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Hybrid optical implementation of discrete wavelet transforms: a tutorial
Vol. 28, No. 2, pp. 51-58, 1996 Copyright 0 1996 Elsevier Science Ltd
Oprics & Laser Technology,
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ELSEVIER ADVANCED
0030-3992(95)00069-O
TECHNOLOGY
Hybrid optical implementation of discrete wavelet transforms: a tutorial C. DECUSATIS,
J. KOAY, P. DAS
Recently, there has been a great deal of interest in the use of wavelets to supplement or replace traditional Fourier transform signal processing. Because wavelet operations can be expressed as either a convolution/correlation operation or a linear filter bank, they are well suited to optical implementations. This paper provides a tutorial on wavelets and the hybrid optical implementation of discrete wavelet transforms. Various approaches for implementing the wavelet transform are reviewed, including traditional Vander Lugt correlators, different types of spatial light modulators, joint time-frequency representations, holographic and interferometric implementations, and acousto-optic correlators. Finally, the concept for a new architecture is proposed using acousto-optic finite impulse response (FIR) filters to implement wavelets with perfect reconstruction properties. KEYWORDS: wavelets, convolution,
Introduction
acousto-optics,
hybrid optics, signal processing
practice, this is overcome by using the so-called windowed or short-time Fourier transform (STFT), which observes only a segment of the input signal to produce the transform (in other words, the STFT uses windowed sinusoids as its basis function, while the continuous Fourier transform uses infinite duration sinusoids). In contrast, wavelets utilize a basis function or mother wavelet, which is allowed both to shift and scale in time. The shifted and scaled mother wavelet produces a family of daughter wavelets, which are able to provide high resolution in both the time and frequency domain simultaneously. This is analogous to a STFT in which the size of the window is allowed to vary; when the window is dilated it accesses low frequency information, and when it is contracted it accesses high frequency information.
to wavelets
Transform domain processing has been shown to be a convenient way to represent filtering, excision, compression, reconstruction, and other signal processing operations. As shown in Fig. 1, by using a linear transformation, it is possible to map a general signal s(t) from the time domain into a different transform domain, where the signal is represented as a function of some other convenient variable such as frequency. We may speak of signal processing in the transform domain, then consider applying the inverse transform, which maps the signal back into the time domain. For practical purposes, the mapping and its inverse should both be unique and unambiguous. There are many possible transformations including the commonly used Fourier, Laplace, Fresnel, Mellin, Hilbert, and Hartley transforms’. As with any linear operator, these can be represented by either a causal linear system, an integral kernel, or in matrix notation.
Let the mother wavelet be represented by s(t); if we translate the window by an amount b and scale it by an amount a, then the daughter wavelets will be given by
The so-called wavelet transform is another possible mapping that can be used for transform domain processing. Until quite recently, most applications have been restricted to the Fourier transform; while this is a powerful signal processing technique, it has inherent disadvantages, such as the inability to represent both time and frequency information with high resolution. In
The continuous wavelet transform is then given by oc I+‘(a,b) =
CD is with IBM Corporation, 522 South Road, Poughkeepsie, NY 12601, USA. JK and PD are at Rensselaer Polytechnic Institute, Troy, NY 12180, USA. Received 27 April 1995.
s -3c
4t)Q’ab*(f)
dt
(2)
Note that the wavelet transform is just the correlation between the signal s(t) and the corresponding daughter 51
52
Hybrid optical implementation
Fig. 1
of discrete wavelet transforms: C. DeCusatis et al.
General transform domain block diagram
wavelets. The wavelet transform maps a onedimensional signal s(t) into a two-dimensional output, W(a, b). Similarly, the inverse transform is defined as s(t) =;
“1 __ ;Ezw(a, b)‘@ab(t)da db ss 00 0
(3)
where C is a finite constant. By setting Q= ag and b = anbo, where n and m are integers, it is possible to
define the discrete wavelet transform, analogous to the discrete Fourier transform W(m,n) = Es(k)*,,‘(y) k
The choice of mother wavelet basis function is very important; many possible wavelets exist, including the Daubechies, Morlet, Haar, and Mexican Hat wavelets’. The choice of mother wavelet can be tailored to the application. Since wavelets do not require a windowed basis function, they offer significantly reduced sidelobes in the transform domain representation. This can improve the ability to filter signals and remove unwanted noise or interference, with a corresponding improvement in signal-to-noise ratio. By providing high resolution in both time and frequency, wavelets can be used to achieve signal compression ratios in excess of 100 : 1, with better fidelity than conventional Fourier domain techniques. Thus, although the concept of wavelets has been known for centuries in various forms2p6, it is only recently that wavelets have been rediscovered and significant progress has been made in this area. There has been an explosion of papers and interest in wavelets and their potential application@‘; some have called them the most significant mathematical event of the past decade’*. Certain implementations of wavelets have the property of perfect reconstruction of a digital data sequence, without aliasing or truncation errors. This is possible because discrete wavelets subdivide the signal spectrum such that errors in the reconstruction of different frequency components cancel each other out. Perfect reconstruction can be achieved using a filter bank structure as shown in Fig. 2(a). The input digital data sequence x(n) is divided in half using a low pass filter Ho(z) and a high pass filter H,(z); each frequency subband is then ‘downsampled’ by a factor of 2 since it can be shown’ that only half of the intermediate outputs are required. Each frequency sub-band is then divided in half once again, using filters that continually decompose the signal into narrower frequency bands as illustrated by Fig. 2(b). This structure is known as a binary subband tree decomposition of the signal; since higher frequency resolution is obtained at each level of the tree structure, this is also known as multiresolution analysis of the signal x(n). The signal can be reconstructed by ‘upsampling’ and using the corresponding filters Go(z) and GI(z) as shown in the second part of Fig. 2(a). The filters H(z) are
known as analysis filters, and the corresponding G(z) are called synthesis filters; these may be implemented using many different techniques, including both finite and infinite impulse response filters’. The analysis and synthesis filter functions are related such that if one set is known, the other may be calculated’. The structure of Figs 2(a) and (b) are known as Quadrature Mirror Filters (QMF). An important property of QMF filter banks is that perfect signal reconstruction using an N-tap digital filter requires the solution of only N/2 equations for the tap coefficients; the remaining N/2 equations can be used to impose some other condition on the filter for a given application. For example, Daubechies13)14 used the remaining expressions to guarantee a minimum ripple over the passband, the so-called regularity condition. Optical signal processing techniques are well suited to the implementation of wavelet transforms because of their inherent two-dimensional nature, high speed, and parallel processing capability. Wavelet-based signal processing has also been demonstrated using digital electronics, which has the advantages of programmability. Hybrid optoelectronic implementations take advantage of the strengths in both approaches; in the following section, we will briefly review some of the hybrid architectures that have been proposed. This is by no means a comprehensive list, since a large number of approaches have been proposed with new ones appearing rapidly. We will then consider a new architecture, which is a direct implementation of the perfect reconstruction QMF filter bank using acousto-optics. Review
of hybrid wavelet
architectures
As noted above, the wavelet transform can be implemented by any type of convolver/correlator architecture. Many of the implementations demonstrated recently are variations of the classic 4F optical correlator and matched filter, first proposed by Vander Lugt15 and illustrated in Fig. 3. An object function is placed at the front focal plane of a lens, which performs a two-dimensional spatial Fourier transform at its back focal plane. Filtering operations can be performed in this plane, and a second lens inverts the Fourier transform (with its coordinates reversed in space) at the output focal plane. The output represents the correlation between the input and the filter function. The implementation of this filter determines the input signal bandwidth limitations, or the maximum number of points that may be computed in a discrete wavelet transform. Since the filter can in general be a complex function, Vander Lugt proposed a holographic implementation containing both amplitude and phase information. This approach has been used to implement a non-real time wavelet transform16, with each daughter wavelet recorded on a separate film plate and loaded sequentially into the optical processor. A more sophisticated design I’ has implemented the Haar wavelet using computer generated holograms, as part of
Hybrid optical implementation
(a>
Fig. 2 (a) Binary sub-band coefficients
of discrete wave/et transforms: C. DeCusatis et al.
