Brightness Distribution Across the Mach Bands Measured with FlickerPhotometry, and the Linearity of Sensory Nervous Interaction*
GEORG VON BPKESY
Laboratory of Sensory Sciences, 1993 East-West Road, University of Hawaii, Honolulu, Hawaii 96822
(Received 15 May 1967)
Since a comparison stimulus produces inhibition, it is difficult to measure lateral inhibition. This is trueespecially for measurements of the brightness of the white and dark Mach bands. Starting with the ob-servation that no Mach bands are formed on the edge of two surfaces of different colors if the bright-ness of the two colors are equal, a marker was then developed which allowed the subject to locate a pointinside the Mach bands under observation. The flicker method was used to match the brightness of theMach bands with the brightness of the marker, without the marker introducing any changes in the bright-ness distribution of the bands. When the brightness of the white band was compared with that of the darkband, it was concluded that the lateral inhibition that produces the Mach bands probably takes place afterthe receptor organs have transduced the stimuli into electrical activity in the nervous tissue. A transfor-mation is, in general, associated with a compression of the range of the transmitted signal magnitude. Thiswould explain why such different sense organs as those for vision and skin vibrations, show very similarlateral-inhibition patterns, since the inhibition occurs mainly in the more central portions of the nervoussystem, which seem to work more or less linearly.INDEX HEADING: Vision.
IT is well known that, for most of the sense or gans,the quality of the sensations does not change much
even with very large changes of the stimulus magnitude.Hue is only slightly dependent upon luminance andpitch of a tone varies little with sound pressure. Suchindependence from the magnitude could be bestachieved if, after compression of the stimulus range,further transmission and interaction in the nervous sys-tem is linear, so that space patterns produced by sum-mation and inhibition are similar for all magnitudes.
The question is: Do any experiments indicate suchlinearity of action of the sensory nervous system, atleast in a certain magnitude range? My opinion is thatthe Mach bands provide an opportunity to investigatethis question for lateral nervous interaction. This ques-tion was discussed in several papers.' 2
Mach found in his investigations' that the localbrightness sensation of an illuminated surface is not
* The research reported here was supported by grant GB-5768from the National Science Foundation, grant NB-06890-01 fromthe Institute of Neurological Diseases and Blindness, USPHS,and grant M-14 from the American Otological Society.
determined only by the local luminance, but is also afunction of the luminance of adjacent sections. If weilluminate a flat surface evenly on one side and then,toward the right side, increase the luminance sharplyto a higher constant value, as indicated in Fig. 1 (A), thenthe brightness (perception) distribution on the surfaceis quite different, as indicated in Fig. 1(B). Mach foundthat to a first approximation the brightness distribu-tion along the x axis, going from the left to the rightside of the surface, is a function of the luminance I andthe second derivative of the luminance distribution,
B=EFI, (d21/dx2)].
Depending on the sign of the second derivative, we havean overshoot or undershoot in the brightness, Fig. 1(B).
If the increase of the luminance along the x axis isconcave, we have a black Mach band; if it is convex,we have a white Mach band. This accentuates the visi-bility of discontinuities in the luminance distribution.
In an earlier paper,4 I showed that discontinuities in astimulus distribution are accentuated not only in visionbut in all the sense organs with a uniform-sensitivityarea in which the magnitude of a stimulus can be varied
FIG. 1. (A) Spatial distribution of luminance across the retina.(B) Brightness overshoot and undershoot, called Mach bands,for the case of linear transmission of the stimulus magnitude tothe section of the nervous pathway where the Mach bands areformed. (C) If during transmission through the end organ a com-pression of the stimulus magnitude occurs, and if lateral inter-action occurs after the end organ, the matching stimulus of theovershoot has to be increased.
from point to point. The Mach bands were demon-strated for shearing displacements and vibratory stimu-lation, for warm and cold sensations of the skin, and fortaste sensations on the tongue. But they can be bestinvestigated for vision, with which they were firstdiscovered.
For these different sense organs, the transducers arevery different but at the same time the Mtiach bands aresimilar even in small details. This brings up the ques-tion whether it is correct in describing the Mach bandsto use the formula in which the bands are attributed tothe second derivative of the stimulus distribution. Oris it more appropriate to compute the transduced distri-bution? That is, let us assume that the transducers firsttransform and compress the stimulus distribution; thenwe can conceive of the second derivative of the magni-tude of the resultant nervous activity at a level in thenervous system where the magnitude range is reducedand the lateral nervous interaction consists of a morelinear summation or subtraction (inhibition) of themagnitude of nervous activity.
