Signal Representations with Important Advantages

B Y M A R K O . F R E E M A N

1047-6938/93/08008-07$6.00 ©Optical Society of America

The goal, of course, is to get the most for the least work. In the signal analysis and processing wor ld , this is strongly inf luenced by how one chooses to represent information. In 1822, Jean-Baptiste Joseph Fourier devised an excellent way to do this: represent a signal as the sum of its frequencies. Power spectra, carrier frequencies and bandwidths, clock fre­quencies, brain activity frequency bands—these global charac­terizations provide much information in a concise manner. Some of the power of this representation begins to fade, though, when one attempts to represent information that changes its character unpredictably dur ing the course of the signal. The standard analogy is a musical score. A global representation l ike the Fourier transform w i l l tell what notes were played somewhere wi th in the piece of music, but the t iming of the notes is bur ied in a complicated frequency repre­sentation. A better representation, l ike the musical score, tells what notes were p layed when.

Wavelets represent a signal i n a way that provides local frequency information for each posit ion wi th in the signal. Whi le they are certainly not the only signal representation that purports to do this, they offer some unique and useful properties w i th a f i rm mathematical foundation. Wave­lets can be used for analyzing a signal based on the posit ion-varying spectra and for adaptive fi l tering where the filter function changes according to local spectral information. 1 , 2 The result ing decomposit ion is we l l matched to fractals and 1/f processes 3 , 4 and shows promise for application in background removal for pattern recognition and texture segmentation. 5 - 7 This article en­deavors to provide an intuit ive picture of the ma in concepts invo lved in signal representations v ia wavelets.

8 OPTICS & PHOTONICS NEWS/AUGUST 1993

WAVE

L O C A L I Z E D S P E C T R A

Probably the most natural way to compute a local spec­trum is to apply the Fourier transform to one local piece of

the signal at a time. This is the philosophy of the windowed or short-time Fourier transform (STFT). In its simplest form,

a rectangular window is used to isolate a portion of the signal, which is then Fourier transformed, i.e.,8

where f is the signal and M is some window function. As the window moves to different positions in the signal, the STFT gives

the spectra of the signal at these positions. Multiplying the signal by a window function, however, results

in convolving (blurring) the signal spectrum with the spectrum of the window. Pair this together with the fact that as the window gets

narrower, its spectrum gets wider, and we have the fundamental trade-off of localized spectra: the better we localize the signal, the poorer we localize its spectrum. The mathematics of the trade-off is exactly the same as the mathematics of Heisenberg's Uncertainty Prin­ciple in quantum mechanics. There is a limit to how well we can know

the frequency content of a signal at a particular position and simulta­neously know what position is being studied.

Dennis Gabor was aware of this uncertainty principle when he intro­duced his own version of the STFT in 1946. The Gabor transform uses a

Gaussian profile for the window in an STFT since the Gaussian is the unique function that minimizes this uncertainty. A comparison of the STFT with a rectangular window and the Gabor transform is shown in Figure 1. On

Figure 1. Comparing the STFT with a rectangular window and the Gabor transform that uses a

Gaussian window. Signal domain functions are on the left and all have the same effective

width. Power spectra are on the right. Notice there is better

frequency localization with the Gaussian window.

FIGURE 1

LETS

W A V E L E T S

Figure 2. The wavelet transform and the Gabor transform are compared for the edge input shown on the left. Examples of the decomposition functions appear in the middle and the magnitudes of the transforms are on the right. The wavelet transform has the nice property of zooming in on the edge location.

the left side of the figure, the decomposition functions are shown in the signal domain. The mean square width (read: effective width) of both the rectangular and Gaussian win­dows were chosen to be the same in this figure. O n the right-h a n d s ide , the f requency d o m a i n vers ions of these decomposition functions are shown. Notice that, in keeping with the uncertainty principle, the frequency localization is better when a Gaussian window is used.

Still, the underlying philosophy of localizing a spec­trum via windowing the signal needs to be re-examined. The window must be chosen very carefully. This is a diffi­cult, if not impossible, task for realistic unpredictable sig­nals. Intuitively, we're dealing with cross purposes here. Frequency is a measure of cycles per unit signal length. H i g h frequency oscillations take much less signal length than do low frequency oscillations. H i g h frequencies can be well localized with a short window, while low frequencies re­quire a longer window. The wavelet transform takes a dif­ferent approach that allows the window size to adapt to the frequency components being analyzed.