analysis
synthesis
tree; (b) bandpass filter responses at different stages of the binary sub-band
an automatic machine-based pattern recognition system or image segmenter. This has the advantage of using graphical computer output to encode wavelets that cannot be easily created by other means. However, the overall system is designed for a specific task and is not easily programmable; filters must be changed manually to perform different tasks. This type of implementation requires the Fourier transform of the wavelet function to be placed at the
53
tree using Daubechies’
4-tap FIR
transform plane of a Vander Lugt correlator. Holographic or interferometric techniques are required if both amplitude and phase information are to be preserved; this is the case for wavelets such as Daubechies’, whose Fourier transforms are complex functions. The Fourier transform of some popular wavelets, including Morlet and Mexican Hat, are realvalued symmetric functions; they can be implemented by intensity-only transmission masks. In some cases, the inverse wavelet transform can also be realized using only
Hybrid optical implementation
54
I
f
I
I
filter plane
input plane Fig. 3
f
f
of discrete wavelet transforms: C. DeCusatis et al.
f output plane
4F optical system
real-valued filters18. By using more complex coding techniques, it is possible to incorporate negative and complex values in the wavelet functions even if the system uses only real positive inputslg; we will not discuss this point in detail. Many different types of spatial light modulators have been proposed for this application, including the magneto-optic spatial light modulator (MOSLM). This device is based on the Faraday effect, which causes the plane of polarization of light to rotate when passing through a magnetic field; this can be converted into intensity modulation by placing the device between crossed polarizers. Commercial devices consist of a magnetic thin film, divided into a 256 x 256 array of pixels which can be addressed electronically to change their magnetization state. Each pixel has three stable states; transparent, opaque, and an intermediate or grey state. This makes it well suited for implementation of the Haar wavelet, which is given by 1 o
(5)
This wavelet produces edge enhancement of images, which is an important tool for pattern recognition, image segmentation, and image classifiers. In the approach shown in Fig. q2’, a MOSLM generates the Haar wavelet under computer control in the transform plane of the Vander Lugt correlator. The wavelet is imaged onto a thermoplastic holographic camera, which contains the input image. Diffracted light from the hologram represents the wavelet transform of the image, which is detected and displayed by a CCD camera and frame grabber card in a PC. Image segmentation of binary images has been reported using this design2’. A similar approach has been demonstratedt6 using a liquid crystal light valve (LCLV) as the spatial light modulator for an optical correlator; this has the advantage of representing wavelets that require both amplitude and phase information. Even if only real, positive values are required, an LCLV can still be useful as a spatial light modulator; one approach uses the LCLV to rebroadcast an image written by an optical scanner at the input of a 4F correlator18. Another scheme required both phase and amplitude information to perform intensity modulation and beam steering in multichannel correlators22, as shown in Fig. 5. Both amplitude and phase information of a complex wavelet are recorded on a holographic plate. The plate may be
Fmmegrabber
Fig. 4 Optical correlator architecture implementation (from Ref. 20)
for single aperture wavelet
subdivided into a set of different regions, with a different wavelet recorded in each region; in this way, different channels can be processed in parallel. This approach takes advantage of the free space interconnectivity of optical beams to avoid the wiring problems associated with pure electronic implementations; the input and output images are still controlled by spatial light modulators and CCDs, respectively. Since this approach can process a joint time-frequency representation of a signal, it is also known as an N4 correlator; output resolution is limited by the diffraction efficiency of the holographic element, and the signal-to-noise ratio achievable at the output plane. It has been noted23 that the connectivity problem can also be addressed using arrays of lenslets placed between the input image and the wavelet filter plane. The Vander Lugt correlator has been applied to both wavelets and short time Fourier transforms, using Holographic
Fig. 5 N4 correlator architecture for a two-dimensional holographic wavelet transform (from Ref. 22)
Hybrid optical implementation
of discrete wavelet transforms: C. DeCusatis et al.
computer controlled s atial light modulators at the input and filter planes !?4. The inverse operation can also be realized with this architecture, which can be modified by placing a phase conjugate mirror (PCM) after the output focal plane. The advantage of using a PCM is its property of reflecting the optical beam along exactly the incident path, so the filters will automatically be aligned for the inverse operation. A PCM can also amplify the reflected signal, if required; this could be used to correct for non-uniform light intensities of different spectral lines. It has also been proposed24 that both the wavelet filters and PCM could be implemented using photorefractive crystals. A modified version of the Vander Lugt correlator has been used together with a Smartt interferometer to implement wavelet transforms25. The Smartt interferometer consists of a pinhole aperture in a partially transparent thin film, which is illuminated by coherent light. The diffracted light from the pinhole and the direct transmitted light through the film combine to form an interference pattern; it is a common path interferometer, which reduces the effect of vibration. This interferometric approach is an alternative to the holographic techniques used to preserve phase information; when a Smartt interferometer is placed in the transform plane of a 4F correlator along with the wavelet transparency, both amplitude and phase information are produced at the output plane. The detection scheme uses two rows of a CCD to preserve this information, rather than a single square-law detector, which is insensitive to phase. An extension to two-dimensional signals has also been proposed25. Acousto-optic devices have been frequently used as spatial light modulators for optical correlators; the wavelet transform has been implemented using both time and space integrating architectures’t26. Because of the lack of available of two-dimensional real-time wavelet transform processors, we have suggested the use of image processors for two-dimensional wavelet transforms’. Additionally, analogue acousto-optic processors have been valuable in the realization of socalled ‘triple continuous’ correlators27. These analogue processors operate on continuously evolving, nonstationary time signals which are impossible to analyse digitally. This is basically a 4F correlator that is made continuous in clock time, separation time, and
t
h,
A0
unit
diodes
Fig. 6
A0 implementation
of QMF bank
55
frequency because it utilizes wavelets rather than Fourier techniques. We shall discuss the use of acoustooptic filters for wavelet transforms in more detail in the following section. Other optical systems have been used to realize hybrid wavelet transforms; because of space limitations, we shall describe only a few here. For example, a hybrid system based on optical scanning has been used to implement the difference of Gaussians wavelet28. In this system, an acousto-optic modulator is used to generate two laser beams with different temporal frequencies. The beams are spatially combined and used to scan an input object; the scanned image gives the wavelet transform which can then be detected and stored in a digital computer for further processing. Because this is an incoherent system, it offers improvements in the signalto-noise ratio over coherent approaches. Other hybrid schemes have been proposed in which neural networks have been applied to the implementation of complex wavelet transforms. Some examples include architectures with electronic feedback and thresholding29 or which utilize spatial light modulators in an optical ring approach to N-wavelet coding30. Since wavelet filters can also be expressed in matrix form, various types of optical matrix processors or hybrid digital optical computers can, in principle, also be used to implement wavelet transforms’. Implementation filter banks
using acousto-optic
FIR
As we have seen, there are many possible implementations of the wavelet transform using optical correlators. In order to take advantage of the perfect reconstruction property, however, it is necessary to implement a QMF filter bank or binary sub-band tree structure. This approach has not previously been implemented optically, although it is more common in digital VLSI architectures’. In this section, we propose a new hybrid optical architecture using acousto-optic FIR filters with electronic feedback. A two channel QMF filter can be realized as shown in Fig. 6; the discrete input signal x(n) is used as the input to an A0 delay line. The delayed signal is sampled by an array of laser diodes or LEDs, which are intensity modulated to provide the filter tap weights. The figure shows a four tap filter, which is the case for implementation of the Daubechies wavelet; the high pass and low pass filters He(z) and H,(z) are implemented as shown. The output from the A0 unit represents delayed samples of the input signal x(n) weighted by the appropriate filter coefficients. This modulated light is collected and summed by a pair of cylindrical lenses, which focus the light onto a pair of photodetectors. By sampling the detector output at half the clock rate, the desire signal is obtained and stored in a digital computer. The computer recursively reuses the filter bank by applying this signal to the A0 device once again; a computer controlled RF signal generator could be used, for example. Figure 7 shows a possible timing diagram for the hybrid processor, assuming that the tap coefficients of Ho and Hi are (1, 0, 1,0) and (1, 0,0, l), respectively. Of course, it is also possible to display the input signal x(n) on the LEDs and the filter coefficients on the A0 device.
