UNIVERSITY of LONDON Imperial College of Science and Technology

UNIVERSITY OF LONDON

Imperial College of Science and Technology

Applied Optics Section

THE VISUAL MACH EFFECT

IN MICROSCOPY

Bohdan Maciej WATRASIEWICZ

Thesis submitted for the Ph.D. degree

1964 2. ABSTRACT

An eye presented with a transition between regions of different luminance levels may perceive bands or fringes parallel to the edge which are purely subjective, i.e. they are not present in the light flux distribution as measured by, say, a photoelectric cell. This phenomenon is called the Mach effect and the bands are called Mach bands. In this thesis we give a historical review of work on the Mach effect in the unaided eye and related topics in the study of the eye itself. Almost no work has been reported on the Mach effect in vision through instruments up to the present time but, as we show, there are fundamental differences from the effects seen with the unaided eye; in particular some remarkable effects are found with the microscope. We describe measurements of the positions and intensities of the Mach bands with the following varying parameters: degree of coherence, direction of polarization, wavelength and retinal illumination level. Also the effect of the object contrast is investigated. Finally, theoretical treatments of the Mach effect are reviewed and theoretical predictions are compared with the experimental measurements. 2a.

PREFACE

As this thesis incorporates some measurements made by other observers, I should like to state explicitly which items were originated and carried out by myself. They are:

1. A method for the preparation of the edge. 2. The visual photometer for intensity measurements on the Mach bands.(resign and construction.). 3. The improvement in stability of the scanning apparatus to enable scans extending over 20 minutes to be made. 4. Physical measurements. a) of _edge profiles at high numerical apeture. b) of edge profiles in the polarised light. c)discovery of hitherto unreported effect of "turned up" fringes due to defocusing in the interferograms of the mi:.Ircscope objective on Twyman—Green interferometer. 5. Subjective measurements of the Mach effect, a)in partially coherent light (at various degrees of coherence). b) at different levels of retinal illumination. 2b.

c)variation of the Mach bands with wavelength. d) the effect of the object contrast on the bmnds. e) the effect of polarised light on the widths and positions of the Mach bands. 6. Realisation of the r6le of diffraction inflections in the Mach effect. 7. the investigation of theoretical models of the Mach effect by numerical computation. 3. CONTENTS Page No. ABSTRACT 2 2REFACE 2a CHAPTER 1: INTRODUCTION 1.1. Introduction 7 1.2. The retina 7 1.3. The Mach effect 14 1.4. Discussion of current theories 33 1.5. The Mach effect in microscopy 35 1.6. Notes on terminology 37 CHAPTER 2: APPARATUS AND TEST OBJECT 2.1. General introduction 41 2.2. The -nhotoelectric microscope 42 2.3. Additions to the apparatus 53 2.4. The photometric arrangement for visual matching 56 2.5. Test object preparation 60 2.6. Preparation of the neutral density wedge ' 63 2.7. Choice of the microscope objective 66 CHAPTER 3: PHYSICAL IMAGES OF A STRAIGHT EDGE AND OF CIRCULAR APERTURES 3.1. Introduction 73 3.2. Method of scanning 74 3.3. Luminance distribution along the line perpendicular to the edge 75 3.4. Images of a straight edge in polarized light 80 3.5. Diffraction images of apertures in opaque films 85 3.6. Conclusions 86 CHAPTER 4: THE DEPENDENCE OF THE MACH EFFECT ON LEVEL OF RETINAL ILLUMINATION AND ON COHERENCE 4.1. Introduction 90 4. Page No.

4.2. Method of measurement 91 4.3. Measurements at different S—values 95 4.4. Mach bands as a function of retinal illumination 107 4.5. Calculation and measurement of retinal illumination 108 4.6. Conclusions 117

CHAPTER 5: THE EFFECT OF WAVELENGTH ON THE MACH BANDS 5.1. Introduction 119 5.2. Method and results 119 5.3. Conclusions 121

CHAPTER 6: THE EFFECT OF EYEPIECE POWER, OBJECT CONTRAST AND POLARIZATION ON THE APPEARANCE OF THE MACH BANDS

6.1. Introduction 134 6.2. The effect of the eyepiece power 135 6.3. The effect of the objeot contrast on the appearance of Mach bands 138 6.4. The effect of the plane of polarization of the illuminating beam on the Mach bands 146 5 .

Page No.

CHAPTER 7: THEORETICAL TREATMENTS OF THE MACH EFFECT

PARTS 7.1. Introduction 148 7.2. Mach's equation 149 7.3. Fry's approach 150 7.4. Treatments of Hartline et al. and Taylor 153 7.5. Approximate conditions for positive roots of equation (4) 158 7.6. Integral equations corresponding to the previous models 162 7.7. Comparison of various mathematical models 163 1. Introduction 163 2. Comparison of descrete models 164 3. Comparison of continuous models 165 4. Two—dimensional models 167 7.8. The solution of the alternate form of the simplified Hartline et al.equation 174 7.9. Solution of Taylor's equation 176

PART II: NUMERICAL CALCULATIONS 7.10. The numerical solution of tl,.o model of L.rtline et al. 181 7.11. The numerical solution of Taylor's equation 184 6 Page No .

7.12. Application of frequency response methods tc visual. problems 189 Appendix I Appendix II 137

CHAPTER 8: DISCUSSION 199

REFERENCES

ACKNOWLEDGMENT S 2 2 7. CHAPTER 1. INTRODUCTION 1.1. Introduction. The Mach effect may be briefly described as a subjective enhancement of contrast in optical images which takes the following form: an edge between light and dark regions, in which the chageover of luminance may be sudden or gradual, appears to the eye to be sharper than it is and it may appear to have light and dark bands parallel to the direc- 2 tion of the edge. This effect, first noticed by Ernest Mach a century ago but it is only recently that physicists and psychologists have begun to study it in detail. Here we are concerned specifically with the Mach effect in relation to images seen in optical instruments, a field which so far does not appear to have been investigated at all. The Mach effect is certainly connected closely with the mechanism of vision and in this introductary chapter we survey briefly the rele- vant parts of the body of known anatomical and physiological knowledge.

1.2. The retina. The optical system of the eye,comprising the cornea, the crystalline lens, et.c., produces a real image of the external world on the light-sensitive screen at the back of the eye which is known as the retina; it is in the trans- 8. mission of signals from this image on the retina through the optic nerve to the brain and their interpretation there that some part of the explanation of the Mach effect may lie. A transverse section through the retina, fig.1.1.1 Shows a ten-layer structure consisting of five different types of cell ; they are: 1. the. receptors. Theae: cells constitute the outer layer of the retina i.e. the layer furthest from the lens et.c. They possess light sensitive portion, the so called rods or cones, which project through an external limiting membrane towards pigment epithelium (layer 1 fig.l.l. ). 2.Bipolar cells.These illustrate. the general structure of a nerve cell or "neuron"; it consists of a cell body with extended nerve fibres or "processes" of which one kind, the dendrites, receives impulses: from other neurons: and the other kind, axons transmit impulses to other neurons; the connections are called "synapses" and they may involve two or more neurons. 1,he bipolar cells (layer 6 in the figure) relay the signals from the receptors to the ganglion cells, the next stage, by synaptic couplings of a wide variety and complexity. They are called bipolar because they have two main processes,' one a dendrite and the other an axon. 3.Ganglion cells. These cells are multipolar and are responsible for relay of signals to the brain; they are 9 01•IfTiZA:k tteif:6MIA

an • ...a . • on. •.. e

Fig. I. 1 The Layers of the Retina6°

I. Pigment Epithelium 6. Inner Nuclear Layer 2. Rod and Cone Layer 7. Inner Plexiform Layer 3. External Limiting Membrane 8. Ganglion Cell Layer 4. Outer Nuclear Layer 9. Nerve Fibre Layer 5. Outer Plexiform Layer 10. Internal Limiting Membrane 10. similar to cells found elsewhere in the nervous system. The short dendritic processes face outwards to the bipolars; their axons are prolonged forming a nerve fibre (layer nr.9) converging towards the optic nerve. 4.Association cells.Functionally these are thought to be responsible for intra—retinal interactions; they allow one part of the retina to modify the response in the adjacent part, or in the same part after a time interval ( spatial and temporal induction).For the most part they intermingle with the bipolars. There are three types: horizontal, amacrine cells and centrifugal bipolars. 5. -Dieuroglial cells. These cells are assumed to insulate and support the conduction cells i.e. reurons in general; in the retina they pass between the two limiting membranes and hold the retina together. The micro—anatomy of the retina, as briefly described above, is well established but there are still unsolved prob— lems, e.g. the details of the cellular interconnections are not well understood. The: physiological processes involved in vision are not understood in detail but briefly some process such as the following is thought to occur. A quantum of light falling on the retina may initiate a photochemical change in a rod or cone, since both these contain phototropic pigments. This change is amplified and transmitted as an 11. electrical signal to the ganglions, Where it appears as a series of electrical pulses. ( spikes); the mode of amplifica- tion and the form in which the signal is transmitted through the bipolar cells are not known. The signals from the ganglions are then transmitted by the optic nerve to the visual cortex; this is the part of the brain which deals with vision and practically nothing is known about its mode of action. The whole process of transmission of signals from the bipolars to the visual cortex can be described in terms of the neuron theory,a theory which has developed as a general view of the functioning of the whole nervous system. As explained above a neuron consists of a cell body with dendrites and axons and there are interconnections with other neurons at synapses; these are thought to be polarised, i.e.signals can only pass in one direction at a synapse, from axon to dendrite. The neuron theory maintains then that the entire nervous system is made of countless neurons of varied shapes and types concatenated in an orderly manner into many different patterns., but that each neuron has its proper place in the network as a whole, and can fulfil its functional role only as part of the whole system. The neurons can be combined to produce structural arrangments of different functional properties, e.g. simultaneous spatial summation, avalanche conduction and facilitation or inhibition between members of the 12. network. A combination of neurons by means of which impulses of the same or diffelent kinds can be added (summation) and the resulting excitation concentrated or intensified is shown diagrammatically in fig.l.2a. Fig.1.2.b. shows the arrangment for the avalanche conduction; the effect here is an increase in intensity of stimulation as each successive link in a system is made up of an increasing number of neurons. The third functional arrangement is shown in fig.l.2.c; the impulse from the neuron on the left travels along a dendrite as indicated to the other neuron and it can either facilitate or inhibit the activity of these neurons. The impulses travelling between neurons are electrical in nature (spikes) and the interaction is based on the frequency modulation principle. An external stimulus inter- acts with a given neuron by modifying the frequency (spikes per second) of discharge and in general for greater stimulus the frequency of discharge is greater. For the interaction between light and the retinal neurons the relation between intensity of illumination and frequency of discharge is logarithmic, i.e. the spikes frequency is proportional to the logarithm of the light intensity. This, of course, is true only if the light intensities lie within a certain range (several log units), if the visual field is not too small 13

eM,

a) summation

b) avalanche

*lank =Mb mon, Mob c) inhibition or facilitation

Fig.i.2. FUNCTIONAL ARRANGEMENTS OF NEURONS 14. and if adaptation and contrast effects are ignored.

1.3. The Mach effect. We now describe in more detail the Mach effect and survey the work which has been done on it up to the present time. When an observer views a more or less sharp change in luminance i.e. an edge between light and dark regions, as in fig.1.3 a and b( full lines) he perceives something different from this, as indicated by the broken lines in the figure. No suitable terminology seems to have been invented to deal with this situation; in the visual photometry there is a well understood distinction between the light flux issuing from an object, which may be specified as so many watts per unit area per steradian per unit wavelenth interval, and its luminance, which is strictly a psychological quantity to be measured in lumens per unit area per steradian. The first of these is a physical quantity, the second a psychological quantity; the first may be measured withhotoelectric cell suitably calibrated, but the second needs in principle a standard photometric observer. In the present case we are dealing not with uniformly lit fields of view but with varying shapes, and th.eselNi general be images seen through a microscope or otheroptical instrument. To make the corresponding distinction we propose to user the term "physical image" to 'denote Is

• distance (a) (b)

Figure 1.3j Diagrammatic representation-of the Mach phenomenon. Objective luminance distribution. Subjective luminance distribution. 16. the image structure as determined by scanning with a linear detector such as a photoelectric cell and "visual image" (or "subjective image") to denote the shapes and luminances perceived by an observer. These terms apply in the first instance to monochromatic light; it is conceivable that the second definition could be extended to polychromatic light by a weighting process, rather like that used in determining trichromatic. colour coordinates by spectrophotometry, but this is not yet possible. Thus the solid and broken lines in figure 1.3 a and b represent physical and visual images respectively. The horizontal scale of visual angle is intentionally left uncalib- rated in this figure because one of the more obscure points about this phenomenon is that the scale seems to vary over a wide range. The bands in the visual image are called Mach bands after 2 Ernest Mach who first observed them in 1865, on artificially produced luminance gradients.Mach rotated rapidly a drum on which he stuck cuttings of black paper shaped as shown in fig.1.4. Four cutting were used to enhance the persistence of vision of the pattern; they were wound round a cylinder such that side AB was parallel to the axis of rotation. Mach observed a bright band near the point C and a fine dark band near D. The luminance distribution during rotation is a function of the height of the black regions (Talbot's law) 17

A M AC-it PAPER CUTTINGS 18. and it therefore contains neither maxima nor minima, as shown in figure 1.3 (full line). From this evidence Mach concluded that the effect must be subjective. After his first discovery he made some more qualitative studies3 of the phenomenon and showed that the bright and dark bands occurred near the points of maximum and minimum rate of change of luminance gradient. These observations lead him to suggest that the subjective stimulus or visual image, could be represented by the following formula:

E = a • log [.BE +k-17 •(d. dx2] 2 (1) where B. is the retinal luminance distribution and a,b and k are constants.

It can be seen from equation (1) that the subjective stimulus E will be either a maximum or a minimum depending 2 on the sign of --d 2B at the point where an extreme value dx occurs. The bright band will occur when the positive sign is. 2 taken with the negative value of --d 2B ; the reverse applies dx to the appearence of the dark band. McDougall (1903)4 and Thouless5 (1922) also made qualitative studies of the effect using similar apparatus. McDougall suggested that the phenomenon could be attributed to an inhibitory effect exerted by more excited receptors 19. 6 on those which are less stimulated.Granit and Schousten and Orstein7 suggested that the mechanism of the inhibitory effect is electrical in character. Working on the above suggestion and on Hering's9 assumption that every visual element depresses the activity in surrounding elements in proportion to itscwnlevel of 8 activity and inversly with the distance between them. Fry deve— loped a mathematical theory (see section 7.2.). This. theory predicts two peaks (maximum and minimum) occurring near the extreme value of rate of change of gradient of luminance distri— bution. 10 In 1953 Ludvigh investigated the relationship between the retinal luminance distribution and the location of subjective contours, or lines of constant visual brightness. He computed the external luminance distribution necessary to give the desired retinal distribution under known experimental conditions. To achieve this it was necessary to take account of the aberrations of the eye.Ludvigh considered the effect of aberra— tions as a blurring and spreading of the image of a point source, including in this diffraction at the pupil. The relative effects of various aberrations were compared by Ludvigh by measuring the diameter of the blur disc. Main aberrations which influence the image for small fields of vision in the vicinity of fovea are listed in table 1.1. It should be noted that all these effects except diffraction will vary from individual to individual; thus if the spread due to the aberrations is made small compared 20. to the spread by diffraction, the individual differences may be neglected as a first order approximation. Table 1.1. shows Ludvigh's estimations of tha aberrations of the image of a point source with a wavelenght range from 5600 to 5850 R.

TABLE 1.1. Approximate diameter of blur disc expressed in seconds of arc subtended at the nodal point of the eye..' 1 mm. diameter pupil, 1. Chromatic aberration 13 2. Spherical aberration 30 3. Diffraction 135 4. Assumed uncorrected refractive error of 0.0625 D. 13 5. Fixation tremor 18 Total 209

In his discussion Ludvigh does not mention factors such as the structure of the crystalline lens and the blood vessels in the retina which scatter light and affect therefore the image formation. In spite of all the precautions taken by Ludvigh one cannot be sure of the exact light distribution at the rods and cones and therefore his predictions, enumerated later, are open to some doubt. He draw the following conclusions: 21.

(a) The loweNhErivative of luminance with respect to retinal distance which has significance for the percep— tion of contours is the second derivative. (b) Two contours will be perceived symmetrically distributed about the point where the second derivative has an extreme value. (c) The separation between the two contours will be smaller the greater the value of the fourth derivative of the luminance with respect to retinal distance. at this point. (d) For the formation of the contours the second derivative at its extreme value must have some absolute minimum value. (d) The attempt to explain Mach bands in terms of the classical concepts of the Weber Fechner fraction, LB/B, would result in a positive intensity gradient producing a negative luminance increment. To explain the subjective contour perception Ludvigh10 '11 assumed the existence of three different kinds of elements in the retina: (a) those. responsible for the absolute response to light, (b) those responsible for the discrimination of luminance, (c) those concerned with the perception of contours. The third set would consist of "on—off" elements stimulated via either ocular or head tremor so that the rate of change of stimulation would be maximal at the points of extreme values of second derivatives. McCollough,1 2 "19;5, used an artifice lly produced luminance gradient to investigate the variation of width and position of the bright Mach band as a function of retinal luminane'e. `She showed that the width of the bright band decreases with

incr'easing' luminance and tends to a constant value at higher luminance levels; also that the band position would change with luminance, "but the nature of this change is determined in large part by factors that vary with the individual." A year later Fiorentini13 investigated Mach bands in the penumbra of a diffusely illuminated opaque screen, and found that there is a minimum gradient of luminance at which bands disappear. The latter findings can be explained by Ludvigh's theory,1° but none of the thoolvies (9nscribe(9. below can explain the results of MeCollough's experiments. Mach bands can be observed after immediate stabilization of the retinal imago, 14but they very soon disappear; they are not visible during a short presentation of the field.13 On the other hand, slow and wide oscillations of the field, perpendicular to the line of sight, enhance the visibility 23. of Mach bands, while rapid motion impairs the visibility.14 In 1958 Green15 postulated the existence of two kinds of elements in the retina: a first set having a response proportional to local stimulation E and a second set whose response is proportional to E, the average cf E over a certain region of retina. The response of the second kind of elements would inhibit the action of the first set, so

ONO that the resulting response is proportional to E — E. A Quantity proportional to E — E was called by Green "contour response" and he assumed that a contour is perceived where the first derivative of 2 — E with respect to distance has a sharp variation. Green proved for the one—dimensional case that with a small averaging region the difference E E can be sufficiently approximated by the sum of two terms proportional to the second and fourth derivatives respectively. The significance of these two derivatives in the formation of contours was previously demonstrated by Ludvigh11 and from Green's proof it follows that they are significant because they determine the value of .2 — E. 15 It was also pointed out by Green that the ratio E E instead of .2 E is better for the description of contour formation when regions of widely differing E are viewed. 16 Ercoles and Fiorentin used the latter expression to interpret their investigation oOlVisibility of Mach bands 24. as a function of field luminance. They reached the following conclusions: (a)At low luminance levels the mechanism responsible for the perception of the subjective contours becomes less sensitive and a higher value of (E 2)/E is required to perceive it. (b)At levels where both bands, dark and bright, are visible, the ratio required for the visibility of the dark band is greater (about four times) than that for the bright band, i.e. the threshold of the contour response for the dark is higher than the threshold for the bright band. (c)The dark band is perceived at levels lower than the minimum level at which the bright band is visible, therefore different mechanisms are responsible for the dark and bright bands. (d)Results at different contrast levels cannot be explained unless, at least at low levels, the perception of contours does not depend only on the ratio (E 2)/E.

In 1956 Hartline, Wagner and Ratcliff,17 using the eye of the Limulus crab, demonstrated the existence of mutual inhibition between two neighbouring receptors (ommatidia). This was done by inserting microelectrodes into two neigh— bouring receptors, A and B, and measuring the frequency of discharge when they were illuminated separately and together. 25.

