Optical realization of the wavelet transform for two-dimensional objects David Mendlovic and Naim Konforti Real-time wavelet transformations of two-dimensional objects are implemented by use of the conven- tional coherent correlator with a multireference matched filter. The different daughter wavelets are spatially multiplexed with different reference-beam directions. Two experiments are described, one of them with a spatial light modulator at the input plane in order to enable the real-time property. Key words: Wavelet transforms, Gabor transform, optical correlators, wavelets. Introduction The wavelet transform (WT) appears to be a natural way of decomposing a signal or scene into a space frequency (or time frequency) presentation. The response curve of the human eye is accurately de- scribed in wavelet terms.' Natural scenes can un- dergo tremendous bandwidth reduction after analysis by wavelet transformations. Other applications of the WT are optical correlators, 2 ' 3 simple presentation of fractal aggregates, 4 and transient signal and image processing. 25 - 7 The one-dimensional (1-D) WT definition of a func- tion f(x) is 89 W(a, b) = J f(x)h*,b(x)dx, (1) where ha,b is a daughter wavelet that is derivated from the mother wavelet h(x) by dilation and shift opera- tions: hab(X) = h( - b) (2) A typical mother wavelet could be h(x) = exp[ - (x/xO) 2 /2]exp(i2'Ti 0 x), (3) where w 0 is a spatial frequency. The authors are with the Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel. Received 17 February 1993. 0003-6935/93/326542-05$06.00/0. 3 1993 Optical Society of America. This Gaussian-envelop mother wavelet h(x) is known as the standard Morlet wavelet.1 0 This mother wavelet is a windowed Fourier transform (FT) with the coordinate transformation x -> (x - b)/a. If the mother wavelet h(x) contains a harmonic structure [e.g., exp(i2Tro 0 x)], the WT repre- sents both frequency and spatial information of the signal. Let us make the consideration between the WT and another windowed FT operation, the Gabor transform. 2 In the Gabor transform the signal is multiplied by a fixed windowed function before the Fourier analysis. The position of the window may be translated along the x axis (as in the WT case), but because of the fixed window, we recognize the follow- ing problem: high-frequency signals are sampled at a high rate and low frequency signals are sampled at a low rate. The WT transform performs at the same sampling rate for each frequency. Returning to Eq. (1), one notices that the WT of a 1-Dfunction has two dimensions. It has been pointed out 2 that the WT is a correlation between the input signal f(x) and the mother wavelet function h(x), scaled by a and shifted by b. On the basis of this consideration, Szu et all' showed how to implement the 1-D WT optically. They used an anamorphic version of the well-known VanderLugt correlator with a matched filter that contains many daughter wavelets. Thus the full WT was obtained at the output plane. Freysz et al. 4 performed a two- dimensional (2-D) WT with optics. Again, the VanderLugt correlator scheme was used. However, this time each daughter wavelet required the whole 2-D plane. Thus their architecture is based on load- ing of the daughter wavelet transparencies sequen- tially. The final result is a non-real-time configura- tion since there is a need for time multiplexing of the different daughter wavelet outputs. 6542 APPLIED OPTICS / Vol. 32, No. 32 / 10 November 1993
Transcript
Optical realization of the wavelettransform for two-dimensional objects
David Mendlovic and Naim Konforti
Real-time wavelet transformations of two-dimensional objects are implemented by use of the conven-tional coherent correlator with a multireference matched filter. The different daughter wavelets arespatially multiplexed with different reference-beam directions. Two experiments are described, one ofthem with a spatial light modulator at the input plane in order to enable the real-time property.
The wavelet transform (WT) appears to be a naturalway of decomposing a signal or scene into a spacefrequency (or time frequency) presentation. Theresponse curve of the human eye is accurately de-scribed in wavelet terms.' Natural scenes can un-dergo tremendous bandwidth reduction after analysisby wavelet transformations. Other applications ofthe WT are optical correlators,2' 3 simple presentationof fractal aggregates,4 and transient signal and imageprocessing. 2 5-7
The one-dimensional (1-D) WT definition of a func-tion f(x) is89
W(a, b) = J f(x)h*,b(x)dx, (1)
where ha,b is a daughter wavelet that is derivated fromthe mother wavelet h(x) by dilation and shift opera-tions:
hab(X) = h( - b) (2)
A typical mother wavelet could be
h(x) = exp[ - (x/xO)2 /2]exp(i2'Ti 0 x), (3)
where w0 is a spatial frequency.
