# Contributed Numbers

### Psychometric function slopes (Burr)

From: David Burr <dave@neuro.in.pi.cnr.it>

What an interesting idea. I'd like to see the list when you've got it.

One obvious number is that the slope of the psychometric function for
almost anything (on log co-ords) is 3. (Weibull function) or a sigma
of about 4 dBs for a cummulative Gaussian. I guess Pelli showed this for
contrast, but it seems to work for orientation, signal-to-noise measures,
vernier or whatever, provided you taker logs, although I don't know if it's
been formally documented.

### Eccentricity (Anderson)

From: cha@shifter.wustl.edu (Charles H. Anderson)

>I am interested in collecting up some experimentally obtained numbers
>concerning human visual performance that people have found useful in
>designing experiments or in thinking about models of human vision.

Brian,

Here are two factors that play a major role in my thinking about human
visual performance. The first deals with how visual acuity changes
with eccentricity, the second with the spatial extent of the window of
visual attention.

The resolution, dE, at the retinal ganglion cell level, primarily Parvo,
changes with eccentricity according to the equation, with all units
in degrees.

dE ~ a(E+E0) ~ 0.01(E+1.3)

These numbers are for the macaque monkey. My guess for human values
is that a~0.006-0.008, and E0 is somewhere between 1.0 and 1.5 for
both monkey and humans. Visual acuity drops by a factor of 2 at
and eccentricity of E0~1.0 degrees. The linear fall off in resolution
reflects the underlying scale invariance of the system.

The anisotropy with angle around the fovea is small except along
the horizontal meridian toward the blind spot, where the resolution
could be a factor of 2 smaller in size, i.e. higher acuity.

As I recall this is quite good out to 40 to 50 degrees.

One has to be careful not to confuse this equation with grating
threshold sensitivity since the Magno system, more specifically the
non-linear subpopulation of the Magno system, can respond to higher
spatial frequencies, especially in the periphery. This response
however is only to the power, all phase information is lost. I have
estimated that information carried in the optic nerve as "dynamic
texture", i.e. the power of high temporal-spatial frequencies,
exceeds that of color. This information is used in manner similar
as color is for both searching for targets as well as aiding in
recognition. It also bypasses the cognitive system altogether and
is used to initiate protective reflexes against in coming objects.

Another interesting numberical factor is the sampling spacing of the
Magno system is about a factor of 3 larger than that of the Parvo
system at all eccentrities. There is some data, which Van Essen
believes, that suggests this rises to about 5 in the center of the
fovea.

Reference:

Van Essen, D.C. and Anderson, C.H., Information processing
strategies and pathways in the primate retina and visual cortex'',
in Introduction to Neural and Electronic Networks ,
eds. S.F.  Zornetzer,    J.L. Davis, and C. Lau,

An updated revised version will be available shortly.

The second factor is that the relative spatial extent of the window of
visual attention spans a diameter of about 30 nodes, where the
absolute spatial size of the nodes depends on the scale at which
covert attention is set. I first settled on this factor using the
observation that grating sensitivity increases with the number of
cycles displayed up to about 10-15 cycles. To first order this is
independent of spatial frequency. Using the Nyquist limit, this means
the number of samples involved in the analysis is about 20-30. This is
much smaller than the number of cycles you can actually perceive. The
equation above would suggest that the number of resolvable points
would be = 2*E/dE ~ 200!

Since then I have found many examples that show there is
something special about a spatial extent of 30 nodes. George
Sperling has shown that sign language can be perceived with
better than 90% accuracy using a display 28 by 20 pixels in size.
Note that this holds irrespective of the distance between the
viewer and the screen. We have summarized some of the supporting
experiments in

Van Essen, D.C., Olshausen, B., Anderson, C.H. and Gallant, J.L.,
Pattern recognition, attention, and
information bottlenecks in the primate visual system.''
In:  Proc. SPIE Conf. on Visual Information Processing:
From Neurons to Chips, vol. 1473, 17-28, 1991.

The machine vision project I initiated at the David Sarnoff Labs
back in 1983 was inspired by these observations. The neurobiological
circuitry for processing visual information using a window of
attention that can be translated and zoomed in scale have been
reported in.

Olshausen, B., Anderson, C. and Van Essen D., A
neurobiological model of visual attention and invariant pattern
recognition based on dynamic routing of information'',
J. Neuroscience, {\bf 13},pp. 4700-4719, 1993.

A second more detailed paper has been accepted for publication in
the Journal of Computational Neuroscience.

"A Multiscale Dynamic Routing Circuit for Forming Size- and
Position-Invariant Object Representations"
Bruno A. Olshausen, Charles H. Anderson, and David C. Van Essen

I believe there will also be a paper appearing in Science shortly by
Ed Conner, Van Essen, and Gallant on experiments showing cells in area
V4 encode information in a local coordinate frame that moves with
covert attention.

### Attention (Biederman)

From: ib@rana.usc.edu (Irving Biederman)

Dear Brian:

A number that I have found useful--and an important constraint in
trying to understand models of visual recognition--is 100 msec.  It is the
exposure duration, followed by a mask, required to recognize an object or
scene.  This value is not noticeablly increased (if at all) if RSVP
presentations are employed rather than single trial presentations.  Also,
in RSVP a negative detection task ("Press the key if you see a picture that
is NOT a mode of transportation") is not much more difficult than
positively specified basic level classes (e.g., "Hit the key if you see a
chair").

References:

Single trial:

Biederman, I., Rabinowitz J., Glass, A. L., \& Stacy, E. W., Jr.  (1974).
On the information extracted from a glance at a scene.  Journal of
Experimental Psychology, 103, 597-600.

RSVP (Good review):

Intraub, H.  (1981).  Identification and naming of briefly glimpsed visual
scenes.  In D. F. Fisher, R. A. Monty, and J. W. Senders (Eds.).
Eyemovements:  Cognition and Visual Perception (Pp. 181-190).  Hillsdale,
N. J.:  Erlbaum.

While we are in the temporal domain, another result that I find
useful is that cells in the anterior reaches of IT show tuned responding
(e.g., if a face cell to a picture of a face) in under 100 msec from the
presentation of the picture.  Reference:

Baylis, G. C., Rolls, E. T., and Leonard, C. M.  (1987).  Functional
subdivisions of the temporal lobe neurocortex. Journal of Neuroscience, 7,
330-342.

Though these cells will continue to fire for about 400 msec, much
of their information (as estimated from a population code of face cells) is
in the first 50 msec.  Reference:

Tovee, M. J., \& Rolls, E. T.  (1995?)  Information encoding in short firing
rate epochs by single neurons in the primate temporal visual cortex.
Visual Cognition, in press (?)

As for a non-temporal constant:

Only two or three simple parts (in their appropriate relation)
rather than the whole object are almost always sufficient for basic level
classification.  Ref:

Biederman, I. (1987).  Recognition-by-Components:  A Theory of Human Image
Understanding.  Psychological Review, 94, 115-147.

I liked the two numbers you mentioned.  (By the way, is the value
for the number of cones per degree of visual angle only at a .5 deg radius
around fixation, rather than throughout the cone regions of the eye?)

If it would not be too much trouble, I would enjoy seeing your
final collection.

Hope all is going well with you.  My best wishes for the New Year,

Irv

### Visual angles (Teller)

From: Davida Teller <dteller@u.washington.edu>

Brian -- I don't know if this qualifies, but here is my favorite "Rule of
Thumb":  your thumbnail at arm's length is a bout a degree of visual
angle!  Also, the sun and the moon each subtend about 1/2 degree.