Since the stimulus range in vision is so great and thecompression probably quite strong, vision seems to be agood sense organ in which to investigate the most prob-able place of origin of the Mach bands. Psychophysicalmethods are convenient to use in vision.
One way to investigate this question (see also Mari-mont5 ) is to measure the brightness increase inside thewhite Mach band and compare it with the brightness de-crease in the dark band. If we assume that the Machbands are consequences of linear summation and sub-
5 R. B. Marimont, J. Opt. Soc. Am. 53, 400 (1963).
Vol. 58
.....
traction before the compression then the decrease andthe increase measured in terms of the equivalentchanges of the stimulus magnitude should be the same.But if the Mach bands are produced after compression,the larger stimulus is more compressed by the sensorysystem than the smaller one; and, therefore, the equiva-lent stimulus change in the white band should be greaterthan that in the dark band.
This is illustrated in Fig. 1. Drawing (A) shows thestimulus distribution. Drawing (B) shows the corre-sponding distribution of the sensation magnitude underthe assumption that inhibition and sensation can beadded linearly as described earlier. In this case, theundershoot and the overshoot are equal in size. If theMach bands are produced in this way, before compres-sion in the nervous system, then the overshoot and theundershoot can be compensated psychophysically byequal local changes of stimulus magnitudes. However, ifthe Mach bands are produced after compression, thenthe stimulus magnitude needed to compensate the localovershoot has to be much the larger, see drawing (C). Inpreparing the drawing, I assumed that the compressionis logarithmic.
VARIABILITY OF THE MACH BANDS
In almost every sensory process, summation andinhibition are involved. This gives to the sensory proc-esses a certain instability. This is true for measurementsas simple as a threshold determination. The same appliesfor the visibility of Mach bands. The border line be-tween two homogeneous surfaces with different lumi-nances may be viewed in several different ways. If all ofthe possibilities are recognized as such, and if our re-sults are represented in different sets, the agreement be-tween the different observers becomes much better. Forexample, if we split a field into two halves with a sharpborder line, and illuminate the two halves to differentluminances, we can look at the border line in at leastfour different ways, as illustrated in Fig. 2. By per-ceiving the entire field as a whole, under a magnifyingglass, we may observe a luminance step as shown indrawing (A) with no Mach-band formation but still withan accentuation of the magnitude of the step. By con-centrating on the area that has the higher luminance, wecan see a white Mach band only, as indicated in drawing(B). By concentrating on the darker area, we can ob-serve the black Mach band, as in drawing (C). By fixat-ing exactly on the edge between the fields, we can ob-serve both light and dark Mach bands. In this condi-tion, as indicated by drawing (D), the width of the Machbands is smaller, by almost half, than the bands ob-served in (B) and (C).
We can give the two halves different colors, such aswhite and yellow; when the yellow is less bright, thedark Mach band appears much more saturated than theequally illuminated yellow surface. In the white area ofhigher luminance, we see an intense white band. When
BRIGHTNESS DISTRIBUTION IN MACH BANDS
Alooking at the
__ - whole f ield
B
focusing onwhite band
C
focusing onblack band
D
focusing on the edge
FIG. 2. Looking in four different manners at the edge betweentwo differently illuminated surfaces produces four differentbrightness patterns.
there is no brightness difference between the yellow andthe white areas, no Mach bands are seen. The sameholds for any color combination.
The inhibition visible in the Mach bands is often con-sidered a small phenomenon. But if we cut a wedge fromyellow gelatin film, as shown in Fig. 3(A), and place iton a grayish background, then we see under a magnify-ing glass a quite large, more-saturated yellow area atthe tip of the wedge. This area is represented black indrawing (A). If the angle of the wedge becomes larger,as in drawing (B), the oversaturated area is less pro-nounced. When we place a reflection gray wedge on ayellow background, drawing (C), the tip of the graywedge becomes brighter when the gray wedge is ofhigher luminance than the yellow background. Ob-servers report that in these experiments with the wedge,when two Mach bands come together, as at the tip ofthe wedge, it is not so much the brightness whichchanges but the widths of the Mach bands.