The authors are with the Faculty of Engineering, Tel AvivUniversity, Tel Aviv 69978, Israel.
Received 17 February 1993.0003-6935/93/326542-05$06.00/0.3 1993 Optical Society of America.
This Gaussian-envelop mother wavelet h(x) isknown as the standard Morlet wavelet.10 Thismother wavelet is a windowed Fourier transform(FT) with the coordinate transformation x ->(x - b)/a. If the mother wavelet h(x) contains aharmonic structure [e.g., exp(i2Tro0x)], the WT repre-sents both frequency and spatial information of thesignal. Let us make the consideration between theWT and another windowed FT operation, the Gabortransform.2 In the Gabor transform the signal ismultiplied by a fixed windowed function before theFourier analysis. The position of the window maybe translated along the x axis (as in the WT case), butbecause of the fixed window, we recognize the follow-ing problem: high-frequency signals are sampled ata high rate and low frequency signals are sampled at alow rate. The WT transform performs at the samesampling rate for each frequency.
Returning to Eq. (1), one notices that the WT of a1-D function has two dimensions. It has been pointedout2 that the WT is a correlation between the inputsignal f(x) and the mother wavelet function h(x),scaled by a and shifted by b. On the basis of thisconsideration, Szu et all' showed how to implementthe 1-D WT optically. They used an anamorphicversion of the well-known VanderLugt correlatorwith a matched filter that contains many daughterwavelets. Thus the full WT was obtained at theoutput plane. Freysz et al.4 performed a two-dimensional (2-D) WT with optics. Again, theVanderLugt correlator scheme was used. However,this time each daughter wavelet required the whole2-D plane. Thus their architecture is based on load-ing of the daughter wavelet transparencies sequen-tially. The final result is a non-real-time configura-tion since there is a need for time multiplexing of thedifferent daughter wavelet outputs.
Recently Freeman et al. 12 suggested a smart opticalconfiguration that performs a real-time 2-D WT.They used a recycled input in an optical correlator,and instead of scaling the mother wavelet, they scaledthe input itself. This approach leads to two maindisadvantages. First, it is against the nature of theWT to scale the input and not the mother wavelet.Second, the recycled correlator requires a high stabil-ity. The immediate effect is the need of a robust andexpensive system. In addition, the presentation ofthe different daughter wavelet outputs is along a line.Thus it is impossible to use the 2-D plane in the mostefficient way.
In the following we suggest another approach forimplementing a real-time 2-D WT. Instead of usinga recycling correlation system with time multiplexingto perform the spatial multiplexing, we use theconventional correlator scheme with a multireferencematched filter (MRMF) and thus spatial multiplexing.Each daughter wavelet is encoded with a differentreference beam. The MRMF is placed in the Van-derLugt correlator, and the full WT is obtained at theoutput plane.
Two-Dimensional Wavelet Transform
For a 2-D image f(x) = f(x1, x 2 ) the definition of theWT is
W(a, b) = f J f(x)h*b(x)dxdx 2 , (4)
where b = (bl, bl). This definition leads to a three-dimensional presentation, which is the reason whythe full 2-D WT cannot be obtained without time orspatial multiplexing.
The function hab(X) is derived from the motherwavelet h(x) by
ha,b(X) = b, a ' b)* (5)
The above definition results in an amorphic WT,since we use one global scaling factor a. An anamor-phic WT can be defined by replacement of the argu-ment a with the scaling vector a = (al, a2). Theactual result of this anamorphic WT is similar to theHaar transform.' 3
In many practical applications the signal is sampleddiscretely so that a discrete dilation factor a is used.It is common to define a = a', where ao > 1, while mis an integer. With this definition the 2-D WT is
Wd(m, b) = f f(x)h*mb(x)dx,
where the daughter wavelet is
1 - b x 2 - b2\hmsb(X) = 7m h( o
am am am
INPUT OUTPUT
f f
Fig. 1. Optical correlator architecture for a 2-D WT. The MRMFis the multireference matched filter.
is affected by the ability to reconstruct f(x) from itsWT Wd(m, b) with the discrete inverse WT opera-tion8 14
(8)f(x) = 7 f Wd(m, b)hmb(X)db.mj _0 _
Another important parameter that affects the selec-tion of ao is the WT computation time. For thespecial choice, a = 2, an ultrafast algorithm forcomputing the discrete WT has been suggested.15
For arbitrary values of ao the costs become large, anda fast optical approach for performing an immediateWT is attractive.