In another area, by behavioral testing, an infant's acuity in
cycles/degree is roughly numerically equal to its age in months (5
months, 5 c/d, etc).  dt

### Visual angles (Brown)

From: dsbrown@hkpcc.hkp.hk (Brian Brown)

Brian

subtenses.... the 'Rule of thumb'.. is that the thumbnail at arms length
covers about 1.25 degrees (or at least mine does), and can be used then to
estimate angular sizes and distances in the real world for any object that
can be aligned with the thumb.
And of course 1cm at 57 cm is 1 degree.

Your human lens (60 Diopters) is wrong... that's the total power of the
eye, cornea, lens and effectivity of the various optical components.

Here are a few more:

Corneal power: about 42D (as you see, leaving only 18D for the lens and
effectivity of the corneal power and lens power)
Corneal diameter: 12 mm
Change in refractive power: about  3D/mm change in axial length

Time taken for a saccadic eye movement in ms:

20 + twice the amplitude in degrees,

Robinson DA. The mechanics of human saccadic eye movement. J Physiol 180,
569-590, 1964.

Carl Zeiss Co
7082 Oberkochen
West Germany

### Visual angles (Knoblauch)

From: Ken Knoblauch <Ken.Knoblauch@cismibm.univ-lyon1.fr>

The constant that I run into most frequently lately is 57.3 cm, which
is the distance at which 1 cm subtends 1 deg, but everyone knows that.

### Units (Mulligan)

From: "Jeffrey B. Mulligan" <jbm@vision.arc.nasa.gov>

candelas per m^2  x area of pupil in mm^2 = photopic trolands

approximate relative luminous efficiencies of typical rgb monitor:
R:G:B = 3:6:1

### Cortex and motion (Britten)

From: khbritten@ucdavis.edu (Ken Britten)

10^6 fibers/optic nerve (OK, I don't use it in my work, but it's such a nice
round number)

30 cortical areas for vision

approx 1 degree spatial scale for "short-range" motion, centrally. (Braddick)

V1 RF size = .1 * eccentricity

MT RF size = .8 * eccentricity (Maunsell \& Van Essen)

OK, that comes from the top of my head, maybe more will bubble up later...

### Chromatic aberration (Howarth)

From: Peter A Howarth <P.Howarth@lut.ac.uk>

2  Diopters    .......  the approximate amount of longitudinal
chromatic aberration of the human eye over the visible spectrum

The reference I like is Howarth and Bradley, Vision Research 26, 361-366, 1986
but probably a better one is Wald and Griffin 1947, JOSA, 37, 321-336

From: mike@monkeybiz.stanford.edu (Mike Shadlen)

Brian,
Here are some #s for you -- I suspect you already know these well.

120,000 neurons per mm^3 in monkey striate cortex
or 200,000 under 1 mm^2 of cortical surface.
(O'Kusky and Colonnier, J Comp Neurol 1982, 210:278)
(see also, Peters' chapter in Cerebral cortex, Vol 6, Joes and Peters
eds. 1987, New York, Plenum)

This one's sort of obscure, but I think about it a lot:
150 neurons in a microcolumn of cortex defined by a cylinder the
diameter of a layer 5 pyramidal cell's dendritic tree.
The real numbers are 143 for monkey and 203 for cat.
(Peters and Yilmaz, Cerebral Cortex 1993, 3:49-68)
(Peters and Sethares, J Comp Neurol 1991, 306:1-23)

### Search (Wolfe)

From: wolfe@search.bwh.harvard.edu (Jeremy Wolfe)

Hi Brian

My favorite number these days is 50 msec/item as an
estimate of the rate at which serial attention moves from
item to item in visual search. There are various theoretical
complications surrounding this but it does useful work for me.

Can I get a copy of the list that you compile? What is it
for, anyway?

jeremy

### Cortex (Kersten)

From: Daniel Kersten <kersten@mach.psych.umn.edu>

Hi Brian,

I don't know if this really satisfies the "usefulness" criteria as
well as #cones/deg. You may be familiar with Cherniak's article below
that was stimulated by the discrepancies he noted in the literature
citations regarding how big the human cortex is (area estimates differed by a
factor of 10, I believe...again this discrepancy probably has more
to do with "who really cares", than any inherent difficult in making
the estimates more precise.)

If you plan on publishing any of these, they should probably be
double-checked. I just pulled them off of an old spreadsheet I filled
out when I first read Cherniak's article.

If one assumes 40% of the connections carry visual information (a
visual scientist estimate, linguists probably put it closer to 1%),
one can calculate that there are 1,990  miles
of visual connections--a number neither very useful or very
believable--but may be fun to quote.

------------------------

Cherniak,  J. of Cog. Neurosc., 1990, vol 2., pp 58-68

Area of cortex	160000	mm^2
Thickness of cortex	2	mm
Volume of cortex = 	320000	mm^3			0.32	liters
Cortex synapse density	4000	synapse/neuron
Cortex connectivity	200000000	synapses/mm^3
connectivity/neuron	5	mm
connection length/mm^3	25000	mm
neuron density in cortex	50000	neurons/mm^3

Total brain volume					1.4	liters

Total neurons in cortex	1.60E+10

From: aadams@garnet.berkeley.edu (Anthony Adams)

Brian,
I'll give it some thought
Incidently 60D is about the power of the human EYE- not the human lens
(which is only about 9-11D IN the eye.
Tony

optic nerve head is 5x7degrees angle (Vertical=7)
20/200 letter )'the big E") is just under one degree angular subtense (50
min exactly)

### Optics (Levick)

From: William.Levick@anu.edu.au (William Levick)

Brian: You might consider including the following formula, although it is
not strictly based on experimentally measured quantities:

B = dE     (approximate, for relatively small E)

where:  B (radians)is the angular diameter of out-of-focus blur circle
d (meters) is the diameter of the entrance pupil
E (diopters) is the out-of focus error of the eye.

It is probably well known but I was not aware of a citation when I presented
it in:

Levick, W.R. (1972).  Receptive fields of retinal ganglion cells.  Pp.
531-566 in: Handbook of Sensory Physiology, Vol VII/2. Physiology of
Photoreceptor Organs, ed. by M.G.F. Fuortes.  Springer Verlag: Berlin.

A simplified form, together with other data is given on pp. 538-539.

Bill Levick

### Visual angles (O’Shea)

From: "Robert P. O'Shea" <R-OSHEA@rivendell.otago.ac.nz>

The visual angle of the first joint of the thumb held
at arm's length is approximately 2 deg.

O'Shea, R. P. (1991). Thumb's rule tested: Visual angle of thumb's
width is about 2 deg. Perception, 20, 415-418.

(By the way, if my memory serves, Helmholtz is the source of the
number: 120 cones per degree of visual angle.

Helmholtz, H. (1873). The recent progress of the theory of vision. In
Popular lectures on scientific subjects (pp. 197-316). London:
Longmans, Green \& Co.)

Cheers,
Robert.

### Retina (Schein)

From: Stan Schein <stan@psych.ucla.edu>

Dear Brian,			1/5/95

A number that is related to 120 cones/deg of visual angle is foveal
retinal magnification, which is about 290 um/deg in human and 210
um/deg in macaque monkey.  (I specify foveal, because the distance
from the posterior nodal point to the retina declines with
eccentricity, so retinal magnification does proportionately.

