Vol. 9, No. 2/February 1992/J. Opt. Soc. Am. A 337
Minimizing and maximizing the joint space-spatialfrequency uncertainty of Gabor-like functions: comment
Stanley A. Klein and Brent Beutter
School of Optometry, University of California, Berkeley, Berkeley, California 94720
Received March 29, 1991; revised manuscript received September 25, 1991; accepted September 25, 1991
We dispute the claim that Hermite functions (similar to derivatives of Gaussians) minimize a joint uncertaintyrelation in space and spatial frequency. These functions are found to maximize rather than minimize the un-certainty of the class of functions consisting of an mth-order polynomial times a Gaussian.
Daugman' argued that one of the beauties of Gabor func-tions is that they are compact in both space and spatialfrequency since they minimize the Heisenberg uncer-tainty in space and spatial frequency. The goal is to mini-mize the product of the variance of the receptive field inspace times its variance in spatial frequency. Stork andWilson2 subsequently showed (correctly) that Daugman'sargument was flawed since it applied only to the complexGabor functions, whereas in vision research real-valuedfunctions are used. Gabor' claimed to show that Hermitefunctions were the appropriate functions for minimizingthe uncertainty relations for real functions. We disagreewith this finding and will argue that the Hermite func-tions produce a maximum of the Heisenberg uncertaintyconditions for a constrained set of functions consisting ofa fixed-order polynomial times a Gaussian. Further-more, we provide evidence that the Gabor functions do agood job of minimizing an alternative uncertainty rela-tion, contrary to what is claimed by Stork and Wilson.2
This paper examines a wide class of functions that arelocalized in both space and spatial frequency. Since allthese functions crudely resemble Gabor functions, we callthem Gabor-like functions.
The enterprise of Gabor3 and Stork and Wilson2 is tofind a function r(x) that minimizes the joint space-spatialfrequency Heisenberg quantity:
U 2 = (X2 )(f 2 ) (1)
= x2r2(x)dxf (d) dx [ r2(x)dx] . (2)
The second integral in Eq. (2) is obtained by usingParseval's theorem to replace the spatial frequency vari-ance fr,, f2R( f) 2df with fr, (dr/dx)2dx, where R(f) is theFourier transform of f(x). Gabor3 attempted to minimizeEq. (2) by using the calculus of variations. He claimedthat the Hermite functions produced the minimum. TheHermite functions are given by
where Dm represents the mth derivative and a is the stan-dard deviation of the Gaussian. These functions have
been normalized:
f Hm2(x)dx = 1.x.
(4)
The orthonormality of the Hermite functions allows anyfunction r(x) that is well behaved at x = 0 to be expandedin a series of Hermite functions:
r(x) = amHm(x).m=O
(5)
By using Hermite polynomial recursion relations, the jointuncertainty, Eq. (2), can be written in terms of the expan-sion coefficients, am,
u2 = [[ am2(m + 0.5)] - [ am-am+,m=O m=l
x ( 2 + m)05] ( am2)2. (6)
To get a clearer understanding of the joint uncertaintyit is useful to restrict Eq. (6) to the case in which only twocoefficients are nonvanishing. We can then make thegeneral transformation:
am = A cos(0),
a = A sin(0).
The joint uncertainty becomes
U2 = [cos 2(6) (m + 0.5) + sin2(0) (n + 0.5)]2
- cos2(O)sin2(0)[(m + 1)nS(m + 2 - n)
+ (n + 1)m3(n + 2 - m)],
(7)
(8)
where the last term with the Kronecker delta contributesonly if n - ml = 2.
