8

M. Rotenberg,

R. Bivins, N. Metropolis,

and J. K. Wooten,

The 3-j and 6-j symbols (MIT Press, Cambridge, Mass., 1959). C. Roth, J. Res. Natl. Bur. Stds., B Math. Sci. 76, 61 (1972). 10 and A. Ginibre (unpublished). J. Bauche, C. Bauche-Arnoult, 11 C. Bauche-Arnoult and J. Bauche, J. Opt. Soc. Am. 58, 704 (1968). 9

12

E. Meinders and P. F. A. Klinkenberg, Physica (Utr.) 38,

253 (1968). P. F. A. Klinkenberg, Physica (Utr.) 57, 594 (1972). 14 P. F. A. Klinkenberg and Th. A. M. Van Kleef, Physica (Utr.)50, 625 (1970). 5I. Shavitt, C. F. Bender, A. Pipano, and R. P. Hosteny, J. Comp. Phys. 11, 90 (1973). "GG. S. Ofelt, J. Chem. Phys. 38, 2171 (1963). 13

P. R. Fields, Phys. 49, 4447 (1968).

7W. T. Carnall, 8

and K. Rajnak,

J. Chem.

J. -F. Wyart, J. Blaise, and P. Camus, Phys. Script. 9,

325 (1974). C. Arnoult and S. Gerstenkorn, J. Opt. Soc. Am. 56, 177 (1966). 20 N. Spector, J. Phys. (Paris) 31, C4-173 (1970). 21 C. Bauche-Arnoult, Proc. R. Soc. A 322, 361 (1971). 22 C. Bauche-Arnoult, J. Phys. (Paris) 34, 301 (1973). 23 J. G. Conway and B. G. Wybourne, Phys. Rev. 130, 2325 19

(1963), and private communication.

24

L. Brewer, J. Opt. Soc. Am. 61, 1101 (1971); 61, 1666 (1971). 25 J. -F. Wyart, Physica (Utr.) 75, 371 (1974). 26 y. Shadmi, J. Oreg and J. Stein, J. Opt. Soc. Am. 58, 909 (1968).

A visual system model and a new distortion measure in the

context of image processing* William A. Pearlman Departmentof Electricaland ComputerEngineering, Universityof Wisconsin-Madison,Madison, Wisconsin53706 (Received 26 August 1977)

Underlying many techniques of image restortation, quantization, and enhancement is the mathematically convenient, but visually unsuitable distortion measure of squared difference in intensity. Squared-intensity difference has an indirect phenomenological correspondence in a model of the visual system. We have undertaken, therefore, an experiment that derives a new distortion measure from an acceptable visual system model and compares it in a fair test against squared difference in intensity in an image restoration task. We start with an eye-brain system model, inferred from the works of current vision researchers, which consists of a bank of parallel spatial frequency channels and image detectors. From this model we derive a new distortion criterion that is related to changes in the per-channel detection probability and phase angle. The optimal linear (Wiener) filters for each distortion measure operate in turn on the same noisy incoherent images. The results show that the filter for the new distortion measure yields a superior restoration. It is more visually agreeable, more sharply detailed, and truer in contrast compared to the squared-difference filter, and impressive in its own right. Its mathematical properties suggest that significantly increased efficiency in the storage or communication of images may be gained by its use.

1.

INTRODUCTION

The image error criterion of average difference squared intensity (squared error) has enjoyed extensive use in image

visual performance under a change in image intensity. One needs to start with a model of the eye-brain system and relate an intensity change to the actual changes in parameters in the

processing because it is convenient mathematically and yields some successful results. Researchers in vision and image

system that affect visual performance. A meaningful error

processing do agree that squared error is unsuitable for measuring degradation in images and a better criterion ought to be found and utilized. Such a criterion would lead to more effective and efficient restoration methods for degraded images. But squared error continues its predominance in both image filtering and data compression. In fact, in the area of image data compression called transform picture coding, nearly all the transform quantization techniques seek to minimize squared error. Because all such techniques of compression and filtering of images claim varying degrees of success any other criterion proposed for image processing must be compared to squared error in convenience, effectiveness, and efficiency.

cant parameters to the effecting intensity change.

criterion for vision would relate the changes in these signifi-

In this report we propose a simple model of the eye-brain system implied by recent vision research and derive from it a visual error criterion that is convenient enough mathematically to use in image processing. In order to evaluate its suitability, we decided to compare it in a fair test against squared error in an image restoration task. The task is to estimate true source images from those images degraded by additive, white Gaussian noise. For the sake of mathematical convenience, optimal linear (Wiener) estimation is chosen.

The reasons governing the choice of squared error as the comparison criterion are as follows: (i) squared error is the commonly used criterion for image processing; (ii) any newly

The problem with the squared-error criterion is that it does

proposed criterion must establish its superiority oversquared

not express, except in an indirect way, the actual changes in

error in some respect; (iii) demonstrated superiority of the new

374

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

0030-3941/78/6803-0374$00.50

1978 Optical Society of America

374

criterion over squared error would strongly suggest it as a more

accurate model of the system's performance change under intensity variation; and (iv) the mathematical properties of the new criterion point naturally to squared error for comparison. The results do indeed establish superiority for the new criterion through the restoration of sharper, truer contrast, and more detailed images than those for squared error. In fact, the results for the new criterion are rather impressive in their own right. The purpose of this report is to describe in detail the visual system model, the derivation of the new error criterion, and the image filtering experiments. We shall present first the motivation for the proposed model of the eye-brain system and the error criterion originating from it. We then derive formulas for optimal estimation using the new criterion. Many interesting properties common to the usual squarederror measure are noted there. These formulas are then applied to the case of estimation of signals in noise. The optimal

linear estimation formulas form the basis of the filtering algorithms for noisy images.