### Stereo, visual sensitivity, many others (Backus)

From: ben@john.Berkeley.EDU (Ben Backus)

Dear Brian,

What a good idea.  I hope you plan to make your collection available soon.

Useful vision numbers:
The stereoscopic threshold for a step in depth on a surface is 3 sec of
arc.  The threshold for detecting that points are not coplanar is 30 sec.
[Stevenson, S. B., Cormack, L. K., \& Schor, C. M. (1989). Hyperacuity,
superresolution and gap resolution in human stereopsis. Vision Res, 29(11),
1597-605.]
The eyes are 6 cm apart
Best contrast sensitivity is at 3 cycles/deg. [Van Nes, F.L., \& Bouman,
M.A. (1967). Spatial modulation tranfer in the human eye. Journal of the
Optical Society of America, 57(3), 401-406.]
Max spatial resolution is 60 cycles/deg [Campbell \& Green, 1965?]
Visible spectrum runs 400-700 nm

Other useful numbers:
A 1 cm wide object at 57 cm distance subtends 1 degree of visual angle;
the width of one's thumb in cm is its angular subtense in degrees at arm's
length.
Luminance in cd/m^2 of starlight is 10^-3, moonlight is 10^-1, indoor
lighting is 10^2, and sunlight is 10^5.  [Hood \& Finkelstein, Ch. 5 in some
book from which I have an unlabeled reprint.  Maybe it's: Handbook of
perception and human performance / editors, Kenneth R. Boff, Lloyd Kaufman,
James P. Thomas. New York : Wiley, c1986.]

Useful numbers I don't remember but hope will be in your collection:
The field of view for both eyes together
The binocular region's field of view
Axial length of eyeball
Rod and cone integration times
Easy way to remember how much light a troland is
Wavelength of peak sensitivities of L, M, and S cones
Fastest simple RT for flash detection;  Typical V1 neuron's RT to a
flash of light.
Size of a V1 foveal hypercolumn in deg visual angle
Number of different V1 hypercolumns used to tile retinotopic visual space

Hope I've helped.

--Ben

### Time, cones, solid angles (Nilsson)

From: Thomy Nilsson <NILSSON@upei.ca>

Dear Brian,
Here is a less common "fact".  The minimum interval at which two brief
(1 ms), small (30') pulses of light can be discriminated from a single
equal energy pulse is about 15- 20 ms and at photpic luminance seems not
to vary with luminance.  (Vision Res '79, ARVO '92).
Your data on the number of cones per degree visual angle is not clear.
Do you mean the diameters of 120 cones across a distance on the retina
subtended by 1 degree  or  an area on the retinae with a diameter equal to
1 degree?
When it come to solid angles, here is what I've found to be a
straight out infront of you; bend the elbo 90 degrees; now rotate the bent
portion as much as possible about the axis of the straight portion.  The
resulting area that would be swept by a 180 sweep with a center at the mid
big.  Smaller or larger people's portions of arm lengths seem to maintain
this principle.
Would you send me list of the "facts' when it is completed?  Thanks,
Thomy

### Numbers (Cormack)

From: CORMACK@PSYVAX.PSY.UTEXAS.EDU (Lawrence K. Cormack)

Brian,

I constantly find myself using:
360/2=PI  or approx. 57.3

I believe many of us use a multiple of this as the viewing
distance in our experiments for obvious reasons.  I
don't really know a ref., but I suspect it would be Euclid.

-Lar

----------------
Lawrence K. Cormack
University of Texas
ph.  (512) 471-1649
fax: (512) 471-5935

### Optics (Moreland)

From: "J.D. Moreland"  <coa09@cc.keele.ac.uk>

Dear Brian,

The use of age-matched controls is good practice in many clinical studies
but is not appropriate for diabetes since lens absorbance changes are
accelerated. Ideally, lens absorbance should be matched individually but,
where the required facilities for this are not available, simple formulae
may be utilised to define lens-equivalent age controls.

Thus:

E = A + 2.54T - 3.8           for 20 <= E <= 60

E = 0.30A + 0.76T + 40.9      for 60 < E  < 80

where E is the age of a normal lens having the same absorbance spectrum as
that of a diabetic patient of age A and with a diabetes duration T > 1.5.

For T up to 1.5, E = A.

E, A and T are all in years.

See:  J D Moreland.  "Lens-equivalent age controls for diabetics".
Invest Ophthalmol Vis Sci  1993, 34, 281-282. (Letter to the Editor)

With best wishes,

Jack Moreland.

### Visual angles (Bach)

From: bach@sun1.ruf.uni-freiburg.de (Michael Bach)

I like the simple equation: at 57cm distance (e.g. to your stimulus) one
degree visual angle covers 1 centimeter.

Good idea, greetings, Michael.

Dr. Michael Bach, Dept. of Ophthalmology, University of Freiburg,
D-79106 Freiburg, Germany. ++049(761)270-4060. bach@sun1.ruf.uni-freiburg.de

From: Al Ahumada <al@vision.arc.nasa.gov>

Model number of eyes per subject: 2

### Visual angles (Tootell)

From: roger@nmr-m.mgh.harvard.edu (Roger Tootell)

Brian:

A conversion factor that I use alot, and which does not seem to be as
widely known as it should be, is:
1 Diopter = 0.57 degrees.  Another handy factoid known to most
physiologists is that at 57 cm, one degree is one centimeter wide; at 57
inches one degree is one inch wide, etc.  If this collection is for your
book, it sounds like a good idea.

1 diopter refracts light by 0.57 degrees.  Useful when calculating the
effect of various lenses and prisms.

Roger

### Binocular vision (Pietsch)

From: reading@ucs.indiana.edu

My favorite one involves the enhancement of visual performance gained by
going binocular.  It applies to absolute light detection thresholds,
brightness discrimination, critical flicker frequency, and visual acuity.
The gain in sensitivity is always [more or less] equal to the square root
of 2!  You can arrive at this result through sampling  theory,probability
summation, or areal summation [an adaptation of Piper's law to binocular
vision]. One of the implications of this approach is that a three eyed
individual should enjoy an enhancement equal to the square root of three!
[as suggested by the late Fergus Campbell}

### Optics (Spillmann)

From: Lothar Spillmann <spillman@ruf.uni-freiburg.de>

Gullstrand's schematic eye puts the refractive power of the
human cornea at 43.05 dptrs and that of the lens at 20.28 (unaccommodated)
and 30.13 dptrs (fully accommodated), respectively. The result of accommodation
is to change the EQUIVALENT power of the eye as a whole from 59.60 dptrs
to 68.22 dptrs, a difference of 8.62 dptrs.

Source: Davson, H. (Ed.): The Eye, vol. 4, page 105 (1962), Academic
Press, New York-London.