56
Hybrid optical implementation
of discrete wavelet transforms: C. DeCosatis et al.
could be used to display a different signal. This would allow one to implement multiple QMF banks located at different stages of the sub-band tree. A common row of diodes can be used to modulate the light since the filter coefficients are the same at each stage. A common clock can also be used since all the clock speeds are integer multiples of the input data rate. As before, this same approach could be used for synthesis with some slight differences, such as: sum the detector outputs and sample the output at the same clock rate as the input.
““)j Fig. 7
n
n
11 ...
Timing diagram of QMF bank
Reconstruction can be performed with the same basic architecture of Fig. 6. This requires upsampling of the input signal, or inserting a zero between alternating data points. The upsampled signal is processed as before by the filter bank, using the tap coefficients of the corresponding synthesis filters G,(z) and G,(z) (usually, but not necessarily, these are the same as the tap coefficients of Ho and HI except in reverse order). When performing signal reconstruction, the detector outputs need to be summed by the digital computer, or by using a lens to sum the optical signals together onto a single detector. Additionally, the detector output during reconstruction is sampled at the same rate as the input data sequence, not half the data rate. Although this implementation is not real time, it does have the potential to be realized using compact integrated optical tapped delay lines, similar to devices that have already been proposed for other applications3’. It is possible to extend this architecture to two dimensions as shown in Fig. 8. In this case, each A0 cell
compufer(timing Fig. 8
Although Fig. 8 shows a horizontal array of input diodes and vertical output detectors, it is of course possible using different cylindrical lenses to have a vertical input diode array and a horizontal detector array. (This architecture could be used as a timeintegrating image correlator. The application described in the previous paragraph does not require timeintegration.) The time-integrating correlator could also be replaced by a space-integrating correlator, which would require a two-dimensional array of input diodes but only a single pair of detectors rather than a detector array. If the stacked array of acousto-optic cells in this architecture proves difficult to implement, we may replace it with any suitable two-dimensional spatial light modulator such as a liquid crystal light valve. By using more complex coding techniques, negative and complex valued wavelets can be incorporated into this design*g)32;IIR filters may also be implemented, although the electronic feedback would be more complex. Since the wavelet transform maps a two-dimensional signal into a three-dimensional transformation, multidimensional architectures could have applications to image processing. We have previously described acousto-optic image correlators33 which perform twodimensional correlations between an M x M pixel reference image g(x, v) and a larger N x N input image f(x, y), where N > M. These image correlations require only a single acousto-optic device and can be used to compute the wavelet transform of images. A schematic
& data storage)
A0 implementation of multiple one-dimensional QMF banks
of discrete wavelet transforms: C. DeCusatis et al.
Hybrid optical implementation
2D output
Fig. 9
Wavelets have the potential to replace Fourier transforms in many applications because of their
correlation
A0 image correlator
diagram of the image correlator design of Psaltis34 is shown in Fig. 9. Both images can be described as a sequence of sequentially scanned rows, f(t, n) and g(t, m). Each row of g(t, m) temporally modulates an element of the LED array shown; there is one LED for each of the M rows of the reference image. The modulated light is focused into the aperture of the A0 device, which is modulated by the other imagef(t,n). The diffracted light is re-imaged by a Schlieren lens system such that each LED is focused onto a separate row of the CCD detector. The system thus functions as a set of parallel one-dimensional time integrating correlators, where the CCD contains the correlation between one row of the input image and all rows of the reference image. To obtain the remaining correlations in the vertical direction, the CCD charge is shifted down by one row and the process is repeated. The CCD accumulates the correlation between a row of the reference and input images, and adds this to the correlation between the previous input image line and the adjacent reference image row. This sum of partial one-dimensional correlations is equivalent to a twodimensional image correlation34. If one of the images is a two-dimensional family of daughter wavelets, it is thus possible to compute the correlations between an input image and the daughter wavelets so that the output will be the wavelet transform of the image. Note that this correlator produces a two-dimensional spatial output, which changes in time to provide the wavelet transform of an input image. The basic architecture has been modified by Molley et al.35>36to increase the throughput by using a larger CCD and masking intermediate rows; without excessively high clock frequencies or A0 cell bandwidth, it is possible to obtain a throughput of better than 1000 frames per second. Conclusions
57
reduced sidebands, multiscale resolution properties, and potential for perfect reconstruction using QMF filters. A fundamental approach to the implementation of wavelet transforms is correlators/convolvers (related approaches such as matrix processors and neural nets are also possible). Many hybrid optical correlators have been proposed, based on variations of the classic Vander Lugt 4F correlator, using LCLVs and MOSLM spatial light modulators. To preserve phase information for certain types of wavelets, interferometric techniques or holographic filters are necessary; for some applications, however, only positive real wavelets are required. Acousto-optic devices are used as spatial light modulators in several approaches. We have proposed a new hybrid system which implements the wavelet transform using acousto-optic FIR filters with recursive electronic feedback; the result is a multichannel QMF filter bank, with the advantage of perfect signal reconstruction. References C., Koay, J., Das, P. Real time implementation of wavelet transforms. Chapter 6 in Subband and Wavelet Transforms: Design & Application (Eds Akansu A. and Smith M.) Kluwer Academic, Norwell, MA (to be published) Chui, C.K (ed) Wavelets: an Introduction to Wavelets, Vol. 1, and Wavelets: a Tutorial in Theory and Applications, Vol. 2, Academic Press, NY (1992) Vetterli, M., Herley, C. Wavelets and filter banks: theory and design, IEEE Tram Signal Processing, 40 (1992) 2207-2232 Smith, M.J.T. IIR analysis/synthesis systems. In Subband Coding of Images, (Ed Woods, J.W.) Kluwer Academic, Norwell, MA (1991) Evangelista, G. Wavelet transforms and digital filters. In Wavelets and Applications, (Ed Meyer, Y .) Springer-Verlag, NY (1992) Resnikoff, H. Wavelets and adaptive signal processing, Opt Eng, DeCusatis,
31 (1992) 1229-1234 Baraniecki, A. Karim, S. Computational algorithms for discrete wavelet transforms, SPZE Proc, 1699 (1992) 408-419 Herley, C., Vetterli, M. Linear phase wavelets: theory and design, Proc Int Conf on Acoustics, Speech, and Signal Processing, Vol. 3 (1991) 2017-2020 Strang, G., Wavelets, Am Scientist, 82 (1994) 250-255
58
Hybrid optical implementation of discrete wavelet transforms: C. DeCusatis et al.
