Wiener analysis of grating contrast judgments

WIENER

ANALYSIS

OF GRATING

CONTRAST

JUDGMENTS

L. A. ABEL’ and R. F. QUICK JR Department of Electrical Engineering. Carnegie-Mellon PennsyIvan~a 15213. U.S.A. (Reccirrd 27

June 1977; in

recised form 3

University. Pittsburgh.

iliOcKmhKr

1977)

Abstrct-Wiener’s

method of nonlinear system identification is modified to alfow it to be used for the first time to analyze data obtained from systems where the stimulus is a spatial waveform and the response is a single number. such as in contrast-matching experiments. Data from models and experiments were analyzed in the spatial-frequency. rather than the spatial domain. First and second order kernels obtained from experimental data agreed with those of a model containing parallel frequency-selective channels. Each channel consisted of a linear filter foilowed by a nonlinear transducer function: the channel outputs were linearly summed. The first order kernels also agreed well with a sine-wave sensitivity function obtained by conventional methods.

tNTRODUCIlON The

problem tem analyzes

of the manner

terms of the iinear kernel. hi(s), and higher order non-

in which the visual sys-

the scene before it has, in recent years. received considerable att~tion. Early descriptions of the mechanism responsible for this analysis viewed it as a single channel structure made up of a series of cascaded linear and nonlinear filters (Schade. 1956; Davidson, 1968: Campbetl. Carpenter and Levinson. 1969). A more recent hypothesis is that there exist parallel channels selectively sensitive to relatively narrow ranges of spatial frequency (Campbell and Robson. 1968; Sachs, Nachmias and Robson, 1971; Graham and Nachmias, 1971; Quick and Reichert, 19753. Much of the research carried out to date in this area has used threshold stimuli. Some work, however, has been done on the perceived contrast of suprathreshold stimuli (Davidson, 1968; Blakemore. Muncy and Ridiey, 1973; Quick, Hamerly and Reichert. 1976). The response to stimuli of these higher contrasts can also be explained in terms of a multiple channel m~hanism. The stimuli used in the experiments mentioned above were gratings. These one-dimensional patterns are easily generated and displayed. They are usually made up of only one or two sinusoidai components to simplify the interpre~tion of experimental results. A stimulus containing additional frequency components would more closely resemble the complex scenes normally encountered. Fortunately, there exist techniques in system theory that can be used to interpret the responses to such inputs. For example, the impulse response, h(s), of a linear system can be obtained by driving the system under study with white noise and cross-correlating the input and output (Papoutis 1965). This provides a complete characterization of a linear system. This concept was extended by Wiener (1958) to nonlinear systems. These can be represented by a series of integrals in 1Present address: Neurology Service (127). Veterans Administration Hospital, University Drive C. Pittsburgh, Pennsylvania 15240. U.S.A.

linear kernels hz (T,, T& etc. Here hi(z) corresponds to the impulse response; the other kernels represent nonlinear elements of the system. Wiener presented a way to derive these kernels using a Brownian motion function as an input to the system. (This function mathematically describes the random motion of a particte in a fluid.) His technique, while theoretically rigorous, proved to be extremely dif%icult to implement. MarmareIis (1972) has shown that to represent even a simple squaring circuit by this approach would require the calculation of 20zo coefficients. A more practical approach was presented by Lee and Schetzen (1961, 1965). They used Gaussian white noise as a stimulus and obtained the kernels through cross-correlations. This approach has been used by Stark (1969), Marmarelis (1972). Marmareiis and McCann (1973) and MarmareIis and Naka (1973) to investigate various aspects of the visual system. The advantage of white noise (or, in practice, noise whose bandwidth exceeds that of the system) as a stimulus is that it contains all frequency components of interest. Hence, the system response can be examined over its full bandwidth much more efficiently using white noise than if sine-wave stimuli were sequentially presented. The white noise approach also permits study of interactions between frequency components that would be missed if only one frequency at a time were present in the input. A method analogous to that of Lee and Schetzen has been developed for use in studying contrast perception, and will be outlined later. EXPERlMEhTAL

METHODS

Stimuli were presented using the apparatus described by Quick et al. (1976). A Tektronix S103N oscitioscopc (P31 phosphor) was used to display the stimuli. Two vertical gratings, each 5” horizontal by 1.5” vertical, were presented simultaneously. The upper pattern was the stimulus g&ting. Below it was the comparison grating, a 6.25 c/deg sinewave. They were separated by a 1” high band of frosted tape with a fixation spot in the center. Both the test and comparison gratings were generated on a Nova I200 computer. which was also used in the display of the stimuli.