Lothar Spillmann, Phone (49-761) 270-9572, Fax (49-761) 270-9500
Arbeitsgruppe Hirnforschung, Institut fuer Biophysik
Hansastrasse 9, D-79104 Freiburg, Germany

### Optics (Enoch)

From: jmenoch@garnet.berkeley.edu (Jay Enoch)

In studies where the Stiles-Crawford effect is of importance, or
in ocular ray tracing, a simple relationship relating displacement
in the entrance pupil of the eye to change in the angle of incidence
at the retina is as follows:

1.0 mm in the entrance pupil of the eye = 2.5 degrees change in the
angle of incidence

I first encountered this derivation (based on the emmetropized
Gullstrand eye) in a monograph produced for the Air Force by
Brian O'Brien and Norma Miller (post 2nd World War).  I have
separately derived it an reported it in many papers (including
my book on Vertebrate Photoreceptor Optics).  Interestingly,
I pursued the issue using the Pommerantzef Schematic Eye
derived for wide angle fundus photography.  This same relationship
holds for the peripheral retina as well for at least 30 degrees
eccentricity (i.e., it may vary slightly, but only by a few tenths
of a degree per mm displacement).

Hope this helps.

Jay M. Enoch  jmenoch@garnet.berkeley.edu

### Optics (Mershon)

From: mershon@unity.ncsu.edu (Don Mershon)

I saw your request (for "useful" numbers) which was distributed via
CVNet.  I am interested in whether you plan to circulate the results of
your survey (perhaps to the same audience).

As a suggestion for your endevour: it would probably be most helpful if
there were more specific identification of the application of such
values.  For example, you mentioned the number of cones per degree ---
should one assume that the number refers to the fovea?  Would it perhaps
be a good idea to indicate the limits (e.g., +/- X deg.)?  Similarly, you
mention that the human "lens" provides about 60 diopters of optical
power.  Actually, the cornea provides about 40 diopters, while the lens
itself provides a variable amount (from 20-32 D), assuming that I am
remembering correctly.  And, of course, again, one should specify that
this would only be correct for a young eye.

Don Mershon  (N.C. State University)

### Squid’s eyes, the brain, subject contours, many (Cavanagh)

From: "Patrick Cavanagh"  <patrick@burrhus.harvard.edu>

Brian,

I share some of your numerophiliac tendancies and will give just a stream of
consciousness recall of the useless numbers which I can recall, some of which
you should verify before using.

Size of giant squid's eye: 19 feet
Percent of all receptors in body which are on the retinae: 70%
Number of moons which will tile the sky: 100,000
Hyperacuity equivalent to: seeing an eye movement at a distance of one mile
Number of neurons in the brain in 1974: 10 billion
Number of neurons in the brain in 1994: 100 billion
Reason mirrors reverse left-right, not up-down: left-right symmetry of body
Earliest subjective contour: 4000 BC
Number of seizures induced by TMS: 2
Number of seizures induced by research fMRI: 0
Duration of Hess positive rod afterimage: 1 minute
Percent of submitted articles which never get published: 0%
Movie image rate: 24 per second
Movie frame rate: 72 per second (each shown 3 times)

And finally, my question of the day:

Why can lobsters survive out of water for several days? How do they breathe?

Number of watts required to power brain: 10 watts
Percent of total body energy required to run brain: 10%
Percent total input energy from breakfast: 10%
Conclusion: don't forget to eat breakfast
Will computers replace brains: No
Why: brains fit in a shoe box, run on 10 watts, and are easily reproduced with
cheap labor

Patrick

### Units, levels, optics (Burns)

From: sburns@vision.eri.harvard.edu (Steve Burns)

Brian:

1.  A troland (td) is produced on the retina when the eye is looking at a
surface of 1 cd/m^2 through a pupil of area 1 mm^2.  This makes it easy to
compute the td value of a surface or TV monitor if you know the approximate
pupil size.

2.  For a standard eye 1 td produces 0.0035 lumens/m^2 on the retina.

3. From experience with a lot of aged individuals, the steady state diameter
of the pupil, even at very high light levels (bleaching), never is less than
2 mm.  Transient constriction very occaisionally will make the pupil smaller
than this.

### Yuks (Groh)

From: groh@monkeybiz.stanford.edu (Jennifer M. Groh)

Hi Brian,

Don't forget my favorite "facts":

1.  The brain contains 10^12 neurons, 10^14 of which are in the
cerebellum.

Reference:  somebody famous.

2.  We only use 10% of our brains.

Reference:  the popular press.

### More on Psychometric function slopes (Eskew)

From: ESKEW@neu.edu (Rhea Eskew)

Dear Brian and David:

I had intended to send you one or two useful vision 'constants', but I
didn't get around to it.   Sorry.  I'm motivated by your last message,
however, to argue against the submission made by David Burr, who said
the slope (beta) of (almost any) Weibull psychometric function is 3.

I (and Stromeyer) consistently find lower betas for the detection of
spots.  Part of the difference is that neither Charles nor I use the
best procedures for measuring beta (we pool data across sessions, for
instance, which is likely to lower the estimate), but I'm fairly
confident that even if we did not do that we'd consistently get betas
lower than 3.

More important than the absolute magnitude of beta, however, is the fact
that we (and others) have demonstrated that beta varies across spectral
conditions and tasks.  I'll mention the three cases I've been most
involved in: (a) chromatic betas are lower than luminance betas for
detecting foveal gaussian blobs (1deg s.d.) (about 1.7 vs 2.4 in my lab
-- not a large difference but a very consistent one). We haven't
published this work yet but some was reported at ARVO last year. (b)
With a 1deg foveal spot, a luminance pedestal approximately halves the
ideal).  Eskew et al JOSA 1991 vol 8 p 394-403.  In that paper we show
that this change in beta isn't simply uncertainly reduction.  (c) The
slopes for detection of chromatic and luminance spots increases in the
periphery (with no pedestals).  See the 91 JOSA paper, Fig. 3, and
especially Stromeyer et al Vision Research 1992 vol 32, pp 1865-1873,
Fig. 7.

I wish the psychometric slope truly was constant -- life would be
simpler.  But I don't believe it.

By the way, one of the constants I should have sent you before is that
red-green chromatic sensitivity is roughly 10X greater than luminance
sensitivity when sensitivity is defined as (1/sqrt[L^2 + M^2 + S^2]),
and L, M, \& S are cone contrasts -- this is for 200 ms flashes.  For
the ratio drops to about about 3X and should stay that way for all
shorter flashes, assuming Bloch's law (Eskew et al, Vision Res 1994 vol
34, pp 3217-3137).  So half a log unit is due to temporal integration,
some is due to spatial integration, and some must be due to other
factors.

Best,

Rhea

### Optics, retina, cortex, more (Wooding)

From: David Wooding <WDRDSW@CARDIFF.AC.UK>

Dr.Wandell,

Thanks for the summary of numbers. A great idea!

I have been a bit busy of late, but was intending to submit
something. Perhaps you would like to include the following, some of
which people suggested should be included. I can expand on
references if you like.

Thanks,

David.