Then follow descriptions of the

comparative experiments with real source images and the salient results. 11. A VISUAL SYSTEM MODEL AND A NEW ERROR CRITERION FOR IMAGE PROCESSING

Many investigators over the past two decades have attempted to apply a linear systems model to the human visual system so that results with grating stimuli could be interpreted

in the light of Fourier theory. The earliest such attempt of Schadel treated the visual system as a linear filter made up of a cascade of individual optical and neural filters. Psychophysically determined functions relating contrast sensitivity to spatial frequency have been taken to be modulation transfer functions of this single visual filter. Since it is apparent that the visual system responds to input intensity in a nonlinear manner there have been more recent attempts to model the processes of the eye as one or two filters in cascade

observer. The existence of spatial-frequency selection of the retinal ganglion cells of cats had already been established then

by Enroth-Cugell and Robson.6 Later, spatial frequency selection was verified in monkeys too by Campbell et al. 7

Further spatial-frequency selection may well exist at higher levels in the brain in cats and monkeys as well as humans. Strong supporting evidence for the existence of similar tuned elements at some level in humans stems from adaptation studies with gratings (Blakemore and Campbell,8 Pantle and Sekular,9 and Tynan and Sekular10 ). The convincing study of Sachs, Nachmias, and Robson" concluded that the single-filter model must be statistically rejected in favor of the multiple-channel one to explain their grating detection results. Further support of this conclusion had previously been given

by Graham and Nachmias.12 The nature and distribution of these frequency channels over the visual spatial spectrum are still not completely known, but have been under extensive investigation. Campbell, Nachmias, and Jukes13 show that discrimination between two gratings of different spatial frequency and equal

luminance contrast depends primarily on the frequency ratio over a broad range of spatial frequencies. This work and the work of Blakemore and Campbell 8 indicate that each channel is sensitive to a narrow range of spatial frequencies and that there are a large number of such channels in a spatial frequency range spanning four octaves. Sachs, Nachmias, and Robson"l inferred somewhat wider channels from their data,

which had poor sensitivity to bandwidth.

Mostafavi and

Sakrison14 succeeded in measuring a bandwidth of 2.5 cycles/degree for a single channel and conjectured that the ratio

of center frequency to radial bandwidth may be constant roughly at 2:1 over the full spatial frequency range. The study

of Sachs, Nachmias, and Robson13 definitively corroborated

the postulate of Campbell and Robson5 that the output of each one of these channels is detected independently. Beyond the multiple spatial-frequency model of the visual

system is a further complementary description proposed and

with a nonlinear element which have been made by Stockham,2 Mannos and Sakrison,3 and Hall and Hall.4 Stockham used a logarithmic transformation to model the visual response to intensity, while Mannos and Sakrison found that

convincingly proven by Schnitzler.15 Schnitzler proposed an image-detector model of the visual system in four basic stages which may be analyzed with the methods of statistical communication theory. First, the incident light is affected by the optical system of the eye consisting of the lens and the pupil,

a 0.33 power law produced better reconstructions of quantized

thereby defining an optical or modulation transfer function.

images. They also found an analytical expression for the visual system's MTF which was used for image encoding. Hall and Hall argue that the action of the eye can be better modeled as a nonlinear transformation sandwiched between two spatial filters that form the visual MTF, a low-pass one corresponding

to purely optical effects and a high-pass one corresponding to neural effects and lateral inhibition. They propose that the bandwidths of these filters vary as a function of input intensity to reflect the signal-dependent and adaptive response of the visual system to the input. Certain phenomena, however, could not be explained these types of models. Campbell and Robson 5 explained detection results for sinusoidal and square-wave gratings a multiple channel model. They offered the explanation

with their with that

the visual system is comprised of a bank of parallel narrowband filters each selectively tuned to a different spatial frequency. The contrast in any channel must reach a certain threshold in order for that frequency to be detected by the 375

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

The retinal elements are image sensors which may be likened to photocells where the incident photons of light are absorbed

and electrical impulses are transmitted to the brain. These electrical signals are then processed in a signal processor so that a decision-making device can be actuated. A random process is involved here as the incident photons from the background are a source of fluctuation noise. Schnitzler

utilized the techniques of statistical communication theory in this image-detector model and presented results which were

highly consistent with the Blackwell16 and Blackwell and Taylor17 psychophysical data. This data pertains to detection of disks of constant, but varying contrast with the surrounding background. The study of Johnson18 provides empirical data

relating the probabilities of detection, recognition, and identification of complex image stimuli to the probability of detection of simple periodic stimuli. The Schnitzler photoreception and detection model ties in perfectly with the multiple-channel model for the visual sysWilliam A. Pearlman

375

NB

SPATIAL

PHOTODETECTORS

TRAINS

I (x, y)-

n

n

FIG. 1. Modelof e re-braindecision system.

tem. There are several spatial-frequency channels associated with the retinal photoreceptors, each one having its own signal

processor and threshold detector. Schnitzler has found that such a model predicts qaite well existing detection data for sinusoidal gratings.19'20 For nonsinusoidal images stimulating

several channels, Mostafavi and Sakrison14 report that detection occurs when the response of one or more channels

exceed threshold. We adopt a simplified model of the visual system consistent

with Schnitzler's. We envision the human eye-brain system as a bank of parallel, narrowband filters each tuned to a different spatial frequency. The output of each filter is independently assessed by the eye-brain decision elements to determine whether or not a stimulus exists in that narrow spatial frequency region. Consistent with this model, we assume that distributed over the retina are receptors which are sensitive only to certain narrow bands of spatial frequencies, corresponding to parallel spatial channels responding to the same incident light distribution, I(x,y). Each channel is assumed to be sufficiently narrowband that a constant intensity and phase at its center frequency charac2 terize its output spatial response over the image dimensions. 1

The retinal receptors for each channel act as a photodetector surface from which a Poisson process of electrical impulses

is emitted for transmission to the brain. The rate of each process is proportional to the light intensity at the output of the associated spatial filter. The decision as to whether a stimulus exists in any channel depends on the reception of enough electrical impulses in a given time. The probability of detection of a complex object is the sum of the probabilities

of detection in each channel, consistent with the assumption of channel independence and the detection results of Mostafavi and Sakrison.14 This simplified visual system model is depicted in Fig. 1. 376