1031

Table

type

Subject

RFQ Within

Between

XSSiOflS

sessions

sessions

sessions

(“,,I

I”,,)

I”,,)

?“I

L.A.4 Within

Periodic: Sinusoidal Nonsinusoidal

7.1

6.7

15.1

10.3

s.1

6.9

9.5

Random: 5.0-7.0 cpd 11.0-13.0 cpd

9.8 14.6

7.2

1I.4

11.0

15.5

8.9

8.9 7.6 9.7

8.5

7.7

IO.0

0.2-12.4cpd All gratings

7.2

10.9

t0 itniljOlC.lI gratings oi oin dj t3 Inorhzr < ? 2 de2 irl:!i:g l,>~c:._ _ _, _.. ,& The results of this sxpsrlment sugges::2 :h:,t :h< r:y,< A.>> ing judgment could bc performed r~~eatab!y essential if the U’ien:r analysis eXpCn?Xntj uerc :\\ i:.ii: ;1 chance to \ieid mearungfu~ results. That rhrb d:d. ;n !,i<[. H.liZ

Between

Subject

Grating

1

X.5

produce such results 1s Illustrated belo% b) the ,iml!arrt> found between sxpcrlmentally deri\:d 6e:nrls 2nd rhosr obtained from anaI!sis of rheoretical models oi the zontrast judgment process. MaJor variations in the ~udgmrnr process itself would probablq habe eliminated an) simple relationship between our data and such models. THEORETICAL

8.4

Each grating waveform was output through an eight-bit D A converter: the resulting analog voltages then went to the display apparatus. which produced the actual oscilloscope patterns. The peak physical contrast of the rest gratings was 20:;. A typicaf grating stimulus is shown in Fig. I. Three subjects participated in these experiments. Subjects LA and FQ were aware of the purpose of the experiments; subject ML was not. All had normal or corrected visual acuity. Subjects viewed the stimuli with natural pupils. A chin rest was provided to aid in the elimination of head movement. The technique used in these experiments was contrast matching. This method has previously been used by several investigators (Davidson, 196X: Blakemore et at, 1973: Quick rr nl., 1976). In the present study the subjects were instructed to adjust a potentiometer that controlled the physical contrast of the comparison grating until its “overall contrast” matched that of the test grating. More specific instructions were avoided, as it was desired to avoid biasing the subjects’ approach to the task. A possible consequence of the vagueness of our instructions is that different subjects may adopt radically different criteria for the setting which constitutes a match. Worse. a single subject may change his criteria within an experimental session or between sessions. If subjects were unable to perform the task repeatably both within and between experimental sessions, the method would be of little use. A series of experiments was carried out to examine this issue. Subjects FQ and LA, both experienced psychophysical observers. participated. A 6.25c;deg sine-wave was matched to various test gratings. These included sinewaves of 6.25, 8 and 12 c/deg: narrow-band noise gratings of 2 c’deg bandwidth and with center frequencies of 6 and I2 c/deg; broad-band noise gratings of 0.2-t2.4 c/deg and periodic gratings containing ten components, ranging from 4 to 40 cideg in 4 c/deg steps. Five trials were made for every test grating in each of five sessions. The results are summarized in Table 1. The within- and between-sessions standard deviations are shown as percentages of the mean for the various classes of gratings. It can be seen that there is no consistent pattern to the errors for the various categories. The betweensessions errors are about the same (although the differences are often statistically significant) as those occurring within sessions. It is also interesting to note that the aperiodic gratings were matched with nearly as much accuracy as the periodic ones, even though the subjects reported that matching the former patterns seemed considerably more difficult. This agrees with Blakemore et al. (1973). who reported that subjects were able to match an 8.3 c/deg sine-

zThis is a discrete form of the Volterra integral series (Volterra, 19.59).