10 Strang, G., Wavelet transforms vs. Fourier transforms, Bull Amer Math Sot. 28 (1993) 288-305
Biographies
11 Freeman, M. Wavele&: signal representations with important advantages, Opt and Photonics News, 4 (1993) 8-14 12 Benvenisie, A.-Multiscale signal processing: from QMF to wavelets. Int Workshov on Altzorithms and Parallel VLSI Archite&res, vol. A, ‘ElseviepPress, (1991) 71-76 13 Daubecbies, 1. Orthonormal basis of compactly supported wavelets, Comm Pure and Appl Math, XL1 (1988) 909-996 14 Daubecbies, I., Lagarias, J. Two scale differential equations, II local regularity, infinite products of matrices, and fractals, AT&T Bell Labs Tech Report (1989)
15 Vander Lugt, A. Signal detection by complex spatial filtering, IEEE Tram Info Theory, IT-10 (1964) 2 16 Freysz E. Optical wavelet transform of fractal aggregates, Phys Rev Left, 64 (1990) 7745-7748
17 Block, P., Rogers, S., Ruck, D. Optical wavelet transforms from computer generated holography, Appl Opt, 33 (1994) 5275-5218
18 McAulay, A., Wang, J. Optical wavelet transform classifier with positive real Fourier transform wavelets, Opt Eng, 32 (1993) 1333-1339 19 Caufield, H., Rhodes, W.T., Foster, M., Horwity, S. Optical implementation of systolic array processing, Opt Comm, 40 20 21
22
23
24
(1981) 86 Burns, T., Fielding, K., Rogers, S., Pinski, S., Buck, D. Optical Haar wavelet transform, Opt Eng, 31 (1992) 1852-1858
Pinski, S.D., Rogers, S.D., Ruck, S.K., Welsh, D.M., Kabrinski, B.M., Warbola, M., Quinn, G., Oxley, D. Image segmentation using optical wavelets, SPIE Proc, 1702 (1992) 1IL19 Sbeng Y., Lu, T., Roberge, D. Optical ti implementation of a two-dimensional wavelet transform, Opt Eng, 31 (1992) 1859-1864 Wang, D.X., Tai, J.W., Zbang, Y.X. Two dimensional optical wavelet transform in space domain and its performance analysis, Appl Opt, 33 (1994) 5271-5274 Yu F.T.S., Lu, G. Short-time Fourier transform and wavelet transform with Fourier-domain processing, Appl Opt, 33 (1994) 5262-5270
25
Zbang, Y., Kanterakis, E., Katz, A., Wang, J.M. Optoelectronic wavelet processors based on Smartt interferometry, Appl Opt, 33 (1994) 5279-5266
26 Szu, H., Telfer, B., Lobman, A. Causal analytical wavelet transform, Opt Eng, 31 (1992) 1825-1829 27 Caulfield, H.J., Ludman, J.E., Hemmer P.R. Optical wavelet transform continuous in time, time shift, and scale, SPZE Proc, 2238 (1994) 166-169
28 Poon, T.C. Real time optical image processing using difference of Gaussian wavelets, Opt Eng, 33 (1994) 2294-2302 29 Szu, H., Telfer, B., Kadambe, S. Neural network adaptive wavelets for signal representation and classification, Opt Eng, 31 (1992) 1907-1916 30 Pbuvan, S. Optical implementation of N-wavelet coding for pattern classification, Appl Opt, 33 (1994) 5294-5301 31 Kim, J., Kenan, R. Programmable integrated optic tapped delay lines using channel waveguide structures, Proc. OSA Annual Meeting, Dallas, Texas, (1994) p. 144 32 Goodman, J., Woody, L. Method for performing complex-valued linear operations on complex-valued data using incoherent light, 33
34 35
Appl Opt, 16 (1977) 2611 Das P., DeCusatis C. Acoustic-optic image correlators, SPIE Proc. 1844 (Acoustic-optics and Applications, Gdansk, Poland)
(1992) 3348 Psaltis, D. Incoherent electro-optic image correlator, Opt Eng, 23 (1984) 12-15 Molley, P.A. An acousto-optic image correlator with a throughtput rate of 1000 templates per second, SPIE, 1295 (Real Time Image Processing II), (1990) 90-101
36
Molley P.A., Stalker K.T., Sweat& W.C. A compact, real-time acousto-optic image correlator, SPIE, 1347 (Optical Information Processing Systems and Architectures II) (1990) 123-130
Casimer M. DeCusatis is a staff engineer for IBM Corporation, System/390 Server Division, Poughkeepsie, New York USA, where he has served as lead engineer for the gigabit data links used on IBM Parallel Transaction and Query Servers, and a principal engineer for ESCON and other fibre-optics products. He received MS and PhD degrees in electrical engineering from Rensselaer Polytechnic Institute (Troy, New York) in 1988 and 1990, respectively, and the BS degree magna cum laude in the Engineering Science Honors Program from Pennsylvania State University (University Park, PA) in 1986. He is co-inventor on four patents and has been invited to speak at international conferences in St Peterburg, Russia and Gdansk, Poland. He is co-authoi of over 40 technical papers, as well as the book Acoustooptics: Fundamentals and Applications(Artech House, Boston, MA 1990); he has also contributed chapters to books on wavelets and fibre-optic data communications, and will serve as editor of the OSA/AIP Handbook of Photometry. Dr DeCusatis is a member of the Optical Society of American, IEEE, SPIE, Sigma Xi Research Society, and 10 academic honor societies including Tau Beta Pi and Eta Kappa Nu; he has also been profiled by Who’s Who in Science and Engineering. J Koay is currently an MS student under the thesis supervision of Professor P. Das in the Electrical, Computer, and Systems Engineering Department at Rensselaer Polytechnic Institute, Troy, New York, USA. His planned graduation date is May, 1995. His thesis is on applications of the wavelet transform in signal processing. P. Das is a Professor in the Electrical Computer, and Systems Engineering Department at Rensselaer Polytechnic Institute, Troy, New York, USA. He was formerly a Faculty Member with the Electrical Engineering Departments of the Polytechnic Institute of New York and the University of Rochester, USA. He has published numerous papers and is currently performing research in wavelets and their applications to spread spectrum communication, ultrasonics, biomedical signal processing, acousto-optics, and nondestructive testing.