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

Note that we have chosen not to precede the spatial filter bank with a singlefilter having either the characteristic of the visual system's MTF or the eye-optical MTF. According to the model it is probably more appropriate to insert a low-pass filter corresponding to the eye-optical MTF, since the retinal detectors act first on the light passing through the optical system. There are subsequent neural mechanisms which affect the response to the lower spatial frequencies. For the sake of simplicity, these neural effects are not represented in

our model. Insertion of any filter at this point will not affect the squared-error experimental results. [The Fourier domain variances of the signal and noise are scaled by the same

amount, leavingthe minimum mean-squared linear estimator, Eq. (3.6), unchanged.] The results with the new criterion will be affected to some degree, however, but it will be small, since the low-pass spatial power spectrum of the signal falls off more

rapidly with increasing frequency than the magnitude of the eye-optical MTF. In view of the fact that insertion of an eye-optical prefilter does not affect results at all in one case and only negligibly in the other, it was decided to omit it. A fair comparison of error criteria can be accomplished without

it. In accordance with the present assumption of many vision researchers, we assume that the spatial channels share many of the same sensitivity or response properties. For any channel, say the ith, we assume that the brain's decision elements count whether or not the number of pulses above the background level in a given time r exceeds a positive threshold

integer Ti. The probability of detection pi can be expressed according to a Poisson probability law, but we prefer to employ a normal approximation

for large rates which has been

experimentally verified from data in visual detection experiments. We follow the development of Schnitzler15,20 who justified his model and approach by deriving equations which William A. Pearlman

376

closely fit the classic Blackwell detection data. The probability of detection pi in response to an intensity Ii incident to the ith channel detector is =f

1

2

e _(X-CoIi)

/21

dx,

(2.1)

where -2= C0 I0 is the variance of the background noise pro-

models more closely the more significant change at the decision level. According to experimental evidence1 cited by Schnitzler, higher-level tasks such as recognition and feature

selection are direct functions of the probability of detection. We have selected the distortion measure d(pi,pi) between the corresponding probabilities of detection in each channel as

duced by the spatial mean value 1Ocommon to each channel

d(pi,p') = (pi - p') 2 , i = 0,1,2,. . . n,

detector. We assume that the background noise is dominant in every channel. The constant C0 is the photoelectric efficiency, which is assumed to be the same for each channel de-

where pi and p' are functions of Ii and i. The average distortion D is taken to be the numerical average of the channel distortions, i.e.,

tector. As a detector responds to the signal relative to the background, the output signal's mean pulse rate is nii = COIi.

D = (d(pi,p)) =

With a change of variables to y = X/rI0, we obtain i=

1

e(yki)

2

/2

(2.2)

(2,j1/2d,

K

The input signal-to-noise ratio ki can be expressed more conveniently when considering the eye in response to an image display of a digitized picture with gray levels 0-255. Let I0

be the mean intensity in these gray scale units. The mean display luminance Lo in candles per meter squared is prorate 7T,in response to the mean display luminance Lo is proportional to Lo, i.e., O = C2L0 = C1C2I0 . The standard de-

viation of the noise is co = (Qo)1/2 = (C 1 C2 1 0)1/2. For a signal luminance Li caused by Ii gray level units at spatial frequency i, the signal pulse rate ni~ is similarly C1 C2 Ii. The signalto-noise ratio ki can now be expressed

Because the per-channel distortion (pi -

p;) 2

is a rather un-

wieldy mathematical function of intensity, we decided to make a first-order approximation as follows: (pi-p;

=

B I \2/ ((2,)1/ l 2 exp [ (kT - B -i)/2] 'of/i

The per-channel distortion now becomes

d(pi,p') = d(IjI)

= B2 exp[_ OT -

Bli) 2] (,i - I',) 2.

27r

1"

(2.4)

1.

This distortion measure does not involve the phase angle of the channel filter outputs at all. Somehow the relative channel phases must be communicated to the brain in order to synthesize the image. Letting Ai = Iiejii and A

Ii-

(Lo)1/2L=.

(2.3)

Once we set the mean display luminance L,,, the ith channel's signal-to-noise ratio is proportional to the ratio of the intensity in the ith channel to the background intensity. The proportionality constant depends on the average unit excitation per lumen of display luminance. It is a physical characteristic of the eye and could only be calculated approximately from incompletely known experimental and physical properties of the eye. The contrast proportionality constant B = (C2 Lo)1/2 was determined

experimentally

by viewing

images. Without calibrating the display luminance exactly, B ranged from 700 to 1414. Our theoretical but approximate

=

I0eii,

'id = IAI

1

|

IAi-

we can use the alternative distortion measure B2 r / BIAiI\1 I2 Ai -A;I 2 d1(Ai,Ai)= - exp -(kT - 1 ] I because a given distortion by dl(AiAi) guarantees that distortion d(pi,p') is no larger. Furthermore, the factors B2/27r are unimportant because we can arbitrarily use any multiplicative constants to scale the distortion. Our final perchannel distortion measure is therefore

dl(AiA;) = KI; 2 exp [-

(kT

-

BIA1 I) 2]

IA -Aij

Now that we have an expression for the probability of detection in each channel, we can describe the proposed fidelity

criterion for vision. When the spatial intensity distribution Ii is distorted to I' it gives rise to a change in the probability of detection pi (I) to p'(I).

We propose to measure the visual

system response to the intensity change in a channel by the corresponding change in probability of detection, because it J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

2

(2.5a)

calculations found B to be 5000 for Lo = 100 mL.

377

Is(-. ' 10

and using the fact that

i= (C2 CO)/21 (1/2

-

d(pi,p').

Such a distortion is proportional to the change in total probability of detection and consistent with the "OR" operation

portional to Io, i.e., Lo = C1J,; and the mean background pulse

=

n + 1 i=O

on independent channels reported by Mostafavi and Sakrison.

where kT = Ti/o0 is the threshold signal-to-noise ratio and ki = C0Iil/cois the ratio of the mean signal pulse rate to the rms background fluctuation. Schnitzler has shown that kT is a constant of approximately 2.6, independent of background intensity I, and spatial frequency. When ki, the detector's output signal-to-noise ratio, equals the threshold signal-tonoise ratio kT, the probability of detection equals 0.5.

ki =

inL

with K an arbitrary constant. The average distortion is I n D= d (Ai,A;). (2.5b) n + i=o 1 The reciprocal factor of I2 is retained to emphasize the dependence of the distortion on contrast ratios. The per-channel distortion measure d, is a squared absolute William A. Pearlman

377

difference between corresponding Fourier coefficients of the true and distorted images multiplied by a function of the true coefficient's amplitude. It is an amplitude-weighted, squared-error distortion measure not to be confused with a common index-weighted measure such as wi IAi - A 2, where wi is a non-negative weight assigned for each i. The ampli-

tude weighting in d1 reflects both the nonlinearity of the channel response to input intensity and the signal dependence of the noise in the visual decision process. We shall now derive general formulas for estimation with an amplitudeweighted distortion measure to further our aim of estimating images with the distortion criterion in Eq. (2.5).