------------------------------------------------------------

THE OPTICAL PATH
----------------
Amount of light lost through scattering in the eye: 3-4%
Lens absorbance peaks at: 365-368nm (near UV)
Said and Weale, 1959; Cooper and Robson, 1969

THE RETINA
----------
Thickness of retina: 350 microns
%age of light incident on photoreceptors which is actually absorbed
by them: 10% at 500nm
Maximum rod density occurs: 10-15 degrees off axis
Osterberg, 1935
Position of cones: central 1-2 degrees of fovea
Number of rods in human eye: 110-125 million
Osterberg, 1935
Number of cones in human eye: 6.3-6.8 million
Dimensions of rod: 30 microns long by 1 micron diam
Dimensions of cone: 50-80 microns long by 5 microns diam
Relative abundance of long and middle wavelength to short cones: 100
Amplification of energy provided by rod: 10e4 - 10e5
Short-wavelength limit of visual sensitivity (human lens): 380nm
Peak absorption of Rhodopsin (rod pigment): 500nm
Extent of macular pigment: up to 8 degrees from fovea

THE CORTEX
----------
approx %age of human cortex occupied by visual system: 60%
approx no.neurons in human visual system: 10e8
V1 surface area devoted to central 10 degs of visual field: 65%

VISUAL FIELDS
-------------
MONOCULAR:
Horizontal dimensions from central fixation: 60 degs nasally,
100 degs temporally
Vertical dimensions from central fixation: 60 degs above,
75 degs below
BINOCULAR:
Horizontal: 200 degrees
Vertical: 135 degrees
Width of area of steroscopic vision: 120 degrees
Height of area of stereoscopic vision: 135 degrees

SENSITIVITY
-----------

COLOUR VISION
-------------
Incidence of anomalous trichromacy in human popn:
1 in 100 (male), 1 in 10000 (female)
Incidence of protanopia and deuteranopia (dichromacy):
1 in 100 (male), 1 in 10000 (female)
Incidence of tritanopia (dichromacy):
1 in 10000
Incidence of rod monochromacy:
1 in 10000
Incidence of cone monochromacy:
1 in 100000

EYE MOVEMENTS
-------------
Slow drifts: velocity 6mins/s, amplitude 0.8-6.0 mins
Microsaccades: duration 10-20ms, amp < 50mins
(elicited by positional error of > 15mins)
Tremors: frequency 30Hz+, amp 20mins
Smooth pursuit latency: 120ms (typical)
Smooth pursuit tracking limit: 30 degs/s
Vergence: velocity 15degs/s, latency 160ms, duration 800ms
Maximum size for natural saccades: approx 15ms
Bahill et al., 1975

### Visual sensitivity (Pelli)

From: "Denis Pelli" <denis-pelli@isr.syr.edu>

Subject:numbers                                       2:39 AM
1/13/1995

hi brian

great list. Here's my contribution:

The contrast threshold (delta L over L) for a static edge at photopic
luminances is 1%. This predicts the visibility of a wide variety of large
sharp edged objects, including letters.

This is an old result, as John explains:

Robson, J. G. (1993) Contrast sensitivity: One hundred years of clinical
measurement. In R. Shapley and D. Man-Kit Lam (Eds.), Contrast Sensitivity
(pp. 253-266), Cambridge, MA: MIT Press.

regards

denis

### Visual sensitivity (Knoblauch)

From: Ken Knoblauch <Ken.Knoblauch@cismibm.univ-lyon1.fr>

Brian,

Cut-off frequency for chromatic gratings: ~10-12 c/deg (e.g., Mullen, 1985,
J. Physiol).
Minimum bandwidth for optimal reading: 2 cycles/character
(relatively independent of size) Legge et al., 1985, Vis. Res.
etc.

Sender: mrbaker@vax.ox.ac.uk (Mark Baker)

Dear Brian,
The list of useful numbers seems a great idea! Have you looked in
Wyszecki \& Styles "Color Science."? It is full of gems of that sort
and might make excellent additional source material. I've only got the
reference for the 1967 edition but I think that there is a fatter even
more comprehensive 1988(?) edition too . . .

Incidentally - the Eye-Movement network keeps a number of useful
files on the Mailbase fileserver - members can have them e-mailed to
them by sending the fileserver a message - would you like to have
a copy of your file put there too? I'll put details of joining on the
end of this message.

Keep up the good work!
Regards
Mark

Dr Mark R Baker  (MRBAKER@UK.AC.OX.VAX)

### Optics (Anderton)

From: (Phil Anderton) <P.Anderton@unsw.EDU.AU>

Hi,
I just read your list on CVNet, and one useful number which is not
there is an approximate rule-of-thumb for deriving angular magnification by
a "spectacle magnifier". This is a lens of high positive dioptric
power held at the normal spectacle plane.  The object of interest placed
close to the anterior focal plane of the lens, and the eye views an
enlarged virtual image of the object. Assuming a normal working distance of
250mm (without the lens), the angular magnification induced by the lens is
about 1/4 the dioptric power. For example, a 16D lens gives a retinal image
about 4 times larger than 'normal'. Beware though, this is very 'rough'.

Philip Anderton.

### Visual sensitivity, optics, sampling (Teller)

From: Davida Teller <dteller@u.washington.edu>

Brian -- and of course there's that wonderful convergence of numbers,
that grating acuity, the cut-off frequency of the optics, and the Nyquist
limit of the foveal photoreceptors are all about 60 c/d.  dt

Return-Receipt-To: "M. Angeles Losada Binue" <iodma15@cc.csic.es>

Hi Brian

One of the main lines here has been modeling the eye and measuring the
ocular MTF using a double-pass method.  I think you may be interested
in these two papers:

1. Accommodation-dependent model of the human eye with
aspherics. R. Navarro, J. Santamaria \& J. Bescos JOSA A 2, 8 pp1273 (85).
2. Monochromatic MTF of the human eye for different pupil
diameters: an analytical expression. JOSA A 11 pp246 (94). Here you can
find an analytical expression for the foveal MTF as a function of
both pupil diameter and normalized spatial frequency.

MTF(sf,p)=(1-C1+C2*p)*exp[-A1*exp(A2*p)*sf]
+(C1-C2*p)*exp[-B1*exp(B2*p)*sf]

where A1=3.53; A2=0.43; B1=1.69; B2=0.28; C1=0.48; C2=0.037
and 2<p<8 mm; sf=sf0/sflim with sf0<50 cpd and sflim being
the cut-off frequency for wavelegth 632 nm.

I hope this will be useful to you.

Cheers

Angeles

### Units (Nilsson)

From: Thomy Nilsson <NILSSON@upei.ca>

Dear Brian,
Thanks for the list.  Thought of 2 more:

retinal illuminance = incident luminance X pupil area in mm sqd X .0016

Note that the above has a correction from Judd's constant of .004.  (See
Applied Optics 1983, p.3462)

When dealing with a round stimulus which is the only scource of light, one
can calculate its luminance from a measure of illuminace as
obtained with a simple light meter:

luminance (cd/m sqd) = illuminance (lumen/m sqd) / Pi X (sine A/2)sqd

where A is the visual angle of the stimulus' diameter.

Regards,
Thomy

Color (Varner)

From DVarner@swri.edu Fri Jan 13 11:31:29 1995

Hi Brian--

I happened to read your posting of the important numbers for vision.
Although I don't do color anymore (or vision either for that matter), I'm
still Dottie and Leo's student.  So I'd like to add:

With these three numbers and a knowledge of the basic color-opponency
relationships, a person can relate spectral wavelengths to color appearance.

denise

Philosophy, numbers, New York running commentary (Graham)

<nvg@psych.columbia.edu> (Norma Graham)

Jan. 14, 1995
Dear Brian,

(1) Re your numbers: To second the point made by one of your
repondents. Most numbers that are useful to me have a limited range of
applicability, and theis range if frequently difficult to succinctly
express.
However, some idea of a number's magnitude is very frequently a
whole lot better than no idea. And I very much enjoyed your list of
replies. And I hope you can manage some numbers of this sort in your book.
And it is great that your book may be getting done!!