The uncertainty in the observations of the Machbands can be very much reduced by placing a fixationpoint at the proper place, right in the edge or a littleto the side, according to what is to be observed. Forthese observations, mostly a light box was used, onwhich thin colored gelatin films were placed. The magni-fying glass has the advantage that it somehow isolatesthe field of interest from the surrounding field of view.Nevertheless, as illustrated in Fig. 2, the brightness andeven the width of the Mach bands depends on the dis-tance of the fixation point from the edge of the stimulusstep. The focusing of the eye lens may also play a role.
MACH BANDS NEAR THE THRESHOLDOF VISION
In the light-adapted eye, the Mach bands seem tobe present over the whole range of vision. Of specialinterest is the question whether they are still present
near the threshold. Earlier investigations' indicate thatevery sharply localized stimulus seems to produce anarea where the sensations are transmitted, but this areais surrounded by an area of decreased sensitivity. There-fore, in the higher-stimulus ranges, sensation is al-ways coupled with inhibition. The fact that Mach bandscan appear which are darker than any section of anilluminated surface, suggests the presence of inhibition.
For near-threshold phenomena, there are two possi-bilities. (1) The stimulus may first produce the area ofsensation, and then as the stimulus magnitude in-creases, the sensation may develop the inhibitory area.Under these conditions, we would expect that with de-creasing stimulus magnitude, phenomena produced byinhibition would disappear first. This seems to be so forskin sensations, where, for vibration amplitudes and fordisplacements near threshold, summation occurs andthe sharpness of localization deteriorates. The samething seems to hold for hearing. When the sound pres-sure is decreased, there is a value below which there isno sensation of tone and yet sound still can be per-ceived. This so-called atonal interval can be quite large,depending on the frequency of the tone.
(2) Another possibility is that inhibition and stimu-lus are interlocked; it may be that inhibition is alwayspresent and that the sensation is more or less a break-through. In this case, we would expect the Mach bandsto still appear near threshold, with no change ofappearance.
To choose between the possibilities we can use equip-ment described earlier.' It consisted of a light box ontop of which an opening could be made. The shape ofthe opening was defined by the edges of four razor
::0E:A : :: t
gray
V i -=V ye'
A
gray
B
C
low
FIG. 3. On the edge of a gray and a yellow area, the overshootwill make the gray lighter and the yellow more saturated. Byletting the two edges run together in an angle, the areas of theover and undershoot can be increased.
6 G. von B&k6sy, J. Opt. Soc. Am. 50, 1^60 (19(0).
3January 1968
I .
GEORG VON B1E KESY
2.5 mm
6 M I % seen from25cm
33x millilambertsabove threshold
lOx
3.3x
thresholdlight adapted
FIG. 4. The Mach bands are the last to disappear near thethreshold of vision when the liminance of the light pattern isdecreased.
blades, the locations of which could be adjusted withmicrometer screws. The shape of the opening is shownin the top drawing of Fig. 4. It was trapezoidal; theupper edge was lowered until Mach bands were ob-served easily. To produce the Mach bands, the trape-zoidal opening was observed through a rotating prismwith the axis of rotation parallel to the base of thetrapezoid. By this procedure, the trapezoid light patternis spread vertically, thereby producing a band of lightwith luminance distribution corresponding to the shapeof the trapezoid.
By proper adjustment of the shape of the trapezoid,two white Mach bands can easily be observed. They areillustrated in the lower drawings of Fig. 4. By reducingthe luminance with a neutral gray wedge and observingmonocularly with the light-adapted eye (room light),the two white Mach bands were the last sensations todisappear when the threshold was approached. This wastrue for white light and other colors. This seems toindicate that, for vision, inhibition and sensation havethe same threshold, which seems to show that a trans-ducer can produce sensation and inhibition at the sametime, over the whole sensitivity range.
Of all the sense organs with a great number of ad-jacent receptor elements, the retina has the most ex-tensive lateral nervous interconnections, which may bethe reason for this extension of inhibition down tothe threshold.