The optimal linear weighted squared-error estimator for a complex random variable X for a single observation variable

Y is derived in the Appendix. Equation (B3) gives the estimator for Y = X + N, where N is independent noise. Here we have an array of Gaussian DFT coefficients, with the nonredundant ones being independent and the rest complex conjugates. In addition, the weighting function for any DFT element is a function only of its amplitude. It turns out that under these conditions, the optimal linear estimator Ai for the DFT coefficient Ai when observing A = Ai + Ni, where Ni is

independent noise, is the same as that obtained from direct substitution into Eq. (B.3), i.e., Ai =

ciA = i

2 I 1 2 Ci -= Eg[ E tAiEg Eg [INi 1] 12 ][JA ±]

c

(3.1)

111. OPTIMAL LINEAR FILTERING FOR IMAGES IN ADDITIVE

NOISE

for i e JN = [0,1,2,... ,N-1].

The chosen experiment is the comparison of optimal linear filtering of images in additive noise using the new and usual

squared-error criteria. The estimation formulas are now

The operator Eg[ ] is a generalized expectation defined as

applied specifically to this case. Let us assume that the source images and additive noise are sample functions from stationary (homogeneous) random ensembles (fields) with known

second-order statistics. For simplicity, the noise is chosen to be image independent, Gaussian, and white. The outputs of the spatial channels are assumed to be the coefficients of

the two-dimensional discrete Fourier transform (DFT) of an image.

Eg[Z] =

f zg(x)p(z,x) dz dx

for a random variable z weighted according to the value of another random variable X. For the new criterion the weighting functions in Eq. (3.1) are

The DFT coefficients are very good approximations

to the complex amplitudes of the outputs of closely packed very narrowband spatial filters. The linear filtering is performed on the DFT process, where the nonredundant coeffi-

g(JAi I) = KJI 2 exp[- (kT- BAi) 2]

cients are assumed to be statistically independent. (The DFT possesses a complex conjugate symmetry because the input

according to Eq. (2.5a). The constant Ki is indexed because it depends on the probability distribution of Ai. When g(IAi I) = 1, Eq. (3.1)yields the linear estimator for the ordinary squared-error criterion. The average error di given by

process is real.) The statistical independence of the spatial 5 channels has been demonstrated experimentally, but, mathematically, it exists approximately as well. For an image of 65 536 elements (256 X 256), the DFT coefficients are very

the above estimator is

3 nearly uncorrelated2 2 and Gaussian in distribution,2 and hence the nonredundant ones are approximately statistically

independent. We assume therefore that our images and noisy DFT coefficients which are images have nonredundant

Gaussian and independent. For an image in white Gaussian noise it is convenient to

di = Eg[|Ai| 2 ](1

-

ci).

The i = 0 and N/2 coefficients are treated separately since they are both purely real and the i = 0 term has nonzero mean.

The above formula is applied to the zeroth term after the image mean Io is subtracted. The Gaussian, white, and

perform the optimal linear filtering on the DFT process for both error criteria. The unitary property of the DFT makes

zero-mean properties of the additive noise produce zero-mean

the average absolute squared error the same in the image and

Gaussian with idential variance -,, the variance of the additive

DFT coefficients Ni, i e JN, which are independent and

DFT domains. Hence, optimal linear filtering by this crite-

noise in the image domain.24 The terms for i 5 0 and N/2

rion is equivalently performed in either domain. Because the

divide their variances equally between their independent real and imaginary parts. The image statistics are similar except that the coefficients have different variances oq2and a nonzero

filtering or the new criterion is more naturally performed in the DFT domain and the coefficients are independent

for all

practical purposes, it is easier to filter the DFT coefficients using both criteria.

mean in the 0th term of I,. We proceed to evaluate E[|Ni

2 E1INI1 ] =

u 21 = f0 W Eg[INi

2]

2 ] as and Eg[INiI

2

jn 1j2g(jai1 )p(lai|)p(lni|)d(lai|)d(jnil)

= a2 JU g(JaiJ|)p(aiJ)d(JaiJ)

378

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

William A. Pearlman

378

| nf

KiIo 2 exp [-(kT-Bx

i0 exp[- (kT - B

an

x]a exp (

I

i)dx,

i F o, N2

2

)](2l/exp

( x2

dx,

1") 7a?2'i

i=O,-,21 (3.2)

for all i in the index set JN. The variances ao are the respective variances of the complex signal DFT coefficients Ai. For the other quantities EIIAi 12] and Eg IIAi 12] in the estimation formulas, we evaluate them to be

E[IA

j

Eg[IA 1 2]

2

1

]

=

IajI2g(|ai )p(Iai|)d(Iaji)

2KI,2

Cxex-(T-(2)

N

2] exp exp[-(kT-Bnx) Box3 (2

2

dx, i s4 0, N

x2 exp[-(kT-Bnx)2 ] (i2Y' 2 exp -

KiI;2f

) dx, i=0

for i E JN, where Bn = B/Io We interpret the above formulas for A, as having the mean 1Osubtracted out to yield a zero-mean random variable. The formulas for Eg IINi 12] and Eg IIAi 12] can be more succinctly expressed in the following forms: 2Kc.r2 (

2

N

2'

2 Eg [ NBkTi 1 1

2 (2)

1/2 K \~ii/

I2i (B

no.

kj )

N

2

(Bn~du / exp(k + ± 6?)G0(6j),

1

(3.4a)

k ) exp(-k2 d+--Y)G

3(yi),

0 2

i 5' 0, 2

2

Eg [JA.

] =

(3.4b) 2)1/2

Ki

\7r

Bki T/3

Io2, BnkT}

U)i

exp(-khr + 6?)G2(0i), i 1

=

0, N 2

where Yi = BnkT(B2 + Cr-2)-1 2 bi = BnkT(B 2 + and the functions Gm(a) are defined by Gm(a)

f-

xme

(x-a) 2

dx,

m

(20cP)-1)-1/2,

= 0,1,2,3.