The joint uncertainty, U of Eq. (8) is plotted in Fig. 1 form = 2 and n ranging from 0 to 5. The horizontal axis is, which goes from -iT/2 to +7r/2 . If In - ml • 2, the
curves in Fig. 1 have the simple form
U = m + 0.5 + sin2(0) (n - m). (9)
One might be tempted to say that at = 0, U is a maxi-mum for n < m and a minimum for n > m. This holds
338 J. Opt. Soc. Am. A/Vol. 9, No. 2/February 1992
6
5
2
- 2 -1.5 -1 -0.5 0 0.5
n=5
n=4
n=3
n=1
n=
1 1.5 2
thetaFig. 1. Plots of the uncertainty U given in Eq. (2), for the sum oftwo Hermite functions: r(x) = cos(0)H2(x) + sin(6)H"(x). Forsmall values of 0, r(x) is dominated by the second-order Hermitefunction, and for 0 7r/2, it is dominated by the nth Hermitefunction. For a single Hermite function, H(x), U = n + 0.5.For n < 2 and also for n = 4, U reaches a maximum value of 2.5at 0 = 0. For other values of n, U has a minimum at 0 = 0. Forthe class of functions that are polynomials of a fixed order timesa Gaussian, this figure implies that the Hermite functions maxi-mize the uncertainty. The dot on the n = 0 function is the com-bination of H2 and Ho corresponding to the second derivative of aGaussian that had been conjectured to be a minimum of the jointuncertainty.
except for n = m + 2. In Fig. 1, the case for m = 2,n = 4 is seen to have a shape different from those of theothers. This is because the negative contribution of theextra term in Eq. (8) is strong enough that 0 = 0 becomesa maximum rather than a minimum. For the case of n =m + 2 the minimum is at
sin2 (0) = (m - 1)m/2(6 + 3m + M2). (10)The value at the minimum is
U2 = 3(1 + m) (2 + m) (1 + 3m + M2)/4(6 + 3m + M2).
(11)
Figure 1 and Eqs. (6) and (8) show that 0 = 0 is a saddlepoint. That means that if a stimulus is a Hermite func-tion to first order,
r(x) = Hm(x) + ef(x), (12)
then the uncertainty is of second order in e:
U = m + 0.5 + e2 K, (13)
where K is a finite quantity that can be either positive ornegative. If the class of functions is limited to a Gauss-ian times an mth-order polynomial, then assuming an/amis of order e Eq. (6) becomes
Equation (14) shows that the mth-order Hermite functionis not merely a saddle point. Gabor was correct that thecalculus of variations implies that the Hermite function isan extremum (minimum, maximum, or saddle point).The funny and surprising result of our analysis is thatinstead of finding a minimum as he had assumed, Eq. (14)shows that the mth Hermite function is the function thatmaximizes the uncertainty! This is because any nonzerovalue of an for n < m will decrease the uncertainty, U
The precise claim of Stork and Wilson2 is slightly differ-ent. They made an arithmetic error of a factor of 2 intheir derivation (as pointed out by Yang4 ) and concludedthat it is the family of derivatives of Gaussians that mini-mize the uncertainty. The derivatives of Gaussians aregiven by
DGm(x) = Dm exp(-X2 /2- 2). (15)
Based on the normalization of Eq. (3), the second Gauss-ian derivative can be written in the form of Eqs. (5) and(7) as a sum of the second and zeroth Hermite functions:
DG 2(x) = A[(2/3)05 H2(x) + (1/3)0 5H0(x)],
where cos2(0) = 2/3.The uncertainty from Eq. (8) is
U2 = (2/3 x 2.5 + 1/3 x 0.5)2 - 2/9 x 2
or = 35/12,U = 1.7078,
(16)
(17)
(18)
which is less than the uncertainty of 2.5 for the second-order Hermite function. This point is indicated by thedot in Fig. 1. However, the fact alone that the Gaussianderivatives have a smaller uncertainty than the Hermitefunctions does not make them special. As seen in Fig. 1,they do not minimize the uncertainty, contrary to theclaim of Stork and Wilson.2 Only the Hermite functionsare singled out in that they are saddle points, as shown inFig. 1.