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ELSEVIER ADVANCED
0030-3992(95)00069-O
TECHNOLOGY
Hybrid optical implementation of discrete wavelet transforms: a tutorial C. DECUSATIS,
J. KOAY, P. DAS
Recently, there has been a great deal of interest in the use of wavelets to supplement or replace traditional Fourier transform signal processing. Because wavelet operations can be expressed as either a convolution/correlation operation or a linear filter bank, they are well suited to optical implementations. This paper provides a tutorial on wavelets and the hybrid optical implementation of discrete wavelet transforms. Various approaches for implementing the wavelet transform are reviewed, including traditional Vander Lugt correlators, different types of spatial light modulators, joint time-frequency representations, holographic and interferometric implementations, and acousto-optic correlators. Finally, the concept for a new architecture is proposed using acousto-optic finite impulse response (FIR) filters to implement wavelets with perfect reconstruction properties. KEYWORDS: wavelets, convolution,
Introduction
acousto-optics,
hybrid optics, signal processing
practice, this is overcome by using the so-called windowed or short-time Fourier transform (STFT), which observes only a segment of the input signal to produce the transform (in other words, the STFT uses windowed sinusoids as its basis function, while the continuous Fourier transform uses infinite duration sinusoids). In contrast, wavelets utilize a basis function or mother wavelet, which is allowed both to shift and scale in time. The shifted and scaled mother wavelet produces a family of daughter wavelets, which are able to provide high resolution in both the time and frequency domain simultaneously. This is analogous to a STFT in which the size of the window is allowed to vary; when the window is dilated it accesses low frequency information, and when it is contracted it accesses high frequency information.
to wavelets
Transform domain processing has been shown to be a convenient way to represent filtering, excision, compression, reconstruction, and other signal processing operations. As shown in Fig. 1, by using a linear transformation, it is possible to map a general signal s(t) from the time domain into a different transform domain, where the signal is represented as a function of some other convenient variable such as frequency. We may speak of signal processing in the transform domain, then consider applying the inverse transform, which maps the signal back into the time domain. For practical purposes, the mapping and its inverse should both be unique and unambiguous. There are many possible transformations including the commonly used Fourier, Laplace, Fresnel, Mellin, Hilbert, and Hartley transforms’. As with any linear operator, these can be represented by either a causal linear system, an integral kernel, or in matrix notation.
Let the mother wavelet be represented by s(t); if we translate the window by an amount b and scale it by an amount a, then the daughter wavelets will be given by
The so-called wavelet transform is another possible mapping that can be used for transform domain processing. Until quite recently, most applications have been restricted to the Fourier transform; while this is a powerful signal processing technique, it has inherent disadvantages, such as the inability to represent both time and frequency information with high resolution. In
The continuous wavelet transform is then given by oc I+‘(a,b) =
CD is with IBM Corporation, 522 South Road, Poughkeepsie, NY 12601, USA. JK and PD are at Rensselaer Polytechnic Institute, Troy, NY 12180, USA. Received 27 April 1995.
s -3c
4t)Q’ab*(f)
dt
(2)
Note that the wavelet transform is just the correlation between the signal s(t) and the corresponding daughter 51
52
Hybrid optical implementation
Fig. 1
of discrete wavelet transforms: C. DeCusatis et al.
General transform domain block diagram
wavelets. The wavelet transform maps a onedimensional signal s(t) into a two-dimensional output, W(a, b). Similarly, the inverse transform is defined as s(t) =;
“1 __ ;Ezw(a, b)‘@ab(t)da db ss 00 0
(3)
where C is a finite constant. By setting Q= ag and b = anbo, where n and m are integers, it is possible to
define the discrete wavelet transform, analogous to the discrete Fourier transform W(m,n) = Es(k)*,,‘(y) k
The choice of mother wavelet basis function is very important; many possible wavelets exist, including the Daubechies, Morlet, Haar, and Mexican Hat wavelets’. The choice of mother wavelet can be tailored to the application. Since wavelets do not require a windowed basis function, they offer significantly reduced sidelobes in the transform domain representation. This can improve the ability to filter signals and remove unwanted noise or interference, with a corresponding improvement in signal-to-noise ratio. By providing high resolution in both time and frequency, wavelets can be used to achieve signal compression ratios in excess of 100 : 1, with better fidelity than conventional Fourier domain techniques. Thus, although the concept of wavelets has been known for centuries in various forms2p6, it is only recently that wavelets have been rediscovered and significant progress has been made in this area. There has been an explosion of papers and interest in wavelets and their potential application@‘; some have called them the most significant mathematical event of the past decade’*. Certain implementations of wavelets have the property of perfect reconstruction of a digital data sequence, without aliasing or truncation errors. This is possible because discrete wavelets subdivide the signal spectrum such that errors in the reconstruction of different frequency components cancel each other out. Perfect reconstruction can be achieved using a filter bank structure as shown in Fig. 2(a). The input digital data sequence x(n) is divided in half using a low pass filter Ho(z) and a high pass filter H,(z); each frequency subband is then ‘downsampled’ by a factor of 2 since it can be shown’ that only half of the intermediate outputs are required. Each frequency sub-band is then divided in half once again, using filters that continually decompose the signal into narrower frequency bands as illustrated by Fig. 2(b). This structure is known as a binary subband tree decomposition of the signal; since higher frequency resolution is obtained at each level of the tree structure, this is also known as multiresolution analysis of the signal x(n). The signal can be reconstructed by ‘upsampling’ and using the corresponding filters Go(z) and GI(z) as shown in the second part of Fig. 2(a). The filters H(z) are
known as analysis filters, and the corresponding G(z) are called synthesis filters; these may be implemented using many different techniques, including both finite and infinite impulse response filters’. The analysis and synthesis filter functions are related such that if one set is known, the other may be calculated’. The structure of Figs 2(a) and (b) are known as Quadrature Mirror Filters (QMF). An important property of QMF filter banks is that perfect signal reconstruction using an N-tap digital filter requires the solution of only N/2 equations for the tap coefficients; the remaining N/2 equations can be used to impose some other condition on the filter for a given application. For example, Daubechies13)14 used the remaining expressions to guarantee a minimum ripple over the passband, the so-called regularity condition. Optical signal processing techniques are well suited to the implementation of wavelet transforms because of their inherent two-dimensional nature, high speed, and parallel processing capability. Wavelet-based signal processing has also been demonstrated using digital electronics, which has the advantages of programmability. Hybrid optoelectronic implementations take advantage of the strengths in both approaches; in the following section, we will briefly review some of the hybrid architectures that have been proposed. This is by no means a comprehensive list, since a large number of approaches have been proposed with new ones appearing rapidly. We will then consider a new architecture, which is a direct implementation of the perfect reconstruction QMF filter bank using acousto-optics. Review
of hybrid wavelet
architectures
As noted above, the wavelet transform can be implemented by any type of convolver/correlator architecture. Many of the implementations demonstrated recently are variations of the classic 4F optical correlator and matched filter, first proposed by Vander Lugt15 and illustrated in Fig. 3. An object function is placed at the front focal plane of a lens, which performs a two-dimensional spatial Fourier transform at its back focal plane. Filtering operations can be performed in this plane, and a second lens inverts the Fourier transform (with its coordinates reversed in space) at the output focal plane. The output represents the correlation between the input and the filter function. The implementation of this filter determines the input signal bandwidth limitations, or the maximum number of points that may be computed in a discrete wavelet transform. Since the filter can in general be a complex function, Vander Lugt proposed a holographic implementation containing both amplitude and phase information. This approach has been used to implement a non-real time wavelet transform16, with each daughter wavelet recorded on a separate film plate and loaded sequentially into the optical processor. A more sophisticated design I’ has implemented the Haar wavelet using computer generated holograms, as part of
Hybrid optical implementation
(a>
Fig. 2 (a) Binary sub-band coefficients
of discrete wave/et transforms: C. DeCusatis et al.
