February 15, 1993 / Vol. 18, No. 4 / OPTICS LETTERS

Optical wavelet matched filters for shift-invariantpattern recognition

Yunlong Sheng and Danny Roberge

D6partement de Physique, Centre d'Optique, Photonique et Laser, Universit6 Laval, Sainte-Foy GiK 7P4, Canada

Harold Szu

Information Processing Group, Naval Surface Warfare Center-White Oak, 19001 New Hampshire Avenue,Silver Spring, Maryland 20903-5000

Taiwei Lu

Physical Optics Corporation, 20600 Gramercy Place, Suite 103, Torrance, California 90501

Received October 1, 1992

We introduce optical wavelet matched filters that perform the wavelet transforms for edge enhancementand perform correlations between the wavelet coefficients for shift-invariant pattern recognition. These newbandpass matched filters show improved discrimination capability with respect to the conventional matchedspatial filter and improved signal-to-noise ratio with respect to the phase-only matched filter.

The wavelet transform (WT) is efficient for mul-tiresolution local analysis of signals. It has beensuccessfully used for image compression.' This Let-ter describes a new application of the WT to opticalpattern recognition. The WT can be used to detectspeech, seismic, biomedical, and other signals by theso-called cocktail party effect.23 Shapes, scales, orpositions of those signals may not be exactly known.For these applications, the WT is more efficient thanthe Fourier spectrum analysis. When the signalsto be detected are known, a priori, automatic tar-get recognition usually involves two major opera-tions, namely, preprocessing for feature extractionand noise suppression followed by correlation witha library of templates. A major difficulty for ap-plications of the orthogonal discrete WIT to patternrecognition is the lack of shift invariance.

Optical real-time WIT based on the explicit WTcorrelation has been demonstrated. 3-5 The opticalWIT is inherently shift invariant. We use the WI'to enhance significant features of patterns and usecorrelations between the WIT coefficients for patternrecognition. Then we introduce the wavelet matchedfilter that performs the WIT and the correlation of theWIT coefficients in a single step. Preliminary resultsare shown to support the usefulness of this approach.

The VVI is a correlation between a two-dimensional(2-D) signal f(x,y) and a set of the dilated waveletsh.(x,y):

Wf(sXsyxy) = f(x,y) * h.(x,y). (1)

The wavelets are generated from a mother waveleth(x,y) by

hs(xY)= y h(s s ) (2)s-S S xSy

where (sr, sy) are the dilation factors.6'7 The waveletsmust satisfy an admissible condition that h(x,y) os-cillates to have a zero mean. The wavelets shouldalso have some regularity so that they are local inboth space and frequency domains. For example,the Gaussian function is local in space and is in-definitely derivable. A derivative of any order n ofthe Gaussian function may be used as the wavelet.The Fourier transform of this wavelet is also lo-cal, indefinitely derivable, and has zero of order nabout the zero frequency. The wavelet admissiblecondition is satisfied. The Mexican-hat wavelet isthe second-order derivative of the Gaussian functionh(x,y) = V2g(xy). As a result of the linearity of thecorrelation and the differentiation operations, we canwrite

V2(g8 * f)(x,y) = [V2g8(Xy)] * f (x,y) = Wf(sX, sy,x,y).(3)

The WT is then equal to the second-order derivativeof the smoothed image f * g,. The smoothing func-tion is the dilated Gaussian function gs(x,y). TheWT coefficient Wf(sx, syxy) is zero when f * g, isconstant or changes linearly but exhibits a positiveand a negative peak in two sides of the edge. Azero crossing occurs at the point of inflection of f * g8.Thus the WT enhances edges of the image. Chang-ing the scales (sr, sy) of the wavelets may change thesize and orientation of the smoothing function. Thisproperty of the WIT is useful because irradiance in-tensity changes in a 2-D image can occur at differentscales and directions. By choosing the scale factors,the WT may enhance the selected edges in a 2-Dimage, as we show in the following for the opticalWT of the letter E.

