76

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-25, NO. 1, JANUARY 1978

A Nonlinear Receptive Field Model of the Visual System TOMOZO FURUKAWA AND SHIRO HAGIWARA

A bstract-A nonlinear model for the visual system is proposed in relation to the dependence of receptive field size on luminance. The model is constructed based on the fact that the exponent of Stevens' brightness function varies depending on the target size. The model consists of two stages; the mechanism of spatial summation and the mechanism of 0.5-power in cascade. The mechanism of 0.5-power is responsible for an exponent 0.5 of the brightness function for the point target. The extent of the spatial summation changes depending on the luminance so that the model yields a brightness function with exponent 0.33 for the infinite target of uniform luminance distribution. The model predicts weli the signal transfer function of vision, the characteristics of the suprathreshold spatial summation and the mean luminance dependence of the spatial frequency characteristics for threshold contrast.

receptive field [8]. Hallett studied the dynamic behavior in the spatial interaction for human rod vision and showed that the brief illumination of the dark adapted eye reduces the extent of spatial interaction [10], [11] . Patel analyzed the luminance dependence of the line spread function of the human visual system at photopic luminances by performing the inverse Fourier transform on the spatial frequency characteristics [5]. Similar analysis at low luminances was reported by van Meeteren and Vos [7]. Several models have been proposed to account for the dependence of the receptive field size on the luminance. Hallett modified the Sparling's model by a bimodal line spread function for the receptive field surround and successfully simulated his experimental findings on threshold disturbance INTRODUCTION V ARIOUS models of the vertebrate visual systems have [11] . Sugie et al. proposed a model in which a stage of forbeen proposed to account for psychophysical phenomena ward shunting inhibition is adopted to account for the nonrelated to the spatial brightness interaction. They are con- linear lateral interaction [12]. In this paper we propose another plausible way of constructstructed based on the neural units known as the receptive field or the lateral inhibition. Since most of these models are based ing a model of luminance dependent receptive field. A logic on linear approximation, their applications are restricted to which we used for constructing the model is as follows. Subpredict psychophysical characteristics for the spatial pattern jective brightness obeys Stevens' power law, that is, brightness grows as a power function of luminance. This fact is a fundaof low luminance contrast. Generally, the psychophysical phenomena of the spatial mental in constructing a model of the luminance dependent brightness interaction show luminance dependent behavior. nonlinearity. Furthermore, it is known that the exponent of For example, the relationship between the luminance and the power function takes different values depending on the visual acuity is expressed by a sigmoid curve [1]. The bright target size. An exponent of 0.5 was obtained for a point and dark bands seen at the contrast edge, which are known as target and 0.33 for a target of 5 degrees of arc [13]. Thus, the Mach bands, are not symmetrical in amplitude and in besides the luminance dependent nonlinearity, a mathematical width [2]. When the contrast of the visual pattern is low model for the luminance dependent spatial interaction could enough to allow a linear approximation, the characteristics of be derived from Stevens' law in brightness. the visual system show the parametric dependence on the CONSIDERATIONS ON STEVENS' BRIGHTNESS mean luminance. For example, the spatial frequency charFUNCTION acteristics for threshold contrast depend on the mean lumiThe subjective magnitude I in sensation is described as a nance, for which intensive works have been carried out by power function of the stimulus magnitude '1 as follows, many investigators [3] -[7] . With respect to the psychophysical facts mentioned above, * K((D (DO)m (1) the response and the size of the receptive field are thought to be dependent on the target luminance. Several investi- where %o is the threshold stimulus magnitude, and K and m gators have studied the luminance dependence of the recep- are constants which take specific values for a specific kind tive field size [8]-[121. By psychophysical experiments on of stimulus. This empirical relationship is known as Stevens' the spatial summation in human vision and physiological power law and holds in almost all kinds of sensation. In the case of vision, Stevens' power law specifies brightness experiments on the isolated frog's retina, Glezer showed that function, the luminance of the test field determines the size of the B =K(L -Lo)m (2) Manuscript received June 25, 1976; revised December 27, 1976. This =

work was supported by Grant 020901 from the Ministry of Education of Japan. The authors are with the Research Institute of Applied Electricity, Hokkaido University, Sapporo, Japan.

where B is the brightness, L is the luminance of the target and

Lo is the threshold luminance. Exponents for targets with the dark surround are as follows [13],