2G0 (a) = V§7[1 + erf (a)], 2G 1 (a) = exp{-a 2 l + V'7a[1 + erf (a)],

2G2 (a) = 3a expt-a 2

1 + V7(a 2 + 1/2)[I + erf (a)],

2G 3 (a) = (1 + 7a2 ) expl-a 2 l + 379

N/7a

(a2 +

Substituting Eqs. (3.4) into (3.1), the estimate for Ai by the new criterion is

These integrals may be expressed in terms of the error function according to

%/2)[1

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

+ erf (a)].

i-~ 0, -

i

1

Ai = ciAi,

¶YiG YiG 3 (Yj)

ci =

3 (-Yi)+

(

N

(BnkTcrn)2Gl(yi)'

2

i =d o,N

tVG2 (0i)

OG 2 (0i) + (BnkToyn )2GO(bi)'

(3.5)

i=02.

For the squared-error criterion, the scaling constant cu(") for the corresponding linear estimator AI(U)is William A. Pearlman

379

= cI2/ CMU)

+ '),(

(3.6)

which agrees with ci for Bn = 0. The complex conjugate symmetry of the DFT of a real array

is always taken into account to reduce the number of calculations. In one dimension, the symmetry yields the equalities =N-i,

ci = CN-i,

cl

=

__

for the linear estimators The scaling constants ci and c MU) of the DFT depend on the known additive noise variance an and the variances of the true image's DFT coefficients, cr?, which are found by estimating its power spectrum. The estimation of the actual two-dimensional

(256 X 256) power

spectrum is performed by averaging modified periodograms of smaller blocks (64 X 64). The method is an extension of 25 The other a one-dimensional procedure reported by Welch.

image-related constant Io, the background or average intensity, is also estimated by a numerical average of the image pixels. The other parameters kT and B are related to the visual system. As explained previously Schnitzler has reported kT, the threshold signal-to-noise ratio, to be close to 2.6 for a large range of contrast ratios.

The parameter B is

the proportionality constant between I, in gray levelunits and the average retinal pulse rate. It is the product of two proportionality constants, one between IOand the image luminance Lo and the other between Lo and the retinal rate. An approximate calculation from known physical constants and a simplified eye model yielded a B of 5000 for Lo = 100 mL. This number was refined experimentally to range between 707

and 1414,the exact value depending on the image restoration properties required. In principle, the estimating coefficients can be calculated before reception of the actual image and noise data, since the background, image power spectrum, additive noise variance, and eye parameters are known beforehand.

FIG. 2.

setupfor comparativeimagefiltering. Experimental

main filtering on the noisy lunar landscape with two different weighting factors, B = 707 and 1414 (lo = 135). Comparing them to the unweighted, squared-error filtered image in Fig. 6, we see that they are both sharper and more pleasing with better restoration of true contrast and detail in the highercontrast regions. In the lower-contrast areas, we notice a mottled appearance probably due to the filter's attempt to pick some detail out of what is mostly noise. The usual squared-error filter produces a fuzzier image and seems to smear out this mottling so that there is less suggestion of detail everywhere, including the lower-contrast regions. The higher

weighting factor seems to enhance the edge detail better than the lower one and thereby produces more mottling in the lower-contrast regions. This suggests that this factor could be adjusted to give a greater or less degree of enhancement to

IV. EXPERIMENTAL COMPARISON OF NOISY IMAGE FILTERING The experiment of comparing filtered images by two dif-

ferent error criteria is diagrammed schematically in Fig. 2. White Gaussian noise is added to a source image in a computer and the resultant noisy image is discrete-Fourier-transformed

to give the noisy DFT data array {a'). The two optimal linear estimates of the image DFT array {aiI and {d(')I are then calculated according to Eqs. (3.5) and (3.6). These arrays are then inverse discrete-Fourier-transformed to give the two estimates of the source image. They are read out to a CRT display for photographing or directly recorded onto film. The two source images for our filtering experiments are shown in Fig. 3. One is a lunar landscape with relatively little

contrast and detail except in the shadowed crater areas. The other, a woman's face with background, contains a high degree

of contrast and detail. In Fig. 4 are the same images in additive noise of the same variance. The signal-to-noise ratios are -9.2 and -2.2 dB for the lunar landscape and woman's face, respectively.

Signal-to-noise ratio is defined as the ratio

image detail. We did not investigate this suggestion further, however, because of limited funds for computer time. Both the weighted and unweighted filters fail to restore the very small craters which are obliterated by relatively high noise amplitudes. The weighted filter, however, restores some craters that remain obliterated in the unweighted filter restoration. A 3 db less noisy lunar landscape, shown in Fig. 7, was re-

stored with a weighted (B = 707) squared-error filter and the usual one. The restorations

are displayed in Fig. 8. Again,

the weighted filter produces a sharper image with truer contrast and better detail. This time, however,there is much less appearance of noise mottling. This result agrees with intuition when we recall our first-order approximation in deriving the new error criterion. It is expected that, for smaller image

degradations, this error criterion will produce even better results since the approximation to per-channel probability of detection is more accurate.

It is quite a happy circum-

stance that very good filtering results were obtained despite the use of very noisy images.

of the mean squared deviation of the image from background

The filtering of the woman's face produced similar results.

to the variance of the noise. The following figure, Fig. 5, re-

The weighted (B = 707, 10 = 99) and unweighted filterings of the noisy image (S/N = -2.2 dB) in Fig. 4(b) are shown in Fig.

veals the results of the weighted, squared-error Fourier do380

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

William A. Pearlman

380

visual system. It therefore relates the consequences of intensity changes over an image more directly to the corresponding signal and decision processes in the eye-brain system

than other error measures such as squared intensity difference. The new criterion is a magnitude-weighted,

absolute

squared difference in the spatial Fourier coefficients of the image. The magnitude weightng approximates the change in the probability of detection caused by a change in intensity.

Probability of detection seems to be a significant quantity in the eye-brain system's action, as verified by psychovisual

(a)

(a)

(b) FIG. 3.

Sourceimages. (a) Lunarlandscape. (b)Woman'sface.

9. Noise mottling is present in both restorations, but is again

more evident in the lower-contrast areas of the picture restored with the weighted filter. Nevertheless, the high degree

of overall contrast and detail is restored more sharply and faithfully with the weighted filter. The usual squared-error filter is, again, somewhat fuzzy in comparison. V.

SUMMARY AND CONCLUSIONS A new error criterion for image processing has been derived

from a parallel spatial channel-image detector model of the 381

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

(b) FIG.4. Sourceimagesin additivenoise. (a)Lunarlandscape,SIN = -9.2 dB. (b) Woman'sface, S/N = -2.2 dB. William A. Pearlman

381

Squared-errorfiltering of noisy lunar landscape, S/N = -9.2

FIG. 6. dB.

