is given in detail in Ref. 1. In thispaper we examine the performance of the lp-norm filtersby measuring the average peak-to-sidelobe ratio (PSR) ofthe output of the filters for different values of p. We alsotest the performance of these filters by placing a target ina scene containing additive noise and a realistic back-ground.
Numerous types of filters have been developed for im-age and pattern recognition, such as the matched filter2
and its variations.3–9 The matched filter maximizes thesignal-to-noise ratio in the presence of stationary additivenoise, but it has been shown to have low discriminationcapabilities.5 However, variations of these filters havebeen shown to be discriminant with good correlationperformance.4,8,10,11 Various other filters based on differ-ent design criteria have been proposed to optimize somecriteria or to provide a compromise between differentcriteria.12–16
In Ref. 1 we developed a family of filters for image rec-ognition that was based on minimizing output due to theinput signal and the output due to the input noise. Weused the lp norm to measure the output. By choosing tominimize both the lp norm of the output due to noise andthe output due to the input scene (or either one of the two)and by using different values of p, we generated a familyof filters indexed by parameters q 5 p/( p 2 1) and s,where q controls the discrimination and robustness of the
filter and s is the standard deviation of the additive noise(see Subsection 2.A).
To examine the performance of lp-norm optimum filtershere, we compute the average PSR of these filters for dif-ferent values of p by computer simulations. Our conclu-sions are that for a fixed and equal set of weights [see Eq.(6)], the average PSR increases as p decreases. Thus fora fixed and equal set of weights, the filters become morediscriminant if we use a lower value of p to minimize thenorm of the outputs.
The paper is organized as follows. In Section 2 webriefly state the minimization problem that is the basis ofthe lp-norm filters designed in Ref. 1. Since the theoret-ical development of these filters appears in Ref. 1, we sim-ply state the final results. In Subsection 2.A these re-sults are stated in forms of the filter equations denoted byHq
s. In Subsection 2.B special cases of the solution ofthe minimization problem are presented, which lead us toa related family of filters denoted by Hq and Hq
0.In Section 3 we present some computer simulations to
confirm our expectations based on the theoretical devel-opment and to test the performance of the lp-norm opti-mized filters. For the images presented here, the filtersdetected the target in the presence of additive noise and arealistic background scene. We have measured the aver-age PSR of the filters for different values of p or q5 p/( p 2 1).
2. REVIEWIn this section we present the necessary background ma-terials to keep this paper self-contained. We refer thereader to Ref. 1 for a detailed analysis.
Let r( j) denote a target to be detected and n( j) the ad-ditive noise, which we assume to be zero mean and whitestationary. Then the input to the system (filter) is
1999 Optical Society of America
B. Javidi and N. Towghi Vol. 16, No. 9 /September 1999 /J. Opt. Soc. Am. A 2147
s~ j ! 5 r~ j ! 1 n~ j !. (1)
Let S(k), R(k), and N(k) denote the Fourier transformsof s( j), r( j), and n( j), respectively.
Let h denote the impulse response function of the sys-tem and H denote the Fourier transform of h. The filterh( j) is designed such that the filter output due to target ris
(j50
J21
h~ j ! * r~ j ! 5 C 5 C~0 !, (2)
where C is a positive constant. To achieve both robust-ness and discrimination capabilities, the filter h( j) is de-signed to minimize a weighted sum of the pth power ofthe lp-norm’s mean of the output due to noise n and thepth power of the lp-norm of the output due to input signals. That is, h is chosen to minimize
a(j50
J21
EU(l50
J21
h~ j 2 l !n~l !Up
1 b(j50
J21 U(l
h~ j 2 l !s~l !Up
,
(3)
under the constraint of Eq. (2). The weights a and b aresuitably chosen positive quantities. If the emphasis is onrobustness, b is the larger of the two. If the emphasis ison discrimination, a should be the dominant quantity.We consider only the case a 5 b 5 1. In Subsection 2.Bthe values (a 5 1, b 5 0) and (a 5 1, b 5 0) are used.