The frequency of discharge of A was decreased when B was illuminated and it stayed depressed until the light was switched off again. Exactly the same effect was observed with receptor B when A was illuminated. When both receptors were illuminated simultaneously, the frequency of each was lower than the corresponding output when each was illumi— nated: separately. Such inhibition occurs between all near neighbours, but its magnitude decreases with distances between receptors. When a third group of receptors X is illuminated in the vicinity of the interacting pair A and B, such that X can onlyinlbit directly B but not A, then the frequency of discharge of A will increase due to partial release from inhibition of B, whose activity was in turn inhibited by X. This effect is called disinhibition.18 Hartline and Ratliff19 in 1958 establEhed a general law of summation of inhibitory influences .of receptors in the eye of Limulus. When the two groups of receptors are. illuminated together, the totpl inihibition exerted by them on a "test receptor" near them depends on the combined inhibitory influence of the two groups. If the two groups are widely separated in the eye, their total inhibitory effect on the test receptor equals the sum of the individual inhibitory effects. However, if the two groups are close 26. enough to interact, their combined effect is usually less than the sum of their separate influences, since each group inhibits the activity of the other and hence reduces its inhibitory influence. Also the test receptor, or a small group illuminated with it, may interact with the two groups and affect the net inhibitory action. 19 Hartline and Ratliff, after studying different con— figurations of receptors, derived an empirical relation for the frequency output of the receptors. For the two receptors, such as shown in fig. 1.5, illuminated simul— taneously this relationship can be written as follows:

fA = CA KAB(fB f1B) for receptor A

for receptor B fB = eB %.k(fA "-RAJ

The response of the receptor is measured by its frequency discharge rate in impulses per second. The response of each receptor is determined by the excitation e, due to an external stimulus, diminished by the inhibitory influence exerted by the other member of the pair. The threshold frequency that must be exceeded before a receptor can exert any inhibition is given by 1'0, thus foAB is the frequency of receptor B at which it begins to inhibit receptor A; fak is the.reverse. K is the inhibitory coefficient, KAB SCHEMATIC REPRESENTATION OF RECURRENT SYSTEM 28. is then the coefficient of. the inhibitory action of receptor 3 on receptor A. In general K decreases and fo increases with the increasing separation between the receptors. When the inhibitory term K(f f°) is greater than e, the corresponding response f must be set equal to zero because negative frequencies have no meaning. Further— more if (f f°) is negative the inhibitory term must be dropped. In such a network the inhibitory influence exerted by each receptor on the other is determined by its level of activity, which in turn is the resultant of the excitatory stimulus to that receptor and the iriibitory influence exerted back upon it by the other. This is called a "recurrent" system. The generalized equation for n receptors in the network is given below: n f. = e. — K. .(f -.e ) i = 1, 2, OW n I 1 j.,_:-1 1 i j j/i th where fi is the frequency output of the i receptor, ei is the response of the ith receptor to the external stimulus (i.e. illumination), Kij is an inhibition constant between 1th and lreceptor:th , and f?j is a frequency threshold which receptor j must exceed to inhibit receptor i, 29. All solutions of the above equation as before must be positive, as negative frequencies have no meaning, and therefore it has to fulfil similar restrictions, namely

0=1 10 0 3-J

If f? f. then f.:=e.. j 0'

A year later Ratliff and Hartline2° by recording the output from receptors demonstrated the existance of contrast bands, in the eye of Limulus crab which were similar to Mach bands inhuman vision. These agreed with the predictions of the above empirical formula. For further discussion of the formula see section 7.4. and review articles ref. 25 and 26. 21 Alpern and David used the method of binocular brightness matching,, to study the effect of inducing targets on the test targets. The arrangement of targets is shown in fig.l.6. Rectangle a is a reference target of constant brightness viewed by one eye, b a test target, c c' and d d' two pairs of inducing targets all viewed b the other eye. At first the effect of each inducing target on the test target is found by separate matching experiments, then the combined effect of all four is measured. Alpern and David21 found that the effect of four targets together was less than the sum of the effects of the individual components taken

alONM ••••• OP= IMMO 41=1•110 .111•1.11 1/11/MA

d b C

Jim

Fig.I.6. ARRANGMENT OF REFERENCE(a) TEST(b) INDUCING (Cc-crd)TARGETS USED BY ALPERN AND DAVID 31. separately.They interpreted these findings as an indication of mutual. inhibition between neighbouring stimuli. This effect is similar to that reported by Hartline and Ratliff in the Limulus eye. Alpern and David also found that calculations based on a model similar to that of Hartline and Ratliff fitted very well with their experimental data. 22 Makavey, Bentley and Casella also used a method of bi— nocular photometry to study the effect of two inducing patches near the test target on its relative brightness. They provided evidence for the existence of disinhibition in the human retina,, but pointed out that they have no evidence for denying the existence of an alternative mechanism further along the visual pathways which can be responsible for the disinhibition effect. However, most of the experimental results; previously reported can be explained on the basis af 4 retinal mechanism. Based on above assumption and on his experimental data of skin inhibition B6k6sy23 proposed a neural unit to explain inhibition effect in the retina. According to it every sti— mulus produces around the sensation area an area of inhi— bition, called the refractory area. A schematic drawing of the so—called neural unit is shown in fig. 1.7,94 fig.l.7b. shows an idealized unit which can be used for computations. B4kesy also determined the actual dimensions of the c7. a) schcmatic

\ sensation \ d1 CMG

%refractory CIFC01 b)simplifizd

c)eye looking from a distance of 25cm.

SiR=I-G .

P12

0 .5 1.0mm Fig.1.7. NEURAL MT 33. approximate neural unit for the human eye; this is shown in fig.l.7c. in the object space at a distance of 25 cm. Such a neural unit, when used to evaluate subjective sensation, produces two Mach bands near the discontinuities of a steep variation of gradient of the presented stimulus.

1.4. Discussion of current theories. Although both theoretical treatments:, the one due to Ludvigh and the other to Green, predict the importance of the second derivative in contour formation there is a very important difference between them. Green's approach is based on the assumption of lateral inhibition in the retina, while Ludvigh postulates the existaace of three different kinds of receptors which, as yet, have not been isolated histolo- gically. The assumption that lateral inhibition in the retina is responsible for the perception of contours is supported by the fact that contours are not seen under conditions which are unsuitable for the inhibitory processes to take place, 'namely short observation times and low illumination levels13. This holds for both contours at sharp edges and at smooth edges (Fiorentini13'14).FUrthermore the work of 21 22 Alpern and David and of Makavey, Bentley and Casella show the existence of lateral inhibition. and 34. disinhibition intuman retina; therefore it may be assumed that disinhibition occurs in the retina,so that Ludvigh's hypothesis of three types of receptors is at least very incomplete and possibly false. In discussing visual perception of fine detail it is convenient to draw a somewhat arbitrary distinction between differential brightness sensitivity (DBS) and contour percep— tion (CP); bath of these come into play in normal vision but under artificial experimental conditions we can emphasise one or the other. Thus DES is used even in viewing brief flashes24, but CP is slow because of the time delay in establishing lateral inhibition. With stabilized retinal images CP ceases after a short period, presumably due to fatigue of the inhibition mechanism,and even large objects tend to disappear, suggesting that DBS is impaired by failure of CP. Mutual inhibitory interaction in the retina not only serves to enhance spatial contrast in the retinal image, but also appears to play a role in generating complex response to temporal change in illumination26. Those produced by movement of the eye, or by movement of the object in the visual field, are of special importance. In all investigations of the Mach effect described above the workers did not take proper account of the fact 35. that all images. on the retina are diffraction images, i.e. possess a certain structure due to diffraction of light at the edges. of the limiting aperture stop, which may be. in the optical system forming the image which is being studied or which may be the iris of the eye, whichever is in effect smaller. As, an example, an image of a straight edge in inco— herent illumination formed by any optical system may be con— sidered;' such an image possesses small inflections in the luminance distribution and, although these are so weak as to be. almost undetectable by ordinary photoelectric scanning techniquea„ yet it will be shown that they are amplified and distorted by the Mach effect into apparent diffraction fringes. Thus. there are two (or perhaps more) dissimilar factors concerned with Mach bands, the lateral inhibition in the retina and the effect of diffraction structure in the image. This thesis includes an experimental study of some aspects of the latter, a topic not previously touched on by other workers.

1.5. The Mach Effect in Microscopy. One of the most important applications of a microscope is to the determination of sizes of small particles. An extensive study of the correlation between physical and visual images of apertures and, discs, under various illumi— 36. nating conditions, was carried out by Chalman and reported in his thesis27. He concluded that no simple relationship exists between the physical luminance distribution in the primary image of apertures and discs. and the subjective luminance distriibution (what we have called visual image) en by an observer using an eyepiece. However, consistent measurements of the visual diameters of non-periodic objects can be directly related to the intensity profile under pre- vailing instrumental and visual conditions. The visual size depends on: (a)Adaptation of the eye. (b)Primary and secondary magnifications. (c)Retinal illimination. In visual sizing, as in physical measurements of diffraca- tion images of non-periodic objects, the most important parameter is the position of the edge image with respect to the geometrical edge. Its position in the visual case is primarily determined by the Mach effect. This effect in turn depen s on the conditions enumerated above, as was shown by Watrasiewicz28 and Charman and Watrasiewicz29. In later chapters of this thesis we describe experiments carried out to investigate the dependence of the Mach effect in microscopy on wavelength, polarisation, contrast and on parameters mentioned before. We note that in microscopy we 37. may expect that the form of the diffraction image will have a particularly strong influence on Mach effect, since the pupil is very small at high magnifications and this makes the elementary diffraction patterns very large in scale on the retina.

1.6. Seine notes on terminology. As. explained in sec. 1.3., there is no terminology in existence to differentiate satisfactorily between an objective luminance distribution or object and the "shape" or collection of contours of apparently constant brightness which the eye and brain perceives. For this purpose we spoke of "physical" and"visual" images. It was also convinient to extend this usage in discussing the determination of "sizes" of microscopic objects. The physical image of a sha -edged object is; not, of course, sharp-edged, because of aberration; and diffraction, so that the physical size must be first de-. fined. Referring to fig.1.8, the light intensity across the image of an opaque object (a) along the broken line would be as at (b) if the microscope worked according to geometrical optics and there were no aberrations, the dimension across the broken line is then unambiguousely determined and we have the "geometrical size". In fact on account of diffraction, and possibly also 38.

+ light I intensity

light intensity

t visual sensation 1

FIG.1.8. 39. aberrations,the image is blurred as at (c); note that objects a few resolution limits or less in size will not even have zero light intensity at the centres of their. images. We can thenebfine the "Physical Size" as the width of graph (c) at the 50% intensity level; this: may be taken as the level at which the intensity is 50% of that of the illuminated background or, perhaps more reasonably, the level which is the arithmetical mean of the background and the lowest level in the image. On account of the Mach effect the eye sees something different again, as at (d). The"Visual Size" then depends on how a setting is made; if a cuLss-line in a micrometer eyepiece is set on an image of this kind it is not yet known for certain just where on the visual brightness scale it will be set; in practice most observers are fairly consistent with themselves in making such settings. t e In discussin Mach erect of images of this kind it is, clear that the basic elementary object to be considered is an edge between light aid dark regions.For this purpose the "Physical Edge PoaitionY is the point of 50% intensity in the physical image. It is known from the diffraction theory of image formation that the shape of the physical image of an edge depends on the mode 'of illumination(coherent, incoherent or partially coherent) and furthermore the physical 40. edge position as abovebfined does not generally coincide with the geometrical edge position. Finally the visual edge position is where an observer scats a crosswire or other pointer to divide the bright from the dark side of the edge.Measurments presented later in this thesis will show how the visual edge postion is related to the geometrical and physical edges in certain oases. 41. CHAPTER 2. APPARATUS AND TEST OBJECTS.

2.1.General introduction. When the same physical image is presented to different observers the physical light distribution, which may be measured, will in general evoke to a greater or smaller extent different subjective responses, i.e. the image will appear different to each observer. Furthermore this subjective response to the same physical luminance distribution, which for convenience has• been termed a subjective, image, more often than not will bear no simple relation to the physical light distribution. Both images could, however, be compared by measuring objective and subjective brightness in the image space from some fixed point and then superim- posing the two images. In this way it would be possible to obtain a quanti- tative assessment of the Mach effect and any other effects which cause differences: between the physical and visual images. In this chapter we discribe the apparatus developed and used to carry out such measurements. in the case where the' physical image is that seen in a microscope. The particular relevance of the microscope is that it is often used at magnification ugh enough to show clearly the diffraction 42. structure of an image, indeed this essential if full advantage is to be taken of its resolving power, so that diffraction effects on the optical image may be expected to be distorted and enhanced by the Mach effect.

2.2. The Photoelectric Microscope.

In the case of an optical microscope, an observer views a primary image of the object through an eyepiece. The physical distribution of light in the primary image can be determined by means of a photomultiplier, the subjective light distribution can be determined by means of photometric matching and the two c) then be compared. A relation between the two images is shown in figure 2.1. The apparatus used is shown schematically in figure 2.2.Pigure 2.3a shows. the general view of the assembled apparatus;figure 2.3b shows the microscope and the block unit. Essentially there are two independent but interrelated parts, a visual photometer and a photo— electric microscope. The later consists basically of a microscope illuminated by means of a monochromator. The 43

(visual angle). Figure 2. 1 . Multiple band structure of the Mach phenomenon observed in the microscope. Objective luminance distribution. Subjective luminance distribution. 44

Field of view edge wi the Mach bandst ah photo - multiplier comparison slit

eYC. piece

slit imaging microscope system

monochromator monochrom at o r unit uni t

Fig. 2.2 . Block diagram of the apparatus. . W Sc r i 4 .2.3o.Ge nera •- ga a 0 W 0 0. .0 C.. 0 0 N Fig.2.3b.The microscope and the block unit 47.

image of the straight edge, which will be described later, can either be focused on a 6p. pihole behind which is a photomultiplier or directed to a Ramsden type eyepiece through which the image is viewed. The light flux passing through the pinhole is measured by the photomultiplier in conjunction with a galvanometer. A displacement of an image achieved by scanning blocks, enables the measurement of light distribution in the image. For visual measurements field of view is divided into two parts. In the top half the image of the edge is seen, to the lower half a comparison slit is projected (fig. 2.2 insert).

As a source of light a tungsten ribbon lamp (6 V, 18 A) with colour temperature of 2850°K was used. The lamp was run from the mains through a voltage stabilizing unit (± 0.5%) and variac auto-transformer to control light output. The 100 cps ripple of the source interfered neither with visual measurements nor with galvanometer readings; the time constant of the latter was about 2 sec. Light from the lamp is focubed by a condenser lens (B),fig. 2.4, on the entrance slit of the Wadsworth type constant deviation

FIG. 2.4 OPTICAL LAYOUT OF APPARATUS

A Tungsten ribbon f ;lament lamp A

B Condenser B

C Monochromator entrance slit C

D Achromatic lens D

E Amidi Prism E

F Achromatic lens F

•G Monochromator exit slit G

H Achromatic condenser H I Object slide J Apochromatic objective J

K Scanning block L Removable mirror NI Micrometer eyepiece

N s;x micron pinhole

O Photomultiplier

PLAN ELEVATION 49. monochromator.30'31 Figures 2.4 and 2.5 give the optical layout of the system. As the • aperture of the microscope condenser must be always illuminated by an axial hea7 of light for all wavelengths, the monochromator (fig. 2.5) was constructed such that the light for all wavelengtilalways emerges parallel to the optical axis of the system, A wave— length between 4000 A and 6600 A may be selected by rotating a micrometer drum, which is calibrated by inserting successively mercury, sodium, and cadmium lamps in place of the tungsten lamp. The entrance and exit slits of the monochromator were adjusted to give an effective bandwidth of 100 Angstroms. The exit slit H is imaged by an achromatic condenser I in the object plane of the microscope objective. The microscope and all associated equipment were mounted on the same optical bench. The angular aperture of the illuminating beam is sufficiently large to illuminate evenly the back aperture of the condenser. The nuterical aperture (NA) of the latter can be varied from 0.2 to 1.3 by means of an iris diaphragm. The object (straight edge) can be imaged either in the plane of the pinhole 0 (fig. 2.4) or in the first focal plane of the eyepieceld. The image can be displaced by the means of scanning block. system L. This unit consists of two glass blocks (fig. 2.7) mounted in such a way that the SO

30° Fixed mirror A

--0 Collimator I/ Telescope

/ / / Rotating mirror {A13=t / BC=1 AD=x D -71C Centre of rotation

FIG2.5. DESIGN OF MONOCHROMATOR 1000-

o4 800- .0

600-

400- 1 mm Entrance slit

200-

Wavelength ( A)

4500 5000 5500 6000 6500 7000 FIG.2.6. VARIATION OF BANDWIDTH WITH WAVELENGTH FIG.2.7. THE SCANNING BLOCK UN I T

FIG.2.8. THE MIRROR UNIT 52. image can be displaced in two mutually rerpendiculaT directions. The rotation of eanh block is controlled, through mechanical levers, by two separate micrometers. One of them, which displaces the image parallel to the slit, is calibrated to give directly readings in microns. Two-stage micrometers were used for the calibration and the mean value taken. Since the two glass blocks work in the low aperture beam, tolerances on their manufacture are not severe and their aberrations are negligible. Behind the blocks and in front of the pinhole, a front aluminized mirror E (fig. 2.7) is placed in such a way that the light from the microscope objective is directed to the eyepiece in the mount H. The mirror can be lifted up and detained by a catch to allow the beam to be focused in the plane of the pinhole. The mirror unit F, when required, is lowered into a kinematic mount; the three steel balls engage in the thPee V—slots in a steel ring G (fig. 2.8) and ensure exact re—positioning. The relative positions of the pinhole and the eyepiece are arranged so that the:first focaJ plane of the latter and the plane of the pinhole are equidistant from the centre of the mirror. To adjust the tube length for different objectives the whole unit (fig. 2.8) can be moved along the optical bench. The detection system of the photoelectric microscope consists of an E.M.I. 13 -stage photomultiplier with 53. antimony—caesium cathode,type 6094, a Pye galvanometer and an EHT stabilized power supply for the photomultiplier.

2.3. Additions to the apparatus.

The apparatus as described above was used by Charman27'32 for scanning apertures and discs not greater than 2.79 p in diameter. As in the present work the distances to be scanned are much greater (about 6p ) and therefore the time required for a single scan is increased from a few minutes to at least 15 minutes, a number of changes were necessary to improve the stability of the apparatus and the accuracy of measurements. With the optical bench horizontal and microscope axis parallel to it, the image stayed in focus only for 3 — 4 minutes. The criterion for defocusing was the measure of transmission through the .814p aperture. If the trans— mission dropped by about 5%, the image was considered to be out of foaus. of Servicing/the microscope etc. made no improvement. When the whole apparatus was bolted to the wall so that it hung vertically the improvement was considerable but as yet not satifactory. It was also necessary to cool the tungsten lamp with a fan to prevent pockets of warm air reaching the microscope. Further a heat shield was placed between the observer and apparatus. This simply:-consisted of an 54. aluminium sheet covering the microscope. As a precaution it was necessary to avoid direct breathing on the microscope. Then afteran initial warming—up period of about 1-i hours the defocusing was less 1% after at least 15 minutes. Despite all the precautions the apparatus was very sensitive to small temperature changes; 0.1°C was enough to upset the focusing. Indeed a person entering the room would cause a temperature change of about 0.100, but for— tunately the apparatus responded to it with 20 min. time lag. The accuracy of focusing was improved by fitting a micrometer screw to operate a lever fixed to the fine focusing knob of the microscope; the condenser was also focused by means of a micrometer screw. When a clear field of view of about 8p, was scanned the transmission at the edges dropped about 10%. This was due to the lateral displacement of the cone of light subtended by the pinhole (610.) at the light—sensitive surface of the photomultiplier. This displacement occurred because the photomultiplier was a considerable distance behind the pinhole (about 15 cm). To remedy this a glass sphere of 6 mm in diameter (fig. 2.9) was used as a lens to image the magnified (x 40) pinhole on the photomultiplier cathode. During the visual observations stray light reduced the contrast. There were mainly four sources of stray light: 55

Photomultiplier.

Glass sphere 4mm.dia.

Pinhole six microns)

Fig. 2.9. 56. (a)Reflections from the surfaces ofreyepiece lenses. (b)Reflections from scanning blocks. (c)Scattering and diffraction effects in the objective itself.

Some improvement was obtained by coating all optical surfaces involved. But of course it was not possible to improve either the objective performance or diffraction scattering by the slit. When the exit slit was slightly increased diffraction occurred then at another limiting aperture.