W H A T A R E W A V E L E T S ?

The wavelet transform begins with a function like one of the decomposition functions used in an STFT or Gabor trans­form. Think of this function as one that oscillates and is localized at some position. A l l the wavelet decomposition functions are then derived by scaling or shifting this funda­mental wavelet. When it is dilated, it accesses lower fre­quency information; when it is contracted, it accesses higher frequency information. By scaling the decomposition func­tion in this way, its effective window size is also scaled and in exactly the manner we would desire. The fundamental wavelet is called Ψ(x ) . Scaling it by a factor of a and shifting it by a factor of b yields the function Ψ a , b (x ) Then the continuous wavelet transform is defined as, 8

The uncertainty relation between the signal and frequency domains still holds and, as the localization becomes better in one domain, it necessarily becomes worse in the complemen­

tary d o m a i n . But the trade-off is varied accord­ing to what type of infor­mation is being extracted.

Figure 2 gives an example of one of the unique and useful prop­erties of the wavele t t ransform that results f r o m this, namely its abil i ty to z o o m in on discontinuit ies, edges, and singularit ies. The wavelet transform and the Gabor transform are compared for analyzing an edge input signal. N o ­tice that, for the Gabor

transform, the localization remains the same for all frequen­cies while, for the wavelet transform, the localization nar­rows as we move toward higher frequencies (lower scales). In this way, the energy of the wavelet transform zooms in on the location of the edge. The edge position can be extracted from a very small number of wavelet coefficients whereas, regardless of the frequency, a large number of Gabor coeffi­cients are required. A second important strength of wavelet transforms is related to the number of pixels or the space-bandwidth product (SBWP) of the transform. Figure 3 shows an input image and four of its wavelet bands. A t first glance, this seems to imply that the SBWP of the output is increased proportionally to the number of scales extracted. This type of SBWP explosion wou ld pose severe limitations to both electronic and optical implementations.

Fortunately, there is an elegant solution. Frequently, scale is sampled at a factor of two intervals, i.e., octave bands, so let's choose this scale sampling to understand how the SBWP problem is solved. A s the scale parameter changes by a factor of two in moving from one band to the next, the bandwidth also changes by a factor of two. Thus, according to the Nyquist sampling criterion, the sampling frequency can also be decreased by a factor of two. Therefore, the real SBWP requirements are depicted in the telescoping sequence shown in the lower part of the figure. The bottom line is that a properly sampled wavelet transform at worst requires only a slight increase in SBWP over the input and at best— which is achieved if orthogonal wavelets are used—has the same SBWP as the input. Of all the t ime/frequency repre­sentations, only wavelets are able to boast strict conserva­tion of SBWP.

M A T H E M A T I C A L F O U N D A T I O N S

M u c h of the contribution of wavelets has been to add math­ematical rigor to existing techniques like pyramid process­ing, multi-resolution image processing, and multi-rate signal processing. We can understand a great deal of this math­ematics by examining a single equation, the resolution of identity, which is given by 9

10 OPTICS & PHOTONICS N E W S / A U G U S T 1993

Figure 3. Four wavelet bands are shown for the input at the top with the corresponding scaled wavelet decomposition functions shown just above each band. The true space bandwidth product (SBWP) requirements are depicted at the bottom. This results from taking advantage of progressively narrower bandwidths of the wavelet bands to reduce the number of samples required at each step.

W A V E L E T S

Figure 4 . Four examples of fundamen­tal wavelets. The Haar wavelet has been around since 1910 . The Daubechies wavelet and the Mal lat wavelet come from Refs. 14 and 12, respectively. One is free to choose these or other profiles for the fundamental wavelet subject to the con­straints discussed in the paper.

T h e l e f t - h a n d s i de o f the e q u a t i o n i s b a s i c a l l y a n e n e r g y m e a s u r e i n the w a v e l e t d o m a i n , w h i l e the r i g h t - h a n d s i de is the s a m e e n e r g y m e a s u r e i n the s i g n a l d o m a i n . C ψ is a p r o p o r t i o n a l i t y cons tan t that d e p e n d s o n the c h o i c e o f the f u n d a m e n t a l w a v e l e t f u n c t i o n Ψ.