APPARATUS FOR THE MEASUREMENTS OFTHE BRIGHTNESS DISTRIBUTION
ACROSS THE MACH BANDS
Much research has been done to measure the bright-ness distribution across the Mach bands. I would likeonly to mention the work done by Fiorentini and
Radici, 7 8 Ercoles,9 Menzel,' 0 Lowry and DePalma," andthe important clectro-physiological observations byIRatliff and Hartline." 2 A summary of all the achieve-ments in this field can be found in Ratliff.13
The Mach bands are very thin lines and it is diffi-cult to define which point of the band is under observa-tion. Even the smallest dot in the field produces a dis-turbance. Since we found that on the edge between twoareas of different but equally bright colors no Machbands are produced, we used, as a marker, a very thinreddish line in a greenish field of equal brightness. Thisreddish line was placed on the side parallel to the Machbands, as shown on the lower right corner of Fig. 5.With a micrometer screw the reddish line could bemoved across the Mach bands. The apparatus for pro-ducing the marker is shown on the left side of the figure.The light from a light source was split by a beam splitter(not shown in the figure) into two parts. One part wentthrough a greenish filter, L2, and the other through a lowsaturation reddish one, L3. By using a Kdhler-type pro-jection, both lights produced a homogeneous illumina-tion on the silvered surface of the prism P2. With a razorblade, a straight scratch was made in the silvered sur-face so that only a very thin line of the reddish light wentthrough the slit and was projected by the lens Li to thesilvered edge of prism P,. Greenish light surrounded theopening of the reddish slit. Its luminance was adjustedwith a neutral gray wedge (not shown in the figure) sothat the brightnesses of the red and the green lights wereequal. A micrometer screw permitted the movement ofthe slit across the Mach bands so that the position of themarker relative to the Mach bands could be measured.The task of the observer was to adjust the brightnessof the reddish marker to equal the local brightness of theMach bands. This was done by using the gray wedge (2)on the left side of Fig. 5. The prism P1 serves to com-bine the marker with the light forming the Mach bands,as seen in the pattern viewed by the observer (on thelower right side of Fig. 5).
Different methods may be used to match the bright-ness of two areas. We tried to use the flicker principle.In the observer's view, the Mach band and the markerreplaced each other alternately nine times per second.The luminance of the marker was adjusted so that noflicker could be observed between the Mach band andthe red slit. We assumed that in this situation thebrightness of the red slit and that of the Mach bandwith which it was exchanged were the same. By movingthe slit across the Mach band to every reading of the
I A. Fiorentini and T. Radici, Atti. Fond. Giorgio Ronchi 12,453 (1957).
8 A. Fiorentini and T. Radici, Atti. Fond. Giorgio Ronchi 13,145 (1958).
9 A. M. Ercoles, Atti. Fond. Giorgio Ronchi 12, 187 (1957).10 E. Menzel, Naturwiss. 46, 316 (1959).11 E. M. Lowry and J. J. DePalma, J. Opt. Soc. Am. 51, 740
(1961).12 F Ratliff and H. K. Hartline, J. Gen. Physiol. 42, 1241 (1959).'3 F. Ratliff, Mach Bands (Holden-Day, Inc., San Francisco,
1965).
Vol. 58
yBRIGHTNESS DISTRIBUTION IN MACH BANDS
-silvered surface
ray wedge
micrometer WuiuiiiUiiC blt.screw
FIG. 5. Optical arrangement to produce a marker which can bemoved across the Mach bands.
micrometer, a corresponding reading on the wedge wasobtained.
The flicker method has the advantage that the bright-ness of the Mach band which is observed is well de-fined without any effort on the part of the observer.In the beginning it was disturbing to see the wholegreenish section flicker. This disturbance can be re-duced by limiting the field of view to a minimum.
The schematic drawing of Fig. 6 illustrates the equip-ment. On the left side is a photographic slide with sec-
tions of different densities with sharp border lines.The corresponding luminance steps are visible to theeye by reflection in the upper, silvered section of thesurface of a prism. The transparent part of the prismtransmits the image of the marker, shown in Fig. 5, tothe eye. The view that the eye sees is shown in the lower-right corner of Fig. 5. The marker and the luminancesteps are side by side. By using an oscillating prismwhich rotates about an axis parallel to the edge of thesilvered surface of the prism, the entire pattern oscil-lates vertically and the marker or the steps alternatelytake over the whole field. Only the small strip which isequal in brightness for both of them will not flicker.