analysis
synthesis
tree; (b) bandpass filter responses at different stages of the binary sub-band
an automatic machine-based pattern recognition system or image segmenter. This has the advantage of using graphical computer output to encode wavelets that cannot be easily created by other means. However, the overall system is designed for a specific task and is not easily programmable; filters must be changed manually to perform different tasks. This type of implementation requires the Fourier transform of the wavelet function to be placed at the
53
tree using Daubechies’
4-tap FIR
transform plane of a Vander Lugt correlator. Holographic or interferometric techniques are required if both amplitude and phase information are to be preserved; this is the case for wavelets such as Daubechies’, whose Fourier transforms are complex functions. The Fourier transform of some popular wavelets, including Morlet and Mexican Hat, are realvalued symmetric functions; they can be implemented by intensity-only transmission masks. In some cases, the inverse wavelet transform can also be realized using only
Hybrid optical implementation
54
I
f
I
I
filter plane
input plane Fig. 3
f
f
of discrete wavelet transforms: C. DeCusatis et al.
f output plane
4F optical system
real-valued filters18. By using more complex coding techniques, it is possible to incorporate negative and complex values in the wavelet functions even if the system uses only real positive inputslg; we will not discuss this point in detail. Many different types of spatial light modulators have been proposed for this application, including the magneto-optic spatial light modulator (MOSLM). This device is based on the Faraday effect, which causes the plane of polarization of light to rotate when passing through a magnetic field; this can be converted into intensity modulation by placing the device between crossed polarizers. Commercial devices consist of a magnetic thin film, divided into a 256 x 256 array of pixels which can be addressed electronically to change their magnetization state. Each pixel has three stable states; transparent, opaque, and an intermediate or grey state. This makes it well suited for implementation of the Haar wavelet, which is given by 1 o
(5)
This wavelet produces edge enhancement of images, which is an important tool for pattern recognition, image segmentation, and image classifiers. In the approach shown in Fig. q2’, a MOSLM generates the Haar wavelet under computer control in the transform plane of the Vander Lugt correlator. The wavelet is imaged onto a thermoplastic holographic camera, which contains the input image. Diffracted light from the hologram represents the wavelet transform of the image, which is detected and displayed by a CCD camera and frame grabber card in a PC. Image segmentation of binary images has been reported using this design2’. A similar approach has been demonstratedt6 using a liquid crystal light valve (LCLV) as the spatial light modulator for an optical correlator; this has the advantage of representing wavelets that require both amplitude and phase information. Even if only real, positive values are required, an LCLV can still be useful as a spatial light modulator; one approach uses the LCLV to rebroadcast an image written by an optical scanner at the input of a 4F correlator18. Another scheme required both phase and amplitude information to perform intensity modulation and beam steering in multichannel correlators22, as shown in Fig. 5. Both amplitude and phase information of a complex wavelet are recorded on a holographic plate. The plate may be
Fmmegrabber
Fig. 4 Optical correlator architecture implementation (from Ref. 20)
for single aperture wavelet
subdivided into a set of different regions, with a different wavelet recorded in each region; in this way, different channels can be processed in parallel. This approach takes advantage of the free space interconnectivity of optical beams to avoid the wiring problems associated with pure electronic implementations; the input and output images are still controlled by spatial light modulators and CCDs, respectively. Since this approach can process a joint time-frequency representation of a signal, it is also known as an N4 correlator; output resolution is limited by the diffraction efficiency of the holographic element, and the signal-to-noise ratio achievable at the output plane. It has been noted23 that the connectivity problem can also be addressed using arrays of lenslets placed between the input image and the wavelet filter plane. The Vander Lugt correlator has been applied to both wavelets and short time Fourier transforms, using Holographic
Fig. 5 N4 correlator architecture for a two-dimensional holographic wavelet transform (from Ref. 22)
Hybrid optical implementation
of discrete wavelet transforms: C. DeCusatis et al.