0018-9294/78/0100-0076$00.75

C 1978 IEEE

FURUKAWA AND HAGIWARA: MODEL OF THE VISUAL SYSTEM

77

m = 0.5 for the case of the point target, m = 0.33 for the case of the target of 5 degrees of arc. It seems that the brightness function states nothing about the spatial interaction in the visual system. But it is noteworthy that the exponent for the 5° target differs from that of the point target. It seems that the difference in exponent reflects the effect of the lateral interaction. It is known that all the sensory nervous systems have the physiological process called lateral inhibition. In the case of the visual system, the extent of lateral interaction is known as the receptive field [14]. The receptive field is an area on the retina which is associated with a single nerve cell at a certain stage of the nervous system. That is, receptor cells in the area are concentrated into a single nerve. A receptive field of a ganglion cell is schematically represented by two concentric circular areas; the center and the surround. The center is excitatorily connected to the single cell and the surround is inhibitorily connected or vice versa, i.e., there are two types of receptive fields; the on-center type and the off-center type. The on-center type is assumed in our discussions. If a retinal image is a point whose size is much smaller than the size of the on-center of the receptive field, the illuminance information of that point is transmitted through the corresponding nerve channel without being subject to the lateral interaction. This situation corresponds to the case of the brightness function for the point source. On the other hand, if the size of the image is much larger than the size of the off-surround and its illuminance is uniform over the area of the image, the illuminance information transmitted through each of the parallel nervous channels is proportional to the integral of the weighting function of the receptive field (the integral of the point spread function of the neural system if the organization of the receptive field is spatially invariant). This situation corresponds to the case of the 5° target. Brightness functions for both cases should have a common exponent so far as the size of the receptive field is independent of the retinal illuminance. Therefore, the fact that the exponent of the brightness function varies depending on the target size can be a basis for a mathematical model for the luminance dependence of the receptive field size.

CONSTRUCTION OF MODEL The spatial characteristics of the visual system are related to the optical and neural functions. Although the receptive field is in the usual sense the neural function, we refer the receptive field to an area on the stimulus target for simplicity in this paper. In other words, assuming that the organization of the receptive field is spatially invariant, we consider a point spread function of the visual system including both the optical and the neural systems. We will use terminologies "the receptive field"' and "the point spread function" interchangeably in the following discussions. As shown in Fig. 1, the point spread function of the visual system R is assumed to have point symmetry and to be approximated by linear combination of two Gaussian functions as follows,

R(r)= KEexp (-

)- KI exp (

2;))

Fig. 1. Organization and response of the on- center type receptive field. The response is assumed to be a linear combination of two Gaussian functions.

where r is a radial distance from the center of the receptive field. The first and the second terms correspond to the excitatory and the inhibitory spatial interactions respectively, where a's are parameters representing the extent of interaction and K's are parameters representing the magnitude of interaction. These parameters should be luminance dependent to account for the dependence of the brightness function on the target size. Several works have been made of the effect of target size on the brightness function. Mansfield reported that the change of exponent between point target (0.050) and extended targets (>0.170) is abrupt rather than gradual [15], while Fig. 1 of Hanes suggests that the exponent changes gradually with target size [16]. We arbitrarily postulate the gradual change of the exponent as explained in the preceding section. There may be several possible alternatives to assign luminance dependence to parameters a's and K's. The following are assumptions adopted in the model construction: (1) The 0.5-power law is an inherent property of each single nerve channel. We call this property the mechanism of 0.5power. (2) To each mechanism of 0.5-power, retinal cells in the corresponding area are concentrated. Signals from the cells add on to each other. We call this the mechanism of spatial summation. (3) The mechanism of spatial summation and the mechanism of 0.5-power in cascade yield the 0.33-power law for the 5° target. (4) The characteristic of the mechanism of spatial summation including the optical system of the eye is expressed in the form of point spread function as Eq. (3). (5) 5 degrees of arc is much larger than the extent of the inhibitory interaction, i.e., the 5° target is considered to be a target of infinite radius with uniform luminance distribution. (6) Parameters KE and KI in Eq. (3) are independent of the target luminance. We express hereafter the point spread function in normalized form as

2 R(r) =exp ( -2%-a exp -cr I(4 2

(

(4)

and a is taken as a normalized amplitude parameter of the model. 0 < aE < a, and 0 < a < 1 should be assumed for the on-center and off-surround requirements.