Actually, the Mach bands were not produced byphotographic transparencies, since the edges did notseem to be well-enough defined. The step of luminancewas produced with an opaque white wedge (whitenedevenly with chalk), which was illuminated from bothsides by the same light source. The chalk wedge was
carefully ground on polished glass and had a very sharpedge. By using neutral-gray filters (see Fig. 7), a knowndifference of the luminances of the two sides of thewedge could be obtained.
The wedge in Fig. 7 was aligned very carefully sothat its edge was perpendicular to the axis of oscilla-tion of the prism. Otherwise the sharpness of the edgewould be diminished. While using a magnifier, the wedgewas rotated until there was no difference of sharpnesswith or without oscillation of the prism. In the sameway, prism P2 in Fig. 5 was rotated until the oscillationsof the prism did not enlarge the width of the reddishslit under microscopic observations.
According to Fig. 5, the Mach bands appeared hori-zontal in the field of view. This is not necessarily thebest orientation. Many observers have astigmatismthat can be corrected by a cylinder lens. The observerswere asked to look monocularly at a set of differentlytilted lines. One of the lines appeared to be sharpest;the Mach bands were rotated to be parallel to that line.This could be done easily with a Dove prism above theoscillating prism shown in Fig. 6. Since several observerspreferred to look straight ahead into the apparatus, onemore total-reflecting prism was used between the oscil-lating prism and the eye in Fig. 6.
Besides the alignment of the reddish slit and theMach bands it was also important to bring them tofocus in the same plane. This was accomplished withlens L1 in Fig. 5. The combination of the two imagesappears in the prism P1. If both images are in the sameplane, then a movement of the head to the left andright does not produce relative lateral displacement be-tween the two images. This is a convenient method forfocusing. Lens L1 also permits adjustment of the ap-parent width of the reddish slit. In general, it was con-venient to make the width of the reddish slit about1/10 to 1/30-th of the width of the Mach bands.
It was necessary to use a head rest and to adjustit for every observer, individually. A photocell was used
A eye
osc iIllating-prism
silveredsurfaceof prism
wedge
FIG. 6. Oscillating square prism used to exchange the markerperiodically with the Mach-band-forming luminance step.
5January 1968
gray
1...n^^nc __
GEORG VON BE3KtSY
beam
chal
[-L1J U Vlight sourcefor the step
side view ofwedge
FIG. 7. A chalk wedge illuminated fromused to produce a well-defined step withnances.
to control the luminance in the differentequipment, and to make sure that the 1iand the marker were exposed to the eyetime intervals.
A final test for the whole equipmentthe magnification of the entire pattern atthe Mach bands are formed in the eyeequipment, there should be no change ofance. If there was one, the edge of the X
sharp enough.
METHOD OF OBSERVAT1F
The flicker method has the advantagemits the monocular measurement of theness distribution across the Mach bandsducing an additional inhibiting field, sin,ance of the Mach bands does not seeUnfortunately, to achieve this, a complic-is necessary.
First, the slit and the surrounding areahad to be adjusted to have exactly equaThe difference of color was kept as smrbut still large enough so as to producedefined fixation line for the eye. Under thithere are no lateral effects if we exchanfthe luminance steps with a marker of eqtthe whole pattern is seen, therefore, as oinance step with the usual Mach band.experiments proved that the intensitiesbands were not decreased by the flickerAt the flicker frequency employed, somereported that the Mach bands becameOver-all, with the pattern oscillating, the cwas of a simple luminance step, with themarker parallel to the edge separatingnances. There was no change in the brigh
tion of the luminance steps when the marker wasintroduced.
k wedge The oscillation of the rotating prism in Fig. 5 was7f carefully adjusted to be the same to the right as to
the left. The luminance steps are, therefore, presentedto the eye for exactly the same time interval as themarker. Because of the equal one-off time, the magni-tude of the stimulus step is reduced to half, independent
ixiliary light of the stimulus magnitude of the marker, see Fig. 8.The one-half reduction holds even when the marker isturned off. The luminance step during flicker is, there-fore, only half of the actual luminance step producedon the chalk wedge in Fig. 7. We, therefore, can change
to the eye the luminance of the marker without changing themagnitude of the stepF 2-F 1 producing theMach bands.