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Fig. 1. Optical wavelet transform. Clockwise from topleft: input image; output with the 1-D Mexican-hatwavelet with s, = 2; output with the wavelet of Eq. (6)with s = 2; output with the wavelet of Eq. (5) withsx = sy = 2.

'Y lWT filter| tsxy

valued and were made as photographic transmittancemasks. Optical WIT outputs obtained in a conven-tional optical correlator are shown in Fig. 1, whereenhancement of the selected edges can be easily seen.

Edges are the most significant features of images.The VWT coefficient of a 2-D image is the edge-enhanced pattern. Pattern recognition by corre-lations between the WI coefficients will havegood discrimination capability. For this purpose,we could holographically record the complex-valuedoptical WIT coefficients Wf(sXsy,x,y) of the librarypatterns f(x,y) as the conventional matched spatialfilters and use a second optical correlator to performthe correlations between Wf(sx,,sy,x,y) and the WITcoefficient Wt(s.,,syxy) for an unknown input t(x,y),as illustrated in Fig. 2(a). However, this approachwould involve two cascaded optical correlators. Thistwo-step correlation would result in more systemcomplexity and low light efficiency.

According to Eq. (4), the correlation Wf * Wt maybe expressed in the Fourier plane as

f J Wt(s., sy,x',y)Wf*(sx, sy,x' - x,y' - y)dx'dy'(a) -

t(x,y) Wavelet matched filters WfrWt_ F(uvj H(sxusyv)j2

(b)

Fig. 2. Two approaches to obtain correlations betweentwo wavelet transform coefficients for pattern recognition:(a) Two steps, (b) one step, where ftx,y) is known, a priori,and t(x,y) is a unknown input pattern.

In the Fourier plane the VVT is expressed as

= ff|T(u, v)H(sxu, syv)F*(u, v)H*(sxu, syv)

x exp[i2 iT(xu + yv)]dudv, (7)

where T(u, v) is the Fourier transform of t(x,y) andthe constants S"

2Sy

2 are omitted. Thus the correla-tion Wf * W, can be directly obtained with a filterequal to F(u, v)IH(sxu, syv)I2 , as described in Fig. 2(b).We call this filter the wavelet matched filters. It isthe product of the conventional matched filter F(u, v)and the square modulus IH(sxu, syv)12 of the WIT filter.

Wf(SX, sy,x,y) = s'sy |f F(u, v)H*(su, syv)

x exp[i2ir(xu + yv)]dudv, (4)

where F(u, v) is the Fourier transform of the signalf(x,y), and H(sxu, syv) is the Fourier transform of thedilated wavelets hs(x,y). Optical WT is based onEq. (4) with H(s.u,syv) as the WT filters.8

In an optical experiment we demonstrated the WIof a capital letter E. The image has step edgesmainly along the x and y directions. We chose,therefore, for the wavelet a one-dimensional (1-D)Mexican-hat function Vy2g(y) that enhanced thehorizontal edges of the image and a separable productof two Laplacians of two 1-D Gaussian functionsVX2g(x)Vy2g(y), expressed in the Fourier plane as

H(u, v) = sX2 u 2 exp(-27r 2sX2 u 2 )sy 2 v 2 exp(-2 7r2sY2 v 2 )

(5)

with sx =ssy, which enhanced the corners8 of theimage. We also used the Laplacian of the isotropic2-D Gaussian function with sx = sy = s, expressed inthe Fourier plane as

H(u,v) = s2(U2 + v2 )exp[-2ir2S 2 (U2 + v2)], (6)

which enhanced the edges of all directions. The WTfilters described by Eqs. (5) and (6) were all positive

PEE E

(a) (b)

(c) (d)Fig. 3. Optical wavelet matched filters. (a) Input, (b)output of the conventional matched filter, (c) output of thewavelet matched filter with the 1-D Mexican-hat waveletwith s, = 5, (d) output with the wavelet of Eq. (5) withsx = s, = 5.

February 15, 1993 / Vol. 18, No. 4 / OPTICS LETTERS

Fig. 4. Three-dimensional plots of the outputs of fourphase-only wavelet matched filters with the dyadicwavelets of Eq. (6) with s = 1/2 (left-hand corner), s = 1(top corner), s = 2 (bottom corner), and s = 4 (right-handcorner). The input was that of Fig. 3(a) degraded by anadditive random noise.