The minimization problem given by Eq. (3) for the case1 , p < 2 can be stated in the Fourier domain asfollows1: Minimize
(j50
J21
uH~ j !uq@sq 1 uS~ j !uq#, (4a)
subject to
(j50
J21
H~ j ! * R~ j ! 5 JC~0 !, (4b)
where q 5 p/( p 2 1) and sq 5 EuN( j)uq.
A. Nonlinear Filters Based on Optimization Using lpNormsThe solution of the minimization problem of Eq. (4) is aconstant multiple of (see Ref. 1)
Hqs~ j ! 5 F uR~ j !u
sq 1 uS~ j !uqG1/~q21 !
exp@iFR~ j !#, (5)
where sq 5 EuN( j)uq and FR( j) is the argument ( phase)of the complex quantity R( j); that is, R( j)5 uR( j)uexp@iFR( j)#.
The case p 5 1 requires a different approach. We re-fer the reader to Ref. 1. However, if we settle on usingthe lower bound estimate, EuN( j)uq > @sAJ#q, whichholds for q > 2 (see Ref. 1), we obtain a crude approxima-tion of a filter equation for the case q 5 ` or p 2 1, givenbelow,
H`s~ j ! 5 F 1
max$AJs,uS~ j !u%Gexp@iFR~ j !#. (6)
We should point out that Eq. (5) requires the values ofEuN( j)uq. With few exceptions, this quantity may be dif-ficult to compute. In Ref. 1 we have given some lowerbound and upper bound estimates for various types ofnoise processes.
B. Subfamily of Nonlinear Filters Based on Minimizingthe lp NormIf we minimize only the filter output that is due to the in-put scene s, subject to Eq. (2), we obtain the followingfamily of filters, which we denote by Hq
If we minimize only the output that is due to the addi-tive noise, we obtain a filter denoted by Hq , where q> 2:
Hq~ j ! 5 uR~ j !u1/~q21 ! exp@iFR~ j !#, (9)
H`~ j ! 5 exp@iFR~ j !#. (10)
Fig. 1. Input scene: (a) two targets ( jet planes) with back-ground scene; (b) scene of (a) with additive zero-mean whiteGaussian noise with standard deviation of 0.7.
2148 J. Opt. Soc. Am. A/Vol. 16, No. 9 /September 1999 B. Javidi and N. Towghi
3. COMPUTER SIMULATIONSTo test the performances of the filters designed in Section2, we performed some computer simulations. In oursimulations our targets are a pair of similar jet airplanes.The size of each target is 107 3 70 pixels. The two tar-gets are placed in a realistic scene containing a realisticbackground [see Fig. 1(a)]. The size of the scene is 2443 164 pixels. We then added zero-mean white station-ary Gaussian noise with standard deviation s 5 0.7 tothis input scene [see Fig. 1(b)].
Figure 2 shows the output of the Hqs family of filters
given by Eqs. (5) and (6). Figure 2(a) is the output of the
Fig. 2. Output of the Hqs nonlinear filters of Eqs. (5) and (6).
These filters are obtained by minimizing the lp norm of the out-put due to input scene and the output due to noise. (a) Outputof the filter when q 5 2 [Eq. (5)], (b) output of the filter when q5 10 [Eq. (5)], (c) output of the filter when q 5 ` [Eq. (6)].
filter when q 5 2, Fig. 2(b) is the output of the filter whenq 5 10, and Fig. 2(c) is the output of the filter when q5 `, given by Eq. (6). The set of filters whose output isgiven by Fig. 2 was designed to optimize both the noiserobustness and discrimination capabilities. We see thatthe correlation peaks are sharper for larger values of q.
Figure 3 shows the output of the Hq0 family of filters
given by Eqs. (7) and (8). Figure 3(a) is the output of thefilter when q 5 2, Fig. 3(b) is the output of the filter whenq 5 10, and Fig. 3(c) is the output of the filter when q5 `, given by Eq. (8). The set of filters whose output isgiven by Fig. 3 was designed to optimize both the noiserobustness and discrimination capabilities. Once again
Fig. 3. Output of the Hq0 nonlinear filters of Eqs. (7) and (8).