However, there is a simultaneous change of the lumi-nance of both steps. Our method of observation was to
bdjth asides lul adjust first the luminance of the marker so that therewas no flicker of the reddish slit at the brightest pointof the Mach band.
sections of the If we now decrease the luminance of the marker touminance step adjust it to the brightness of the two steps and the
during equal dark Mach band, as can be seen from Fig. 8, F1 and F 2
will drop their luminance simultaneously withoutwas to change changing the difference between F1 and F2.the eye. Since To avoid this decrease of the luminance of the whole
tnd not in the step during flicker, we can add, simultaneously to thetheir appear- luminance Si and S2 of the wedge, the same auxiliary
vedge was not luminance S on both sides of the wedge and adjust itsvalue so that it is equivalent to the decrease of theluminance of the marker M. By doing so, the whole
ON visible step remains unchanged during flicker, except thebrightness of the superimposed reddish slit. By chang-
local brig ht- ing the luminance of the marker and compensatinglocal simultaneously the loss of luminance by an increase of
without intro- the luminance of S1 and S2, the flicker of the slit could bece the appear- eliminated at different points across the Mach blands,m to change. without changing the appearance of the luminance stepted apparatus during flicker.
The manner by which the auxiliary light was pro-Iof tbhe marker jected from above by a separate light on both sides of1brightnessles. the chalk wedge can be seen in the lower drawing inta as posslue,
FiC. 8. The magnitude of the luminance step seen on the chalkwedge changes during flicker.
6 Vol. 58
BRIGHTNESS DISTRIBUTION IN MACH BANDS
Fig. 7. The amount of auxiliary light is determined bythe luminance of the marker. The auxiliary light is ad-justed by maintaining the over-all light, from the entirepattern, at a constant value by the use of a photocell.The luminance of the auxiliary light which is necessaryto keep the photocell output constant was recorded forthe different values of the marker. This was corrobo-rated in our observations by visual comparison of theflickering step with a standard.
RELATIVE BRIGHTNESS CHANGE PRODUCEDBY THE WHITE AND THE DARK
MACH BANDS
In the following observations, no attempt was madeto investigate the many aspects of the Mach bands.The only aim was to observe the relative brightnessesof the white and the dark Mach bands with a precisionnecessary to decide the locus of the lateral interaction.For a more complete investigation, the luminance rangeof the equipment should be increased and its handlingsimplified. We investigated only photopic vision ata medium luminance where the Mach bands are easyto observe.
In Fig. 8, S2 was adjusted to a luminance of 20 mL,and SI to 5 mL. The actual luminance step duringflicker was 7.5 mL. The flicker frequency was 9 persecond.
To match the maximum of the white band, the auxili-ary light was shut off; it was then, in general, enoughto make the luminance of the marker M 20% higherthan the luminance of S2. For this reason, the luminanceF2 during the flicker became about 10% higher than 52.With the help of the auxiliary light it was kept constantat this level during all of these measurements. The con-stant level of F1 was 14.5 mL.
The distance of the virtual image of the luminancestep seen through the magnifier was adjusted to 25 cmfrom the eye. A mm scale was placed at the same dis-tance from the observer and on it the movements of the
25
20-
dis
-2 -I aedge of the
step
FIG. 9. Brightness change across the
FIG. 10. Apparatus to superimpose on the luminance step amarker with different color, which could be moved acrossthe step.
marker across the step could be seen and compared withthe readings on the micrometer screw in Fig. 5, a cali-bration which had to be made only once for eachobserver.
A representative curve of the brightness distributionacross the Mach bands for two observers is shown inFig. 9.They represent the meanvalue of 10 observations.The precision of each single observation is about ±t 10%.However, as shown in Fig. 2, some observations arecompletely off the curve; these were left out, since it isquite usual that while one Mach band is being observedthe other one is suppressed.
These observations indicate that each Mach bandis about 1.5-mm wide when seen from 25 cm. Further-more, the figure shows that the overshoot and theundershoot measured in a linear scale of the luminance(the stimulus) are different. The fact that the overshootis so much greater indicated that lateral inhibition ofthe Mach-band type occurs in human vision at mediumsensation magnitudes to a large degree after the trans-ducer elements have compressed the stimulus.