W h e n the r e s o l u t i o n o f i d e n t i t y i s sa t i s f i ed , i t essen t i a l l y says that the set of w a v e l e t f u n c t i o n s c o v e r s L 2 ( R ) , the space o f squa re i n teg rab le f u n c t i o n s . T h e r e are n o ho les . If C ψ = 1, t h e n the w a v e l e t s f o r m a n o r t h o n o r m a l b a s i s a n d the r e s o l u ­t i o n of i d e n t i t y i s s i m p l y P a r s e v a l ' s r e l a t i o n . If C ψ > 1, t h e n there is r e d u n d a n c y , i.e., o v e r l a p o f the w a v e l e t s i n L 2 space . CΨ i s a m e a s u r e of the r e d u n d a n c y .

T h e a d m i s s a b i l i t y c o n d i t i o n f o r w h a t cons t i t u tes a v a l i d f u n d a m e n t a l w a v e l e t Ψ c o m e s d i r e c t l y f r o m the r e s o l u t i o n o f i den t i t y . T h a t a d m i s s a b i l i t y c o n d i t i o n i s 9

w h e r e Ψ is the F o u r i e r t r a n s f o r m of the f u n c t i o n Ψ. G e n e r ­a l l y , th is a d m i s s a b i l i t y c o n d i t i o n is sa t i s f i ed i f the f u n c t i o n Ψ i s r e a s o n a b l y s m o o t h a n d Ψ(0) = 0 . 8 , 9 In o the r w o r d s , w a v e l e t s are i n h e r e n t l y b a n d p a s s i n n a t u r e . S o m e e x a m p l e s o f w a v e l e t f u n c t i o n s are s h o w n i n F i g u r e 4.

T h e w a v e l e t t r a n s f o r m m u s t b e s a m p l e d to b e p r a c t i c a l . T h e c o n s t a n t - Q p r o p e r t y o f w a v e l e t b a n d s — t h e cen te r f re ­q u e n c y d i v i d e d b y the b a n d w i d t h i s the s a m e fo r a l l b a n d s — leads to a s a m p l i n g i n the sca le d i m e n s i o n that is e x p o n e n t i a l . S a m p l i n g i s d o n e at p o w e r s o f s o m e bas i c sca le p a r a m e t e r ao

j, w h e r e j i n d e x e s the w a v e l e t b a n d . S ince the b a n d w i d t h s are c h a n g i n g b y th is fac tor o f ao as w e m o v e t h r o u g h sca le , t h e n i n a c c o r d a n c e w i t h N y q u i s t s a m p l i n g , the s a m p l i n g o f the p o s i t i o n v a r i a b l e b s h o u l d a l so c h a n g e b y th is s a m e sca le fac tor . S a m p l i n g i n p o s i t i o n the re fo re o c c u r s at i n t e r v a l s o f kboα0

j w h e r e k is the p o s i t i o n i n d e x .

In the d isc re te w a v e l e t t r a n s f o r m , the r e s o l u t i o n o f i d e n ­t i ty c h a n g e s i n f o r m s l i g h t l y i n t o a n e q u a t i o n fo r w h a t i s k n o w n as f r ames . T h e e q u a t i o n n o w b e c o m e s 9

t hey are cons tan ts of p r o ­p o r t i o n a l i t y b e t w e e n the e n e r g y o f t he w a v e l e t d e c o m p o s i t i o n a n d the e n e r g y o f the s i g n a l i t­self . T h i s is r e a l l y a re ­m a r k a b l e e q u a t i o n . It s a y s that as l o n g as p r o ­p o r t i o n a l i t y b e t w e e n the e n e r g y o f the s i g n a l a n d the e n e r g y o f i ts d i sc re te t r a n s f o r m ( w a v e l e t o r o t h e r w i s e ) i s b o u n d e d b e t w e e n s o m e t h i n g greater t h a n z e r o a n d less t h a n i n f i n i t y fo r a l l p o s ­s ib l e ( square in teg rab le ) f u n c t i o n s , t h e n the r e p ­

r e s e n t a t i o n i s c o m p l e t e . N o i n f o r m a t i o n i s l os t a n d the s i g n a l c a n b e p e r f e c t l y r e c o n s t r u c t e d f r o m i ts d e c o m p o s i t i o n .