METHODS

The analysis technique described here is not restricted to contrast matchmg. We desire a procedure for idenrlf:lng the kernels describing any system in u hich the inpu: is a spatial waLeform and the output a smgle point. The procedures devrfoped by Wiener (19%) and Lee and Schetzen are not suited for use with data of this sort. In the work of Lee and Schrtzen the Wiensr kernels of a system were obtained by cross correlating-bb means \lf time averaging -the white noise input and the corresponding output. In the contrast matching experiments presented here. however. the stimulus is a spatial waveform of fixed spatial extent: the response is a single number. corresponding to the perceived contrast. Clearly these cannot be averaged in the same fashion as two time-varying waveforms. In this section we will outline our method of analysis of this type of system. It is given in greater detail in Abel (1976). The gratings used in these and other experiments are. of necessity. bandlimited. As a result. the spatial waleforms can be completely represented by a discrete sequence of sample values whtch are the values of the waveform taken at uniformly spaced locations. If both the mput and output are sampled in this manner, the system response can be written

as:

Here. rlr is the response to the !&’ random grating presented. and rk (i) the ilh sampled luminance value of the k’” input record: ho. k, and Jr2 are the zeroth. first and second order kernels of the series. Here. h, is the linear term corresponding to the impulse response of a linear system. The term hl(i, j) and corresponding higher level terms represent the contribution of nonlinear interactions of increasing order. A detailed discussion of how these kernels can be interpreted may be found in Marmarelis and McCann (1973) and Marmarelis and saka (1973) for the continuous case. The preceding discrete equation can also be viewed as a multidimensional Taylor series approximation of the system being represented. Since the input to the system being discussed here is a spatial waveform and the output is a single point. timeaveraging is not possible. However. since the random process used to generate the stimulus records was assumed to be ergodic (meaning that its statistical properties may be estimated either by averaging down the ensemble of member Functions generated by this process or by averaging along one such function), it becomes possible to replace the time averaging of a single input and output with ensemble averaging over a family of stimuli. It is possible. using this approach. ro derive expressions for the kernels that are analogous to those given for the continuous case by Lee and Schetzen. The equations for the zeroth. first and second order kernels are: h&G = $$

2

*_

r,e,@)

I Za)

IO.:3

Fig. 1. Typical ten-component,

2-20 c/deg stimulus grating.

1035

Wiener analysis of grating contrast judgments

Here. r,(p) indicates the p’” sample value of the Ph stimulus waveform. of which there are K. The term rt is the k’” contrast judgment value. corresponding to the k’” stimulus. M1 and M, are the second and fourth moments of the input process et. In our work ek was uniformly distributed over the interval (- 1. I): in this case MI and M, are 0.333 and 0.200. respectively. The kernels are computed by forming products such as rter(. ) and averaging over the K waveforms of the ensemble for each p and Q. This method allows us 10 analyze systems in which the input is a waveform of fixed extent and the output is a single number. We can. with this method. compute kernels which provide information about both the linear and nonlinear characteristics of the system responsible for the contrast judgment process. This can be done in a number of wa)s. One approach is to compute kernels using the random luminance profiles; in this case they are specified as functions of spatial position relative to the fixation point. This corresponds to the method used by Marmarelis and his co-workers. with time being replaced by spatial posirion. It is not necessary. however, to perform the kernel computation using the stimulus waveform itself. Some transformation of the stimuli can be made before the kernels are generated. as long as the auto-correlation properties of the transformed stimuli are those of quasi-white noise. For example. one can use the magnitude of the Fourier transform of the spatial waveform thus obtaining kernels expressed as functions of spatial frequency. Such alternative approaches may help lo provide further insight into the nature of the mechanism responsible for the judgment of conrrast. RESULTS

As has been discussed. one can eenerate kernels using not the experimental stimulus;tself. but some transformation of it instead. such as the magnitude spectrum. If we start with the premise that the visual system is organized into parallel channels selectively sensitive to relatively narrow ranges of spatial frequency. a stimulus composed of several widely separated frequency components would be well suited for studying it. The luminance profiles. e(x). would be replaced in the kernel computation by their corresponding magnitude spectra. m(f). where f is spatial frequency. This technique was used in several model and experimental studies. The stimuli were generated by using ten pairs of uniformly distributed random numbers as the magnitude and phase of ten spatial frequency components covering the range 2-20 c/deg in 2 c/deg increments. These stimuli can be viewed as generalizations of the two component gratings used by Graham and Nachmias (1971). Quick er al. (1976) and many others. This method of analysis allows us to examine the response to more complex stimuli. rather than being forced to study the interactions of only two frequency components at a time. After the random magnitude and phase spectra were generated, the inverse fast Fourier transform was used to generate luminance profiles from them. The magnitude spectra of the gratings were stored for later use in the analysis. Both the models and the experiments used the same inputs and were studied with the same analysis program. In these investigations. the frequency domain kernel h,(f) now corresponds to the modula:ion transfer function (MTF). while h2(j,,f2) indicates the nature of interactions between the different frequency components of the stimuli. The model study will be described first. 1. Models

The results reported here were all obtained by means of calculations performed in the spatial-frequency domain. Those models and experiments studied directly in the spatial domain yielded either negative or inconclusive findings. These are presented in detail. along with their implications. in Abel (1976).