METHOD OF THE DIFFERENCE LIMEN
According to Weber's law, the just noticeable in-crease of stimulus magnitude divided by the stimulusmagnitude (AI/I) is quite constant in the medium rangeof vision. It is, therefore, reasonable to assume that wecan measure the brightness distribution across a Machband by superimposing a thin beam of light on the
;placement of the luminance step and increasing its luminance so that themarker superimposed slit can just be noticed (see also Fioren-
'1 2 mm tini el al.t4 This is illustrated on the left side of Fig. 10.If the color of the beam is a little different from the
Mach bands. 14 A. Fiorentini ce al., Atti. Fond. Giorgio Ronchi 10, 371 (1955).
10' ,-
7January 1968
A
GEORG VON B9K1t:SY
10-
A/1/
/ I o/ I
-4 -3 -2 -I 0 +1 +2 +3mm from the edge of the step
.4
7-
5-
3 -
I _
ovzD
O -O
o -UO 00 )0
.- -a-0) a2-c2
0'P
0E o
millilamberts
2 5 '1o 20 50 100FIG. 11. In the region of overshoot AI is high as expected. In
the region of the dark Mach band a superimposed light must bemade quite strong to be noticed, since inhibition also reduces theeffect of the superimposed light.
color of the step, the location of this marker can beeasily detected. The drawing shows how, with the helpof a half-transparent mirror, the marker was superim-posed on the luminance step. The marker's luminancewas adjusted by a gray transparent wedge. Moving themarker across the edge of the step made it necessaryto change the luminance of the marker, as shown inFig. 11. The ordinate represents the luminance of themarker in millilamberts. The luminance step was 1:4.The values obtained are quite different from the differ-ence limenwhich is usuallyobtained for brightness, sincethey depend on the magnitude of the marker, its color,and especially on its width.
A more interesting difference is that the curve showsa very well-developed overshoot, as expected, but noundershoot at all, in spite of the fact that the darkMach band is quite visible. That is, we might expectthat at low brightness M would also be low.
This finding is not new. It was already shown forvibratory sensation" that, at the location of the over-shoot, a summation of all the stimuli occurs, so thatany increase of the stimulus magnitude is easily rea-lized. On the other hand, near the location of the under-shoot, an inhibitory effect occurs which very effectivelysuppresses sensations. As a consequence of this, Weber'slaw does not hold during the crossing of the inhibitedarea, and the luminance of the marker has to be in-creased to make the marker visible. Figure 11 shows thatat the location of the black Mach band, a superimposedluminance greater than the higher luminance level of thestep is almost needed. The difference-limen method,therefore, does not measure the brightness of the Machbands.
However, since it determines the width and the mag-nitude of the overshoot, in agreement with the flickermethod, it can be used to investigate the overshoot.
15 G. von 1e'k6sy, Psychol. Rev. 66, 1 (1959).
FIG. 12. An increase of the over-all luminance of the stepwill produce a much slower increase of the magnitude ofthe overshoot.
Since, with this method the white Mach band is notdisturbed, we used it to determine whether the Machband is produced before or after the compression. If itis produced before compression, the overshoot shouldincrease linearly with the stimulus magnitude.
We produced, therefore, a 1:4 luminance step whosebrighter surface had a luminance shown by the ab-scissa of Fig. 12 and we observed the luminance whichjust made the marker visible.
The luminance increment necessary for visibility ofthe marker at the brightest part of the overshoot wastaken as a measure of the magnitude of the overshootand is shown by the ordinate of Fig. 12. The magnitudeof the overshoot increases much more slowly than theluminance of the step. This indicates that the summat-ing process, which is assumed to be linear, takes placeafter the compression of the stimulus magnitude. Theslope of the curve does not change with the color of thelight, even for deep red, but it can change with differ-ent observers.
CONCLUSIONS
The assumption was made that, as a first approxima-tion, the lateral inhibitions and summations add to-gether in a linear way. This assumption seemed to besuccessful in describing the different Mach bands invision, skin sensations, and taste. The Mach bandsrepresent relatively small sensation-magnitude changes,but they occur only where lateral spread and magnitudeare well defined.
The question was asked, at what level of the sensorynervous network does the lateral interaction occur whichproduces the Mach bands? Of main interest is whetherthe lateral interaction occurs in the receptor cells them-selves or later along the nervous track. The results ob-tained indicate that the receptor cells first compress thelarge stimulus range to a much smaller activity range.Lateral interaction seems to occur in this smaller range.