There is a second error in the paper of Stork andWilson2 that is quite interesting. They examine differentmetrics for the uncertainty relationship. One of theirmost interesting metrics can be written in the frequencydomain as
Uf 2 = - f0)2 R(f)12 df
x f [xr(x)]2dx [f JR(f)2df]
X [| r()dx]
= | (f - f)2 IR(f)12dff |d f[ JR(t'2df]
(19)
where R(f) is the Fourier transform of r(x). We will callEq. (19) the displaced uncertainty metric. Equation (19)is the same as Stork and Wilson's Eq. (A12).2 This metriccalculates the frequency variance around a specific spatialfrequency rather than around zero. Before reading Storkand Wilson's paper, we assumed that the Gabor functionminimized this metric. Stork and Wilson,2 however, ar-gue that instead of the Gabor function, the function thatminimized Eq. (19) has the following tuning in frequency:
Gs.w(f) = exp[-(IfI - f)2a-2/2], (20)
where the carrier frequency, f is equal to the displacedfrequency in Eq. (19). We worried that these functionshave a sharp cusp at f = 0 that might cause broadening inspace. We therefore numerically calculated the uncer-
- -LOB_ I - In ru
l l l i] l l l
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..
X f JR( f)12 d ,
Vol. 9, No. 2/February 1992/J. Opt. Soc. Am. A 339
0.6 -
0.5 C~~~~
Q~~~~~~0.1/
a) 1 -
O ~ ~ Nraie 0.4quency Gao
CI
CL
a) 0.2
0.0* 0 1 2 3
Normalized frequency, gFig. 2. Pair of plots of the displaced uncertainty Ug as a func-tion of the normalized center spatial frequency, g = f, on theabscissa. The dashed curve is for the symmetric Gabor functionGg(f) = exp[-(f - g)2/2] + exp[-(f + g)2/2]. The solid curveis for the function suggested by Stork and Wilson: Gsw =exp[-(lfl - g)2/2]. It is seen that, contrary to Stork andWilson's claim, the uncertainty for Gsw is larger than that for theGabor function. The joint uncertainty is always less than U =0.5, the value that pertains to a Gaussian and to a complex Gaborfunction. The value of 0.5 is the minimum uncertainty of theundisplaced uncertainty, U,.
tainty, Eq. (19), of both the function of Stork and Wilsonand a symmetric Gabor function. In space, the symmet-ric Gabor function is given by
Figure 2 shows plots of the uncertainty, Ug, for the tuningcurves given by Eqs. (20) and (22), with normalized fre-quency units, given by g = f, being used for the ab-scissa. We have taken the carrier frequency, f,, to equalthe displaced frequency, fo, as was done by Stork andWilson. The plots show that, contrary to the suggestion ofStork and Wilson, the original Gabor function of Eq. (22)does a better job of minimizing the uncertainty rela-tionship than does their suggestion in Eq. (20). The plotsdo not give the global minima. We know that forcing theGabor carrier frequency, f,, to equal the displaced fre-quency, f 0, does not produce the minimum uncertainty.In fact, the minimum is achieved when the displaced fre-quency is given by the mean frequency of the mechanism,as given by
f= fJR(idi IR(f)Idf. (23)
This mean frequency is always above the carrier fre-
quency. Further research is needed to determine whatfunctions produce a global minimum of Eq. (19).