(a)

~'

After investigating the mathematical properties of weighted measures for estimation, we sought to compare our new error

criterion to the usual squared difference in intensity in an image restoration experiment. The reason for choosing squared intensity difference for comparison is that it underlies &

practically all estimation and data compression methods in image processing on account of its mathematical convenience.

If w canpproveasuperiorityfor our new criterion in a fair test against squared intensity difference, we have shown that our new criterion does indeed represent more closely the eye-brain

system's action under image degradation. Then we can proceed further with special adaptive estimation and data

'

.4

4

? 4

dB) filtered by new error criFIG. 5. Noisylunar landscape(SIN =-9.2 tenion. (a) B=707,/4,= 135. (b) B=1414,/Io= 135.

testing. The weighting of the magnitude causes intensity differences near threshold to be weighted more than the same ones farther away from threshold. This corresponds to the relatively large changes in probability of detection near threshold compared to those farther away given the same intensity change. Imbedded in the absolute-squared difference of the complex Fourier coefficients is an unweighted measure of channel phase difference, which is also important for characterizing the image. 382

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

'W

-'r4

U*.-.;cr:

-.-

'y.&

,

& r-4

'%-,

Z

~

,

,~*

~

,

;e'e

£ '-~.

''''

'

w.l

~

Iv

.

"

-.

.J4

>,

K" ''

/

.~~K

FIG. 7. Lunar landscapein additive noise, SIN =-6.2

dB.

William A. Pearlman

382

more uniformly than with the new filtering, which produced a mottling effect. This effect is attributed to the filter's attempt to extract signal from what is mostly noise. For moderately noisy images, the mottling decreased significantly, but was still present. Since the new criterion involves a first-order

approximation to probability of detection, we expect even better results in lowernoise situations. The fact that sharper, more pleasing restorations are obtained for all the noisy images is proof of the effectiveness of our new criterion.

(a)

(a)

(b) FIG. 8. Comparativefilterings of lunarlandscapein noise(S/N =-6.2 dB, to = 135). (a)Newerror criterion, B = 707. (b)Squared-errorcriterion.

compression techniques with the confidencethat it will again be an improvement over existing methods. The experiment chosenis the optimal linear filtering of the same images in noise using the new and usual squared intensity difference criteria. The filtering by the new criterion produced sharper, more detailed, truer contrast images in all cases. The usual filtering produced a comparatively fuzzy image. This fuzziness was an advantage for very noisy images

in the low-contrast regions where the noise was averaged out 383

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

(b) FIG.9. Comparativefilteringsof woman'sface in noise(S/N = -2.2 dB, /0 = 99). (a) Newerror criterion, B = 707. (b) Squared-errorcriterion. William A. Pearlman

383

The gains achieved in estimating images in noise with our new criterion come at the expense of an increase in mathematical complexity and computational load over the usual criterion. We must discrete-Fourier-transform the image, calculate the estimating constants for each Fourier coefficient,

and invert the DFT to obtain our estimated image. Although the squared-error filter can be implemented in the image domain, it is usually performed in the Fourier domain for the convenience of having approximately uncorrelated coeffi-

cients. The estimating constants can be calculated more easily, however, for the squared-error

criterion.

Tables for

fSg(x)[x

-

h(y)]2 p(x,y) dx dy =

fp(y)rfg(x)[x

-

h(y)] 2p(x/y) dx}dy. (A2)

Because p (y) and the bracketed integral are non-negative for every y, we can minimize the full expression by minimizing

the bracketed integral for each y. For any fixed y, we must determine the number a = h(y) that minimizes

fg(x)(x

-

a)2p(x/y) dx.

A necessary condition for a minimum is that the derivative of the above integral with respect to a must be zero. Carrying

the difficult integrals involved in the constants for the new criterion could be stored beforehand and accessed to avoid real-time computations. (The storage of the integrals is the

out the differentiation, we obtain

same for any image, so on a per-image basis is an insignificant

Solving we obtain the extremal solution for a as

-2fg(x)(x - a)p(x/y) dx = 0.

increase in memory.) The increase in memory or computational load is really not large enough to be a problem in view of current computer technology. In conclusion, we have derived a new error criterion whose

suitability has been established through experiments in filtering noisy images. Although it is more complicated mathematically than the usual squared-intensity difference,

=h(y) =fxg(x)p(x/y) dx

(A3)

Sg(x)p(x/y) dx

The assumption of integrability of g(x) assures the finiteness of the denominator of x. A second differentiation

yields

2fg(x)p(x/y) dx,

it is also more effective in restoring detail and sharpness for

which is non-negative for non-negative g(x). The solution

images degraded by additive noise. Moreover, the criterion is tractable enough to lead to closed-form formulas for linear estimation with a moderate, but insignificant increase in computational burden over other criteria. The filtering results are significant enough to demonstrate savings in re-

gous to the usual one as

quirements

of bit rate or signal-to-noise

ratio for a given

subjective reconstruction quality in communications. Moreover, in data compression, the relatively low weighting of amplitudes of Fourier coefficients farther away from threshold deems that these amplitudes be encoded much less accurately than those near threshold.26 This new criterion, therefore, also offers the promise of substantial savings in information rate (bits) in the storage of images. Data compression with the new criterion is a fruitful ground for further research.

xi is therefore a global minimum for every observed value y.

If we define a generalized expectation operator Eg analoEg[Z] = f fzg(x)p(z,x) dz dx

(A4)

for any random variable Z weighted with respect to another random variable X, we can write our estimator X = h(Y) more succinctly as =

h(Y)

X

Eg[1/Y]

(A5)

which for g(x) = 1 reduces to the ordinary minimum meansquared-error estimator. In order to be consistent with the interpretation of the operator Eg as an expectation, we may require that for ay con-

stant b, ACKNOWLEDGMENT

Eg[b] = f bg(x)p(x) dx = b,

I wish to acknowledge my gratitude to Mr. Kelly A. Miller who contributed to the success of this endeavor by performing

much of the initial computer programming.

so that

Sg(x)p(x) dx

=

1.

This requirement normalizes the scale of the weighting function in accordance with the probability density of the weighted random variable.