The basic organization of the models examined in this part of the study is shown in Fig. 2. The magnitude spectra of the stimulus records served as the input. The models were organized inro ten parallel channels, with each channel operating on only one input frequency component. This restriction. which

inverse transducer function )I/*

MTF coefficients

transducer functions

Fig. 2. Block diagram. frequency domain model. Input: magnitude spectrum. incorporated in weighting factors at channel inputs. “A.

I.3 8-l

> matching contrast

Frequency

sensitivity

L. .A. ABEL and R. F. OLICX JR

1036

process. In the matching experiments. the tina! resuit was the physical contrast of the 6.X c. deg grating that the subject felt corresponded to the perceived contrast. It was necessary to pass this physical contrast through the inverse of the 6.25 cideg transducer function. This stage could be omitted from models of other experimental procedures that yield a result which directly corresponds to perceived contrast without involving any such inverse transformation. such as magnitude estimation (Hamerly. 1976) or recording of occipital potentials (Campbell and Maffei, 1970). In neither of these paradigms is the result expressed in terms of a comparison grating. The model results presented here were obtained from a model of the sort previously described. The transducer functions were square-law transformations between physical and perceived contrast. The MTF of the initial spatial filter was an average of the first order kernels obtained from the experiments with human subjects described in the next section. The first and second order kernels obtained from analysis of the model responses are shown in Fig. 3. It can be seen that h,(f) closely resembles the MTF of the initial spatial filter. Two features of interest in h,(f,. f:)

ensured channel independence. was equivalent to having a narrow bandwidth filter at the input to each channel. At each channel output there was a nonlinear transducer function. The outputs of the channels were then linearly summed This transducer function organization was first suggested by Nachmias and Sansbury (1974) and Stromeyer and Klein (1975). and was extended to suprathreshold contrasts by Hamerly (1976). It was suggested that the transducer functions for the various channels were of similar shape, being strongly nonlinear for low contrasts and approaching linearity for constrasts further away from threshold. The variation of contrast sensitivity with frequency was included by shifting the transducer functions along the physical contrast axis so that higher frequencies, for example, required greater contrast for a given response level. The well-established logarithmic compression at the input to the visual system was not included in the models, inasmuch as Hamerly, Quick and Reichert (1977) found it not to be a factor in contrast judgment at the relatively low contrasts used in both their study and ours. The models contain a final stage that must be included in order to simulate the contrast matching

25

-b-

26-2

2 26

1 23

Fig. 3. Results of frequency domain model with quadratic transducer functions. (a) Normalized first order kernel h, (f). Vertical bars in this and succeeding figures arc &l standard error. Maximum value of h,. 0.2461. (b) Normalized second order kernel hl (ft. 1;). Maximum value of h2, 0.1237. Average diagonal term standard error, 19.78; average off-diagonal standard error, 1.38.

Wiener analysis of grating contrast judgments

0

0 h,.subicct

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I

2

4

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I

6

8

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I

10

12

14

1037

LA

OQ I 16

*

20

r.c daq Fig. 4. Normalized

first order kernels from 2 g’deg experiments. shown with MTF obtained for normal subjects replotted from curve given by Fiorentini and Maffei. 1976.

125

I

100 t 75.-

= z-

25-

0

-25-

f.='deg

-SOL

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._

z

3

=

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100-10

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85

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-2 1 -20

3

-14

3

18

10

-5

5.5

0

-lo

-9

40

13

-1

33

9

14 65

Fig. 5. Results of 2 c/deg increment frequency domain experiment, subject LA. (a) h, (f); maximum f2); maximum of ha, 0.2371. Average diagonal standard error, 27.81; average off-diagonal standard error, 5.01.

h,, 0.4140; (b) h, (f,,

1038

L. 4. ABELand R. F. QI,ICK

1

2

4

6

8

10

12

JR

14

16

18

20 1. C deg

-2+

L

-50

-b-

21 41 6 I8

IlOll2ll4lltlll8t2Q

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-7

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100

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-3

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-8

1

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-1

6 8 10

80-10-S 56

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2

3

7

5

1

5

0

$:

12

=14

44

2 44-4

16

50

18 20 i

-4

8

36

4 47

Fig. 6. Results of 2 c/dcg experiment. subject ML. (a) h, (I); maximum of h,. 0.5946. (b) h2 (/,. fJ: maximum of hl, 0.3189. Average diagonal standard error, 50.76; average offdiagonal standard error. 8.86. are the resemblance of the terms on the diagonal, h2(J f) to the first order kernel. and the appreciable

negativity of the first several low frequency offdiagonal terms. It is interesting to note that these negative terms are present in the absence of any inhibition between adjacent frequency channels which might have been expected to be the source of such terms. When the inverse transducer function at the model output is omitted the negative off-diagonal terms vanish. When inter-channel inhibition is added to the model the off-diagonal terms simply become more negative. If the transducer function exponent is changed, the first and second order kernels are both affected. If it is increased, h,(f) and hz(J f) both fall off more sharply from their peaks. The negative off-diagonal terms also increase in magnitude. If the exponent is decreased, the opposite changes occur. If the exponent is set to 1 the model becomes linear. and only the linear ta &(f), is nonzero. 2. Experiments Several experiments were performed using the same ten-component stimuli that were used in the mod& just discussed. Figure 4 shows the first or&r kernels from two experiments, along with an MTF for normal subjects replotted from Fiorentini and Maffei (1976).

It can be seen that the first order kernels are in excellent agreement with the contrast sensitivity function derived from sine-wave thrushold measurements. The kernels obtained from the data of subject LA are shown in Fig. 5, and those for subject ML are seen in Fig. 6. Unlike LA, the latter subject had not previously participated in psychophysical experiments. DISCUSSlOFi The application of frquency domain white noise analysis to contrast matching data yielded a number of interesting results, as opposed to our earlier efforts at analysis directly in the spatial domain. The first order, kernels were in clear agreement with MTF’s obtained through other mahod+ The second order kernels closely res&bit tWse obtaWd from a model consisting of paraW oanow-band channeis containing nonlinear transdua% functions, with the channel outputs b&g linearly summed. The question of interchannel inhibition is left open by t&se experiments, since the negative terms of the second order kernel tit miagt appear to it&&ate ini&iitioa oan aIso be expI&ed by tht ps#ez%e of an hirarr transducer function as the last stage in the contrast matching process. The model that yielded results closest to

Wiener analysis of gratirsg contrast judgments

those produced by the experimental data was one containing such an inverse transformation at the output and containing transducer functions with an exponent of approximately 2. As a further check that the similarity of model and experimental kernels was not due to some artifact of the analytical method the terms of the multi-dimensional Taylor series representations of the models were also calculated. These had the same form as their corresponding kernels. thus providing an independent check on our results. The frequency domain analysis is a rather specialized adaptation of the general Wiener method of system identification. What has. in effect, been done is to make certain assumptions about the organization of the contrast judgment process and to use these in generating a more nearly optimal stimulus. Here, the assumption was made that the judgement mechanism utilized the magnitude spectrum of the input stimulus. By including this step in the generation of the stimuli, the analysis was left with a lesser degree of nonlinearity to be accounted for in the remainder of the system. In contrast with this approach, the more general adaptation of Wiener analysis. made in the spatial domain. yielded almost no information. This illustrates a basic difficulty with this approach to system identification: while the Wiener method is potentially extremely powerful capable of analyzing a wide variety of nonlinear systems. practical limitations often prevent its actual application to such analyses. Frequently. more data are required to obtain convergence of sufficient terms to characterize the system than can practically be generated. In the present case genearlity was sacrificed in favor of convergence. By placing these limitations on the analysis we were able to compute first and second order kernels consistent with the hypothesis that the suprathreshold contrast judgement process can be represented in terms of parallel, independent spatial frequency channels, operating on the magnitude spectrum of the stimulus and possessing linear input filters followed by nonlinear transducer functions. and whose outputs are linearly summed. These findings, while the result of a totally different approach, agree with those of Hamerly ef al. (1976). Another, more general, result of this investigation is the finding that Wiener analysis can, in fact, be adapted to psychophysical problems. The modified form of the analysis used in the work presented here could also be adapted for use in studying other systems where a single response point is elicited by a complex stimulus. This adaptation, coupled with more efficient means of stimulus generation and data collection. may serve to extend the range of processes that can he examined through this potentially powerful technique. Acknowle&emenr--This

work was supported

by NSF

pant ENG-75-10241. REFERENCES Abel L. A. (1976) An application of Wiener analysis to human visual psychophysical response. Ph.D. thesis. Carnegie-Mellon University.

1039

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