(2) Not mentioned in your email summary of other replies you
received is one I do find useful (not only quantitatively but also it
reminds me of the qualitative relationship).  Namely

0.5 db per period is the loss of contrast sensitivity moving away from the
fovea (relatively independent of spatial frequency, at moderate temporal
frequencies, and for photopic luminances, etc. etc. and so forth).

How good is this number? Asking how the loss depends on number of periods
is a whole lot more useful than asking how it depends on distance in visual
angle. Actually,  there is a loss of between 0.5 and 1.0 log units contrast
sensitivity (10 to 20 db) in thirty periods in John's and my data (Fig. 4
from Robson and Graham, 1981, reprinted as Fig. 5.6 in Graham, Visual
Pattern Analyzers, 1989).  Also see see Fig. 13.5 (section 13.4.3.in the
book) showing that this "number-of-periods" rule of thumb is consistent
with size-scaled models of the retinal inhomogeneity.

(3)  A related point, made with regard to Charles Anderson's window
of attention, in particular with regard to his using grating sensitivity as
one support for it:  Figuring out the window of attention from the number
of cycles over which grating sensitivity increases is problematic unless
you get the grating someplace where there is sufficient homogeneity.  And
if you get someplace where there is uniformity, grating sensitivity
increases out to at least 30 (or 60) than 10 or 15 cycles.  And we did not
test higher.   Of course, it may only be a factor of 2 (or 4) different
from his estimate,  but I am not sure that it wouldn't increase even out to
greater distances (for conditions where the observers have reason to spread
their window of attention as wide as possible). See FIg. 3 from Robson and
Graham, 1981 reprinted as Fig. 5.7 of Graham, Visual Pattern Analyzers,
1989.)

(4) Of course another number which appears here and elsewhere is
that the amount by which sensitivity would be predicted if you double the
number of things involved. A notice one response dealt with it in the
binocular situation. But you might try something more general. Is it dealt
with in the text of the book anyway?

(5)  I find that remembering contrast sensitivity at the peak of
the sensitivity vs. spatial or temporal frequency  in typical conditions is
something between 100 and 1000 is useful. I usually say 300 for moderate
temporal frequencies and moderate photopic but this really does depend on a
lot. (Again, however, compared to total ignorance, this is a very useful
number!!)

### A challenge (Sivak)

From: "Dr. J. Sivak" <jsivak@sciborg.uwaterloo.ca>

To Brian Wandell,

The giant squid eye cannot be 19 feet in diameter ( see P. Cavanagh).
Even 19 inches sounds big.

### A response (Cavanagh)

From: "Patrick Cavanagh"  <patrick@burrhus.harvard.edu>

Hmm, well I can't vouch for the accuracy of the eyeball size estimate but the
source is "How Animals See" from Facts on File -- I forget the author's name.
She has been criticized before for errors but the giant squid is an elusive
character (as opposed to the giant squid axon) and I have no idea how to check
her information.

Patrick

### And a resolution (Cavanagh/Backus)

From: "Patrick Cavanagh"  <patrick@burrhus.harvard.edu>

> Dear Professor Cavanagh:
>
> I have "How Animals See" (picked it up for \$4, says the sticker still on
> it).   "Squid," in the index, leads to page 9, which, I regret to report,
> gives 370 cm (12 ft), not 19 ft, as the diameter of the giant squid's eye.
>
> Sandra Sinclair, 1985
> How Animals See: Other Visions of Our World
> New York: Facts On File Publications.

What cruel tricks time plays on memory. I had remembered the giant squid eye
from "How Animals See" as being 19 feet, perhaps I did the conversion wrong.
Nonetheless, it is a big eye.

I have been searching for a copy of "How Animals See" for the last 4 years,
copy for me if you see another one.

> Re: the bilateral symmetry of the body is responsible for the apparent
> left-right,  without up-down, reversal in a mirror.    I give you this
> creature in return:  shaped like an upright wine bottle, the creature is
> capable of spinning itself about the vertical axis by means of a bent arm
> that sticks out from the side of the bottle, with a jet nozzle at its end.
> Male arms are bent one way, and female arms the other, so that males and
> females spin in opposite directions.  In the mirror, a female would see not
> a left-right reversal of herself (though such an identification could be
> made), but surely herself as she would look were she male.
>
> Sincerely,
> Ben Backus
> UC Berkeley graduate student (Marty Banks' lab)

Mmm, a real gender bender. But why make her spin? If she stops spinning and
looks at her nozzle organ in the mirror, it will be bent in the male direction
(a left-right reversal to us but an embarrassing male-female reversal to her).
But rather than a wine bottle body which has an obvious top, imagine a perfectly
cylindrical body. Now in the mirror "she" could see herself as a male or "she"
could equally well see herself as an upside down female. Which would you pick?

On the other hand, so to speak, up and down would have to be labeled
before the gender-specific nozzle bend could be established so there
still isn't true up-down symmetry. If sex were designated by, say, the
color of the body, the mirror images would only produce an up-down
reversal and their species would have lost an endless source of good
jokes (eg. No, I don't like your theory but turn over and I'll kiss
you).

Patrick

### Handsome units (Kooijman)

From: "a.c.kooijman" <a.c.kooijman@med.RUG.NL>

Dear Brian,

I have a paper in front of me with some handsome numbers and
relations on light units:

some examples:
*
1 photopic lumen = 1 scotopic lumen for monochromatic light of 555 nm

'lumen' can be replaced by:
lux, candela, nit, troland, apostilb, footlambert (though you have to
avoid this non-metric unit)

* in 'free' viewing to a scene or surface:
retinal illuminance [in troland] =
luminance of the scene [in candela/sq m] * pupil area [in sq mm]

* in a maxwellian view condition:
place a lux meter at 10 cm behind the focus of the maxwellian focus
(thus about in the middle of the brains of the subject)
retinal illuminance [in troland] =
10E-4 * the illumination on the lux meter (full illuminated
sensor)

best wishes

Aart

### Philosophy (Nelson)

From: jnelson@ln.nimh.nih.gov (Jerry Nelson)

17 Jan 95
Dr. Wandell,

What an outpouring you got in response to a request for useful
constants in vision.  The wonderful replies included much more
-- people's pet relationships, perhaps their pet ideas.

Is their an unsatisfied need here?  Very good people will
complain privately of their inability to pursue questions they
consider fundamental.  At meetings some very important
intellectual work is done outside the formal constructs
intended to house them.  There is a richness and creativity in
all of us which is ill-housed by the usual professional
structures.

What did you tap?  It was clever to ask for only one number.
That kept replies down to lengths under 3 long paragraphs.
You tapped a playfulness.  And acceptance was contingent on
how HELPFUL TO OTHERS the contribution might be -- zero
pomposity here.

But without review the contributions were uneven, and without
a ritualized structure (INTRO, METHODS, etc) the format is
chaos -- charming, but a lot of editing work for you and your
book.

Anyway, what fun.  After your book is over, fight postpartum
letdown by starting a MODERATED newsgroup -- entries come to
you, and if a single glance can't quickly reveal humor or
helpfulness, then it doesn't go in.  Even in mean times,
American science has a mutual helpfulness hard to match
elsewhere.

Thanks again, good luck with the text.
--jerry

### Angles, eye movements, and more (R. Boynton)

From: ps45@sdcc12.UCSD.EDU (robert m boynton)

I was off CV-net for a while for some reason, and missed
the request for magic numbers.  I would like to add
a few.