Optical Implementation of the Two-DimensionalWavelet Transform
The 2-D WT [Eq. (6)] is exactly a correlation betweenf(x) and hmb(x); thus hmb(X) could be considered as amatched filter. The system for obtaining a 2-D WTis shown in Fig. 1. It is a conventional VanderLugtcoherent optical correlator with a multireferencematched filter. For a certain input object f(x), this4-f processor provides the correlations with severaldaughter wavelets according to the discrete WT defi-nition [see Eqs. (6) and (7)]. Each daughter wavelethmb is encoded with a different reference beam otm;
(6)
(7)
Because of the sampling theory, the selection of ao
Fig. 2. Typical response for the improved matched filter. Thecentral area contains the zero-order information of the outputsignal.
Fig. 3. (a) Mother wavelet function, (b) its impulse response.
thus the final composite matched filter is
MRMF(x) = E ({FT[hmb(X)] + exp(iqamx)lc.c.). (9)m
FT is the Fourier-transform operation and c.c. is thecomplex conjugate. Note that am is a vector sincethe reference beam has two tilt directions, and it isnecessary to use I I > amin in order to ensure theseparation of the different diffraction orders. Accord-ing to Eq. (9), the filter generating is done by amultiexposure process (one exposure per referencebeam). Another way to encode the compositematched filter is to place the daughter wavelets at
Fig. 4. Magnification of the multireference matched filter withfour daughter wavelets.
111 1~~~~~~I I I III ' 2I I I 2 3111 * III=
6 III I1 =1 =15 =IIIII
III I
III =2III= 3illI-4II ol 4
111-5111_ 6
I II _ I(a)
(b)Fig. 6. (a) Test mask as the input pattern, (b) system output withfour different WT orders.
different lateral locations and then to perform theFourier transform (optically or by computer evalua-tion and plots):
MRMF(x) = FT hmb(X - bm) + exp(iaex)}c.c.
(10)
bm is the lateral shift that is related to the referencebeam direction m; a is the global carrier spatialfrequency of the hologram for ensuring the separa-tion of the different diffraction orders.
Figure 2 shows a typical response of the compositematched filter. In this example we encoded fourdifferent daughter wavelets. The symmetrical struc-ture of Fig. 2 is caused by the amplitude-only informa-tion of the MRMF.
The matched-filter generating process could bedone in one of two ways: (i) Use optical holographicmeans following the VanderLugt matched-filter prepa-ration steps. For example, one can plot the differentdaughter wavelets with the same output arrange-ment as Fig. 2 and then record the Fourier transformof the plot with a reference beam and a lens [see Eq.(10)]. An important assumption for using Eq. (9) or
(10) processes is that we work in the linear region ofthe t-E curve; thus the amplitude transmittance ofthe hologram can be supposed to be proportional tothe intensity of the interference pattern. (ii) EncodeEq. (9) or (10) directly, using one of the computer-generated-hologram techniques.
Placing the input object at the input plane of thecorrelator as in Fig. 1, we get at the output plane adistribution similar to that in Fig. 2. Each sectorcontains the WT of the input with one specific scaleparameter.
Experimental Results
The optical setup of Fig. 1 is used as a real-timeoptical wavelet transformer. The input object ispresented at the input plane with a transparency or aspatial light modulator (liquid-crystal TV) that worksat a TV rate. In the MRMF, the mother wavelet wasthe inverse Fourier transform of a band-pass filter.Figure 3(a) shows a three-dimensional plot of theselected mother wavelet, and Fig. 3(b) shows itsFourier transform (a ring). The aspect ratio of thisring (the ratio between the outer and the innercircles) is J'2. The daughter wavelets are differentscale versions of the function Fig. 3. In this case weused a = 2_. The hologram was encoded by thecomputer according to Eq. (10) by use of a high-resolution plotter. Figure 4 shows a magnificationof the MRMF. One can recognize the four differentrings (owing to the four daughter wavelets) as well asthe four different carrier spatial frequencies. The im-pulse response of this matched filter is shown in Fig. 5.