The wavelet matched filter is shift invariant. Whent =f, an autocorrelation peak of Wf(ssy, x,y) willappear at the location of the t(x,y).

In the experiment, the input scene contained threeletters E of different sizes and a letter P, as shown inFig. 3(a). The square moduli IH(sxu, syv)I2 of the WTfilters were made as the photographic transmittancemasks. The optical wavelet matched filter was asuperimposition of the mask IH(su, sv)J2 with a con-ventional computer-generated holographic matchedfilter F(u, v) for the letter E in the bottom right-hand corner of Fig. 3(a). The wavelets were a 1-DMexican-hat wavelet with sy = 5 that enhanced hor-izontal edges of the images and a separable productof two 1-D Mexican-hat wavelets described by Eq. (5)with s -sy = 5 that enhanced corners of the images.Figures 3(c) and 3(d) show the optical correlationoutputs of the two wavelet matched filters wherethe high cross-correlation peaks in the output of theconventional matched filter shown in Fig. 3(b) wereeliminated.

In comparison with high-pass filters, such as theGrad operator,9 '10 the wavelet matched filters arebandpass matched filters owing to localization of thewavelets in the frequency domain. The bandpass fil-ter has better output signal-to-noise ratio (SNR) thanthat of the high-pass filter. Moreover, the adaptivechoice for the wavelet function and its scale factorsallows enhancement of the selected features of theimage.

The phase-only filter (POF), described by(D(u,v) = F(u,v)/lF(u,uv), is also a high-pass fil-ter that has high light efficiency and good dis-crimination." The wavelet matched filter can alsobe generated with the POF as '(u,vu)IH(sxu,syv)I2,where the bandpass function IH(s.u, sV)I 2 may im-prove the output SNR. Mallat and Hwang6 recentlyproved that if the regularity of f(x,y) is Lipschitza at (x0,yo), then in a neighborhood of (xo,y0) themodulus of the WVT, IWf(s,x,y)I, will decay as O(sa)when the scale s tends to zero, on the condition

that the wavelets have n vanishing moments for0 < a <n. Thus,ifthef(x,y)isdiscontinuousandisLipschitz a - 0 at the edges, the IWf(s,x,y)A remainsapproximately constant over a large range of scales.White noise has negative Lipschitz exponent every-where, and its WT modulus IWn(s,x,y)j will increaseproportionally to s-' when the scale s tends to zero.Mallat eliminated the noise by an ad hoc algorithm,removing maxima of the WT modulus whose ampli-tudes increase when the scale decreases. For a noisyinput t + n, the wavelet matched filter will generatethe output (W, + Wj * Wf, where the output noiseWn is correlated and smoothed with Wf. The outputnoise Wn * Wf still increases when the scale s tendsto zero, as shown in the following experiment.

Figure 4 shows the correlation outputs obtained bycomputer simulation. The wavelet matched filterswere combinations of the POF, CD (u, v), of the letterE in the bottom right-hand corner in Fig. 3(a), withthe dyadic wavelets described by Eq. (6) with s =1/2, 1, 2, 4. The input was that shown in Fig. 3(a),degraded by an additive random noise with the SNRequal to 2. The output SNR was equal to 21 for theconventional POF and increased from 28 to 45 and63 for the wavelet matched filters with s = 1/2, 1, 2,respectively, but decreased to 26 with s = 4. Whenthe scale of the wavelet was too large, the waveletmatched filter lost its discrimination capability andhigh cross-correlation peaks appeared, as shown inFig. 4. These results suggest a possible optimizationof the wavelet scale factors.

In summary, we have introduced wavelet matchedfilters that perform in a single step both the VVTfor edge enhancement and correlation between theWT coefficients for optical shift-invariant patternrecognition. The bandpass wavelet matched filtershave shown improved discrimination capability com-pared with the conventional matched spatial filterand higher output SNR compared with the phase-only filters.

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