2.4. The photometric arrangement for visual matching.

The photometric arrangement used for visual matching is shown in fig. 2.10. Light from the tungsten ribbon lamp S is focused onto the slit B, which is in the focalilane of the lens C. A parallel beam of light passes through the Amici dispersing prism D and is then focused by the lens E on the slit P. Lenses C and E are both achromatic. A diminished image of the slit F is formed on the neutral density wedge H by means of a low power (x 9) microscope objective. As the neutral density wedge is fairly short (3 cm) demagnification of the slit F ensures that the variation of density of the wedge does not create a luminance variation across the image of the slit. Finally 57. the lens I brings the real image of the slit F, through the mirror system J, K, Z and the photometer cube M, into the first focal plane of the eyepiece N. The diagonal of the photometer cube, half of which is aluminized to form a reflecting surface, divides the eyepiece field of view into two parts (insert fig. 2.10). In the bottom half the image of slit F appears; the top half contains the image of the edge formed by the micros— cope objective. The arrangement is shown in fig. 2.11. The neutral density wedge is moved by pulling two strings. The distance moved by the wedge, perpendicular to the optical axis, is calibrated to give transmission factor. Calibration was carried out by directing the beam into the pinhole of the photomultiplier and noting the galvanometer deflection at different positions of the wedge. The wavelength of the photometer beam can be changed by shifting bodily the source S9 condenser A and the slit B in the plane perpendicular to the optical axis of the system and the edge of the slit. The size of the slit B was chosen to give the bandwidth of 100 Angstroms. It was found that when the wavelength of the photo— electric microscope and the photometer were made equal by means of a spectrometer, the comparison slit appeared to be Fig.2.1o.Visual photometer. A58

Monochromator entrance slit

Achromatic lens

Amici prism

Achromatic tens

Monochromator exit slit

Achromatic objective G Neutral density wedge H

Projecting lens

Photometer cube M

N eyepiece field of view 59

Eye-piece focal, plane

from the microscope i•

reflecting surface A

projection lens

I neut ral density wedge

slit

monochromatic light Fig.2.11.0ptical system used to image the comparison slit in the focal plane of the eyepiece. 60. somewhat different in colour from the microscope field of view. To compensate for this subjective effect the wave— length of the photometer beam was altered until the sub- 0 jective colours matched. This shift was about 100 A towards the blue end of the spectrum. To minimize diffrac— tion effects at the edges of the slit, it was made fairly large.

2.5. Test Object Preparation. Most of the investigations were carried out on a straight, sharp edge; some measurements were also made on circular apertures and discs. Circular apertures were prepared by the method described by Slater.33 It is based on the shadowing of polystyrene spheres of known diameter with aluminium film. The Dow Chemical Co. of Michigan, in 1948, made a large batch of size—monodisperse polystyrene latex, which soon became a well known substandard for calibration of electron micros— copes. More recently the company has prepared latexes with spheres of different sizes; these are listed below. 61. Table 2.1.The diameters of the polystrene spheres used

in the test—object preparation.

Average diameter Standard Number of deviation readings 0 0 A A 880 80 1164 1380 62 526 1880 76 1065 2640 60 577 3400 52 415 3650 79 438 5120 74 359 5570 108 373 11720 133 315 18300 180 ) not 27900 1500 ) known

A microscope slide is first thoroughly cleaned with Teepol and alcohol, then a small amount of dilute solution of latex is smeared on the slide and allowed to dry. Spectroscopically pure aluminium is then evaporated at normal incidence on to the slide, at a pressure of about 5 x 10-5 mm Hg. To minimize penumbra effects, the slide is placed 40 cm from the evaporation source, which is about 5 mm wide. After the evaporation the spheres are removed from the slide by a jet of distilled water directed at a grazing incidence. Finally a cover slip is cemented with 62.

Canada Balsam aid baked for 24 hours to remove all air bubbles. The diameters of the apertures thus produced in an otherwise opaque film are equal to the diameters of the original spheres to within 1.5% (Slater, ref. 33). The 0 0 thickness of the aluminium film is between 600 A and 800 A. It was measured on a Zeiss—Linnik interference microscope. Opaque discs were produced by a three—stage evaporation process described by Charman.34 Polystyrene spheres were first shadowed with a rock—salt film and then with a silver film. The slide was then gently slid into distilled water, when the rock—salt film dissolved and the silver film floated off, to be caught on a second clean slide. It was allowed to dry and the remaining spheres were gently brushed off. Then a second opaque layer of silver was evaporated onto it. This new layer adheres strongly to the slide where apertures or scratches existed, but weakly to the first silver layer. When a gentle jet of water is directed onto the slide the first film comes away, taking with it the second film except where it adheres directly to the glass. The final thickness of the opaque disc is 0 about 1000 A, and the error in diameter not greater than 3% (Charman, ref. 34). In order to carry out measurements of a straight edge a reference point is required near the edge; it was 63. provided by an 0.814p, aperture as explained later (see Chapter 3). Filaments of Formvar (type 1595E) dissolved in chloroform were laid on the slide; also some polystyrene spheres 0.814 IL in diameter were sprinkled on the same slide. Then aluminium was evaporated as described above.. After the evaporation the filaments were gently pulled off and the Theres washed away. A cover slip was cemented as before. In this way several edges were formed on the same slide with apertures in the vicinity. An edge was considered suitable for measurements only if it appeared straight and smooth for about 200p, when viewed with an oil immersion objective; also the aperture whose centre served as a reference point had to be 4v;to 6v away from the edge. Only one particular concentration of Formvar solution allowed the production of suitable edges. At this particular concentration the filaments assumed ribbon-like shapes with very flattened edges and with contact angle greater than 90°.

2.6. Preparation of the Neutral Density Wedge.

Again evaporation techniques were applied to make a neutral density wedge. An Archimedes spiral (fig. 2.12) was placed between a clean microscope slide and the source. The slide was rotated in the jig shown in the photograph, fig. 2.13, at about 100 r.p.m. and at the same time 64