These filters are obtained by minimizing the lp norm of the out-put due to input scene. (a) Output of the filter when q 5 2 [Eq.(7)], (b) output of the filter when q 5 10 [Eq. (7)], (c) output of thefilter when q 5 ` [Eq. (8)].
B. Javidi and N. Towghi Vol. 16, No. 9 /September 1999 /J. Opt. Soc. Am. A 2149
we see that the correlation peaks are sharper for largervalues of q. Since in the derivation of Hq
0 filters only thelp norm of the scene input is minimized, we notice thehigher values of peaks at nontarget locations.
Figure 4 shows the output of the Hq family of filtersgiven by Eqs. (9) and (10). These filters are linear. Fig-ure 4(a) is the output of the filter when q 5 2, Fig. 4(b) isthe output of the filter when q 5 10, and Fig. 4(c) is theoutput of the filter when q 5 `, given by Eq. (10). Theset of filters whose output is given by Fig. 4 was designedto optimize both the noise robustness and discriminationcapabilities. Once again we see that the correlation
Fig. 4. Output of the Hq linear filters of Eqs. (9) and (10).These filters are obtained by minimizing the lp norm of the out-put due to noise. (a) Output of the filter when q 5 2 [Eq. (9)],(b) output of the filter when q 5 10 [Eq. (9)], (c) output of the fil-ter when q 5 ` [Eq. (10)].
peaks are sharper for larger values of q. Note that whenq 5 2, Hq filter is the usual matched filter.
To evaluate the performance of the filters and to con-firm the theoretically predicted performance of the filters,we computed the average PSR of the output of differentreceivers when the input scene contained the target in thepresence of additive white Gaussian noise. PSR is de-fined as the expected ratio of the output peak to themaximum-output-noise sidelobe. We placed a referencetarget of size 32 3 21 pixels in a scene of size 1283 128 pixels and added zero-mean white Gaussian noiseof standard deviation s 5 1. We then found the ratio ofthe output peak ( peak at target location) to themaximum-output-noise sidelobe for the filters designed inthis paper. To obtain the average PSR, the statistical av-erage was obtained over more than 100 trials.
The average PSR for Hqs receivers of Eq. (5) for values
of q 5 2, 3, 4, 5, and 10 are 35, 70, 98, 106, and 130, re-spectively. The average PSR for Hq
0 receivers of Eqs. (7)and (8) for values of q 5 2, 3, 4, 5, 10, and ` are 120, 160,165, 168, 169, and 170, respectively.
PSR is a measure of discrimination capabilities of thefilter. Hq
0 filters of Eqs. (7) and (8) were designed to op-timize the discrimination capabilities. Thus we see that,for each fixed q, the PSR values for Hq
0 filters are higherthan the corresponding PSR values of the Hq
s filters.We see that higher values of q give us a higher PSR.
Finally, the average PSR for Hq receivers of Eqs. (9)and (10) for values of q 5 2, 3, 4, 5, 10, and ` are 1.2, 1.5,1.6, 1.9, 2.1, and 2.3, respectively. These filters were op-timized to be robust with respect to noise. Consequently,the PSR values of these filters are less than the corre-sponding PSR values of Hq
0 and Hqs filters.
4. CONCLUSIONSWe examined the performance of the lp-norm optimumfilters. These filters were derived on the basis of usingan lp-norm metric for arbitrary values of p . 1 ratherthan the standard mean squared metric. The lp-normcriterion is used to derive filters to obtain greater freedomin adjusting noise robustness and discrimination capabili-ties.
These filters were obtained by minimizing the outputdue to noise and output due to the input signal, with thelp norm used as the metric.
We tested the performance of the filters with computersimulations. In our tests we used the average PSR( peak-to-sidelobe ratio) as a metric to compare the dis-crimination capabilities of the filters for different valuesof p. The tests that we conducted show that the filter’sperformance (discrimination capabilities) improves whenp decreases. This result is shown by sharp peaks at thetarget location and higher PSR values for smaller valuesof p.
ACKNOWLEDGMENTSWe are grateful to S. Hong for his assistance with thecomputer simulations.
2150 J. Opt. Soc. Am. A/Vol. 16, No. 9 /September 1999 B. Javidi and N. Towghi
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