For a zero peak frequency, g = f = 0, the displaced un-certainty is U = 0.5. This was expected since the uncer-tainty relation of Eq. (19) is the same as the originaluncertainty given by Eq. (2) and since the Gabor functionfor f0 = 0 is a Gaussian. The fact that the uncertainty is0.5 for large values of fg is a consequence of two ingredi-ents. (1) Since the first integral in Eq. (19) goes overonly positive frequencies and since f, is large, the negativefrequency contribution becomes negligible, and so the fre-quency tuning curve is essentially a Gaussian [the secondterm of Eq. (22) is negligible]. (2) The spatial varianceis given by the variance of the envelope of the Gabor func-tion, which is a Gaussian [Eq. (21)]. These two factorsimply that the relevant aspects of the function are Gauss-ian in both space and spatial frequency so that the uncer-tainty is 0.5, as for a Gaussian. For g = = 0.8 theuncertainty has a minimum value of U = 0.296 for theGabor function. This seems to violate the minimumvalue of U = 0.5 associated with the Heisenberg uncer-tainty relation. There is no real violation, however, sincethe displaced uncertainty relation of Eq. (19) differs fromthe Heisenberg relation of Eq. (2). The reduced uncer-tainty results from the oscillating receptive field's causingthe spatial variance to be less than the variance of theGaussian envelope.
The above integrations were done with a linear fre-quency axis. Klein and Levi' showed that for log axes(as might be appropriate for vision modeling) the Gaborfunctions had an infinite variance in spatial frequency.Therefore many functions with sharp low-frequency at-tenuation would do better than the Gabor functions whenthe uncertainty is calculated by using logarithmic axes.
Minimizing the time-frequency uncertainty has been ofinterest to researchers in signal analysis from the time ofGabor's paper' to the present.6 One might wonder whythe enterprise of minimizing the joint space-spatial fre-quency uncertainty, Uf,, is of interest to the vision com-munity. The viewprint calculations of Klein and Leviprovide some justification. A viewprint is a joint space-spatial frequency plot of the activities of mechanisms thatare localized in both space and spatial frequency. If theywere not localized, then the mechanisms used both by thevisual system and by modelers would be susceptible tomasking by stimuli that are well separated in positionor in spatial frequency. This localization in space andspatial frequency is precisely what is measured by thedisplaced uncertainty metric. The Heisenberg metric[Eq. (2)], on the other hand, measures the spatial fre-quency, variance around zero frequency rather than mea-suring the mechanism bandwidth. Research in imagecompression provides even stronger reasons for minimiz-ing the joint uncertainty. Localization in space is neededto avoid masking by adjacent features, and localization inspatial frequency is needed to guarantee that the high-pass filters do not have significant low spatial frequenciesthat would aid their visibility.7 Most of the informationin a scene is stored in the tiny high-spatial-frequencymechanisms, and the main compression savings comesfrom coarsely quantizing their response (high spatialfrequencies have poor visibility). We had hoped thatthe Gabor function with g = 0.8, corresponding to
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the minimum point of Fig. 2, would be a good candidatefor an image compression high-pass filter, but it turns outto have barely any inhibitory side lobes and thus is not ahigh-pass filter at all.
ACKNOWLEDGMENTS
We thank Jian Yang for pointing out the error made byStork and Wilson in confusing the Gaussian derivativesand the Hermite functions. We also thank Yang andRichard Young for comments on the manuscript. Thisresearch was supported by grants EY-04776 from theNational Eye Institute and AFOSR-890-2038 from theU.S. Air Force Office of Scientific Research.
REFERENCES
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JOSA Communications
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4. J. Yang, "Do Gabor functions provide appropriate descriptionsof visual cortical receptive fields?: comment," J. Opt. Soc.Am. A 9, 334-336 (1992).
5. S. A. Klein and D. M. Levi, "Hyperacuity thresholds of 1 sec:theoretical predictions and empirical validation," J. Opt. Soc.Am. A 2, 1170-1190 (1985).
6. C. Dorize and L. F. Villemoes, "Optimizing time-frequencyresolution of orthonormal wavelets," in Proceedings of the In-ternational Conference on Acoustics, Speech and Signal Pro-cessing (Institute of Electrical and Electronic Engineers, NewYork, 1991), pp. 2029-2032.
7. S. A. Klein and T. Carney, "'Perfect' displays and 'perfect' im-age compression in space and time," in Human Vision, VisualProcessing, and Digital Display II, B. E. Rogowitz, M. H. Brill,and J. P. Allebach, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1453, 190-205 (1991).