Fig.2.12.Archinwdes spiral.

~~~~i~J:~:,; ~~~~~ hhild~ill `:;~ ,' 66. aluminium was evaporated. The evaporation was cut off by a shutter when one end of the slide was opaque. A large cover slip was cemented to protect the film. The transmission factor decreased linearly with distance from 100% to 15% and then tailed off logarithmically. A typical calibration curve is shown in fig. 2.14. The length of the wedge was limited to 3 cm by the jig. Originally this apparatus was used by Dobrowolski35 to produce aspheric surfaces.

2.7. Choice of the microscope objective.

Two apochromatic objectives were available for measure— ments: Beck 1.3 NA (1.28 NA measured value), magnification x 95, and Zeiss objective 1.32 NA, magnification x 100. The appearance of the Mach bands differed somewhat when viewed with these two objectives. When viewed with the Beck objective the bands appeared more diffuse, not so well defined and less bands were seen. These differences in appearance were assumed to be due to the difference in the axial aberrations between the two objectives, and therefore it was decided to measure these aberrations in the plane perpendicular to the observed edge. The two interferograms are shown in figure 2.15. The measurements were carried out on the Twymann and Green interferometer at the National Physical Laboratory 67

0 I 2 3 4 5 6 7 8 9 10 II 12 13 14 IS 16 17 Distance (mm)

i g. 2.14.Calibration curve fof the neutral density wedge. 67a

Zeiss

Fig.2-15. Interferograms A=5461

Beck 68. in green lignu of 5461 X. The two interferograms are shown in figures 2.15 a and b. The following figure (2.16) shows the corresponding aberration curves with the appropriate tilt and defocus terms subtracted. It can be seen that the axial aberrations of the Beck objective are significantly greater than those of the Zeiss objective at about 1.1 NA. In the course of measuring the aberrations of microscope objectives on the Twyman—Green interferometer a curious source of error was observed which does not seem to have been noted before. Objectives in general frequently have what is known as "turned up edge! on the axial wavefront: this is a more or less sharply turned up region of the wavefront round_the edge amounting to about a quarter wavelength and producing a diffuse flare round the star image; it is usually due to the fact that all optical surfaces are low at the edges because of the extra polishing action there and this produces a total advancement of the wavefront. This effect can be the most important defect in a well—corrected micros— cope objective used in monochromatic light. It was found in photographing the axial fringe pattern that the same objective sometimes appeared to have sharply turned edges and sometimes not, and this variation was traced to differences in focus in the photography. Ordina- rily in photographing fringes on the Twyman—Green one is 69. not very careful about the focal setting of the camera since the fringes are visible over a great depth; however, in principle one should focus on the aperture stop of the objective, which is usually a ring at the top end of the the mount, near/R.M.S. screw and clear of the glasswork. For the perfectly corrected objective when the camera is not focused exactly on the aperture stop, a plane wavefront returning from the objective is delimited by this diaphragm and returns to interfere with the comparison wavefront from the other arm of the interferometer: as this plane wavefront passes beyond the diaphragm it spreads sideways by diffrac— tion and the spread parts are slightly retarded, since they are in fact Huygens wavelets from the edge. Now, if it is focueedat some distance from the diaphragm, we are actually testing this wavefront with turned edges. This was verified experimentally by setting up a Twyman—Green interferometer (fig.2.17): two plane mirrors, one in each arm, and with a small diaphragm (about 2 cm. diameter) in one arm. Fig. 2.18a shows the fringes when the stop was in focus and figure 2.18b when the plane of focus was several centimetres away; the sharp turn—up on the fringes can be clearly seen on the latter figure. I Beck

-)%14

A=0 5461 microns

X14 Zeiss

)14

I II I I I I —.5 -.4 2 -.1 0 .2 3 .4 .5

13n.a. diameter n.a.

Fig.2.16.Aberration4 curves for objetives. 71

diaphragm

camera lens

film

Fig.2-17. Twyman-Green interferometer with diaphragm in one arm.

a) focussed

b) defocussed to show spurious turned edge. Fig.2-I8.Twyman-Green fringes through an aperture.. 73.

CHAPTER 3.

PHYSICAL IMAGES OF A STRAIGHT EDGE

AND OP CIRCULAR APERTURES

3.1. Introduction.

Before any visual measurements can be made it is necessary to investigate the physical image of the edge under different conditions and to determine the distance of the geometrical edge from the centre of the reference aperture: Theoretical calculations of edge profiles by Welford36 predict that for incoherent illumination the 50% trans— mission points should coincide with the geometrical image of the edge. To obtain approximately incoherent illumi— nation of the edge it was necessary to stop down the numerical aperture of the objective from 1.32 to about 0.33, so that the ratio (S) of the condenser NA to the objective NA was fairly large, in this case about 4. The value of this ratio is a measure of coherence; when S tends to infinity the illumination tends towards perfect incoherence. Thus the edge and the aperture were scanned the with/ stopped down Zeiss objective, taking S = 4, and the average distance of the 50% transmission point from the '7 centre of the aperture was found. All distances along the edge were then measured from the geometrical edge thus located.

3.2. Method of Scanning.

The improved version of the photoelectric microscope was used to scan the edge, the procedure being as follows. After the initial warming—up time of 12 hours, the micros,- cope was focused on the reference aperture and the galvano— meter deflection was noted; it was observed every few minutes and focusing readjusted if the deflection decreased by more than 2%. Usually, after 15 minutes of readjustment deflection became very steady, and if it remained the same for at least 5 minutes, scanning commenced. The image of the edge was displaced until the galvanometer registered almost full deflection on the clear side of the edge. Then readings of the galvanometer deflection were taken every 0.04µ until the aperture was reached again and passed. In this way a profile of the edge together with the reference aperture was obtained. The profile of the aperture was plotted and the position of its centre found. Now with the previously determined position of the geometrical edge, the profile of the edge can be plotted with distances measured from the geometrical edge image. Five separate scans were 75. made to determine a mean profile. All measurements were performed in green light of 5300 ± 50 I wavelength, except as otherwise stated.

3.3. Luminance distribution along the line perpendicular

to the edge.

Figures 3.1 to 3.4 show experimental results obtained the witli1.32 NA x 100 Zeiss objective for various values of S, 36 together with the theoretical calculations of Welford, and experimental measurements of Charman32 with 0.12 NA, x 13 objective. It can be seen that the results for the low NA objective agree reasonably well with scalar theory predictions, except perhaps at S = 1, where the predicted diffraction fringes were not confirmed experimentally. This discrepancy may be due to objective aberrations. However, scans with the high NA objective differ significantly from scalar theory. To determine whether objective aberrations are responsible for this discrepancy, the objective was stopped down to NA = 0.8, and scans were repeated for S = 1; the results are shown in fig. 3.6, Curve A. With the objective NA = 0.8 the transmission on the clear side is generally lower than before, probably due to decrease of scattered light; also there are some indications of a minimum at about 4 z—units, as scalar theory predicts, but otherwise a significant 76

120 Jr I E - s • I • •s • 100 • • ...... % NC • i • i ,..... TA i

80 I 1 IT . I sconn in; apert u re :7z M i

S I

60 I I R AN

40 1 T

20 -' ) / Z — U N ITS ...... ,,.., -6 -4 -2 0 2 4 6 8 10

Fig.3.1 The image of an opaque straight edge. 1.Present results n.a.=1.3 2.Charman n.ags0-12 S=0-08. 3-Theoretical curve (Welford) S=0. 77

1 100 1 CE N

1 I

75 TA T I M

I 50 I S

RAN 25 I 1 T Z-UNITS

1 I I -6 -4 -2 0 2 4 6 8 10 Fig.3.2. n.vr1•3 S=0.5. (present result 78

100 2 , .1/41 •

CE , 80 3 TAN i 60 T

40 SMI

20 TRAN : Z -UNITS .... 1 i -4 0 4 8 12 Fig.3.3. I. n.arg1.3 S;;1-0. 2.n.a.=0-12. S=0-95. 3.Theoreti cal curve S=0-95.

i I0

E i_ NC 80 1 TTA

60 I M

40 I I RANS

20 I T

lZ-1‘.3 -UNITS kI ..._

[ I I -4 0 4 8 12 Fig.3- 4. I. n.a.=0-65 S=2-0. 2. n.a.= 0- 12 S=5.7. 3. Theoretical curve S=00. 80. discrepancy exists.

3.4. Images of a straight edge in polarized light.

Due to the significant differences between the high aperture measurements and theoretical predictions, it was decided to investigate image profiles in light polarized in different planes. A difference might be expected because of the failure of scalar theory to agree with the experi— mental measurements. Two planes of polarization were used, one with the electric vector parallel to the edge (E11), the other with the electric vector perpendicular to the edge E The results are shown in figures 3.5 to 3.8. The edge profiles with light polarized perpendicularly the to the edge cut the geometrical edge below/other two images and their transmission is generally lower. For non—polarized light the point of intersection agrees very well with scalar theory predictions;36 for parallel polarization the point of intersection lies. somewhat bcanw t very near the image with non--polarized light. These results are mmarized in the table below. 100 •

80 0 _tn U)

60 tn

w (.9 40 z _w 5=1 N A:=1.3 U cr 20 0..

Z-UNITS

1 I I -6 0 2 4 8 Fig.3.5. A unpolarised li ht. B polarised in E11 plane. C polarised in E1 plane. ----scalar theory/ -rafter Welforcr 82 I0 I I I ,1"i Al theory,„ ,I •..,/ -•

, i 80 Z i 0 A&13

2 o 60 z < cc —1- w . 40 (9 4 i- z —w S=1 N A=0.8 — u ct w I 20 i

I , , , ,, ,- ,, , Z -UNITS ..-r -r 1 1 1 1 -6 -4 -2 0 2 4 6 Fig.3.6. A unpolorised illumination. B polarised in Ell plane. C polarised in E1 plane. 120

I 0 0

- z 0 80 A 1) 2 — z cc 60 w o S=0.2 N. z 40 U cc

Fig. 3.7.

Z— UN IT S I I -4 -2 0 2 4 6 8 -iio

-ioo B A Ag13 E

-80- I C

I TTAN

-60 I MI S -50-

AN S=0.5 N.A=1•3

-40 II TR

-20

-10 Z-U N IT S I 1 -4 -2 0 2 4 6 8 10 Fig.3.8. 85. Table 3.1.

Mode of polarization of light S NA Theoretical predictions None Parallel to Perpendicular the edge to the edge • 0.2 1.32, 24 20.5 20.5 16.5 0.5' 1.32 — 23.5 23.5 16.0 1.0 ' 1.30 33.3 30.0 31.0 28.5 1.0 0.8 33.3 35.5 35.3 33.5 4.0 0.33 50.0 50.0 - 48.5 33.0

The agreement for S = 4 is always good because the objective NA is stopped down considerably, to 0.33 approxi— mately, and therefore the objective is then effectively of low NA. The "non—polarized" light actually consisted of light polarized in the plane perpendicular and parallel to the edge in the ratio of 1:1.15.

3.5. Diffraction images of apertures in opaque films.

Charman's37 measurements with high NA objective of diffraction images of apertures show large deviations from the scalar theory calculations of Osterberg and Smith,38 Smith,39 Slansky40 and De and Som.41 The same measurements were repeated using a better corrected 1.32 NA Zeiss apochromatic objective. It was found that in general 86. closer agreement was achieved. Figs._3.9 and 3.10 show present measurements along with those of Charman and a 38 theoretical curve due to Osterberg and Smith for S = 1. The experimental measurements of maximum transmission of the apertures (fig. 3.9) lie very near the theoretical curve and even indicate the existence of the kink as predicted by Osterberg and Smith.38 The measurements of half width (fig. 3.10) lie on the opposite side of the theoretical curve from those of Charman9 and are even nearer the geometrical size than scalar theory predictions •

3. 6 . Conclusions.

It was shown by Charman32 that the predictions of scalar theory are confirmed by experimental measurements at low objective NA. There is, however, one exception; the image of a straight edge with S = 1 differs considerably from the theoretical calculations of Welford.36 Measurements at high objective NA show significant deviations from scalar theory. Images of a straight edge differ considerably from the theoretical images; also differences in profiles occur for differently polarized light; this confirms the failure of scalar theory. Diffrac- tion images of apertures are very sensitive to objective aberrations and show better agreement with scalar theory 100"1--Scc=r11.ieory 87 a • c 80- o - after Cha r man E 60- In Experimental results • 0 40- F ig. et; 20- c Aperture diameter (z-units)

5 10 les 20 25 30 25

Scalar theory // after eml harman

Experimental results •

Fig. 3.10.

Aperture diameter (z-units)

...... rot.....massesararamaan 0 5 10 15 20 25 3 88. than profiles of straight edges. The similarity of l'nonr. polarized" and "parallel polarized" images is probably due to the fact that the content of parallel polarized light in the "non—polarized" light exceeds that of perpendicularly polarized light. Curve A represents the image in the non—polarized light, in light polarized with the electric vector parallel to the edge, and Curve C in light polarized perpendicular to the edge. The dashed curve in figures 3.5 to 3.7 is the theoretical curve obtained by Welford36 on the basis of the scalar theory. For all values of S (1, 0.5, 0.2) and for the objective working with numerical aperture 1.3 and 0.8 respectively images in the non—polarized light (A) and light polarized parallel to the edge practically coincide, while curve C shows always a significant difference. Curve B for both-.numerical a:ercs- shows some flattening at z = 4, i.e. near the position of the theoretically predicted minimum; it is somewhat more significant at 0.8 NA, fig. 3.6, also the shape aprroximates better to the theoretically predicted curve, except for the fact that the maximum transmission is lower. Curves in figure 3.5 (objective working at 1.3 NA) have greater transmissions but less steep slope. No maximR, or minima were detected, however, for S = 1 as predicted by the 89. scalar theory. The only significant difference between the parallel polarized and non—polarized light occurs at S = 1 (1.3 NA); the non—polarized image has no indication of change of slope at z = 4. For the almost coherent illumination S = 0.2, 1.3 NA, a small difference exists between the images in differently polarized light; there is also a significant discrepancy between scalar theory predictions and the experimental measurements. 90. CHAPTER 4.

THE DEPENDENCE OF THE MACH EFFECT ON LEVEL OF RETINAL ILLUMINATION AND ON COHERENCE.

4.1. Introduction. The: image of an edge of a small opaque disc will have a_ luminance gradient which bears no simple relation to thp positan of the geometrical edge, but if the correlation is once established for a particular set of microscopical conditions the geometrical si.zea of small objects can be found. In visual sizing, however, the non—linear response of the human eye to luminous flux and the occurence of the Mach effect wherever the luminance gradient has extreme. values are two complicating factors,. It was found by Oharman27 ' 44that visual size is a function of level of retinal illumination, adaptation, magnification and coherence of the microscops inumination. The laler factor is, possibly mainly concerned with the physical images,, in the sentie that if two physical images obtained under different coherence conditions are identical and are presented to the eye at identical retinal illumination levels,, etc., they will probably look the same, but we do not know if this is true and in the pre— 91. sent study coherence has been assumed to influence the visual image. In the measurements of small particles mentioned above, the visual sizing was performed by placing a cross—wire at the point where: the subjective brightness of the image just differed from that of the surrounding field. This point is determined by the occurrence of the Mach banda. In what follows the effects of the level of retinal illumination and of coherence on the appearence and the position of the Mach bands will be described.

4.2. Method of measurements.

After the complition of the physical measurements described in Chapter 2, there are three sets of subje:citve 92.

parameters to be measured, namely: widths of bands, rela, tive- brightness distribution in the bands and the position of the bands with respect to the geometrical edge (i.e. the determination of the position of the visual edge). First the widths of the Mach bands were determined using the scanning block system described previously. The image was displaced until the pointer appeared to be touching the borderline "a", figure 4.1, between two adjacent bands. This position was determined by moving the eye along the border in question and judging whether the pointer was in line. This procedure was repeated at each border. It was found convenient to start at the border a, figure 4.1, and then at borders b, c,. d and e. With the oil immersion objective the band to the right of the border

71 was usually very faint and therefore it was not measured. In this way five readings were obtained; at least ten sets of five readings were taken. The whole procedure was time— consuming and very fatiguing to the eye, so that a break of at least a few minutes was necessary after two sets. The positions of the bands were then determilied by measuring the distance between the centre of the reference aperture Fig.4.1. Bands andthei

LUMIN ANCE D GEOMETRICAL reference letters. I C DIST ANCE EDGE 0 b 1 eft r 93

94. and the border b. This border line was chosen because (a) it was the best visible border, (b) it was found that the measurement of this distance gave smallest standard deviation from the mean value and (c) it represents the position of the visual edge. The average distance was calculated from five different readings.. To find the relative light distribution in the bands, the eyepiece field of view was divided into two parts (see Chapter 2) and the brightness of the reference slit in the bottom half was first matched to the clear field with the edge removed from the field of view, then in turn to the two bright bands. No matches were made with the dark bands because it was not possible to reduce the brightness of the reference slit enough due to the presence of scattered light. Five readings were taken for each band and the relative brightness was then expressed as percentage of the clear field transmission. With the above measurements it was 1 . possible to plot the subjective image of the edge.This was done by marking the off position of each band and then drawing -.a profile consisting of vertical and horizontal lines; finally the sharp corners were smoothed. The measurements of the widths of the bands were rather tiring to the eye and usually a short rest of about one 95. minute after each set of five readins3was necessary. The position of the pointer did not affect the subjective appearance of the bands provided it was fairly thin. However when the pointer was set on any border the eye was somewhat confused and to ensure that the pointer coincided exactly with the border line,it was necessary to move the eye along the border line to judge whether the pointer was in line.

4.3. Measurement at different S—values.

Measurements were carried out with two different objectives, a 1.32 NA,x 100 Zeiss apochromatic and a 0.28 NA, x 13 Cooke chromatically corrected objective, in green light of 5300 ± 50 1 and approximlely pt the same retinal illumination (104 trolands). For alit measurements a x 20 Ramsden eyepiece with exit pupil of 0.23 mm was used the except for one measuremuatat ES = 2 witlYx 100 objective;, in this case ax10 Ramsden eyepiece was used to retain the 28 same angular magnification1MBfirst measurements were carried out with the oil immersion objective at four different S—values (S = 0.2, 0.5, 1.0, 2.0) and these are shown in the figures 4.2a to 4.2d. Dashed lines represent the physical image of the straight edge, full lines the corresponding subjective luminance distribution.

96 B. M. Watrasiewicz

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Fig.4.2. Luminance distribution in physical and subjective images of a straight edge at different modes of illumination. - -- Objective luminance distribution. Subjective image distribution. 97. The following two figures 4.3 and 4.4 summarize the subjective measurements. The latter shows diagrammatically the relative positions of the bands as they were measured. The appearance of the subjective visual image with narrow cone illumination (S = 0.2) is the most striking feature that emerges from these measurements. The dark and bright bands seem to have been assumed by microscopists to be diffraction bands due to the coherent illumination of the object. However, figures 4.2 and 4.4 show that the eye does not perceive exactly the physical diffraction bands but the Mach bands in displaced positions; also bands were seen on the dark side of the edge which were much more intense than the physical diffraction bands. Furthermore the position of the visual edge moves steadily towards the brighter side of the edge as the condenser is stopped down.

In order to see how much of this shift is subjective, the shift of the physical j- --light distribution must be taken into account. To do this tie concept of the physical edge, as defined in. chapter 1. sec.1.6., must be used to determine how much the 50;- transmission point shifts with S value. The, consecutive positions of the. 50% point were found from fig. 4.2; also the position of the • visual edge was deterMined from the 'tame figure. 98.

With a corresponding definition the position of the visual edge and physical edge were both measured ftom the position of the geometrical edge and are tabulated below (table 4.1) along with the comiesponding S-values. It may be seen from this table that both edges show some shift from the bright side towards the dark side as S changes from 0.2 to 2.0, but the shift of the visual edge ILS substomtially greater and that there is no simple relation between them. Similar results were obtained by other observers; these results are summarised and discussed in chapter 8.

Table 4.1. Shift of edge with coherence. ' S Physical edge Visual edge z-units w-units 0.2 1.05 2.4 0.5 0.95 2.0 1.0 0.75 0.75 2.0 0 0.20

The measurments with the low power objective (x13), stopped down to 0.12 NA, confirm the" above, conclusions. The subjective image profile with the corresponding physical images are shown in figure 4.5a to 4.5e. and 9.9

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microns 1 1 -6 -4 -2 o 2 4 6 8 Fig.4.6.Visual edge shift with `S'. (0.12 NA) 107. diagrammatically in figure 4.6. The physical images were obtained by Charman32 and the subjective images by the 29 author using apparatus described in Chapter 2. Thefull and ope4 circles in the last two figures represent the points 2 where d B is either maximal or minimal respectively. Their ctic significance will be discussed later.

4.4. Mach bands as a function of retinal illumination.

For the sake of convenience and briefness the banda will be referred to by letters as shown in figure 4.1. Measurements were carried out for a fairly large range of retinal illumination,(15.100) between 15 trolande'and 3.6x104 trolands,upDartop limit being determined by the output of the lamp; the lower limit represents the border between photopic and mesopic vision. The measurements were made at S = 1 and with the Zeiss 1.32 NA objective. Each individual band changes in width when the retinal illumination is altered; also the position of the visual edge shifts towards the darker side at lower levels of retinal illumination. This is shown clearly in the figure 4.7. The distance between the two edges is measured in in the object space microns4 positive values indicate a shift away from the geometrical edge towards the clear field. The vertical lines passing through the points on the graph represent standard deviation from the mean value. The shift of the 108. . visual edge in this range of retinal illumination is about 0.1 micron ; this would introduce a significant error into visual sizing of small particles.