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. BME-25, NO. 1, JANUARY 1978

78 optical system

nervous system

B

zI1

=K =

Cc

.c_-

K

SR(r)r dr d0}

{2r(uE2 aui2)}0 5 S 0- 5. -

(8)

From Eqs. (5), (6), (7) and (8), we have

C

K(1 target receptor

-

a)0-5

=

C1

(9) (10)

mechanism of mechanism of spatial summation 0.5-power

KVAHT(kE - aa])°5507 = C2 point spread function R(r,S) It is evident from the assumptions that Eq. (9) obtained for the case of point target gives only a relationship between linear Fig. 2. Functional representation of the luminance dependent model of the visual system. coefficients. Thus, Eq. (10) is a fundamental equation which determines the luminance dependences of GE and ap. For of the analysis, we adopt normalized parameters convenience (7) Changes of oE and a1 depend on the luminance in such referred to the values at an arbitrarily chosen are which a a way that the brightness function for large target with uniluminance SO. of form luminance distribution has an exponent 0.33. A schematic diagram of the model mentioned above is shown S = S/So in Fig. 2. The point spread function, Eq. (4), is defined for r = rIaE(SO) the system from the target to the output of the mechanism of spatial summation. The parameters aE and a, depend on GE(S) = GE(S)/GE(So) the luminance Si at the corresponding point on the target. The luminance dependences of aE and a, can be derived from sI,(S)= aI(S)IaE(So) the above assumptions as will be shown below. (11) {aI(So)IaE(So)}2 = b. Suppose that luminance distributions in Stevens' experiment are given as Then, the point spread function becomes = S(r, 0) S6(r) for the point target, (12) R(r, S) = exp ( 2&E )2) - a exp ( 2a6(S)2) and layer

S(r, 0) = S for the large and uniform target, where 6(r) is a Dirac delta function and is defined as 6(r)=0 for r>0

and Eq. (10) is simplified as

E2- aJ])/(l - ab)=

3

and further restrictions

2E abj> 6aI2 > ° 1E0

f2 f 6(r)rdrd0 = 1. 0

(13)

(14) (15)

1 - ab > 0

0

Then, if the luminance is much higher than the threshold, the brightness functions for the respective target have forms B = C S0'5(5) and B=

C2SO33

are assumed. As it is obvious from Eq. (13), we should further assume an interrelation between -E and -I to reduce a degree of freedom. The following are mathematically simple interrelations to be considered. Type I aEIGI = constant (6)

where Cl and C2 are constants which can be obtained

experimentally.

The output of the model should predict these functions. The output of the model corresponding to the center of the point target becomes

Type II Type III Type IV

a, = constant

&E = constant aE *AI = constant

The above expressions are assumed only for simplicity of analysis. Then, we will choose a particular type by considering its qualitative correlation with experimental evidences. Fig. 2 B=K {J 56(r)R(r)r dr do} of Patel [51 suggests that Type II is plausible at photopic levels. seems that Fig. 3 of van Meeteren and Vos [71 also K(1 - a)0 5s055 (7) supportsIt Type II at mesopic levels. However, Smith reported where K is a proportional constant. The output for the center that the size of the center field does not change with luminance at low luminance levels, but the decrease in sensitivity of the large target with uniform luminance becomes

79

FURUKAWA AND HAGIWARA: MODEL OF THE VISUAL SYSTEM 1.0 ix

.8

wn

0 2

s~1 I

a3

-.2

(a)

0.0

4 radial

6

a, -a

8

distance

(c )

.0

a,

.2-i

.8 .6

2 -i1

A.

d

.2

-5distarnce

0

-b

4

6

8

°! 5(d)

( b)

0

Fig. 3. Luminance dependences of the point spread function ((a) and (c)), and corresponding line spread function ((b) and (d)); parameters are a = 0.01 and b = 50. (a) and (b) are for Type I, and (c) and (d) are for Type II. Curves 1, 2, 3, 4 and 5 correspond to normalized luminances of 0.01, 0.1, 1, 10 and 100 respectively.

is resulted from the increase in the amount of inhibitory interaction [17]. Kornfeld and Lawson estimated the luminance dependence of the receptive field size from the experimental results by various authors [18]. Table II of their report suggests that Type II is plausible at high luminance levels and Type I at mesopic levels. As for Type III and Type IV, the extent of the inhibitory surround increases with the increase of the luminance. No supporting evidences are found within the scope of our investigation. In any case, change of the receptive field for whole luminance levels cannot be simulated by a particular type. Our primary concerns are changes of the characteristics of the spatial brightness interaction at mesopic and photopic luminance levels. Thus, the luminance dependences of -E and I are obtained for Type I and Type II; Type

I

2

=

S 0.34

,6.2 b §-0-34 =

Type II

OE = ab +

(1 - ab)S 3

I2 = b §S>Smin a2 = AI2 = b S