If A=B=1, t h e n the set o f f u n c t i o n s { Ψ j , k } f o r m a n o r ­t h o n o r m a l b a s i s a n d th is i s s i m p l y P a r s e v a l ' s r e l a t i on . In th i s case , r e c o n s t r u c t i o n i s s t r a i g h t f o r w a r d . W h a t is s u r p r i s i n g i s that the s a m e r e c o n s t r u c t i o n a p p r o a c h w o r k s e v e n w h e n the d e c o m p o s i t i o n f u n c t i o n s are n o t o r t h o g o n a l to e a c h o ther . If A a n d B a re e q u a l b u t t he i r v a l u e i s n o t u n i t y , the r e p r e s e n t a ­t i o n i s c a l l e d a ' t i gh t f r a m e . ' T h e r e c o n s t r u c t i o n f o r m u l a is g i v e n b y

A a n d B are t e r m e d f r a m e b o u n d s . A s i n the c o n t i n u o u s case ,

I n the g e n e r a l case , A a n d B are n o t e q u a l . Per fec t r e c o n ­s t r u c t i o n c a n s t i l l b e a c h i e v e d b u t the i n t e r p o l a t i o n f u n c t i o n is n o l o n g e r the s a m e as the d e c o m p o s i t i o n f u n c t i o n . D e s i g n o f the i n t e r p o l a t i o n f u n c t i o n fo r th i s case i s d i s c u s s e d i n D a u b e c h i e s . 9 W h e n A a n d B are c l ose i n v a l u e , a v e r y g o o d a p p r o x i m a t i o n to the r e c o n s t r u c t i o n i s a c h i e v e d s t i l l u s i n g the s a m e f u n c t i o n f o r d e c o m p o s i t i o n a n d r e c o n s t r u c t i o n . F u r t h e r m o r e , s i nce Ψ m u s t b e a v a l i d c o n t i n u o u s w a v e l e t f u n c t i o n , A a n d B c a n a l w a y s b e m a d e a r b i t r a r i l y c lose to e a c h o the r b y s a m p l i n g at a h i g h e r rate. T h e r e c o n s t r u c t i o n e r ro r v e r s u s s a m p l i n g ra te i s a l s o q u a n t i f i e d i n D a u b e c h i e s . 9

Thus, there is room to trade off between some desired shape of the wavelet function Ψ, the accuracy of the reconstruction, and the SBWP of the representation.

IMPLEMENTATION: ELECTRONIC OR OPTICAL?

W a v e l e t t r a n s f o r m s es s en t i a l l y p r o v i d e a t i m e / f r e q u e n c y o r s p a c e / s p a t i a l - f r e q u e n c y rep resen ta t i on w i t h the c o n v e n i e n c e o f a F o u r i e r t r a n s f o r m . T h e F a s t - F o u r i e r - T r a n s f o r m (FFT) r e v o l u t i o n i z e d d i g i t a l s i g n a l p r o c e s s i n g a n d the o p t i c a l F o u ­r i e r t r a n s f o r m l ies at the hear t o f m o s t o p t i c a l s i g n a l p r o c e s s ­i n g sys tems . N o w , there are d i g i t a l a l g o r i t h m s a n d i n teg ra ted c i r c u i t s that c a n ca l cu la te a w a v e l e t t r a n s f o r m e v e n fas ter t h a n a n F F T . A n u m b e r o f a rch i t ec tu res h a v e b e e n s u g g e s t e d f o r p e r f o r m i n g the w a v e l e t t r a n s f o r m o p t i c a l l y . 4 , 5 , 1 0 , 1 1

T h e b a s i c i d e a fo r the fast d i g i t a l i m p l e m e n t a t i o n o f the w a v e l e t t r a n s f o r m is s h o w n i n the tree s t ruc tu re of F i g u r e 5. T h e o r i g i n a l d e r i v a t i o n f o r th i s w a s p r o v i d e d b y M a l l a t . 1 2