The variation in width of the banasA, B, C and D is shown in figures 4.8a and 4.8b. The variation of the band A is somewhat erratic but it can be seen that its width diminishes at low retinal illumination. Band B, on the other hand, decreases uniformly in ',idth (on logarithmic scale) with the increasing retinal illumination. The remaining two bands C and D are seen to vary very little with the retinal illumination in the range from 3.6 x 104 to 300 trolands but they vanish completely at about 180 trolands. Bands A and B also disappear at somewhat lower retinal illmnination, namely at about 15 trolands. This level of the retinal illumination is the threshold between photopic and mesopic vision. The model, shown in the photograDh- (fig. 4.9), attempts to summarize the measurements just described. It shows very clearly the shift of the visual edge and the disappearance of each pair of Mach bands.

4.5. Calculation and measurement of the retinal illumination.

When an observer views an image through the microscope the eyepiece alailight from the image is directed towards the POSITION OF THE VISUAL EDGE AS A FUNCTION OF RETI NAL ILLUMINATION

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01 SP u0/oil 01 01 113. exit pupil of the eyepiece, and, if the latter is smaller the than the iris of the eye, all/light will reach the retina. Thus if the light flux emerging from the eyepiece is measured the retinal illumination can then be calculated. The flux is measured by replacing the eye with a perfectly diffusing screen and measuring its luminance. Let S1 (figure 4.10) represent the image viewed through the eyepiece r mm away, represented by its exit pupil A, and let S2 be a perfectly diffusing screen placed at a distance R mm from A. then the flux dP through If LI is the luminance of S1, 1 the element of area dA of the pupil and from an element dS1 of SI is dA dF1 = LI dS1 2

The total flux through the pupil is

and this is the total flux that falls on the screen S2.

The illumination of S2 is given by F1 II1SI A -12 = 0 2 w2 r hence the luminance of S2 is diffu sing screen S2

image plane S i

Fig. 4.104. 115. L S A L 1 1 2 r2 21cS2 since a perfectly diffusing screen scatters with a solid angle 2n. Therefore L22TTS2 L1 A - 2 r 31 but

hence

L1A = 2itR2L2. But the unit of retinal illumination, the troland, corresponds to a luminance of 1 cd/m2 seen through a pupil of 1 mm2 in area, thus L1 A represents the retinal illumination in trolands. The luminance of the screen S 2 was measured by means of an S.E.I. photometer for one particular lamp output, but at the same time the collector voltage of the photomultiplier and the galvanometer deflection were noted. From this one measurement, repeated a few times, the corresponding retinal illumination was calculated. All other values were found on the assumption that the luminous flux is proportional to the photomultiplier current and hence to the galvanometer 116. deflection, provided that the anode voltage is kept constant. A typical calculation is shown below:

R=150 mm 12=0.2 cd/mm2

Va=1.75. kV Galvanometer deflection 10 cm (x 0.001 range) Retinal illumination. = 0.2 x 2 x 1502 .3 x 104 trolands.

In this calculation the Stiles-Crawford effect, which amounts to a decreased visual effectiveness cif the light passing through the outer parts of the pupil, has been neglected. This may be. justified. first, becuase in microscopy at high magnifications the area of the eye pupil used is small anyway and one assumes that the central part would automatically be used, and secondly because we are here interested only in the order of magnitude on the retinal illumination. 117.

4.6. Conclusions. Meaaurments with low and high power objectives show similar variations and indicate that when an observer is presented with a diffraction pattern he will perceive bands at somewhat displaced positions from those predicted by physical optics; these bands are subjective Mach bands. The relative displacement of the Mach bands from the diffraction bands and the fact that there are usually more. Mach bands than diffraction bands may be explained in terms of the theory ( see Chapter 1 section 1.2) that Mach bands are formed where the rate of change of luminance gradient is stationary. When the illumination of an object changes from coherent to incoherent, the visual edge shifts from the bright side towards the dark side of the edge. Thus the visual size of an aperture should - increase while the visual size of a dark disc should decrease when the mode of illumination is changed from coherent to incoherent. This is indeed the case, as was shown by Charman.27'45 The inveatiagations at various levels of retinal illumi 118.

nation show the following interesting points: (a)the two faint bands C and D disappear at about 180 trolands, (b)the other two bands A and B also disappear at lower retinal illumination at about 15 trolands (beginning of mesopic vision), i.e. there is no Mach effect at low levels of retinal illumination, (c)the visual edge position shifts towards the dark side of the geometrical edge as retinal illumination is reduced. It is difficult to draw any conclusions about the relative variation in size of apertures and discs as the effect of geometry on Mach bands is not known, and it is probably very significant. Circular symmetry may be especially significant as the inhibition and disinhibition regions tend to have concentric symmetry in the retina (Wagner, MacNichols and Wolharsht, ref. 46). The effect of orientation of the edge was n't etudied because this would involve considerable changes in the apparatus. 119. CHAPTER 5.

THE .EL,r.tECT OF WAVELENGTH ON THE MACH BANDS.

5.1. Introduction.

All previously described measurements were carried out 0 in monochromatic light with a narrow band—width of 4" 50 A; however as the response of the human eye to different wave— lengths varies considerably in the visual spectral range some variation in the appearence of the Mach bands might be expected. The change of position of the visual edge for example will affect the visual size of a given object.

5.2. Method and results.

The position and size of the Mach bands were determined in exactly the same way as previously described for various wavelengths,. Photometric matching was carried out with the reference slit having the same colour as the microscope field of view. All these measurements were carried out with approximately the same subjective brightness of the field of view. The light output from the lanip was adjusted until the subjective brightness at a given wavelength appeared to a be the same as that at 5300 A corresponding to 3 x 104 trolands. The Zeiss objective (x 100) at S = 1 was used for these measurements. a. This was done by memory matching. 120.

Fig. 5.1 shows the variation of the position of the visual edge defined as in chapter 4 with wavelength. It can be seen that the distance between the vusual and geometrical. edge decreases as wavelength increases; the visual: edge_ shifts towards the dark side of the edge since it lies on the bright side of the geometrical edge. This result cannot be explained in simple terms4 the scale of the physical image is constant in z-units, so it increases linearly in size with wavelength, i.e. we should expect a shift away from the geometrical edge with increasing wavelength. The dependance of all four Mach bands on wavelength is shown in figures 5.2 to 5.4; all show a somewhat complicated variation between 4700 A and 6500 2.. Band A figure 5.2, shows a pronounced minimum at about 5100 and a maximum at about 6000 2.. The two dark bands B and B show little variation with wavlength, but band C increases: in width as the wavlength is increases and attains a maximum 0 at about 6300 A. These results are summarized diagrammatically in figure 5.5 which shows very clearly the shift of the visual edge towards_ the geometrical edge as the wavelength is increased. 121.

The relative subjective brightness distribution in the Mach bands, for five different wavelength, is shown in figures 5.6 to 5.10. These figures do not show any very significant features. Band C is relatively bright at the low end of the spectrum (4700 51) and decreases down to 20% (about half of the maximum value) at the wavelength of about 6250 The first bright band (A) follows more or less the same pattern starting from 5000 2., but it is fairly faint at 4700 a.

5.3. Conclusions. As has been seen from figures 5.2. to 5.5, the variation of the bands with wavelength is very complicated, nonetheless it was expected that there would be some relation between the widths of the bands and the spectral response of the observer's eye. Therefore, the response was measured; it is shown in figure 5.11 superimposed on all the other curves. The maximum of the band A occurs at about 6000 R, at the point Where the eye response drops down to about 50%. This suggests that when the effective brightness sensation decreases, the width of the band A increases. 122.

This conclusion seems toEgree with the previously measured variation of bands widths with retinal illumination in monocharomatic light. It can be seen in figure 4.8a that when the retinal illumination decreases from 104 to 103 trolands the widths of band A increases to a maximum. 0 Band C also has a maximum at about 6100 A, however its variation with retinal illumination (fig.4.8b) is too small to draw any significant conclusions. The variation of both dark bands; with wavelength is very small and probably insignificant in view of errors involved in these measurements. POSITION OF THE VISUAL EDGE AS A FUNCTION OF WAVELENGTH .12 I ..- II - I I I I 1 I I I II II

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Fig.5.11. Mach bands and spectral response A of the eye. 134. CHAPTER 6. THE EFFECT OF EYEPIECE POWER, OBJECT CONTRAST AND LIGHT POLARIZATION ON THE APPEARANCE OF THE MACH BANDS.

6.1. Introduction. The visual size of a microscopic object depends among other things upon, the overall magnification44and therefore for a given microscope objective on the eyepiece power. Charman44 found that the visual size of circular apertures and discs increases, with the eyepiece power until the overall magnification with an oil immersion objective is about 1500 times. In practical applications.objecta are ::not completely opaque but semi-transparent and therefore the effect of the object contrast on the appearance of the Mach bands is of practical interest.Again as for different modes of illumination, there will be two factors influencing the appearance of the Mach bands: change in the physical image of such an object in partially coherent illumination and the subjective effect due to the lateral inhibition in the retina. Furthermore, the mode of polarization of the illuminating beam may modify the visual image because, at least 135.

at higher numerical aperture systems, the physical image will appear somewhat different (see Chapter 3). It has also been suggested that canes in the retina behave as waveguides50thus they may not have cylindrical symmetry in their behaviour with respect to incident light and this may have• some effect on the appearance of the Mach bands. Some preliminary measurements were made to investigate the influence of the factors enumerated above on the subjective image.

6.2. The effect of the eyepiece power.29

Three Ramsden type eyepieces were used with magnifica— tions x10, x20, and x50. The first two conveniently focused On the same pointer in the first focal plane of the eyepiece so that the optical tube length was the same for both eye— pieces. The latter x50, had its first fodal plane very near the front component, inside the casing. Therefore a new pointer was necessary; it was introduced inside the eyepiece casing, the meachanical tube length was readjusted and the scanning block micrometer recalibrated. Also it was not possible to make photometric measurements with the x50 eye— piece, so the relative brightness of the Mach bands was only estimated. Measurements with the x10 and x20 eyepieces were carried out in exactly the same way as was previously described. 136.

Figure 6.1 shows results with a loW numerical aperture objective (x 13, NA = 0.12); for all three curves shown S = 1 and the retinal illumination were kept con— stant, at about 104 trolands. To attain the latter condition the light output from the light source was varied with the square of the eyepiece power, It can be clearly seen that the visual edge shifts away from the geometrical edge towards the clear field as the eyepiece power is increased. Increase of the eypiece power increases the scale of the retinal image; the same image contours cover greater numbers of receptors and therefore a greater number of neural units. In this way the network of receptors responsible for the transmission of the message to the brain is more complicated and the brain receives more clues. Further— more when a greater number of retinalreceptors is involved in the interaction the bright bands should appear less bright because of the greater inhibitory effect. This is indeed the case as shown in figure 6.1. With the present knowledge it is not possible to explain the shift of the visual edge, but the effect of pupil sire (see ch.8) and changed light scatter in the retina due to different angle of incidence may be responsible to some extent for the change in the appearance of the image. 1 J 1 I „ , . ,

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20 M 1CRONS . .,... -5 -4 -3 -2 0 I 2 3 4

Fig.6.1. Effect of eyepiece magnification on Mach bands . 138.

6.3. The effect of object contrast on the appearance of

the Mach bands.

Three slides, with edges, were prepared in the same way as previously described except when shadowing with aluminium the film was not made completely opaque but of varying transmissions. These edges were scanned with1.3 NA. oil immersion objective at S = 1 and S = 2. The latter scan, as before, was necessary to determine the position of the geometrical edge with respect to the reference aperture. It can be proved that for ' incoherent illumi— nation the inge of a step superimposed on a, background of constant luminance intersects the geometrical edge half— way between maximum and minimum luminance of the stop irrespective of the constant background luminance this is illustrated diagrammatically in the figure 6.2. Thus the distance of the geometrical edge from the centre of the reference aperture can be measured. Determination of the physical and subjective images at S = 1 was carried out in the same way as before. For these lower contrast objects it was possible to make photo— metric matches on the dark bands. In the following figures the bands which were seen but were not measured either because they were too narrow or too faint are shown in dashed line. 139

100 ITTANCE

M 70 S N TRA

DISTANCE

Fig. 6.2. Incoherently illuminated edge and Its image (diagrammatically).

140. A contrast factor was calculated by measuring percentage transmission of the darker side and then substituting the value into the formula:

Imax Lmin Contrast (0)- ''max + min luminances, where Zmax and I,min are on the clear and dark side of the edge respectively. This is, of course the equivalent of the expression used by Michelson in defining contrast of interference fringes. Fig. 6.3 to 6.6 show geometrical, physical and subjective images (full line) of a straight edge for different contrast factors. at S =1. Figure 6.7 summarizes, schematically shifts in the subjective image.. As the contrast decreases the secondary bands become more faint and then disappear. If the contrast is further decreased, the bright band A becomes narrower and more faint; at 0=0.21 it is just about visible, on the other hand the dark band becomes wider and less dark; eventually both disappear. The position of the visual edge, figure 6.7, seems to remain more or less constant up to C =0.47, but at C=0.21 it shifts away from the geometrical edge towards the bright side. 141

120

E I I C\.. 100 f I 4.•• ......

C t 10 ° #...... N

/ I 80 I TA

i IT

I I 60 M 1 1 S 1

I 40 I

TRAN /

1 20

... .. /0 ZwUNITS

-4 0 4 8 12 Fig.6.3.1mages of the opaque edge. C=1 N.A.=1.3. 1 .42

100 ADNID MED MP .0 ilF.

A 80 / NCE TTA I 60 /111 SM N

40 TRA ....,••

20 - - - • . • • " j \I)

Z - UNITS

—4 -2 0 2 4 6 8 Fig.6.4. Images of low contrast cdgc. C 0-55. S=1 N. A.2--1-3.

• I43

100 f\-.....— E

C / N I 80 TA / /

MIT /

S /

N / 60 / / RA . / / T . - , 4 I % 4/ 1 ..------°' 1

2 1

Z-UNITS 1 -4 -2 0 2 4 6 8 Fig.6-5. Images of low contrast edge. C =0.47. S=1 N. A.=1:3. 144

100 -. ---- .-----

E ,

I I - 60-* NC

....- ..., ..... ITTA

I I

M 60 S N

I I 40 TRA

Z - UN ITS i. I -6 -4 -2 0 2 -6

Fig. 6.6. Low contrast edge Image C =0.21. S=1 N. A.= 1. 3. 1 I 1 1 I I I i% 1 1.0 VilIfinA VIWIII//1/ /1/11/1 I IMI

.8 i— tn

-6 < .II VIM //Mr/ //II cc - - - V/ I/I /11111/1111 t— r.. MP 4=1111=1 z awl 0 U .2 tit/Mil/MIA 1111•111 Z - UNITS II I 111 I i i 4 3 2 1 0 I 2 3 4 Fig.6.7. Variation of Mach band with contrast. S =I N.A.=I.3 146.

6.4. The effect of the plane polarization of the

illuminating beam on the Mach bands.

As previously described in Chapter 3 opaque edge profiles were determined for plane polarized light, with the electric vector parallel (curve B, fig. 3.6) and perpendicular (curve 0, fig. 3.6) to the edge. The position and width of the Mach bands corresponding to these two modes of polarization were determined using a x20 eyepiece; the results are presented diagrammatically in figure 6.8. It is interesting to see that the small shift of curve C (fig. 3.6) towards the clear side away from the geometrical edgewas. detected asa corresponding shift of the position of the visual edge (C, fig. 6.8). Furthermore the width of the first dark band has increased a little as one would expect due to the deterioration of the contrast in the physical image, curve C figure 3.6. The much smaller difference between the physical image in non-polarized light and polarized in the plane parallel to the edge has not been "sensed" by the eye as the Mach bands appear almost identical. There is no indication that the polarized light has any direct effect on the retina. receptors.

1 I

polarisation:

Vt/A WZZZ/ZZA 1 E. vec tor Ir tothe edge

‘Ni FAX\NX\N

VZIZZ_ZZA I No polaristat ion

I I I I •3 • 2 •I 0 .1 -2 .3 microns Fig. 6.8. Shift of the Mach bands with polarisation. S=I N. A.= 1.3. 148.

CHAPTER 7.

PART I.

THEORETICAL TREATMENTS OF THE MACH EFFECTS.

7.1. Introduction.

All subjects, however difficult, will always attract some theoreticians and contrast effects in human vision, such as the Mach effect, are no exceptions. Mach himself was the first to propose a mathematical equation to describe qualitatively this effect. Since then more approaches have been suggested. All theoretical trattents, those which were mentioned in the introduction (bh.l) and those which will be discussed in this chapter, with full mathematical details, can only be re— garded us ruther inadequate and representing a very crude approximftion to reality. These models are based on only one functional property. of the retina, namely on the assumption of lateral inhibition. Other neuron pr_operties, which were mentioned in sec1.1, such as avalanche and summation processes to name only two, are ignored.But,however inadequate these models may appear to be, it is useful to have them because they can give some insight into the relative importance of v,rious functional properties of neurons and also because they provide a starting hypothesis which,with improved knowledge, may develop into a more sophisticated model. 149.

All the treatmenta are qualitative except one by Hartline, Wagner and Ratliff17, and this allows no arbitrary adjustments of constants involved in the equation. These workers studied the retina of the Limulus crab quantitatively by recording outputs from receptors using microelectrodes. Employing different patterns of illumination of the receptors, they were able: to6btermine the constants, involved in the interaction between receptors; using their equation (,sec.7.4), they were: able to calculate the response of the crab retina to a step-like luminance distribution and thus demonstrate a contrast effect which resembles the Mach affect in human vision.Furthermore, by recording the excitation caused by such luminance distirbution they were able to confirm their predictions. Generally it is assumed that complete processing of visual information takes place at the retinal level, the signal being then transmitted to the btain without further modulation. However,, there is no satisfactory proof that this is the case although the work of Hartline et al. quoted above: is in agreement with this hypothesis.

7.2. Mach's Equation. To describe, the subjective sensation distribution in response to an external stimulus, Mach3 developed a

150. mathematical expression (see also Chapter 1, 1.2); this equation is

r(x) = a log 1L(21) c (d:B(xl b B(x) dx2 where x is a distance measured along the surface of the retina, r(x)7subjective brightness distribution,B(x) is retinal illumination and a, b and c are constants. The sign of the second term is to be chosen according to the sign of the second derivative. The abovp equation will predict only two bands in response to a step function; it cannot predict any other effect such as variation of band—width with retinal illumination.

7.3. Fry's approach.

Fry8 started from Hering's assumption9 that every visual element depressed activity in the surrounding elements to an extent varying directly with the intensity of its own activity and inversely with the distance from the element undergoing depression. Thus elements near the edge of a uniformly illuminated area will display higher activity than the elements receiving the same amount of stimulation near the centre of the area. Fry assumed that the retinal illumination I of an 151

Fig.7.I . Analysis of inhibition in the retina.

VISUAL ANGLE 0

32 4 U w

GC L4J30 2.0

cr) w 1 28 1.6 0 cC

26 12

-4 -2 0 2 4 X (m

Fig.72.The F-distribution for a blurred border.

(after Fry) 152 element of area dA at a point Q, figure 7.1, sets up an electric field in which the potential difference dE, at a point P at a distance r from Q, is some function of r, i.e. dE = f(r). This potential will be responsible for the inhibition of an element at point P. Thus the total poten— tial difference produced by all elements surrounding P is: + E = I(r).f(r) dA

Or in Cartesian coordinates:

E = I(x,y), f(x,y)-dx-dy

- C. This inhibiting potential will affect the frequency of impulses in bipolar cells, and according to Fry the output frequency will be given by:

F = F' + k2 log p where F' represents the frequency as it would be when unimpeded by E and k2 is a constant; p is a function of E and is given by: 1 = 1 + k E 5 F' can be eliminated and then F is given by:

F = k log 1 + k 8 1 k E 7 5 153. On the assumption that 4ao 2/ 2 E = 1(x) e-x /2G dx 013 Fry calculated the subjective response of the eye to a blurred edge; his results are shown in figure 7.2. These bands resemble the Mach bands reasonably well, but they seem somewhat broader than the bands observed in practice. There are only two bands: one bright and one dark.

7.4. Treatments of Hartline et al. and Taylor.

The equation proposed by Hartline et al17919920925 was previously mentioned in Chapter 1; it is:

f. = ei z .(f. f°.) - 3=1• 1.0 0 j/i = 1, 2, N where fi and f are the responses of the ith and 3.th receptors respectively pnd ei is an excitation of the ith receptor due only to an external stimulus; both f and e are measured in impulses per second coming from a given is the threshold frequency which receptor receptor.f.ij jmustexceedinordert0inhibitreceptor i, Kij is an inhibition coefficient, i.e. a coefficient of action of receptor j on the receptor i. Both f° and K are only 154

functions of distance between the receptors; the former increases and the latter decreases with distance. There are two more restricting conditions, namely: when the inhibitory term K(f - f0) is greater than e the corresponding response f must be set equal to zero because negative frequencies have no meaning and if f f° is negative the inhibitory term must be dropped. This equation, because of the latter conditions, has no simple analytical solution. Allowing for the restrictions this equation can be re-written in the form: N f. = e. - 2 K. .max(0,f.-e.) j=1 10 i = 1, 2 .... N

where Kij:> 0, Kij = Kii and qj =

Melzak49 obtained the conditions necessary to satisfy equation 3; they are:

(a) <1, i = 1, 2 N. j=1 1J JI j/i f. N K.% f. 1;1 3 (b) + z , - 1 ei j=1 (e. + j/i 3_0 13 i = 1, 2 .... N Must have positive solutions

Unfortunately this does not in any way help to obtain the solution of equation 3 because equation 4 is very difficult to solve. Taylor48 proposed a somewhat similiar equation there is no threshold frequency but instead the limits of summation are restricted, that is the summation is not carried over all receptors but only round the receptor under consideration in a restricted area. For a one-dimensional model the equation takes the form: n+h f(n) = e(n) + K E f(m) n-h where IC is the magnitude and 2h the spread of the interaction. For the two-dimensional lattice f is a function of two parameters and therefore the equation becomes:

f(m,n) = e(n,m) + K2 E f(p,q) ....(6) A where A is the area over which summation is taken. Taylor has solved both equations and found the subjective response to a unit step. Fig.7.3.a shows an intensity distribution for a one-dimensional model; horizontal axis represents distance along the line of receptors; in this particular case 21 receptors were assumed to be "illuminated"; for numerical computations Taylor assumed as a relation between the- magnitude and the spread of interaction,K -1-. The, response of the model to this intensity distribution is shown in the fig. 7.3b. 156.

The solution of the two-dimensional model is shown in the figure 7.3d.. This time 441 receptors arranged in a square lattice having one, side of the square equal to 21 receptors ware "illuminated". Figure 7.3c. shows the input intensity distribution along the center of the square perpendicular to one side; figure 7.3d. shows- the corresponding response along the same central line. The difference between the one and two dimensional models is striking: the latter shows multiple band structure at each edge,, while the first shows only two fairly smooth bands near each edge. This model,, simple from the physiological point of view, and based on the assumption of lateral inhibition between receptors, points to a very significant fact,, namely that one should expect differences. between one- and two- dimensional:models,. Furthermore because of this difference only two-dimensional models should be compared with practical measurements as in practice the intensity distribution is; always two-dimensional. Finally one more criticism of these models is the assumption of a uniform lattice; this has no anatomical justification, but it simplifies enormously the computation problems.

157

4—s

(a)

2 2 s)

LL (b)

Fig.7.3. Taylor's solutions.

1/c1 2/d3 -2w•w w 2w 3w 41w -!1 x 'Pw NEURAL UNIT 1.05 134 -P/d .124 3 1 1.00. . 0 w 2w 3w 4w Sw INHIBITION FUNCTION Fig.7.5.A finite approximation to the spatial impulse response function for the continuous Fig.7.4.A five receptor model. model. 158

7,5. pproximate conditions for .ositive roots of e.uation 4.

Approximate conditions for positive roots of equation 4 can be written in the form N f. K. f ••-• 1 j=1 e j e . + f .K. 3_0

1 i = j where . . and K.. = K.. = 0 63_ 0 = 0 i / j II 33

K.ij • 0 f = 1 j=1 e. + f K j ij ij

This is of the form

MK. a. . = b. 3-3 I

K 1 K12 ln where 9 e 1 o e + f12 K 12 9 el + flnK ln

K21 1 a..10 = e2 + f21K21 2

•

Knl 1 + fn,o K e en , n,

469

ei can be taken outside the determinant from each row, thus: e K e K 1 1 12 1 1n e + fo o 1 12 K12 el + flnK •

1 1 1 . aid x • 0 • x 1— e e e • l 2 n .

•

enICn1 o en + fnln1

e.IC. 4 Now . Kid for e.;, K. .f. e + f. 1 lj 1j

hence

K 12 Kln 1 aid = x e e2 1

• • 1

When the constant term is omitted, the determinant can be regarded as the sum of two matrices: I 0

0 0 0 K12 • • Kin

K21 0

aid =1 A + II

Thus the expansion of theaii 1 along the diagonal elements gives:

d= 1 • • • • lai K12K21 K13K31

+ K12 K24 K41 +

+ KKKK+12 21 34 43 ""

As Kid "1, the most significant terms are the first 1 then two; if Kid = Ko K2 ,2 .o + .... higher orders of Ko la. .1= 1 ° 16

The determinant will be positive if: K2 K 1 o 16 ,2 16 or o 17 4 (.1

It can be seen that for cases when Kid varies more slowly with distance the assumption that K.(2)

The development of laik l will have the same form as that of IaijI with Kik = bk. As b. = 1 its terms will be greaterthaalthecorrespondingtermofyand therefore the last determinant will decide the sign of the root. Thus the approximate conditions can only be used as a guide, these are: (1) ej > Kid fclj

( 2) Ko2 < 2

Generally the smaller I<:0 the greater the probability that all roots will be positive. It is obvious that for a large number of redeptors 162/ the evaluation of both determinants will take a very long time and therefore the algebraic approach must be abandoned.

7.6. Integral equations corresponding to the previous

models.

The integral equation corresponding to equation 2 for a large number of receptors takes the form: +a)

f(x) = e(x) k(x 9 s) Cf(s) — fetx,$) ....(7) -co It is now only valid for the one-dimensional lattice of receptors; it takes a simplified form if f°(x,$) is put equal to zero,

f(x) = e(x) - k(x-s) f(s)-ds ....(8) as in this particular case k(x0) = k(x-s). In the same way the integral equation corresponding to Taylor's equation is: x+h

f(x) = e(x) - 4114k(x-s),f(s).ds ....(9) s=x-h Methods of solution of these equations will be described in section 7.7. It should be noted that somewhat different integral equations will be obtained if equation 2 is, at first, written in the following form:

e. = E k. .f J 1 -co 1J 166 where, by definition, kii = 1, f°(x,$) = 0. This gives then the following equation:

e(x) = fk(x-s).f(s)-ds ....(11) -07) 7.7. Comparison of various mathematical models.

1. Introduction.

All models previously mentioned can be expressed in either summation form, i.e. descrete model, or in the form of an integral equation which represents then a continuous 23 model. For the purpose of mathematical analysis Bekdsy expressed his neural unit model in the following form:

r(x) = h(x-s).f(s)-ds 0, = h f or r. j=„7, i-j j where h(x) is the neural unit and r(x) is subjective response to the external stimulus f(x). Bliss and Macudy51 coguiared this model with the simplified version of Hartline,s equation; the simplification was achieved by the omission of the threshold frequency, thus: n f. = e E k.. f. ....(12) i j=1 3-3 J/i The comparison was made for one-dimensional lattices in two cases: 164

(a)for a discrete model, (b)continuous models.

2. Comparison of the discrete models.

For the purpose of the comparison equation 12 can be written in the following form: n e. = E k. f. j=1 ij 0

where, by definition, kii = 1. For large values of n this becomes: co k.3.-- f j=—' j as k. . is only a function of the separation between the 3.0 receptors. 51 Bliss and Macudy used the so—called "two—sided" z transform defined by:

F (w) = f e—iwny

(w) represents a quantity f located at x = ny Fn n

n or Gn(z) = fnZ where z = e—iwy

They proved that the t-transfdrm of the Bgkesy neural unit is the reciprocal of the .i.—transform of the inhibition if function, i.e.

H(z) = 1/k(z) ....(13)

Assuming that k = e-aw1131 for aw> 0, they found K(z),

K(z) - 1 2 b2 b b(z+z-1 ) where b= e aw and 0< b <1. From the equation (13), H(z) Can be found, which is: 2 H(z) -b 1 1 + b -b .z 1 -b2" z + 1 -b—7 + 1 - b2

The spectra of both the neural unit and the inhibition function are compared in figure 7.4; they were calculated for 1 p = 0 hp = c-1/10 p = ± 1 ± 2 0 1p I >2

t ip' and k = .2 tIPI + 10 P s s where s = T/3, t = V3 — 2), p = 0 ± 1, ± 2

3. Comparison of the continuous models.

The continuous analogue of equation 3 is:

e(x) = k(x-s)• f(s) ds ....(14) -alxl again on the assumption that k(x) = e ICI. ,

51 Bliss and Maaudy, using Fourier transforms, have shown that the following relation holds in the transformed space: 2 F(u) a + u E(u) 2a where as E(u) e(x), dx and co f(u) = f(x).e-iux dx f If e(x) is a point stimulus represented by a delta function (5), then f(x) will be a function corresponding to a neural unit in the Bek6sy model, hence: 2 2 H(u) a + u ....(10) 2a The inverse transform of the above expression gives:

h(x) = 2a 5(x) - (2a)611 (x) where 0 represents a second derivative of the delta function. The finite approximation to this function is shown in the figure 7.5. It is similar to the neural unit of Bekesy (see Chapter 1). If f(x) is a continuous function of x, the inverse transform gives: ,2, f(x) = -1--ae(x) - (2a) -Ldx1-7 te(xi 167,

This equation shows the importance of the second derivative; also it is similar in form to that of Mach.

4. Two—dimensional models.

To estimate fully the validity of the simplified version of the equation of Hartline et al, i.e. equation 14, it is necessary to find out how a two—dimensional model behaves and whether there will be some new features. It can be seen that the integral equations corres— ponding to the continuous models are of the Fredholm type with the unknown under the integral sign, In this case the kernel is of the form k(x—s) and therefore Fourier transform methods can be used to obtain the solution. When the equation is transformed into the Fourier space, the Fourier transform of the unknown function can then be expressed explicitly in terms of other known quantities and then by taking the inverse Fourier transform of the rearranged equation the unknown function can be found. For the two—dimensional models the two—dimensional-Fourier transform is required. This approach will be used through— out. The one—dimensional equation can be generalized into a two—dimensional equation in the following way: 49

e(x,y) = )( k(x—s y—t)-f(s,t).ds ,dt ....(15) ..43o again assuming an exponential variation of the inhibition function, i.e.

k(x,y) = e—alx+YI substituting into equation 15 gives CO CO e(x,y) = 1.)(e—alx—s+y—ti f(s,t).ds.dt

-CO -CO or co m e(x,y) = i(le+a(x—s+Y—t) f(s,t).ds.dt x y x y + 67.4ix—s+y—t) f(s N .. 1 T), ds-dt.

...MI --CO

Let u = x—s and v = y—t, then: co m e(x,y) = e f(x—ut y—v)-du.dv 0 0

ea (u+v) f(x—u,y—v)-du.dv

Taking Fourier transforms with t and 1 as coordinates in the transformed space,

c'? cif7 e+a(u+v) 4i(Cx+ly)d E(t71) = f(x—uty—v) u.dv dx dy .43off co 0

. +11 • Au+v) f(x-u,y-v).a v e—i271(;x+11Y) dx.dy waV Interchanging the order of integration,

a(u+v) (I -2Tri(U+T)y) E( . e jj f(x—u,y—v) e dxdy . dudv +

y) e—a(u+v ) r271f (x-19 y—v) e (x+r) dxdy dude

Now let w and z be dummy variables given by w = x u and z = y v:

co (a-21-TiOu.e(a-27ii)v i f ) -27i(tw+T)z) .dudv ) ) f(w,z e dwdz

0 — ( a+21-ri — ( a+2Tri w+11z ) 11 e e dvdu if f ( w , z) e-27j- dwdz

But by definition F(C,1) = 4f(wlz).ew+7-)z) dwdz. J.-co Substituting this into the equation (16) and evaluating other integrals gives

E(,,,) _ (a2 .1. 2(4:J) )472,02) °F("1))

Solving for F(0) and taking the inverse transform, pages 170 and 172 not used

171.

2 2 2 2 )(a 44a 41 ) cagi(S x-Fly) ,ni) .d5 f(x,y) 7) 2 2(a -41E Pr' ) .... (17)

If the equation (17) is compared with corresponding equation (10) for the one-dimensional model, it can be seen that the two equations, for a given value of a, will be equivalent only if high spatial frequency Fourier components are considered in the case of the two-dimensional lattice. Tha latter assumption rules out sharp variations in luminance distribution. This means that the product is small compared with a, and then the equation (17) becomes:

ao 2 2 2 2 2 f(x,y) .1/((a 4,4a2I )(a ±-4a1 ) x+4137- e ) ,1) d5 of)? 2a2

GO. (3.7a)

To find the inverse Fourier transform of the equation (17), the right hand side can be treated as a product of two, functions:

173

f(xly) = IrG(&911).E( e2 7d- X+11 dri -+W or writing T-1 for the inverse Fourier transform 00 T— 1 [G“,i).E(tT))] = [[ g(x1 ,y') dx' dy' -co ....(18) But e(x,y) is known, and therefore only g(x,y) is needed. Now, ,-, 2 ,2)(a2 4. 47t2,02 1 C4-,i) . (a + _ --75 .G.,(0.G (n) 2a2 2a,' ' 2 2 2 2 where GI = a + 47; and G2(1) = a + 4%r) g1(x)g2(Y) Let g(x,y) = 2a2 co a2 4.7t2c2)e-27tivc where g1 (x) = p

and (a2 + 4712.02)e-12"Y

Now g1 (x) can be found from

(a2 4712c2)e-27tix gel& g1 (x) = 07.

a2)( —2Td_x d +4712f e e-27Eivc dA

= a25(x) + if 4 It2 2 e-2 7ti)a tt = a25(x) 6 (x) (see Appendix 1)

where 1(x) is a Dirac delta function.

1 - 7 4 Substitutng the above results into equation (4) gives:

GA (0 eG2(11 •E(t 911) T-1 L 2a2

1 j 2 6,(x1 )-5"(x1 ).1+25(y')-6"(y') e(x—x'„y—y')dxidy' 2a2 {a Multipling out and integrating (Appendix 2) yields: 1 f(x,y) = 2a2e(x,y) xxle (x,y) — 2ert (x,y) + .017 (x,y) yy 2a2 xyyx

where eXX(x,y) denotes the second derivative of e(x,y) with respect to x etc. From the last equation it can be seen that the two— dimensional model will not have any new features if the stimulus, e(x,y), is constant along the y—direction because then the last two terms will be equal to zero. Equation (17a), which is much more complicated, will yield different solution for the two—dimensional model and may lead to a multiple band structure of the Mach effect.

7.8. The solution of the alternative form of the

simiplified equation of Hartline et al.

The equation is

CO f(x) = e(x) ik(x—s).f(s).ds 175

Taking the Fourier transform it yields:

F(u) = E(u) K(u)F(u) solving for F(u),

F(u) 1 + K(u) or F(u) = E(u) E(u) + E(u) 1 + K(u)

E(u) - E(u), 1 + K(u)

The inverse transform of the last equation yields:

f(x) = e(x) T-1I E(u).Geul ....(2o) where G(u) = "(u) 1 + K(n)

Now T-1 [G(u)-E(u)] = jr e(x xl)g(xl)dx' where g(xl) = T-1[G(u)] Again assuming that k(x) = e-alxl then It(u), as before, is given by K(u) - 2a a2 + 472u2 and G(u) therefore,

G(u) = 41. 2 u2 +2a a2 + 2a 114

58 From Campbell and Foster tables (pair 632)

g(x') = a— e —pix*1 I where p = 2 + 2a)N 2 then it follows that: 0 -CO 1 T LE(u).G(u)] = aje(x—xl ), e+Px' • dx, + pjfe(x—xi) -e —px1 dx P P 0. The evaluation of the above integrals yields the following solution: f(x) = (1 2a. , ) e(x) — e"(x) eiv(x) 0140

7.9. Solution of the Taylor equation.

In the equation (5) ]c will be assumed to be a function of the distance between the receptors, i.e. the kernel is again equal to k(x—s), then the equation becomes: x+h e(x) = ir k(x—s), f(s)-ds ....(22) x—h where k(x—s) = e

First method.

Let x s = z, then the equation (22) becomes:

e(x) = k(z).f(x—z).dz ....(23) 177

By defining a blanking function b(z) such that

1 -h < z < h b(z) = 0 -h > z > h the last equation yields:

CO e(x) = jk(z)•b(z) •f(x-z)• dz -00 Taking the Fourier transform,

E(u) = F(u).TN(z).b(z)i

Now it only remains to evaluate T[k(z).b(z)]; this can be done as follows:

TCk(z).b(z)] = P(u-u1 ).B(u1 ).dul 00 ..‹10 where K(u) = rk(z).e2uiuz dz -(0 and 00 h B(u) = b(z)02-Tiuzdz= e2uiuz dz

Therefore oa T[k(z)b(z)] = 1 sin 2uhu' du ' la2 + 472(u-u')2 • uu'

2a 1 + (27u sin 2uhu - cos 2uhu) a2 + 472 u a (by contour integration) 178

Second Method.

Taking the Fourier transform of equation (23) gives: co h E(u) = f k(z).f(x-z) earixu,dz. dx -co -h substituIng for k(z) and putting x z = y, where y is a dummy variables h _alzi e E(u) = f e fur) ariu(y+z) dz dy

Changing the order of integration gives: co E(u) = e-alzi e2niuz dz ff(y) e2niYu.dy

The second integral represents the Fourier transform of f(x); the first can be evaluated as was done in paragraph 7.7, this yields:

E(u) = F(u) 2a [1 + -ah /81(2nu sin 2Trhu- cos 2rrhun a2 + 4n2u 2

From this last expression it can be seen that the two independent methods give the same result. Thus F(u) can be found: 2 2 F(u) (a + 47 u) E(u) 2a[1 + v-ah/4( 2nu sin anhu - cos 2nhu)] ....(24) 179 It can be seen that for h tending to infinity the last equation will reduce to:

F(u) - a2 472u2. E(u) ....(25) 2a

It is extremely hard to evaluate the inverse transform of the equation (24) and therefore for the purpose of comparison G1(u) and G2(u) will be plotted as functions of u. ,2 4172122 Gi(u) = °"' 2a and „-ah (217u sin 27Thu -- cos 2nhu)] G2(u) = G1 (u)/[1 + aa These two plots are shown in figure 7.6. for a = h = 1. From the shape of G2(u) it is obvious that it is an oscil- latory function, amplitude depending on ah, and that in the x-space it will show a similarly complicated behaviour. 180

/ 40 / •• / /

/ all 1 ) / /

lal 20 r

.4'

al /

, 1 I 1.0 u (ra d)

-20 a

• l a I

Fig.7.6. ------G1(u)

Giu) IS/

PART II.

NUMERICAL CALCULATIONS.

7.10. The numerical solution of the model of Hartline et al.

The numerical work described below was carried out on two electronic computers: a National Elliot 803A and a Mercury (Ferranti). The programme for the solution of the Hartline et al, equation was developed on the Elliot 803A computer and then, to increase the speed of computation, it was translated into Mercury autocode and run on this computer. The programme takes all the limiting conditions into account and uses an iterative process to obtain the solution. The first approximation to the solution is assumed to be equaltoe.,thus after substitution we have: N f.1 = e. E k..(e. f? .) 1 1 j=1 1° J 10 di A second step gives: N e =e.—E k. — f.c).) j=1 13 3 di thus the nth approximation becomes:

e) f. = e. E k. .(f r!'-1 ij 1 3--1- 10 ji 182,

With a moderate number of receptors, the number being limited by computation time, it was found that five to seven iterations were enough to yield a solution accurate to the second decimal place. Probably the number of iterations necessary will increase with N. Unfortunately the time necessary to obtain a solution for a moderate number of receptors, about 100, was very considerable: 50 to 60 minutes. This limited severely the numbers of runs. For the numerical calculations it was assumed that the inhibition coefficient varied inversely with the separation between the receptors, i.e.

kij. . = B/(i j)2 and 2 f?ij = C(i j)

Figure 7.7 shows the solution for a two—dimensional lattice of 39 x 7 receptors; the change in the luminance gradient was along the longer side of the lattice. In this figure the secondary bands are very clearly visible. The run was then repeated for a one—dimensional lattice with 100 receptors. Although the same inhibition constant and the same frequency threshold were used the solution (fig. 7.8) shows no secondary bands. However, if the actual numbers are examined carefully, some indication of secondary bands can be found; it is so small that when I I 1

—100 •••

Mb • NNW .11/ MM. 4•1111P mg, ow, eon mr =VI 110 1110 -I

• -4 -3 -2 -1 0 I 2 3 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 receptors receptors Fig.7.8.0ne dimensional model 7.7.Two dimensional model with 10Ix I receptors 39x7 receptors 184 plotted it can not be seen. This seems to suggest that the secondary bands are features of the two—dimensional lattices only. It is hard to find any analytical reason why the model of Hartline et al. should give differing solutions for one and two—dimensional lattices. It was shown previously (7.7) that if this model ii4:approximated to a linear system, there is no difference between the two solutions.

7.11. The numerical solution of Taylor's equation.

(a) One—dimensional lattice.

The equation was written in the integral form x+h f(x) = e(x) k(x—s).f(s).ds x—h and again an iterative process was used to obtain the solution. In the same way the first approximation to the solution is x+h f(sds.Y f1(x) = e(x) jr k(x_s) x—h and the nth is given by: x+h fn(x) = e(x) k(x—s) fn_1(s).ds sc

The numerical integration was carried out by the trapezoidal rule. All numerical work was done on the Elliot 803A computer as for small values of h and E moderate .gout 40 number of receptorsAhe time for about ten iterations was only about 20 to 35 minutes. In this case the inhibition constant k was assumed to decrease exponentially with the separation between the receptors: k(x—s) = be—alx—s1

For large values of a, about 5, six iterations were enough to obtain a solution accurate to the third decimal place; for small values of a, less than unity, about 20 iterations were necessary. Figures 7.9 and 7.10 show the response to a step input. As before all curves were normalized to 100%. These curves show more than two bands; it is interesting to note that the largest bright band occurs at the point of dis— continuity, and the dark band is shifted. When the input is not a step function, but has a finite slope, figures 7.11 & 7.12, then the two primary bands occur at the discontinuities and the two secondary bands follow.

(b) Two—dimensional la-Aice.

One more programme was written, to solve Taylor's equation for a two—dimensional lattice; again the same -100

I hr2 a=0 b=.2

a a I I 0 1 2 3 4 -5 -4 -3 -2 -1 0 1 2 31 receptors

Fig.79. 40 receptors Fig.7.10. 40 reteptor3 27 iterations 11 Iterations 100 I / / / h=2 a=1 b=.2 50

I I I I it- I I I I I I I I t I I I I I -5-4-3-2 -1 0 1 2 3 4 -5-4 -3 -2 -1 0 1 2 3 4 receptors receptors Fig.7.11.40 receptors Fig.712. 40 receptors. -100

—50

ma own ow. S

a -4 0 2 4 retinal distance receptors Fig.7.13.A solution of Taylor equa- F1G.7•14. tion for a two dimensional lattice with 21x7 receptors. 189 process of iteration was used and the integration was carried over an area of 4h2. Such a solution, after six iterations, is shown in figure 7.13 (constants used in the calculation are shown on the figure); here again the secondary bands are clearly visible and are somewhat larger than in the one—dimensional case. All solutions have a feature in common, namely symmetry of the bands, while measured Mach bands are asymmetrical. However it must be remantered that these solutions were obtained in terms of frequency output (pulses per seer) from the receptors, but not in terms of subjective bright— ness. If the usual relation between brightness B and frequency f, log B m f, is used to convert frequency output into subjective impression of brightness, the bands become asymmetrical (fig. 7.14); the degree of asymmetry is a function of the constant of proportionality between log B and f. Thus the calculated and measured bands appear, at least, qualitatively similar. It was not possible to make a fit to the experimental data as the times involved in the computations are too long.

7.12. Application of frequency response methods to visual

problems.

For a number of years frequency response methods have been used for the assessment of image quality. The approach 00 is based on the assumption of linearity of the system (H.H. Hopkins, ref. 54), which leads to the following convolution integral:

Bt(u)) = f B(u)-G(ut—u) du

where B(u) and B/(u') are the luminance distributions in the object and image space respectively, and G(u') the spread function of the optical system, i.e. its response to a unit pulse. The above equation can be applied to any system which satisfies the necessary conditions. If B(u) and Bt(ut) are known, for example measured, the spread function G(ul) can then be calculated using Fourier Transform methods. The above equation is then transformed into:

bf(s) = b(s)•g(s)

where 101(s) is the Fourier transform of B,(u1 ) and so on; g(s) is known as the transfer function of the system and it is given by

g(s) = bt(s)/b(s) 52 The last equation was applied by Lowry and DePalma to the human visual system to calculate what they called the psychological transfer function. The convolution of the psychological spread function with the physical image 101 presented to the eye should correspond to the appearance of the subjective image as seen by the observer. Thus the transfer function for the visual system of an observer, starting with the eye and the retina and ending in the brain, can be calculated by dividing the Fourier transform of the subjective image (i.e. the image containing the Mach bands) by the Fourier transform of the physical image presented to the eye. However, before any significance is attached to such results, it must be realized that the Mach effect is a non—linear phenomenon, as it was proved by Marimont,53 and that the most that can be obtained is some indication of the degree of non—linearity in the visual system. The inherent non—linearity in human vision is due to (a) logarithmic relationship between the stimulus and the subjective perception of brightness, and (b) the lateral inhibition in the retina, i.e. the stimulation at one point affects the output at another. The non—linearity of the first kind occurs, for example, in film emulsion and some image converter tubes, but the non—linearity of the second kind is only found in animal nervous systems.

(1) Calculation of the Fourier Transforms.

The step function, like the image of an edge, is an unbounded function and therefore the numerical methods for 192. evaluation of Fourier Transforms cannot be applied unless the function is converted into a bounded function. As the microscope field of view is limited by the eyepiece, the image of an edge viewed through an eyepiece can be con— verted into a bounded function by making the luminance equal to zero at the edge of the field of view. Once the function is bounded, evaluation of the Fourier transform amounts to finding a Fourier series for the function. If the complex form of the Fourier expansion is considered, then by definition, the coefficients are given by: x 1_ 2 f(x),eiarnx dx Fn = P J x1 where x1 — x2 = P, the chosen period or the interval in which the function f(x) will be expressed as a Fourier series. Making the following substitution the above equation can be written in a different form. If 1/P = Su, then Fn = &Unnem) where 2 F(n8u) = I f(x).eiaanbux dx

Now if x1 — co and x2 +co, i.e. P —.›m' and Su —0, the Fourier transform of f(x) is given by:

F(u) 1rf(x).ei2nux dx 193

It follows, therefore that if the Fourier series is calculated with constantly increasing period, the coeffi , aents thus found will tend to the Fourier tansform; with a sufficient number of points they will define the curve which will be the Fourier transform of the function.

(2) Results.