In the first experiment we made the input was atransparency, as in Fig. 6(a). Figure 6(b) shows theoutput plane when the MRMF in Fig. 4 is usedas a matched filter. The different wavelet orders(m = 0, 1, 2, 3) can be recognized according the ar-rangement of Fig. 2. The m = 0 section presents thelowest spatial frequencies, and m = 3 contains thehighest frequencies.
In order to check the real-time capability of thesystem, we conducted a second experiment by replac-ing the input transparency with a spatial light modu-lator (liquid-crystal TV panel). Figure 7(a)is pre-sented on the TV panel, and the output is shown inFig. 7(b). Figure 8 shows a magnification of thedifferent wavelet outputs. The spatial frequency istwo times higher than in the original input because ofthe obstructed zero-frequency area in the matchedfilter (see Fig. 4).
Conclusions
We have applied the wavelet transform (WT) to a 2-Dreal-time optical configuration. A set of daughter
wavelets with different reference beams was used toencode a multireference matched filter in a conven-tional VanderLugt correlator. The suggested sys-tem is advantageous over other 2-D WT implementa-tions mainly because of the following: (i) It is areal-time configuration; i.e., the system provides sev-eral daughter wavelet responses simultaneously.(ii) Following the spirit of the WT, we applied themultiscale operation on the mother wavelet itself andnot on the input object. (iii) The optical setup issimple and stable. Two experiments have been donewith a matched filter that contains four daughterwavelets. Further investigation can be carried outin order to improve the efficiency of the matched filter(e.g., by use of phase-only blazed methods) and toextend the number of daughter wavelets.
References1. H. J. Caufield, "Wavelet transforms and their relatives,"
Photon. Spectra (August 1992), p. 73.2. H. Szu, Y. Sheng and J. Chen, "Wavelet transform as a bank of
the matched filters," Appl. Opt. 31, 3267-3277 (1992).3. X. J. Lu, A. Katz, E. G. Katerakis, and N. P. Caviris, "Joint
transform correlation using wavelet transforms," Opt. Lett.18, 1700-1703.
4. E. Freysz, B. Pouligny, F. Argoul, and A. Arneodo, "Opticalwavelet transform of fractal aggregates," Phys. Rev. Lett. 64,7745-7748 (1990).
5. H. Szu and J. Caulfield, "The mutual time-frequency contentof two signals," Proc. IEEE 72, 902-908 (1984).
6. J. Caulfield and H. Szu, "Parallel discrete and continuouswavelet transforms," Opt. Eng. 31, 1835-1839 (1992).
7. D. Gabor, "Theory of communication," Proc. Inst. Electr. Eng.93, 429-457 (1946).
8. A. Grossmann and J. Morlet, "Decomposing of Hardy functioninto square integrable wavelets of constant shape," SIAM J.Math. Anal. 15, 723-736 (1984).
9. I. Daubechies, "The wavelet transform time-frequency localiza-tion and signal analysis," IEEE Trans. Inf. Theory 36, 961-1005 (1990).
10. J. M. Combes, A. Grossmann, and Ph. Tchamitchian, eds.,Wavelets: Time-Frequency Methods and Phase Space(Springer-Verlag, Berlin, 1989).
11. H. H. Szu, B. Telfer, and A. W. Lohmann, "Causal analyticalwavelet transform," Opt. Eng. 31, 1825-1829 (1992).
12. M. 0. Freeman, K. A. Duell, and A. Fedor, "Multi-scale opticalimage processing," presented at the IEEE International Sym-posium on Circuits and Systems, Singapore, June 1991.
13. A. Haar, "Zur thorie der orthogonalen Funktionen-systeme,"Math. Anal. 69, 331-371 (1910).
14. I. Daubechies, "Orthonormal bases of compactly supportedwavelets," Commun. Pure Appl. Math. 41, 909-996 (1988).
15. S. G. Mallat, "A theory for multiresolution signal decomposi-tion: the wavelet decomposition," IEEE Trans. Pattern Anal.Mach. Intell. 31, 674-693 (1989).