12 OPTICS & PHOTONICS N E W S / A U G U S T 1993

The b a n d p a s s w a v e l e t f u n c t i o n is sp l i t i n to complementary lowpass and h ighpass funct ions that f a l l in to the c lass known as quadrature mir­ror filters. Assume that the continuous signal has been sampled appropriately for its frequency content. The highest frequency wavelet band is extracted w i th f i l ­ter G, whi le its complementary lowpass information is ex­tracted wi th filter H. N o w the bandwidths of each channel have been effectively reduced and can be resampled at a coarser rate determined by the scale parameter a0. In a d ig i ­tal system, this is usual ly a factor of two, since that means simply comput ing the filter function for every other output point. The process then iterates. The lowpass information again passes through the fi lters G and H fo l lowed by subsampl ing, fo l lowed by another pass of the result ing lowpass information through the filters, and so on. The discrete wavelet transform is the information emerging f rom the G channels.

W i th this approach, if there are L N mult ipl ies and adds for the highest frequency wavelet band where N is the num­

ber of samples i n the signal, and L is related to the length of the filter, then there are L N / 2 for the next band, L N / 4 for the next band, and so on. This geometric progression never exceeds 2 L N regardless of the depth of the tree, i.e., we have an O(N) computation. The factor L is typical ly much less than N due to the fact that we are extracting local informa­tion only. The wavelet transform therefore is computed w i th fewer operations even than the FFT, which, due to its global nature, requires O(N log 2 N) mult ipl ies and adds.

This should raise a warning flag for all optical signal process­ing researchers considering optical implementation of wavelets. The increase in speed offered by the optical Fourier transform over the FFT has not been a sufficient advantage alone to warrant the use of optical rather than digital systems. It wou ld

Figure 5. Block diagram of the

algorithmic approach used for the fast digital

computation of a wavelet transform. H

and G are complemen­tary low- and high-pass

filters. The arrow following each filter

block indicates subsampling by a

factor of two.

OPTICS & PHOTONICS N E W S / A U G U S T 1993 13

W A V E L E T S

Figure 6. An optical system for computing the 2D wavelet decomposition of an input image. The computation at any given scale occurs on the path directly from the input to the output. The HOE splits the function into four channels, each of which is filtered by a wavelet oriented at a different angle. The feedback loop scales the input by the factor a for each pass around the loop.

be bl ind to assume that this wou ld change for wavelets.

There are, however , operations related to wave­lets that are more natural in the optical domain and may give an optical system an advantage for some ap­plications. First, they add the important class of effi­cient space-variant opera­tions to the repertoire of optical processing systems. Second, scaling operations have been the bread and butter of optical systems (telescopes, microscopes, cameras) since the f ield be­gan. Opt ical wavelet sys­tems may be able to exploit this natural abil ity when a

scale parameter other than a0 =2, or particularly where the abil ity to vary the scale parameter arbitrari ly provides an information advantage. Th i rd , the speed of the 1D algo­rithms for digital wavelets is only easily extended to two or more dimensions through the use of separable wavelet func­tions. This places restrictions on the types of information, for example, directional information, that is extracted. There is no such l imitat ion i n an optical system. Fourth, by work ing in the discrete domain, a translation or rotation of the signal can result in a radical change in the result ing wavelet coeffi­cients. 1 3 Opt ical systems that perform the wavelet transform in continuous coordinates and then sample the output have much nicer shift and rotation properties.

One optical system that seeks to exploit some of these advantages is shown in Figure 6.5 This system performs the wavelet fi ltering using a VanderLugt style optical correlator system on the direct path f rom input to output, and a feed­back r ing to perform the scaling operation. The holographic optical element (HOE) i n the fi l tering path splits the beam into four channels, each of wh ich is operated on by a differ­ent 2D wavelet filter in the Fourier plane. In this case, the filters extract information over different wedges of angles in frequency space. The feedback r ing is arranged to mini fy the input by the scale factor a each time it passes around the loop. This shifts a new band of frequencies into the passband of the filters for each scale. By using one of the feedback mirrors to introduce a tilt i n the Fourier plane, al l images of the Fourier plane remain stationary for al l scales. Thus al l scales remain properly al igned w i th the filters in the Fourier plane, but the results in the output plane are shifted so that adjacent scales appear i n adjacent locations and not on top of each other. In addit ion, the wavelet information mov ing from one frequency band to the next lower band in the output is also mini f ied by the same factor of a. This makes efficient use of the output SBWP by dropping the sampl ing rate correspondingly for each band, if we assume a un i ­formly sampl ing detector array at the output.