The rule of thumb varies remarkably from one submitter to
another, doesn't it?  I really liked Al Ahumada's modal
number of eyes per subject.

Anyway, here are mine:

.3 mm on the retina = 1 degree of visual angle

number of eye movements  per second during very intensive visual search:  3 to 5

Nodal point of eye is 7 mm behind corneal vertex
Axial length of the eye is about 1 inch or 25 cm
Index of refraction of eye media (roughly) 4/3
Wavelengths of peak sensitivity:  505 and 555 nm

Maximum density of cones at foveal center: About 130,000,000/mm^2

Time for complete dark adaptation after exposure to a bright
light: 30 to 45 minutes

Cheers,

### Receptors, units, behavior (Ross)

From: jr@psy.uwa.edu.au (John Ross)

Brian: Some more (approximate) numbers, all useful I find.

1.     	Number of rods per (human) eye 120m

Number of cones per eye  6m

Number of fibres/optic nerve 1m

Ratio of receptors to fibres = 125:1

2.      30'' of arc = minimum separable; closest spacing of cones (approx.)

3.      	100ms = critical duration for Bloch's law (can be less); period of
frequency at which flicker and motion are seen best.

4.      	3 or 4 = number of cone types.

5.      	4 = Jevons number: maximum number of objects countable without
error at a glance.

6.      	4 or 5 log units: improvement in sensitivity with dark adaptation.

7.      	7 = Miller magic number; minimum number of 'stations' for best
apparent motion.

8.      	500 = best contrast sensitivity.

John Ross

### Optics, fun, retina, many (Tyler)

From: cwt@skivs.ski.org (Chris Tyler)

Dear Brian,

Well, I see you received a goodly supply of numbers.  Some others that I
have come across are:

The human eye is exactly the same size as a quarter.

The rod-free, capillary-free foveola is 1/2 deg in diameter, same as
the sun, the moon, and the pinky fingernail at arm's length.

One deg is about .3 mm (300 u) on the retina.

Diameter of smallest cone or rod is 1 u, packed at up to 300 per deg.

Diameter of largest human cone is 10 u, as large as those of any amphibian.

There are 10^8 photopigment molecules per rod. Each one can be
photoisomerized in 100 femtosec.  They are packed into 800 disks

Density of cones falls with E^-2/3 out to 20 deg eccentricity (not
reciprocally).

There are about 4000 cones in the central (foveolar) half degree.

The same number in a 5 deg diameter patch at 40 deg eccentricity.

The range of visible light is 1 octave at around 1/2 the diameter of the
smallest cone.

Each human striate cortex is 4 x 8 cm, making a total of 64 sq cm.
There are about 6400 hypercolumns therein.

You can think of this area as spanning 8 octants around the visual
field and 8 doublings from 1/2 to 64 degrees eccentricity.

The viewing distance at which an object is projected at veridical size
on the foveal representation is 20 cm (I think; you can work it out
for your own best foveal magnification estimate).

Foveal cone detection threshold occurs at an average of 1 quantum per
cone for extended stimuli.

A comment on Dave Burr's law of Psychometric functions.  Mayer and I
found that the slope (beta) varied from 2-6 across observers.  It is
well known that it falls to 1 for suprathreshold discrimination tasks.

Optimal contrast threshold is 1/4%.

Visual apparatus needs to be engineered to 1 part per thousand.

The bandwidth of the visual system is 1 teraHerz (e.g., the raster
bandwidth for a panoramic wall panel that looked like a full-field
dynamic view of the real world).

The number of possible brain states that we can experience in a lifetime is
10^10^10 (a googol).

The Ferry-Porter Law.  In the absence of flicker adaptation, CFF
increases directly with log I over 6 log units. (25 Hz per decade in
the periphery, 10 Hz per decade in foveola)

The maximum CFF ever reported is 107 Hz from my right eye at 300,000
Td.

The finest Vernier acuity reported was .8 arc sec in Dennis Levi's
right eye

Tyler's Law of disparity scaling: The maximum disparity for depth
resolution scales linearly with stimulus size.  Horizontal binocular
fusion follows the same law.

The best stereoacuity reported is 2 arc sec.  Contrary to the limited
statements in most textbooks, this is sufficient to discriminate the
distance of 2 miles from infinity (e.g., distant mountains) or resolve
the depth of a human ovum (100 u) at 10 cm.

The vertical horopter is tilted at an average of two degrees, just
enough to ensure that it falls in the plane of the ground for an erect
observer who is fixating anywhere in the ground plane.  Pettigrew
showed that the tilt was 20 deg for the cat and ground owl, and zero
for the tree owl, so that the horopter fell in the ground plane for
all four species.

At a pupil diameter of 3.6 cm, the pupil area is 10 cm, so for normal
viewing conditions, Trolands = 10 x cd/m2.

Speaking of units, it should be arc sec, arc min and Td for
consistency with the physics and astronomy communities.  All units
from proper names are capitalized (e.g., Hz, V, etc).

0.1 log unit is 27%

The standard error of a Poisson process (such as light) is equal to
the square root of its mean, which is why taking more readings
improves S/N ratio.  This fact is unknown to most undergraduates, in
my experience.

The eyes are half-way down the head - a fact known to most artists.

If the sun's light was blocked, it would be eight minutes before the
moon went dark.

I am happy to dig out the references for any of these that interest you, so
let me know if you want them.  Christopher

From: P Thompson <pt2@unix.york.ac.uk>

aftereffects reach their maximum 'strength' after an adaptation period of

Peter

### Visual sensitivity (Klein)

From: klein@adage.Berkeley.EDU (Stanley Klein)

Brian, here are some numbers from Dennis Levi and Stan Klein,

The following numbers are give or take a factor of 2 (or so) to keep the
numbers simple:

Detection thresholds:
line (a pair of adjacent opposite polarity edges) - about 1 %min

1 Td is about 1 absorbed photon per cone per integration time (E100 msec).

The best hyperacuity is equivalent to a retinal distance of one tenth of the
wavelength of red light (beat that, Patrick).

One can localize a peripheral point to within one tenth of a hypercolumn.

One hypercolumn is .1 times eccentricity

The best temporal hyperacuity for vernier acuity of moving targets (no

The Weber fraction is crudely about 10% for almost anything (except for
position (or size) where one must remember to also divide by 2pi). Too bad
the presence of multiple channels makes this number difficult to use.

For a monochromatic display to be able to present perceptually lossless
images without halftoning it must be able to display about 100 bits/min^2.

The perceptual information in a monochromatic image is up to about 20
bits/min^2.

People are quite happy with monochromatic images compressed to about 1
bit/m^2.

One can derive Levick's formula (blur angle=pupil size times diopters of
defocus) from the following geometric construction. 1) Focus at infinity. 2)
Look at something at a finite distance. 3) The linear blur projected onto the
object will be the size of the viewer's pupil (about 3 mm) independent of the
object distance.

Many psychophysical functions vary with eccentricity according to
Th =k(E+E2)
where Th is the threshold, k is the slope, E is the stimulus eccentricity,
and E2 is the eccentricity at which the threshold is twice the foveal value:
For resolution and contrast senssitivity, E2 is about 2-3 degrees.
For position judgements E2 is about .5 -1 degree.
For absolute motion, E2 is about 5-10 degrees.
We like the position judgement E2 and are starting to call it L2 (the E2
characteristic of the local sign). It is the value that we expect to be similar
to the anatomical cortical magnification factor.