computer controlled s atial light modulators at the input and filter planes !?4. The inverse operation can also be realized with this architecture, which can be modified by placing a phase conjugate mirror (PCM) after the output focal plane. The advantage of using a PCM is its property of reflecting the optical beam along exactly the incident path, so the filters will automatically be aligned for the inverse operation. A PCM can also amplify the reflected signal, if required; this could be used to correct for non-uniform light intensities of different spectral lines. It has also been proposed24 that both the wavelet filters and PCM could be implemented using photorefractive crystals. A modified version of the Vander Lugt correlator has been used together with a Smartt interferometer to implement wavelet transforms25. The Smartt interferometer consists of a pinhole aperture in a partially transparent thin film, which is illuminated by coherent light. The diffracted light from the pinhole and the direct transmitted light through the film combine to form an interference pattern; it is a common path interferometer, which reduces the effect of vibration. This interferometric approach is an alternative to the holographic techniques used to preserve phase information; when a Smartt interferometer is placed in the transform plane of a 4F correlator along with the wavelet transparency, both amplitude and phase information are produced at the output plane. The detection scheme uses two rows of a CCD to preserve this information, rather than a single square-law detector, which is insensitive to phase. An extension to two-dimensional signals has also been proposed25. Acousto-optic devices have been frequently used as spatial light modulators for optical correlators; the wavelet transform has been implemented using both time and space integrating architectures’t26. Because of the lack of available of two-dimensional real-time wavelet transform processors, we have suggested the use of image processors for two-dimensional wavelet transforms’. Additionally, analogue acousto-optic processors have been valuable in the realization of socalled ‘triple continuous’ correlators27. These analogue processors operate on continuously evolving, nonstationary time signals which are impossible to analyse digitally. This is basically a 4F correlator that is made continuous in clock time, separation time, and
t
h,
A0
unit
diodes
Fig. 6
A0 implementation
of QMF bank
55
frequency because it utilizes wavelets rather than Fourier techniques. We shall discuss the use of acoustooptic filters for wavelet transforms in more detail in the following section. Other optical systems have been used to realize hybrid wavelet transforms; because of space limitations, we shall describe only a few here. For example, a hybrid system based on optical scanning has been used to implement the difference of Gaussians wavelet28. In this system, an acousto-optic modulator is used to generate two laser beams with different temporal frequencies. The beams are spatially combined and used to scan an input object; the scanned image gives the wavelet transform which can then be detected and stored in a digital computer for further processing. Because this is an incoherent system, it offers improvements in the signalto-noise ratio over coherent approaches. Other hybrid schemes have been proposed in which neural networks have been applied to the implementation of complex wavelet transforms. Some examples include architectures with electronic feedback and thresholding29 or which utilize spatial light modulators in an optical ring approach to N-wavelet coding30. Since wavelet filters can also be expressed in matrix form, various types of optical matrix processors or hybrid digital optical computers can, in principle, also be used to implement wavelet transforms’. Implementation filter banks
using acousto-optic
FIR
As we have seen, there are many possible implementations of the wavelet transform using optical correlators. In order to take advantage of the perfect reconstruction property, however, it is necessary to implement a QMF filter bank or binary sub-band tree structure. This approach has not previously been implemented optically, although it is more common in digital VLSI architectures’. In this section, we propose a new hybrid optical architecture using acousto-optic FIR filters with electronic feedback. A two channel QMF filter can be realized as shown in Fig. 6; the discrete input signal x(n) is used as the input to an A0 delay line. The delayed signal is sampled by an array of laser diodes or LEDs, which are intensity modulated to provide the filter tap weights. The figure shows a four tap filter, which is the case for implementation of the Daubechies wavelet; the high pass and low pass filters He(z) and H,(z) are implemented as shown. The output from the A0 unit represents delayed samples of the input signal x(n) weighted by the appropriate filter coefficients. This modulated light is collected and summed by a pair of cylindrical lenses, which focus the light onto a pair of photodetectors. By sampling the detector output at half the clock rate, the desire signal is obtained and stored in a digital computer. The computer recursively reuses the filter bank by applying this signal to the A0 device once again; a computer controlled RF signal generator could be used, for example. Figure 7 shows a possible timing diagram for the hybrid processor, assuming that the tap coefficients of Ho and Hi are (1, 0, 1,0) and (1, 0,0, l), respectively. Of course, it is also possible to display the input signal x(n) on the LEDs and the filter coefficients on the A0 device.
56
Hybrid optical implementation
of discrete wavelet transforms: C. DeCosatis et al.
could be used to display a different signal. This would allow one to implement multiple QMF banks located at different stages of the sub-band tree. A common row of diodes can be used to modulate the light since the filter coefficients are the same at each stage. A common clock can also be used since all the clock speeds are integer multiples of the input data rate. As before, this same approach could be used for synthesis with some slight differences, such as: sum the detector outputs and sample the output at the same clock rate as the input.
““)j Fig. 7
n
n
11 ...
Timing diagram of QMF bank
Reconstruction can be performed with the same basic architecture of Fig. 6. This requires upsampling of the input signal, or inserting a zero between alternating data points. The upsampled signal is processed as before by the filter bank, using the tap coefficients of the corresponding synthesis filters G,(z) and G,(z) (usually, but not necessarily, these are the same as the tap coefficients of Ho and HI except in reverse order). When performing signal reconstruction, the detector outputs need to be summed by the digital computer, or by using a lens to sum the optical signals together onto a single detector. Additionally, the detector output during reconstruction is sampled at the same rate as the input data sequence, not half the data rate. Although this implementation is not real time, it does have the potential to be realized using compact integrated optical tapped delay lines, similar to devices that have already been proposed for other applications3’. It is possible to extend this architecture to two dimensions as shown in Fig. 8. In this case, each A0 cell
compufer(timing Fig. 8
Although Fig. 8 shows a horizontal array of input diodes and vertical output detectors, it is of course possible using different cylindrical lenses to have a vertical input diode array and a horizontal detector array. (This architecture could be used as a timeintegrating image correlator. The application described in the previous paragraph does not require timeintegration.) The time-integrating correlator could also be replaced by a space-integrating correlator, which would require a two-dimensional array of input diodes but only a single pair of detectors rather than a detector array. If the stacked array of acousto-optic cells in this architecture proves difficult to implement, we may replace it with any suitable two-dimensional spatial light modulator such as a liquid crystal light valve. By using more complex coding techniques, negative and complex valued wavelets can be incorporated into this design*g)32;IIR filters may also be implemented, although the electronic feedback would be more complex. Since the wavelet transform maps a two-dimensional signal into a three-dimensional transformation, multidimensional architectures could have applications to image processing. We have previously described acousto-optic image correlators33 which perform twodimensional correlations between an M x M pixel reference image g(x, v) and a larger N x N input image f(x, y), where N > M. These image correlations require only a single acousto-optic device and can be used to compute the wavelet transform of images. A schematic
& data storage)
A0 implementation of multiple one-dimensional QMF banks
of discrete wavelet transforms: C. DeCusatis et al.
Hybrid optical implementation
2D output
Fig. 9
Wavelets have the potential to replace Fourier transforms in many applications because of their
correlation
A0 image correlator
diagram of the image correlator design of Psaltis34 is shown in Fig. 9. Both images can be described as a sequence of sequentially scanned rows, f(t, n) and g(t, m). Each row of g(t, m) temporally modulates an element of the LED array shown; there is one LED for each of the M rows of the reference image. The modulated light is focused into the aperture of the A0 device, which is modulated by the other imagef(t,n). The diffracted light is re-imaged by a Schlieren lens system such that each LED is focused onto a separate row of the CCD detector. The system thus functions as a set of parallel one-dimensional time integrating correlators, where the CCD contains the correlation between one row of the input image and all rows of the reference image. To obtain the remaining correlations in the vertical direction, the CCD charge is shifted down by one row and the process is repeated. The CCD accumulates the correlation between a row of the reference and input images, and adds this to the correlation between the previous input image line and the adjacent reference image row. This sum of partial one-dimensional correlations is equivalent to a twodimensional image correlation34. If one of the images is a two-dimensional family of daughter wavelets, it is thus possible to compute the correlations between an input image and the daughter wavelets so that the output will be the wavelet transform of the image. Note that this correlator produces a two-dimensional spatial output, which changes in time to provide the wavelet transform of an input image. The basic architecture has been modified by Molley et al.35>36to increase the throughput by using a larger CCD and masking intermediate rows; without excessively high clock frequencies or A0 cell bandwidth, it is possible to obtain a throughput of better than 1000 frames per second. Conclusions
57
reduced sidebands, multiscale resolution properties, and potential for perfect reconstruction using QMF filters. A fundamental approach to the implementation of wavelet transforms is correlators/convolvers (related approaches such as matrix processors and neural nets are also possible). Many hybrid optical correlators have been proposed, based on variations of the classic Vander Lugt 4F correlator, using LCLVs and MOSLM spatial light modulators. To preserve phase information for certain types of wavelets, interferometric techniques or holographic filters are necessary; for some applications, however, only positive real wavelets are required. Acousto-optic devices are used as spatial light modulators in several approaches. We have proposed a new hybrid system which implements the wavelet transform using acousto-optic FIR filters with recursive electronic feedback; the result is a multichannel QMF filter bank, with the advantage of perfect signal reconstruction. References C., Koay, J., Das, P. Real time implementation of wavelet transforms. Chapter 6 in Subband and Wavelet Transforms: Design & Application (Eds Akansu A. and Smith M.) Kluwer Academic, Norwell, MA (to be published) Chui, C.K (ed) Wavelets: an Introduction to Wavelets, Vol. 1, and Wavelets: a Tutorial in Theory and Applications, Vol. 2, Academic Press, NY (1992) Vetterli, M., Herley, C. Wavelets and filter banks: theory and design, IEEE Tram Signal Processing, 40 (1992) 2207-2232 Smith, M.J.T. IIR analysis/synthesis systems. In Subband Coding of Images, (Ed Woods, J.W.) Kluwer Academic, Norwell, MA (1991) Evangelista, G. Wavelet transforms and digital filters. In Wavelets and Applications, (Ed Meyer, Y .) Springer-Verlag, NY (1992) Resnikoff, H. Wavelets and adaptive signal processing, Opt Eng, DeCusatis,
31 (1992) 1229-1234 Baraniecki, A. Karim, S. Computational algorithms for discrete wavelet transforms, SPZE Proc, 1699 (1992) 408-419 Herley, C., Vetterli, M. Linear phase wavelets: theory and design, Proc Int Conf on Acoustics, Speech, and Signal Processing, Vol. 3 (1991) 2017-2020 Strang, G., Wavelets, Am Scientist, 82 (1994) 250-255
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Hybrid optical implementation of discrete wavelet transforms: C. DeCusatis et al.