APPENDIX: ESTIMATION WITH A WEIGHTED, SQUARED-ERROR CRITERION

A. Nonlinear estimation We shall now investigate optimal estimation with a general

weighted, squared-error criterion and apply the results to the distortion measure dl in Eq. (2.5). We observe the real random variable Y and wish to estimate a related real random variable X by a function h ( Y) = X such that

For complex random variables X and Y with a real, nonnegative, and integrable weightingfunction g with domain in the complex plane, the same result (A5) obtains with the obvious application of (A4) to complex random variables.

B. Linear estimation Instead of finding the best function of Y that minimizes the

(Al)

average-weighted squared error, let us now attempt to find the best linear function of Y. Assume that X and Y are

is a minimum. We assume that the weighting function g is non-negative and integrable. Writing out the expectation in terms of p(x,y), the joint probability density of X and Y, we have to find the function h that minimizes

zero-mean complex random variables with generalized means

E[g(X)(X -

384

g)2]

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

of zero, i.e.,Eg[X] =Eg[Y]= OandE[X] =E[Y] =O. The weighting function g is a real, non-negative, and integrable function of a complex variable. For example, the amplitude William A. Pearlman

384

dmin =Eg[JXI2 (1 -c)

weighting function in (2.5) meets these conditions. Furthermore, for jointly Gaussian zero mean X and Y as in our case, the generalized means are zero. We seek the complex constant c = a + jb in the estimator of X, X = c Y that minimizes 2

d = E[g(X)IX-cyl

]

= E[g(X)IX -

12]

-

cE[g(X)X Y]

c*E[g(X)XY*] + |c| 2E[g(X)l y 2] Eg[jXI 2] + IcJ2Eg[1 YJ2] - cEg[X*Y] - c*Eg[XY*].

(Bi)

Letting c = a +jb and differentiating in turn with respect to a and b and equating to zero, we obtain

2aEg[JYJ2 ] - a* - a= 0,

-=

da

ad b = 2bEg[I YJ2] - ja*+ ja =O, with a

(B2)

EgtXY*].

The extremal solution is (a* +a)/2Eg[YJ2 ]

a

b=j(a*

-

=

Re[a]/Eg[lY

2

2

],

a)/2Eg[IYJ ] = m[aO.Eg[lY ]. 2

Therefore, c = a + jb = Eg[XY*]/Eg [I YJ 2 ]

(B3)

isthe extremal solution for c. It is obviously a minimum since the second derivatives of d with respect to a and b are both 2 Eg [IYI 2], which is non-negative by the assumption of nonnegative g(x). For g(x) = 1, the constant c in (B3) reduces

to that for the usual (unweighted) squared-error criterion. Interestingly enough, the weighted linear estimator obeys an orthogonality principle in the form

Eg[(X-cY)Y*]

(B4)

= O.

The minimum average error can be written down easily having observed this fact: dmin = Eg[IX-cyl

2

]

= Eg[(X - cY)X*] 2 =Eg[X 2 ] - Eg[XY*]J /Eg[Yj2 ].

(B5)

cEg[Y].

We can apply these linear estimation formulas to the case of Y as the sum of the complex random variable X and the independent complex noise N. Let X and N have zero means in the regular and generalized senses. By substituting Y =

X+ N in (B3), the constant c in the linear estimator X

= cY

is

Eg[X(X+N)*] Eg[1X1 ] , Eg[JX + NJ 2] Eg[JXJ 2] + Eg[JNJ2]()

B

since Eg [kN*] = 0 by the assumption of independence and zero mean for X and N. Similarly, the minimum average error from (B5) is 385

This work was supported by NSF Grant No. ENG75-10545 and The Graduate School, University of Wisconsin-Madison, and was presented at the Annual Meeting of the Optical Society of America in Toronto, Canada, October, 1977. 0. H. Schade, Sr., "Optical and photoelectric analog of the eye," J. Opt. Soc. Am. 46, 721 (1956). 2 T. G. Stockham, Jr., "Image processing in the context of a visual model," Proc. IEEE 60, 828-847 (1972). 3 J. L. Mannos and D. J. Sakrison, "The effects of a visual fidelity criterion on the encoding of images," IEEE Trans. Inf. Theory 20, 525-536 (1974). 4 C. F. Hall and E. L. Hall, "A nonlinear model for the spatial characteristics of the human visual system," IEEE Trans. Syst. Man and Cybern. 7, 161-170 (1977). 5 F. W. Campbell and J. G. Robson, "Application of Fourier analysis to the visibility of gratings," J. Physiol. 197, 551-566 (1968). 6 C. Enroth-Cugell and J. G. Robson, "The contrast sensitivity of retinal ganglion cells of the cat," J. Physiol. (London) 187, 517 (1966). 7 F. W. Campbell, G. F. Cooper, J. G. Robson, and M. B. Sachs, "The spatial selectivity of visual cells of cat and the squirrel monkey," J. Physiol. (London) 204, 120 (1969). 8C. Blakemore and F. W. Campbell, "On the existence of neurons in the human visual system selectively sensitive to the orientation and size of retinal images," J. Physiol. (London) 203, 237-260 (1969). 9 A. Pantle and R. Sekular, "Size-detecting mechanisms in human vision," Science 162 1146-1148 (1968). 'OP. Tynan and R. Sekular, "Perceived spatial frequency varies with stimulus duration," J. Opt. Soc. Am. 64, 1251-1255 (1974). "M. B. Sachs, J. Nachmias, and J. G. Robson, "Spatial-frequency channels in human vision," J. Opt. Soc. Am. 61, 1176-1186 (1971). 12N. Graham and J. Nachmias, "Detection of grating patterns containing two spatial frequencies: A comparison of single-channel and multiple-channel models," Vision Res. 11, 251-259 (1971). 13 F. W. Campbell, J. Nachmias, and J. Jukes, "Spatial-frequency discrimination in human vision," J. Opt. Soc. Am. 60, 555-559 (1970). 4 1 H. Mostafavi and D. J. Sakrison, "Structure and properties of a single channel in the human visual system," Vision Res. 16,957-968 (1976). 5 1 A. D. Schnitzler, "Image-detector model and parameters of the human visual system," J. Opt. Soc. Am. 63 1357-1368 (1973). 16H. Richard Blackwell, "Neural theories of simple visual discrimination," J. Opt. Sc. Am. 53, 129 (1963). 17 H. R. Blackwell and J. H. Taylor, in Proceedings of the NATO Seminar on Detection, Recognition, and Identification of Lineof-Sight Targets (unpublished); also, University of Michigan Engineering Research Institute Report No. 2455-10-F (1958) (un*

18 J. Johnson, Image Intensity Symposium, Fort Belvoir, Va., October 6-7, 1958, AD220160 (unpublished). 19A. D. Schnitzler, "Analysis of noise-required contrast and modulation in image-detecting and display systems," In Perception of Displayed Information, edited by L. M. Biberman (Plenum, New York, 1973). 20 A. D. Schnitzler, "Theory of spatial-frequency filtering by the human visual system. I. Performance limited by quantum noise," J. Opt. Soc. Am. 66, 608-617 (1976). 21 The spatial bandwidth of the channels is in dispute. Generally, experiments without grating adaptation produce narrow channels and, with adaptation, wide channels. We assume here that adap-

tation has not taken place.