The true promise for optical information processing sys­tems w i l l only be realized when we have the capability to

bu i l d compl icated mul t i ­stage operations from sim­pler bu i ld ing blocks. The Fourier transform, the opti­cal correlator, and now, the optical wavelet system rep­resent some of these bui ld­i ng b locks . The current research interest in the de­velopment of miniaturized optical correlators may pro­vide the necessary compo­nent to move to these more sophisticated optoelectronic process ing systems. The wavelet transform, by vir­tue of its fundamentally be­i n g a co r re la t i on - t ype filtering operation, should map wel l into these new components.

Regardless of the particular implementation, wavelets pro­vide relatively low overhead access to a rich signal representation.

A C K N O W L E D G M E N T S

M u c h of the understanding of this subject was gained through work ing w i th m y students Kenneth A . Due l l and A d a m S. Fedor at the Universi ty of Colorado—Boulder . I am grateful for their insights, and hard work, inc luding Adam 's help w i th the figures.

MARK O. F R E E M A N is a senior engineer with the Optoelectron­ics and Systems Laboratories of the Industrial Technology Re­search Institute in Hsinchu, Taiwan, R.O.C.

R E F E R E N C E S 1. K . A . D u e l l a n d M . O . F reeman, " A d a p t i v e noise f i l ter ing i n the wavelet

d o m a i n , " i n preparat ion. 2. H .L . Resnikof f , "Wave le ts a n d adapt ive s igna l p rocess ing , " Op t . Eng .

31, 1992, 1229-1234. 3. G . W . W o r n e l l , " A Ka rhunen -Loeve - l i ke expans ion for 1/f processes v ia

wave le ts , " I E E E Trans. Info. Theory 36, 1990, 859-861. 4. A . A r n e o d o et al., "The op t ica l wavelet t rans fo rm, " Wavelets and Thei r

App l i ca t i ons , M .B . R u s k a i et al., eds., Jones and Bartlett, Boston, Mass . , 1992, 241-273.

5. M . O . F reeman et al., " O p t i c a l wave le t processor for p r o d u c i n g spat ia l ly l oca l i zed r ing-wedge- type in fo rmat ion , " Proc. SPIE 1772, 1992, 241-250.

6. C . P . V e r o n i n et al., " O p t i c a l image segmentat ion us ing neura l -based wave le t f i l te r ing techniques," Op t . E n g . 31, 1992, 287-294.

7. D .P. Casasent et al., "Wave le t a n d Gabo r t ransforms for detect ion," Op t . E n g . 31, 1992, 1893-1896.

8. O . R i o u l and M . Vet te r l i , "Wave le ts and s igna l p rocess ing , " IEEE SP M a g a z i n e , Oct. 1991, 14-38.

9. I. Daubech ies , "The wavelet t ransform: T ime- f requency loca l i za t ion and s igna l ana lys is , " IEEE Trans. Info. Theory 36, 1990, 961-1005.

10. Spec ia l Issue on Wave le t Transforms, Opt . Eng . 31, Sept. 1992. 11. H . S z u et al., "Wave le t t ransform as a bank of the matched f i l ters," A p p l .

Op t . 31, 1992, 3267-3277. 12. S . G . Ma l l a t , " A theory for mu l t i reso lu t ion s ignal decompos i t ion : The

wavelet representat ion," IEEE Trans. P A M I 11, 1989, 674-693. 13. E.P. S imonce l l i et al., "Shi f tab le mul t i -sca le t ransforms," IEEE Trans.

Info. Theory 38, Special Issue on Wavelet Transforms and Mul t i reso lu t ion S igna l A n a l y s i s , 1992, 587-607.

14. I. Daubech ies , " O r t h o n o r m a l bases of compact ly suppor ted wave le ts , " C o m m u n i c a t i o n s o n Pure and A p p l i e d Mathemat ics X L I , 1988, 909-996.

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