10 Strang, G., Wavelet transforms vs. Fourier transforms, Bull Amer Math Sot. 28 (1993) 288-305
Biographies
11 Freeman, M. Wavele&: signal representations with important advantages, Opt and Photonics News, 4 (1993) 8-14 12 Benvenisie, A.-Multiscale signal processing: from QMF to wavelets. Int Workshov on Altzorithms and Parallel VLSI Archite&res, vol. A, ‘ElseviepPress, (1991) 71-76 13 Daubecbies, 1. Orthonormal basis of compactly supported wavelets, Comm Pure and Appl Math, XL1 (1988) 909-996 14 Daubecbies, I., Lagarias, J. Two scale differential equations, II local regularity, infinite products of matrices, and fractals, AT&T Bell Labs Tech Report (1989)
15 Vander Lugt, A. Signal detection by complex spatial filtering, IEEE Tram Info Theory, IT-10 (1964) 2 16 Freysz E. Optical wavelet transform of fractal aggregates, Phys Rev Left, 64 (1990) 7745-7748
17 Block, P., Rogers, S., Ruck, D. Optical wavelet transforms from computer generated holography, Appl Opt, 33 (1994) 5275-5218
18 McAulay, A., Wang, J. Optical wavelet transform classifier with positive real Fourier transform wavelets, Opt Eng, 32 (1993) 1333-1339 19 Caufield, H., Rhodes, W.T., Foster, M., Horwity, S. Optical implementation of systolic array processing, Opt Comm, 40 20 21
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Pinski, S.D., Rogers, S.D., Ruck, S.K., Welsh, D.M., Kabrinski, B.M., Warbola, M., Quinn, G., Oxley, D. Image segmentation using optical wavelets, SPIE Proc, 1702 (1992) 1IL19 Sbeng Y., Lu, T., Roberge, D. Optical ti implementation of a two-dimensional wavelet transform, Opt Eng, 31 (1992) 1859-1864 Wang, D.X., Tai, J.W., Zbang, Y.X. Two dimensional optical wavelet transform in space domain and its performance analysis, Appl Opt, 33 (1994) 5271-5274 Yu F.T.S., Lu, G. Short-time Fourier transform and wavelet transform with Fourier-domain processing, Appl Opt, 33 (1994) 5262-5270
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Zbang, Y., Kanterakis, E., Katz, A., Wang, J.M. Optoelectronic wavelet processors based on Smartt interferometry, Appl Opt, 33 (1994) 5279-5266
26 Szu, H., Telfer, B., Lobman, A. Causal analytical wavelet transform, Opt Eng, 31 (1992) 1825-1829 27 Caulfield, H.J., Ludman, J.E., Hemmer P.R. Optical wavelet transform continuous in time, time shift, and scale, SPZE Proc, 2238 (1994) 166-169
28 Poon, T.C. Real time optical image processing using difference of Gaussian wavelets, Opt Eng, 33 (1994) 2294-2302 29 Szu, H., Telfer, B., Kadambe, S. Neural network adaptive wavelets for signal representation and classification, Opt Eng, 31 (1992) 1907-1916 30 Pbuvan, S. Optical implementation of N-wavelet coding for pattern classification, Appl Opt, 33 (1994) 5294-5301 31 Kim, J., Kenan, R. Programmable integrated optic tapped delay lines using channel waveguide structures, Proc. OSA Annual Meeting, Dallas, Texas, (1994) p. 144 32 Goodman, J., Woody, L. Method for performing complex-valued linear operations on complex-valued data using incoherent light, 33
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36
Molley P.A., Stalker K.T., Sweat& W.C. A compact, real-time acousto-optic image correlator, SPIE, 1347 (Optical Information Processing Systems and Architectures II) (1990) 123-130
Casimer M. DeCusatis is a staff engineer for IBM Corporation, System/390 Server Division, Poughkeepsie, New York USA, where he has served as lead engineer for the gigabit data links used on IBM Parallel Transaction and Query Servers, and a principal engineer for ESCON and other fibre-optics products. He received MS and PhD degrees in electrical engineering from Rensselaer Polytechnic Institute (Troy, New York) in 1988 and 1990, respectively, and the BS degree magna cum laude in the Engineering Science Honors Program from Pennsylvania State University (University Park, PA) in 1986. He is co-inventor on four patents and has been invited to speak at international conferences in St Peterburg, Russia and Gdansk, Poland. He is co-authoi of over 40 technical papers, as well as the book Acoustooptics: Fundamentals and Applications(Artech House, Boston, MA 1990); he has also contributed chapters to books on wavelets and fibre-optic data communications, and will serve as editor of the OSA/AIP Handbook of Photometry. Dr DeCusatis is a member of the Optical Society of American, IEEE, SPIE, Sigma Xi Research Society, and 10 academic honor societies including Tau Beta Pi and Eta Kappa Nu; he has also been profiled by Who’s Who in Science and Engineering. J Koay is currently an MS student under the thesis supervision of Professor P. Das in the Electrical, Computer, and Systems Engineering Department at Rensselaer Polytechnic Institute, Troy, New York, USA. His planned graduation date is May, 1995. His thesis is on applications of the wavelet transform in signal processing. P. Das is a Professor in the Electrical Computer, and Systems Engineering Department at Rensselaer Polytechnic Institute, Troy, New York, USA. He was formerly a Faculty Member with the Electrical Engineering Departments of the Polytechnic Institute of New York and the University of Rochester, USA. He has published numerous papers and is currently performing research in wavelets and their applications to spread spectrum communication, ultrasonics, biomedical signal processing, acousto-optics, and nondestructive testing.