Figure 7.15 a and b shows typical Fourier Transforms of the physical and subjective images. Figure 7.16 shows the calculeted transfer functions for three different retinal luminance distributions. It is very obvious that these three curves are very different from each other, thus the transfer function in each case is different and it can be concluded that there is a considerable non— linearity in the system. all

I I . . ■ I a I I I I A A • .1 I .2 Iincs/z•units lines/ z-units physical image subjective image. Fig.7.15. Fourier transforms S=0.2 z- units Fig.7.16.Transfcr functions. 10‘

Chapter VII - APPENDIX I.

co Evaluation of ji4n22 e &E.

Let us consider the following equation: oo f(x) = JF(u)- e-iarTux du

Differentiating on both sides with respect to x, co f' (x) = )(F(u)(-ariu) e-127rux du ....(2) -CO hence T-1 [uF(u)] = + 1 f' (x) 2rr Differentiating equation (2) again gives:

f" (x) = )1(-21-riu)2.F(u). e-ariux du -co or 2 T-1 Eu2F(u)] 1 f" (x) A-iar/ In the same way the nth derivative can be obtained, and this will give the following formula:

T-1[unF(u)] = fn(x) -ari From this equation the transform of u2 can be obtained by making F(u) = 1 i.e. 197

—1r 2 = 2 T LU . —i27)2 dx '['.8(x).] Thus 44T22 e-2rriCx,g= —4.72 2 8 (x) = — vi(x) ,47

Chapter VII — APPENDIX II.

Evaluation of co co 2 = 121 o(xl)-6"(x') {a 6(30)-5"(Y1 e(x—xt ly—y!)dxedy' 2a

Multiplying this out gives:

CX) CO

1 E1•48(xt )84Y')+8"(x')8°(Yi )—a28(xl )8"(Y i )—a2CY1 )8"(xlx 0 2

e(x—xt,y—yt)-dx'dyl

Now taking term by term we get:

8(xt)8(y!) e(x—x';y—y?) dxidyi 1 •-4:0 -CO coy° = , e(x—xt,y—y° dyt dxt

=.1 8(x') e(x—x',y) dx' = e(x,y) ....(1)

198 Using the above result,

m p(x1)6"(yl-e(x—xi ly—Y') dlcidy°

8n(Y1 ) e(x7Y-30) dY1

CCI = [61(yi) e(x—x')10 + PO(Y')-0(xly—yl) dyl

= j s'(Y') eT(x,y—y') dyr

co co = [8(30)-0(x,y —30)] + )16(y1 )-e"(xly—yt).dyi

= en(xty)

From this it follows that a) dnn o(x i ) . f(x—xl) dx' = an x = 0 -a) {dx n Thus the value of the integral I is

1 I 1 It 1 4V I, ,v) I 1 e(x,y) veyy(x,y) ff Gloc;yy'—'41 = 2a2 - 'exx(x'Y) --2a CHAPTER 8.

DISCUSSION.

From the measurements described in the previous chapters there emerged a significant feature of the Each phenomenon, namely the multiple band structure of the subjective image. There cannot be any doubt that all normal observers can see more than the two bands usually reported. Figures 8.1 (Charman and Watrasiewicz, ref. 29) shows the appearance of the same physical image to three different observers. The qualitative appearance of the subjective image is more or less the same, except that observer R.L. could not see the dark bands on the bright side of the edge. Quantitatively bands differ somewhat in size and position; this is not surprising as one must allow for experimental errors and for the fact that no two eyes are identical. Furthermore, four observers carried out measurements of the subjective image using the same edge and the same oil immersion objective at S = 1, and obtained very similar results, as can be seen in the figure 8.2. The measurements of the observer T.M. differ significantly from all other measurements, probably because this observer has an astigmatic eye. 200. Both figures show that the position of the visual edge does not differ much from observer to observer; the differ— ences just exceed the experimental error involved in the determination of its position. The measurements of the relative brightness of the bands, table 8.1, show much greater discrepancies; this is partlly due to the matching arrangements: non—linearity of the neutral density wedge and the steep rate of change of the transmission factor with distance along the wedge.

Table 8.1: Subjective brightness of the Mach bands for aifferent observers.

Observer Percentage transmittance

BAND A BAND B 1 T.M. 174.4 ± 34 36.5 ± 6.2 W.T.W. 154.5 f 30 11.2 ± 1.7 1 ,L. 141.3 ± 23 35.3 ± 10.5 B.M.W. 123 f 10 14 ± 2

The fact that the subjective image has a multiple band structure gives rise to a more basic question, whether the multiple bands appear in response to a step—like luminance '5values 0 • aos obs. 0 • 0.32 O • 0.60 R. L. O 1.00 0 5.70

0 0.08 obs. o 0 0-3 2 yf o 0 0.60 W.N.C. 1-00 re•%%.1i /",‘NI! 0 ° 5.70

I 0 0.08 obs. 0 0. 32 0 O • 0.60 O 0 1.00 5.70 o MICRONS I -8 -6 -4 -2 2 4 6 FIG.8.1 0 —A -.3 -.2 -4 o .1 .2 •3 .4

Fig.8.2.Position of bands as seen by different observers. 203. gradient or whether they are due to the presence of maxima and minima of the second derivative of luminance with respect to the retinal distance. According to Ludwigh's theory (Chapters 1 and 7) the latter assumption should be correct. As previously described, numerical differentiation was carriea out and positions of maxima and minima of the second derivative were found and marked in figures 4.5 a to 4.5 e, by clear and black circles; black circlesbnote minima and clear circles maxima. The agreement between the positions of centres of bands and circles in these figures is reasonably good, considering the difficulties in the experimental measurements and the even greater errors involved in the notoriously inaccurate process of numerical differentiation. It must be noted, however, that numerical differentiation of the phySical image in incoherent illumination, i.e. with. S = 5.7, yields only two points where the second derivative has an extreme value.; but nonetheless four subjective bands were visible. Charman32 has shown that with objectives of low numerical aperture measured images and calculated images agree reaso- nablyu well, thus the positions of extreme values of the second derivative may be located more accurately by calm- lefcion, on the assumption that S = 5.7 is equivalent to S =03. 204.

For completely incoherent illumination (S = co) the well;known classical formula for the image of the edge can be used. It is

1(z) = dz

function of the first order. The where S1 is the Struve 56 StruvefunctionS.(z), of order it is defined by the equations: 2 Si(z) = 2(1-z)1 ( (1 t ) - sin zt.dt r(i + 2)r(1) or (_)m i+2m+1 S.(z) = E (Iz) m=0 r(m i)r(i m

The expression for I(a) can be expanded as the power series: 2z) 3 I(4-z)=. — i2. 2z rit .1.3..3.5

for small values of z and sa the asymptotic expansion e7 for large values. of z:

2 1+ _1_ 1 cos 2z •IC/4) I(z) _ E• 12Z] ea z5 •••••• 205.

Thus the positions of maxima and minima of the second derivative of I(z) with respect to a should correspond to the positions of the points of inflection of the integrand. These points can be accurately located by determining the positions of zeros of the third deriva— tive of I(z) with respect to z. The second derivative is given by:

S o(2z) 31(2z) I" (z) =77 72 —717

and the third derivative is:

I"'(z) = a (6 - 2z. )31(2a) — 5zS0(2z] 8 a7 206.

The graph of the function in square brackets was plotted to find the approximate positions of zeros and then these positions were determined more accurately by calculating the numerical value of the function in square brackets using the tables of Jahnke and Emc4i9 section VIII, 10. The points of inflection occur at positions given below, accurate to the second decimal place: z = 1.43, 4.10, 5.59, 7.32. The calculations were carried out only on one side of the edge as the curve is antisymmetrical about the geomet— rical edge position for S =co. Figure 8.3 shows the calculated edge intensity on the dark side of the geometrical image (curve A). The inflec— tions are only perceptible on the curve B with ordinates magnified ten times. Their positions are marked by short vertical lines. On the same figure the first derivative, I'(z), is also plotted, i.e. curves C and D; they show the inflection points much more clearly. All these curves were computed using the above mentioned tables. These points of inflection were marked on figure 8.4 showing the physical and subjective images, at S = 5.7. Clear squares denote maxima and black squares minima of the 1,07 Fig.8.3.COmputed edge image for S=oo on dark side of edge. A and B the edge- image, C and D gradient of A and B.

1.0 .20-

0.8 .15-

I- z w 0 a cc (D -10 -

C A

-05- 0.2

---.....,

2 4 6 8 10 12 14 Distance from geometrical image, z-units 120 1/4

100 0 U) U) 80

60 4 w 0 5=5.7 N A=0.12 40

I w •20 a MICRONS

I I -8 -6 -4 -2 0 2 4 Fig. 15,4. 209. second derivative with respect to the retinal distance. As can be seen from this figure, the agreement between the positions of the centres of the bands and the clear and black squares is staggering. The points of stationary second derivatives correspond to extraordinarily small inflections in a smooth curve; the curve of the intensity in the incoherent edge image at distances greater than z = 4 from the central point is mathematically a smooth curve with small variation superimposed on it and the amplitude of this variation is less than 1% of the smooth curve, and yet these small irregularities seem to be amplified by the visual mechanism into bright and dark bands. These small vari— ations have only points of inflection but the gradient never falls to zero.

The above applies, cf course, to measurnments from one observer. It was not intended in this work to carry out extensive survey but nevertheless We note that. the measurei- mentssummerized in figs..8.l and 8.2 suggest that there could be an element common to other observers, i.e. there might be a "standard observer" of the Mach effect and effects such as those found above may be the rule, not the exception. 210.

The other important parameter in all these measurements is the size of the pupil. The effect of the pupil diameter on the Mach bands waa investigated by Leelawongs55 on a macroscopic scale. A razor blade edge was illuminated by a large source, i.e. incoherent illumination, throught a diffusing screen. The edge was viewed through an artificial pupil of appropriate size placed at a distance of 25 cm. behind the edge, on the opposite side to the source. The results are shown in the figure 8.5. At first glance it appears that pupil diameter has a very considerable effect on the Mach bands but this is somewhat misleading. The change in the pupil diameter affects the physical image i.e. the spread of the diffraction image on the retina: for larger pupil diameters, the inflection points in the image profile come closer together and therefore the Mach band pattern shrinks. To check this, the widths of bands and their relative distance from the geometrical edge were concerted into z-units (diffraction units) and tabulated (table 8.2.); this table shows that all corresponding bands are of the same size within experimental error, whatever the pupil diameter. There however, one significant feature shown by figure 8.5, namely the .disappearence of the secondary Mach bands for larger pupil sizes, at about lmm. This effect TABLE 8.2. "- Pupil:- diameter ..BAND WIDTHS (diffraction units) Distance (mm ). , from the edge The dark field. to the 1st. Bright bawl on the 1st. dark band Bright band 2nd.dark band dark band bright field (z-its) - / .29 3.90* .24 2.12 t .17 2.17 + .15 1.15+.10 2.01+ .10 .36 4.41+ .25 2.52 + .31 2.48 4 .15 1.27 +.11 2.26 +.18 .52 5.15+ .43 2.95 + .50 2.62 + .2.5 1.59 +.21 2.50 +.30 .64 4.55+ .56 2.58 4 .34 2.02 + .34 2.72 +.20 .72. 4.35+.21 2.66 + .42 2.25 + .30 2.57 +.30 .85 4.60+ .50 2.62 + .84 2.06 + .30 2.66 +.40 1.00 3.96+ .76 2.86 + .64 2.16 +.58

The knife edge was incoherently illuminated. The retinal illumination was kept constant at 682 trolands. The distance between the eye and the edge was 250 mm.

H 212..

may be purely subjective, i.e. due to retinal interconnections. As the pupil size increases, the infledion points come closer together and possibly at about 1 mm. pupil diameter they are too close to be effective. On the other hand, with larger pupils: there will be more scattered light reaching the receptors, because more light scattering structure will be illuminated and thus the effect of inflections in the. luminance distribution curve may be diminished or even annulled at larger pupils. Thus for different pupil sizes the Mach- bands still appear at the inflection points. It can also be seen from figure 8.5 that the Mach bands represent details smaller than the formal resolution limit of the eye; in this case it may be appropriate to take the contour resolution of the eye, this being of order 10 to 12 sec. of arc, corresponding to about 0.6 z-units; this is about of the same order as the smallest band measured which was just under 1.0 z-units. The neat explanation of the Mach bands in terms of higher derivatives of the edge image is, unlikely to be the whole story. Most theoretical models and the empirical equation of Hartline et al. suggest that the multiple band structure should appear in response to a step-like luminance 213. distribution. Unfortunately it is difficult to obtain such a distribution on the retina; one might expect to get it by viewing an edge with a very large pupil,, so that the diffraction structure was below the inter-receptor distance, but the aberrations of the eye are not negligable for pupils larger than 1 mm. and thus an edge with a finite gradient and almost certainly with inflections due to diffraction would be formed. An attempt was made to diminish the diffraction effects by using white light. The microscope condenser was illuminated by a filament lamp. In this way the effective bandwidth• was considerably increased and therefore diffraction effects should have been smoothed out. Nevertheless three Mach bands were clearly visible. It was not possible to measure the exact effect of white light on the image of the straight edge because the photomultiplier sensitivity varies considerably with wavelength. It seems that the eye is in some ways, more sensitive than the apparatus used and can easily see changes of intensity of less than 1% of the immediate surroundings. Thus an improvement of the sensitivity of the photoelectric microscope is necessary to increase the accuracy of comparison between the physical and subjective images.Also the 214

t 3

BRIGHT SIDE OF THE EDGE

E 0 0 LLI Posi I 0 n of geometrical edge

—4- 2 0 U-cr

-8 E Resolution of the eye C AN T

— 12 IS D

DARK SIDE OF THE EDGE —16 AR GUL

AN PUPIL DIAME TER (mm.

0.2 0.4 0.6 0.8 1.0 1.2

Fig.8.5. The influence of eye pupil diameter on

the width and position of Mach bands; observed

in incoherent light, retinal illumination 682 trolands. A4g photometric matching technique needs improvement as the present accuracy in matching is only about 10%. Most of the error is due to the non—linearity and steep rate of change of transmission with distance of the neutral density wedge. Further improvement in the sensitivity of the photo— electric microscope is not very easy as the photomultiplier and the light source are run at their maximum. The most direct way is to increase the length of the galvanometer scale by projecting the galvanometer spot onto a scale situated at some considerable distance. However, such an increase of sensitivity without further increase of period of the galvanometer would also increase noise level. With the present arrangement it is not very practical to increase the period as then the time necessary for a scan would be increased and the defocusing would become more severe. This further improvement can only be gained by very sophisticated Methada such as automatic focusing arrangements. An improvement would help in investigations of the influence of the polarization mode on the appearance of the physical image; it would also allow the further investigation of the discrepancy between the scalar and vector theories of diffraction, although as yet there are no theoretical calculations of the edge profiles based on the vector theory of light. 2,14,

All proposed mathematical models, except one due to Hartline et ale, to describe the Mach effect give only a qualitative description and,as shown in Chapter 7, some of them predict multiple band. structure in response to a step— like luminance distribution. The model of Hartline et al, empirically deduced from the measurements carried out on the eye of the Limulus crab, predicts Mach bands quantits— tively with great accuracy, As was pointed out previously this model predicts multipic band structure in response to a stell function, however i. all publications there are only two bands shown. This model is incompatible with Ludvigh's findings, but it is not unlikely that the human and crab retinae differ functionally. The potential importance of the Mach effect in the particular case of the visual microscope is easily under— stood. Diffraction and aberration cause sharp edges of areas of different but uniform transmittances to be imaged as zones of finite width and varying luminance between areas of different but uniform luminance; the precise distributions of luminance in these zones depend upon the optical con— ditions in each case. These luminance distributions resemble edge profiles and therefore bright and dark Mach bands would be seen which can be mistaken for the physical detail in the image. In general, the itiach effect results in 217. subjective sharpening of the edge gradients; bright and dark bands may be seen if the luminance gradients across the edge change rapidly enough. From the point of view of the microscopist, such an subjective enhancement of the edge sharpness may be beneficial; it may aid the detection and recognition of details in the image. It will, moreover, improve the precision, though not accuracy, of measurements of object size made with the microscope. Disadvantages arise, however, if purely subjective bright or dark fringes are seen, since these may be falsely identified as representing genuine detail in the object such a surrounding motbrane; considerable care is therefore necessary in interpreting the image. Striking examples of the importance of these physiological effects have been encountered in investigations of the image of high-contrast objects under various conditions of illumination. For example when the eye observes circular discs or apertures in partially coherent or incoherent light they seem to be surrounded by very prominent bright and dark bands. However, theoretical calculations by Welford47 and Slansky40 show that the bright bands exceed the surrounding field by much less than 1%. Charman44,45 could not even measure this small change in the luminance 218. distribution phctoeloctrically, but the human eye picks them out very easily and makes them very visible. Unfortunately, as was shown by Watrasiewicz,28 the subjective bands are shifted in position and therefore the Mach effect in this case would improve the precision but not the accuracy of measurement. Furthermore the visual measurements of the Airy discs pattern should not agree with the physical measurements or theoretical calculations, as indeed was found by Smith.39 The results described indicate the strong influence of the physiological effects during visual observations with the microscope. The importance of such effects may be expected to extend to many other optical instruments in which the image is observed visually. It is suggested that techniques similar to those described, in which the distri— bution of retinal illumination is known, might prove useful in further elucidating the Mach effect. Finally, in further investigations of the Mach effect distinction must be made: between macroscopic luminance distribution viewed through large pupils, grater than 1 mm. and between microscopic images viewed through pupils less than 1 mm. In the latter case the image has structure of 219, its omn, i.e, the inflection points, which significantly alter the appearance of the Mach bands. In the former it is the broad overall luminance distribution, which can be deduced from geometrical optics, which can give rise to Mach bands, but those are,, of course, those whiCh have been seen by observers since Mach's first observations a hundred years. ago. REFERENCES

2. E. Mach, Sitz Wien Akad,Wiss. 52, abt II, 303 (1865). 3. E. Mach, Sitz .Wien A::ad,Tiss. 54, abt II, 131 & 393 (1866); 57, abt1I, 11 (1867; 115, abt IIa, 633 (1906). 4. W. McDougal, Proc.Physiological Soc. 1, 19 (1903). 5. R.N. Thouless, Brit.J,Psychol. 11, 110 (1922). 6. R. Granit, J. of Physiolo'y 77, 207 (1933); §1, 359 (1935); 105, 24 (1946,. 7. J.F. Schousten and L.S. Ornstein, J.Opt.Soc.Am. 29, 168 (1939). 8. G.A. Fry, Am.J.Opthalm. 25, 162 (1948). 9. E. Hering, Grundzuge der vom Lehre Liohtsinh,:.Berlin (Springer: 1920). 10. E. Ludvigh, Perception of Contours I and II (U.S. Naval School of AviatiOn, Med. Reports, N.M. 001.075.01.04 and N.M. 001.075.01.05). 11. E. Ludvigh, Arch.Opth. 43, 397 (1940). 12. C. McCollough, J.Exp.Psychol. 49, No. 2, 141 (1955). 13. A. Florentini, Prob.Contemp.Optics Sept. 600 (1956). 14. A. Florentini &A.M.Ercoles,Opt.Acta 4, No. 4, 150 (1957). 15. P.H. Green, Factors in Visual Acuity, Aeromedical Div. U.S. Air Force Office of Sci. Research, AF 18(600), Project 9777, Univ. of Chicago (1958). 16. A.M. Ercoles and A. Florentini,:Atti,Fond.G.Ronchi 14, 230 (1959). 17. H.K. Hartline, N.G. Wagner and F. Ratliff, J.Gen.Physiol. 39, 651 (1956). 121

18. H.K. Hartline and F. Ratliff, J.Gen.Physiol, 40, 357 (1957). 19. H.K. Hartline and F. Ratliff, J.Gen.Physiol. 41, 1049 (1958). 20. F. Ratliff and H.K. Hartline, J.Gen.Physiol. 42, 1241 (1959). 21. M. Alpern and H. David, J.Gen.Physiol. 43, 109 (1959). 22. W.R. Machavey, S.H. Bentley and C. Casella, J.Opt.Soc.Am. 52, 85 (1962). 23. G.V. Bekesy, J.Opt.Soc.Am. 50, 1060 (1960). 24. A, Florentini, Progress in Optics, ed. E. Wolf (North-Holland Fulaishing Co., Amsterdam) Vol.2, p.255 (1961). 25. H.K. Hartline, F. Ratliff and W.H. Miller, Nervous Inhibition, ed. E. Forey, Pergamon Press, N.Y. (1961). 26. F. Ratliff, H.X. Hartline and W. Miller, J.Opt.Soc.Am. 53, 110 (1963). 27. W.N. Charman, Ph.D. Thesis: "The Visual Sizing of Non- Periodic Objects by Microscopy", University of London (1961). 28. B.M. Watrasiewicz, Opt.Acta 10, 209 (1963). 29. W.N. Charman and B.M. Watrasiewicz, J.Opt.Soc.Am. June 1964 (to be published). 30. F.L.O. Wadsworth, Astrophys.j. 2, 264 (1894). 31. J. Strong, "Modern Physical Laboratory Practice", p.380, Blackie (1938). 32. W.N. Charman, J.Opt.Soc.Am. 21, 410 (1963). 33. P.N. Slater, Optics in Metrology (Pergamon Press, N.Y.) p.269 (1960 . 34. W.N. Charman, Applied Optics 1, 249 (1962). 35. J.A. Dobrowolski, Ph.D. Thesis: "Optical Aspherising by Vacuum Deposition", University of London (1955). 36. W.T. Welford, Optics in Metrology (Pergamon Press Inc., N.Y.), p.85 (1960). 37. W.N. Charman, J.Opt,'Soc.Am. 53, 415 (1963). 38. H. Ostenberg and L,W. smith, J.Opt.Soc.Am. 50, 362 (1960). 39. L.W. Smith, J.Opt.Soc.Am. 50, 369 (1960). 40. S. Slansky, Rev.Opt. 39, 555 (1960). 41. M. De and S.C. Som, Opt-Acta 9, 17 (1962). 42. R. Grant, J.Physiol, (Tuf.-,nd.) 77, 207 (1933). 43. T. Tomita, J.Opt.Soc.Am. 53, 49 (1963). 44. W.N. Charman„ J.Opt.Acta 10, 129 (1963). 45. W.N. Charman, J.R.Micros.Soc. 82, 81 (1963). 46. H.G. Wagner, F.F. MacNichols Jr. and M.L. Wolbarsht, J.Opt.Soc.Am. 53, 66 (1963). 47. W.T. Welford, J.Am.Opt.Soc. 45, 1006 (1955). 48. W.K. Taylor, 3rd Symp. Information Theory, p.314, ed. C. Cherry (1955). 49. Z.A. Melzak, Information & Control 5, 163 (1962). 50. J.M. Enoch, J.Opt.Soc.Am. ILI, 71 (1963). 51. J.C. Bliss; and W.B. Macudy, J.Opt.Soc.Am. 51 1373 (1961). 52. E.M. Lowry,and J.J. de Palma, J.Opt.Soc.Am. 51, 740 (1961), 53. R.B. Mhrimont, J.Opt.Soc.Am. 21, 400 (1963). 225.. 54. H.H. Hopkins, Proc.Phys.Soc. 79, 889 (1962). 55. R. Leelawongs, M.Sc. Thesis: "The Mach Effect in Microscopy and in Vision of Macroscopic Objects", University of London (1964). 56. "A Treatise on the Theory of Bessel Functions", G.N. Watson, p.318, Cambridge University Press (1962). 57. "A Treatise on Bessel Functions", A. Gray and G.B. Mathews, p.2180, Macmillan & Co. (1952). 58. "Fourier Integrals for Practical Application", G.A. Campbell and R.M. Foster, Bell Telephone System Technical Publication, U.S.A. (1931). 59. Tables of Functions, E. Jahnke and F. Ernde, Dover Publications, Inc., N.Y. (1945). 60. J.D.Spooner "Ocular Anatomy" The Hatton Press Ltd.1957. 2Q4.

ACKNOWLEDGEMENTS

I should like to thank Dr. W.T. Welford for suggesting the problem and for his very helpful guidance throughout, Professor W.D. Wright for the use of his laboratories and Dr. H.H. Hopkins for his help with mathematical problems. My thanks are also due to Dr. C.G. Wynne for permission to use the Elliot Computer to the University of London Computer Unit for alloting time on the Mercury Computer, and to Dr. W.N. Charman for the use of some of his drawings. I should also like to thank Miss R. Leelawongs, Dr. T. Moss and Mr. I. Leifer for taking some measurements. Thanks are due to Dr. K.H. Ruddock for measuring the spectral response of my eye and to my wife for help during final stages of editing and for taking some measurements. Finally I am indebted to the National Coal Board and Safety in Mines Research Establishment (Ministry of Power) for financing this research. Reprinted from OPTIeA ACTA, Vol. 10, No.3, p. 209, July 1963

Measurements of the Mach effect in microscopy