22

2

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

(B7)

published).

If Eg [X] and Eg [Y]are nonzero, the best linear estimator is

= cY +r, where r =EgX]-

Eg[1XJ 2 ]Eg[JNJ 2] Eg[IXJ 2] + Eg[1N1 21

W. A. Pearlman, "A limit on optimum performance degradation in fixed-rate coding of the Discrete Fourier Transform," IEEE Trans. Inf. Theory IT-22, 485-488 (1976).

23

W. A. Pearlman, Quantization Error Bounds for Computer-Generated Holograms, Technical Report No. 6503-1, Information

System Laboratory, Stanford University, Stanford, Calif. August 1974 (unpublished). 24 The DFT of the Gaussian noise process alone is exactly Gaussian William A. Pearlman

385

with statistically independent nonredundant coefficients due to the linearity of the DFT. 25 P. D. Welch, "The use of Fast Fourier Transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms," IEEE Trans. Audio and Electroacoust.

26

15, 70-73 (1967).

G. Senge, Quantization of Image Transforms with Minimum Distortion, Technical Report No. ECE-77-8, Dept. of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, Wis., May, 1977 (unpublished).

Refractoriness in the maintained discharge of the cat's retinal ganglion cell* Malvin Carl Teich, Leonard Matin, and Barry I. Cantor Columbia University, New York, New York 10027 (Received 20 September 1976; revision received 16 September 1977) When effects due to refractoriness (reduction of sensitivity following a nerve impulse) are taken into account, the Poisson process provides the basis for a model which accounts for all of the first-order statistical properties of the maintained discharge in-the retinal ganglion cell of the cat. The theoretical pulse-

number distribution (PND) and pulse-interval distribution (PID) provide good fits to the experimental data reported by Barlow and Levick for on-center, off-center, and luminance units. The model correctly

predicts changes in the shape of the empirical PND with adapting luminance and duration of the interval in which impulses are counted (counting interval). It also requires that a decrease in sensitivity to stimulation by light with increasing adapting luminance occur prior to the ganglion cell and is thus consistent with other data. Under the assumptions of the model, both on-center and off-center units appear to ex-

hibit increasing refractoriness as the adapting luminance increases. Relationships are presented between the PND and PID for Poisson counting processes without refractoriness, with a fixed refractory period, and with a stochastically varying refractory period. It is assumed that events unable to produce

impulses during the refractory period do not prolong the duration of the period (nonparalyzable counting). A short refractory period (e.g., 2% of the counting interval) drastically alters both the PND and PID, producing marked decreases in the mean and variance of the PND along with an increase in the ratio of mean to variance. In all cases of interest, a small amount of variability in refractory-period duration distinctly alters the PID from that obtainable with a fixed refractory period but has virtually no effect on the fixed-refractory period PND. Other two-parameter models that invoke scaling of a Poisson input and paralyzable counting yield predictions that do not match the data.

1.

INTRODUCTION

The sequence of impulses recorded from the retinal ganglion cell of the cat remains irregular even when the retina is

thoroughly adapted to a steady stimulus of fixed luminance. Though this maintained discharge has been observed in a variety of vertebrates, its main characteristics have been described in greatest detail in the cat.'-1 0 Barlow and Levick 6

pulse-counting distribution (see Fig. 1). These properties have led workers to consider that scaling and refractoriness mediate between absorbed quanta and the maintained dis-

charge.1,4-6,8- 3 While we employ the term "refractoriness" in its usual sense-to imply a reduction in sensitivity for some period following the occurrence of a neuronal spike-it is

important to note that we do not imply that the refractory period we deal with be identified with those normally found

and Sakmann and Creutzfeldt7 have noted that the average maintained firing rate of on-center units increases as the adapting luminance is increased from zero up to a point at

in axons following electrical stimulation (see Sec. V for further related considerations). We shall also often employ the term "dead time" to refer to the period following an impulse (spike)

which the increase is either slowed or reversed with further increases in luminance. At still higher adaptation levels, the firing rate again rises.3' 6 Both Sakmann and Creutzfeldt 7 and Barlow and Levick 6 attribute the plateau or fall in firing rate to increasing contribution from the inhibitory surround. The

roots in the nuclear physics literature.' 4 The terms refractory period and dead time will thus be nearly interchangeable. We use the term "scaling" to refer to a particular kind of reduction in sensitivity in which a system requires r input events in order to produce a single output event; r is then the scaling parameter. Barlow and Levick 6 considered such a simple scaling

average firing rate of off-center units tends to decrease with increasing adapting luminance, though the change is neither as consistent

nor as large as that observed for on-center

units. The most striking properties of the maintained discharge are the large and increasing compression between numbers of quanta absorbed and nerve impulses generated, and the substantial increase in mean-to-variance ratio of the empirical pulse-number distribution (PND) with increasing luminance. 5 The pulse-number distribution is also often referred to as the 386

J. Opt. Soc. Am., Vol. 68, No. 3, March 1978

during which another impulse cannot occur;this term has its

model for the case in which the input events represented Poisson-distributed quanta of light and the output was the discharge of ganglion cell spikes, but found that the required parameter values were seriously wrong. Barlow' 2 subsequently criticized this scaling model on other grounds as well,

though Levick9 has recently pointed out that some modifications introduced by Stein' 5 "16 may improve matters.

We

develop the simple scaling model further and arrive at a conclusion similar to that arrived at by Barlow.'2

0030-3941/78/6803-0386$00.50

1978 Optical Society of America

386