by B. M. WATRASIEWICZ Technical Optics Section, Physics Department, Imperial College, London, S.W.7

(Received 23 May 1963)

The subjective light intensity distribution in microscopical images has been measured and compared with the physical images under varying illumi nation conditions. I t is shown that certain fringes which resemble diffraction fringes are not present in the physical image and are in fact manifestations of the Mach effect.

1. INTRODUCTION During the last decade or so it has been shown that contour perception plays a very important role in human vision. For instance, the disappearance of edges during observation with stationary images makes the whole field appear uniform in spite of objective variation of luminance across the field. Further more, contours play an important role in sizing small particles in microscopy. The visual size is very much dependent on the Mach effect occurring at the points of steep variation of luminance gradient. In discussions of the Mach effect it is generally assumed that an observer is presented with luminance gradients of certain steepness (figure 1 (a) and (b), full lines) and he will then perceive a bright band where the variation of second derivatives of luminance

, .. I ' .. ' ......

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Figure 1. Diagrammatic representation of the Mach phenomenon. Objective luminance distribution. ----- Subjective luminance distribution. with respect to distance is maximal and a dark'band wllerethe second derivative is minimal (figure 1 (a) dashed line). Bands will also opcur at the disconti nuities of luminance gradient (figure 1 (b), dashed line). These subjective 210 B. M. Watrasiewicz luminance distributions (dashed lines figure 1 (a) and (b» are referred to as the Mach phenomenon; bright and dark bands are called the Mach bands, after E. Mach [1] who first observed them in 1865. Many measurements of the subjective light distribution and band width have been made on a macroscopic scale [2-10]. These measurements were carried out either in the penumbra formed by the edge of an opaque screen or on artificially produced luminance gradients.

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Figure 2. Multiple band structure of the Mach phenomenon observed in the microscope. Objective luminance distribution. ----- Subjective luminance distribution.

Field of view edge Wi@t the Mach bands 0 photo - multiplier compari~on sl i t

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Figure 3. Block diagram of the apparatus used.

As yet there are no measurements reported of the Mach bands on a micro scopic scale. When the present investigations described below were started, it Measurements of the Mach effect in microscopy 211

was obvious from the very beginning that the Mach phenomenon on the micro scopic scale is different from that observed by other workers. The basic difference is the multiple band structure as shown in figure 2 ( dashed line). At least one bright and two dark bands were always observed on the dark side of the edge. A bright band, similar to the one reported on the macroscopic scale, was seen on the bright side of the edge; occasionally a very narrow dark band appeared adjacent to it, away from the geometrical position of the edge. The present measurements were carried out on the apparatus shown diagram matically in figure 3. The same microscope could be used either for visual observations or for photoelectric scanning of the image. The measurements of light distribution in the Mach bands were made with the help of an auxiliary system imaging a bright comparison slit in the lower half of the field of view (figure 3, insert). Both photoelectric and visual measurements were made at different numerical apertures (NA) of condenser and objective, i.e. at different values of s, where s is the ratio of condenser NA to objective NA. Thus s is a measure of the degree of coherence. The experimental results show clearly that the bands seen in the microscopic images with coherent illumination are not diffraction bands but Mach bands occurring at different positions from diffraction bands.

2. ApPARATUS AND TEST OBJECT The apparatus consists of a monochromator unit, of which the exit slit is imaged by the microscope condenser in the object plane. The geometry is such that the entrance pupil of the condenser is always uniformly illuminated. The photoelectric scanning of the image was carried out by the method described by Charman [14]. For the visual observations no graticule or cross-wire available commercially was found suitable. As soon as an object of an appreciable size was placed near the Mach bands, the observer experienced discomfort and was not able to set the position on the border between the two adjacent bands. All measurements were carried out with a very fine pointer made of wire 1 fL in diameter, placed in the focal plane of the eye-piece, perpendicular to the bands. Light distribution in the Mach bands was measured by photometric matching. The comparison slit, of which the luminance could be varied, was projected into the focal plane of the eyepiece (figure 4). The slit was parallel to and below the image of the edge (figure 3, insert). A straight and sharp edge was prepared by drawing filaments of Formvar (type 1595E) dissolved in chloroform and laying them on the slide. Also some polystyrene spheres of 0·814 f-t in diameter were put on the slide to provide reference points as explained in § 3. Then all was shadowed with an aluminium film as described by Slater [11] and Charman [14,15]. The edge was considered suitable only if it was straight and smooth for about 200 f-t when viewed through an oil-immersion objective; also the reference aperture had to be 4 to 6 microns away from the edge. 3. METHOD The physical image was determined from an average of at least five scans. As there is no convenient reference point on the edge all distances were measured relative to the centre of 0·814 f-t aperture about 5·8 fL away from the edge. 212 B. M.Watrasiewicz

Next the widths of the Mach bands were determined using also a micrometer block system (Charman [14]) and the pointer described previously. The image was displaced until the pointer appeared to be touching the border line between the two adjacent bands. This position was determined by moving the eye along the border in question and judging whether the pointer was in line. This

Ey~.pit:c~ foca \ plane I

projec tio n lens

c:=l==::::J neut ral density wedge

C==:::::J slit t ' monoc hroma ti c Ii gh t

Figure 4. Optical system used to image the comparison slit in the focal plane of the eyepiece.

procedure was repeated at each border. It was found convenient to start at the border A, figure 2, then at borders B, C and D. In this way a set of five readings was obtained. The whole procedure was time-consuming and very fatiguing to the eye, so that a break of at least a few minutes was necessary before taking the next set of readings. The positions of the bands were determined by measuring the distance between the centre of the reference aperture and the border B. The widths of wider bands were measured with an accuracy of 5 per cent to 10 per cent ; the narrower with an accuracy of 7 per cent to 15 per cent (see figure 7 and table). The distance of bands from the centre of the reference aperture was measured with an accuracy of 0·05 {J, in about 5·8 {J,. The average of at least ten readings was taken to determine widths and an average of five to determine the position of bands with respect to the centre of the aperture. Finally photometric matching was carried out by varying the luminance of the comparison slit. A neutral density wedge (figure 4) was used for this purpose. At first the slit was matched to the clear field, then with the bright bands and at the end with the dark backgro:und. No matChing was possible with Measurements of the Mach effect in microscopy 213 the dark bands, due to the presence of scattered light in the system. The average of five or six consecutive readings was taken; the standard deviation was just under 10 per cent.

4 . EXPERIMENTAL RESULTS All measurements were made with a 1·32 NA x 100 Zeiss apochromatic objective, using green light of wavelength 5300 ± 50 A. It was found necessary to select this objective with some care as being very well corrected on axis ; other objectives with poorer aberration corrections did not give consistent appearances in the experiment. For visual observations x 20 and x 10 Ramsden eyepieces were used; the latter was only used for a condenser aperture corres ponding to s = 2. All measurements were carried out at approximately the same luminance level. Normalized transmissions (with respect to clear field transmission) were plotted as ordinates; distances from the position of the geometrical edge were plotted as abscissae. Distances perpendicular to the edge can be given in terms of three different but equivalent units; these are (a) linear units in the object space, (b) dimension less diffraction units and (c) angular subtense in the image space of the microscope. For convenience all three are given here; the lengths are given in microns; the diffraction units are defined by the relation 27T z = X- (NA)l), where 1} is the linear measure in the object space; angular subtenses are given in minutes of arc. Figure 5 (a) to (d) shows the physical and the corresponding subjective luminance distributions in the image of the straight edge, for four values of s(s =0·2, 0'5, 1·0, 2·0). Figure 6 shows only the subjective lumiI',ance distribu tion for the same values of s. All results are summarized diagrammatically in figure 7 and table. Percentages refer to the maximum transmission of the band ; tolerances are standard deviations.

5. DISCUSSION The appearance of the subjective visual image with narrow cone illumination (s = 0·2) is the most striking feature that emerges from the measurements described above. The dark and bright bands seen have been assumed by micro scopists to be diffraction bands due to the coherent illumination of the object. However, figures 5 and 7 show that the eye does not perceive exactly the physical diffraction bands; also bands are seen in the dark side of the edge and these could not occur in the physical image. As yet it is not possible to draw any definite conclusions about the position of the Mach bands with respect to the geometrical edge because the Mach phenomenon is essentially non-linear and depends on many parameters. A simple proof of non-linearity was given recently by Marimont [16]. There are three important parameters that will influence the position and widths of the Mach bands. Namely, luminance level, contrast and luminance gradient. Band widths, for instance, will increase with the increase of luminance; this was shown by McCollough [10]. 214 B. M. Watrasiewicz

140 140

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Z-UNITS

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Figure 5. Luminance distribution in physical and subjective images of a straight edge at different illuminations. Objective luminance distribution. Subjective image distribution. Measurements of the Mach effect in microscopy 215

The multiple band structure of the Mach phenomenon shown in figures 5 and 6 has been seen by many observers with the apparatus described above. The subjective luminance distribution was measured by one other observer ; the agreement was within 10 per cent. The retinal inhibition mechanisms, which were assumed by many workers to be responsible for the contour perception, and which were applied by Fry [13, 12] and Bekesy [17] to explain the simple Mach phenomenon can be used to explain the multiple band structure of the Mach phenomenon provided that the approach of Ratliff et al. [18] is followed.

140

130

120

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Figure 6. Subjective images of a straight edge at different illuminations.

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-.9 -·8 -;J ,,6 -.5 -,4 -:3 -:2 -.1 0 .2 .3 .4 .S .6 .7 .8 .9

Figure 7. Schematic representation of the relative position of the Mach bands at different illuminations with respect to the geometrical edge. Shaded areas represent the dark bands. 216 Measurements of the Mach effect zn microscopy

Band widths (microns) Relative brightness Bands (per cent) " s=0'2 s=0'5 s=l s=2 s=0·2 s=0'5 s=l s= 2 ------Dark band oil bright field 0·090 ± 0·018 Bright band on bright field 0·327 ± 0·022 0·235 ± 0·021 0·168 ± 0·031 0-463 ± 0·029 146·7 119·3 123 149 Dark band on dark field 0·150 ± 0·022 0·157 ± 0·014 0·104 ± 0·005 0'324± 0·066 Bright band on dark field 0·154 ± 0·024 0·175 ± 0·019 0·180 ± 0·034 0·344 ± 0·016 68 37·7 14 39 Dark band on dark field 0·117 ± 0·006 0·064 ± 0·021 0·038 ± 0·005 0·250 ± 0·040 Dark background 13·9 14·5 6 6· 6

ACKNOWLEDGMENTS I should like to thank Dr. W. T. Welford for suggesting this problem and for constant help during the progress of this work; also Mr. I. Leifer for taking some observations. Acknowledgment is made to the National Coal Board for· a grant in aid of this research. The opinions expressed are those of the author and not necessarily those of the Board.

La distribution subjective d'intensite dans les images donnees par Ie microscope a ete mesuree et comparee aux images physiques dans des conditions d'eclairement variees. On montre que certaines franges, qui r,~ssemblent aux franges de diffraction n'existent pas dans l'image physique et sont en fait cies manifestations de l'effet Mach. Es wurde die subjektive Helligkeitsverteilung in mikroskopischen Bildern gem essen und mit den physikalischen Werten verglichen, wobei die Beleuchtungsverhaltnisse verandert wurden. Gewisse Streifen, welchewie Beugungsstreifen aussehen, fehlen dem physikalischen Bilde und erweisen sich als Erscheinungen des Mach-Effektes.

REFERENCES [1] MACH, E., 1865, S.B. Wien Akad. Wiss., 52/2, 303. [2] FIORENTINI, A., 1956, Prob. Con. Opt. (Instituto Nationale Di attica), 600. [3] FIORENTINI, A., JEANNE, M., and TORALDO di FRANCIA, G., 1955, Opt. Acta, 1, 192. [4] McDoUGAL, W., 1903, Proc. physiol. Soc., 1, 19. [5] FRY, G. A., and BARTLEY, S. H., 1935, Amer. J. Physiol., 112, 414. [6] TORALDO DI FRANCIA, G., 1952, Atti Fond. ROl1Chi, 7, 324. [7] FIORENTINI, A., and ERCOLES, A. M., 1957, Opt. Acta, 4, lSI. [8] ERCOLES, A. M., and FIORENTINI, A., 1959, Atti Fond. ROllChi, 14, 230. [9] FIORENTINI, A., and ERCOLES, A. M., 1956, Prob. Con. Opt. (Institllto Nationale Di attica), 624. [10] MCCOLLOUGH, C., 1955, J. expo Pyschol., 49, 14I. [11] SLATER, P. N., 1960, Optics in Metrology (New York :- Pergamon Press Inc.), p. 269. [12] FRY, G. A., 1948, Amer. J. Opt., 25, 162. [13] FRY, G. A., 1963, J. opt. Soc. Amer., 53, 94. [14] CHARMAN, W. N., 1963, J. opt. Soc. Amer., 53, 410. [15] CHARMAN, W. N., 1963, J. opt. Soc. Amer., 53, 415. [16] MARIMONT, R. B., 1963, J. opt. Soc. Amer., 53, 400. [17] BEKESY, G. V., 1960, J. opt. Soc. Amer., 50, 1060. [18] RATLIFF, F., HARTLINE, H. K., and MILLER, W. H., 1963, J. opt. Soc. Amer., 53,110. Reprinted from JOURNAL OF THE OPTICAL SOCIETY OF AMERICA, Vol. 54, No. 6, 791-795, June 1964 Printed in U. S. A. Mach Effect Associated with Microscope Images

W. N. CHARMAN* AND B. M. WATRASIEWICZ Technical Optics Section, Imperial College, London S. W. 7, England (Received 14 November 1963)

The distributions of subjective brightness in the microscope images of an opaque straight edge under various conditions of illumination are compared with the corresponding physical luminance distributions. The importance of the contour-enhancing function of the eye during visual observations with the microscope is shown.

INTRODUCTION luminance distribution. In general, the Mach effect N recent years, considerable attention has been results in a subjective sharpening of the edge gradients; I devoted to the role of the Mach effect in the visual bright and dark bands may be seen if the luminance perception of contours (see, for example, the review by gradients across the edge change rapidly enough. From Fiorentinil). Due to this effect, blurred boundaries be- the point of view of the microscopist, such a subjective tween regions of differing luminance appear much enhancement of the edge sharpness may be beneficial; sharper in the subjective image seen by the observer it may aid the detection and recognition of detail in the than they are in the external luminance distribution or image. It will, moreover, improve the precision, though in the distribution of light on the observer's retina. not the accuracy, of measurements of object size made Most investigators of the phenomenon have attempted with the microscope. Disadvantages arise, however, if to correlate a known, macroscopic, external luminance purely subjective bright or dark fringes are seen, since distribution (shown typically by the dashed line in these may be falsely identified as representing genuine Fig. 1) with the distribution of subjective brightness detail in the object, such as a surrounding membrane; perceived by the observer (full line, Fig. 1). The cor- considerable care is therefore necessary in interpreting responding spatial variation in retinal illuminance the image. Striking examples of the importance of these (dotted line, Fig. 1) has, at best, only been known physiological effects have been encountered in investiga- approximately. tions of the images of high-contrast objects under various conditions of illumination. The influence of the Mach effect during visual observations with optical instruments has received less comment. Fleury' has, however, stressed the possible METHOD AND EXPERIMENTAL RESULTS role of the effect in connection with the general problem The object used for most of the experiments was a of contrast perception with the microscope and very thin, high-contrast, straight edge between transparent recently Watrasiewiez' has shown that certain fringes and opaque fields. It was imaged in light of wavelength which are observed visually in microscope images are X=0.53±0.05 ,u by an almost perfectly corrected ob- due to the Mach effect and are not genuine diffraction jective of numerical aperture (N.A.) 0.28, which was fringes present in the physical image. stopped down to an N.A. of 0.12 and used to give a The potential importance of the Mach effect in the primary magnification of about 13X. particular case of visual microscopy is easily understood. The experimental technique used to determine the Diffraction and aberration cause sharp edges of areas distributions of light flux in the edge images formed by of different but uniform transmittances to be imaged as the objective at its long conjugate when various ratios zones of finite width and varying luminance between areas of different but uniform luminance; the precise distributions of luminance in these zones of the image depend upon the optical conditions in each case. These luminance distributions resemble that shown in Fig. 1. When such a distribution is viewed with, for example, an eyepiece, the perceived distribution of subjective brightness will not correspond in any simple way to the

* Now with the Applied Physics Division, National Research Council, Ottawa, Ontario. A. Fiorentini Progress in Optics, edited by E. Wolf (North- Holland Publishing Company, Amsterdam, The Netherlands, 1961), p. 282. DISTANCE 2 P. Fleury, Contraste de Phase et Contrasts par Interferences, FIG. 1. The Mach effect associated with a changing luminance edited by M. Francon (Editions Revue d'Optique, Paris, 1951), gradiant. --- luminance distribution; • • • distribution of retinal p. 15. illuminance, subjective brightness distribution; M Mach 3 B. M. Watrasiewicz, Opt. Acta 10, 209 (1963). bands. 791

June 1964 MACH EFFECT IN MICROSCOPE IMAGES 793 slit being parallel to the edge; the slit width subtended about 2° at the eye. The luminance of the slit necessary X 10 to give a brightness match between the slit and each 120 co Mach band in the edge image was taken as a measure of co 100 the subjective brightness of that band; the brightnesses were normalized with respect to that of the clear field X 20 22m 80 remote from the edge. As the angular sub tenses of the Mach bands were very small (about 5') it was only 60 possible to make a brightness match at the central por- X 50 tion of each band where the brightness was approximately constant.' The standard deviation of repeated 40

measurements of this type was about 10% of the mean 20 and a similar degree of precision was found in the measurements of the widths of the bright and dark

bands for both observers. The full lines of Fig. 2(a) -5 -4 -3 -2 0 2 3 to (d), show the subjective brightness distributions MICRONS deduced from the observations of B. M. W. FIG. 4. Change in subjective brightness distribution in the image of an opaque straight edge when the viewing magnification is It appears from the results found by the two observers changed. Objective N.A.= 0.12, S=1.0, X=0.53 p, primary mag- that, although the distributions of subjective brightness nification 13X, eyepiece power as indicated, observer B.M.W. perceived by different individuals when viewing the same physical microscope image under the same condi- measured by these observers. Variations in the appear- tions are qualitatively the same, considerable differences ance of the Mach bands to different individuals, cor- in the detail of the subjective image occur. This is responding to those found in the present experiments, illustrated by Fig. 3, which indicates diagrammatically have been reported in earlier macroscopic studies of the the positions of the bright and dark bands seen by effect.6•7 Since in the present case the angular subtense W.N.C. and B.M.W. under the same conditions. It of the edge images is very small, the presence of in- originally seemed possible that these differences might dividual variations of the same type as are found in be due to the slightly different measuring techniques acuity experiments is not surprising. used by the two observers but careful checks showed that this was unlikely. A third observer (R.L.) therefore One feature of Fig. 3 deserves comment. The results repeated the same measurements using the same tech- of all three observers show that the main dark to light nique as that employed by B.M.W. Her results differed edge, which would normally be chosen by an observer from those of both the other observers, as is shown as representing the edge position in the image, moves in Fig. 3. Both R.L. and W.N.C. had difficulty in steadily towards the lighter side of the true geometrical measuring the positions of the bands on the dark side image of the edge as the condenser is stopped down. This of the edge; the dashed lines in Fig. 3 indicate qualita- appears to be in agreement with results reported by tively the positions of bands which were seen but not Charmang who found that reducing the value of S decreased the visually measured diameter of bright disks in an opaque surround and increased it for opaque I I disks. R.L. Measurements were made by one observer (B.M.W.) r „t o , o 1 • 0.08 of the subjective brightness distribution in the edge o i o 1 • 032 L",k, \'' o p o I • 0.60 image at a constant level of retinal illuminance (104 L tt 01 1.00 V...s R\ N 0 1 5.7 trolands) when a value of S= 1.0 and eyepiece powers W.N.C. 10X, 20X, and 50X were used. The results found are o 10.08 shown in Fig. 4. They indicate that the main edge be- o Ak .. 0.32 O x'. 0.60 tween the predominantly dark and light halves of the o I.00 o 1 5.7 field moves towards the bright side of the edge as the B.M.W. eyepiece power is increased. 00 av 0.08 0-32 0 1 • 0.60 0I I.00 DISCUSSION o 1 5.7 1 I 1 I During these experiments, the diameter of the exit -6 -4 -2 0 2 4 6 8 MICRONS pupil of the eyepiece (about 0.23 mm for the,_20X FIG. 3. Schematic representation of the positions of the Mach bands seen by the three observers under the various conditions of illumination. The shaded areas represent dark bands, the full and 6 C. McCollough, J. Exptl. Psycho'. 49, 142 (1955). open circles points where the second derivative of transmittance 7 A. Fiorentini, Atti. Fond. G. Ronchi, 12, 180 (1957). is a maximum or minimum, respectively. W. N. Charman, J. Roy. Micr. Soc. 82, 81 (1963). June 1464 MACH EFFECT IN MICROSCOPE IMAGES 795

The results described indicate the strong influence of ACKNOWLEDGMENTS physiological effects during visual observations with the Our thanks are due to Dr. W. T. Welford for much microscope. The importance of such effects may be constructive criticism and advice and to Miss R. expected to extend to many other optical instruments in Leelawongs for assistance with the observations. Ack- which images are observed visually. It is suggested that nowledgement is made to the National Coal Board for techniques similar to those described, in which the a grant in aid of this research. The opinions expressed distribution of retinal illuminance is known, might prove are those of the authors and not necessarily those of the useful in further elucidating the Mach effect. Board.