What a great idea to do this.
Stan

### Genetics (Piantanida)

From: Tom Piantanida <pianta@unix.sri.com>

Brian,  Here are a few more numbers for your survey.

As a rule of thumb, for image stabilization to occur, spatial noise needs
to be 1 minarc or less, and latency needs to be 1 msec or less.

The genes encoding rhodopsin and cyanopsis have five exons, those
encoding chlorolabe and erythrolabe have six.

The opsin proteins of rhodopsin and cyanolabe consist of 348 amino
acids, those of chlorolabe and erythrolabe consist of 364.

The gene that encodes rhodopsin is on chromosome 3 (3q21- 3qter), the
cyanolabe gene is on chromosome 7 (7q22-7qter), and the genes for
chlorolabe and erythrolabe are on the X chromosome (Xq28).

Panum's fusional area can be as large as about 2 degrees under
stabilized-image conditions.

When the fusional limit is exceeded, diplopia occurs, but fusion can
be reestablished at disparities close to 2/3 of the fusional limit
(not 6minarc as Fender and Julesz reported.)

This is a great idea.
Tom Piantanida

### Units, references, sensitivity (Meese)

From: tim@vis.psg.bham.ac.uk (Tim Meese)

Here are a few numbers for your list, if your still collecting them (they
are mainly obvious, but I find them useful):

1) I prefer to report stimulus contrast in dB (20*log(c)), though my
referees often seem to prefer percent or proportion! The conversion is
easy however: an increase of 6dB represents a doubling of contrast (or
whatever other variable you may have converted to dB). In absolute terms
0dB is a contrast of 1% (if c is in the range 0 to 100%), or 1 (ie. 100%,
if c is in the range 0 to 1).

2) Useful stimulus spacing for detection/discrimination experiments etc:

3) An approximation I find useful: To convert from the spread (SD) of a
cummulative Gaussian to the spread (logits) of a logistic curve
(1/(Exp[-L]+1)), multiply the former by 0.6. Graphical analysis shows this
to be a fair approximation, and it has been reported in a foot note by
Taylor and Creelman (1967; JASA, 41, pp782-)

4) There are some more good numbers relating alternative measures of
Gaussian widths in Norma Graham's book (p50).

5) Some undergraduates have problems in visualising large numbers. An easy
way of making 1,000,000 seem small is to imagine (or build) a cube with
sides of 10 cm (ie. something quite small that can easily be held in one
hand). However, this cube is also 1,000,000 cubic mm.

6) Percentage of 2AFC trials in which you guess: 0%
(the feedback beeps must be wrong)

sincerely

Tim Meese

### Binocularity (Pietsch)

From:	PO2::"reading@ucs.indiana.edu"  9-JAN-199

My favorite one involves the enhancement of visual performance gained by
going binocular.  It applies to absolute light detection thresholds,
brightness discrimination, critical flicker frequency, and visual acuity.
The gain in sensitivity is always [more or less] equal to the square root
of 2!  You can arrive at this result through sampling  theory,probability
summation, or areal summation [an adaptation of Piper's law to binocular
vision]. One of the implications of this approach is that a three eyed
individual should enjoy an enhancement equal to the square root of three!
[as suggested by the late Fergus Campbell}
***

PIETSCH, P. AND SCHNEIDER, C. W., Two-eyed Versus One-eyed Salamanders: Does
Binocularity Enhance the Optically Evoked Skin Blanching  Reactions  of
Ambystoma Larvae?   PHYSIOL BEHAV 48(2) 357-359, 1990.  A wide variety of
visual functions show increases attributable to binocularity, and the question
pursued here was whether a second eye enhances the visually stimulated skin
blanching reaction of the larval salamander.  Dermal melanin spots (melansome
aggregations within dermal melanophores), which contract or expand to lighten
or darken the skin reflectance, were measured in eyeless (controls), one-eyed
and two-eyed Ambystoma punctatum  larvae after chronic exposure of subjects to
a white background  (i. e., stimulus conditions for maximum blanching).   The
eyeless subjects showed no blanching (thus remained dark) in white cups, and
they exhibited melanin spots 7 or 8 times the size those of the other two
groups.   All one-eyed or two-eyed subjects exhibited blanching reactions;
planometric comparison revealed a significantly larger melanin spot area for
one-eyed than for two-eyed animals; i. e.,  the binocular condition permitted
greater contraction of  the pigment spots than did the monocular condition.
Analytical data compared favorably with independently ascertained pigmentation
indices.   The results indicate that a second eye quantitatively elevates the
blanching maximum of a larval salamander.

IF YOU'RE INTERESTED, I'LL SEND YOU A REPRINT OF THE ENTIRE ARTICLE.

PAUL PIETSCH
INDIANA UNIVERSITY, BLOOMINGTON, IN 47405 USA

### Philosophy (Cohn)

From: tecohn@mindseye.berkeley.edu (Theodore E. Cohn)

Who could have guessed that numbers would be so much fun?  I've been
thinking a bit about numbers and it occurs to me there are two categories of
same.  The first category includes those which represent pschyological (or
anatomical or physiological) constants such as the number of foveal cones
per square mm. The other category includes numbers we use to divide up
continua into easy-to-describe pieces.  I heard one of the latter cited the
other day...its the distance beyond which we accept the view that image
disparity doesn't contribute to depth perception.  (a ref is C. Schor and M.
Flom "Relative value of stereopsis as a function of viewing distance, Am J
Optom 46:805-809, 1968).  With respect to the second category, universal
agreemnt on the value of a given number would neither be achievable nor
expected.  In the case of the former, agreement could be reached to within
the irreducible measurement error implicit in operations used to define the
number.
Ted

### Numbers to facts (Ross)

From: jr@psy.uwa.edu.au (John Ross)

Brian:
I would like to add a few more numbers to the beginning of my list:

1 = the number concurrent perceptual interpretations of visual
input (as in rivalry, ambiguous figures, but more generally scene
interpretations)

2 = Ahumada's modal number (of eyes), but also of receptor systems
(rods and cones), major pathways (magno and parvo), and dimensionality of
retinal images.

3 (or 4?) = dimensionality of perceived space.

John Ross

### Sensitivity, eye movement, color (G. Boynton)

From:  boynton@blue.stanford.edu  (Geoff Boynton)

One quantum is sufficient to activate a rod. (Cornsweet, Hecht et al)

Riggs. L.A. (1950) Involuntary motions of the eye during monocular
fixation.  J Exptl. Psychol 40 687-701

Pupil size ranges from 3mm under 2.5 log cd/mm^2 to 8mm in darkness
(Wyszeki \& Stiles, Color science. New York: Wiley, 1982)

There are 11 basic color names (Boynton, Robert M. Olson, Conrad
X. Vision Research.  1990 Vol 30(9) 1311-1317.)

### Neural waves (Drosler)

From:  <Jan.Drosler@psychologie.uni-regensburg.de>

The velocity of signal propagation in the retina (in radial direction )
is one degree of visual angle per millisecond.

It was measured behaviorally by letting the observer synchronize two
LEDs flashing at identical periods, one of which is fixated in the
center of the fovea. The temporal lead of the peripheral LED necessary
for a judgement of synchrony permits the calculation of the velocity
stated above. It appears to be constant over a range of zero to 55
degrees of visual angle. The standard error is 0.46 ms at all
meridians tested.

Reference: Arch. Psychol. 131, 249 -266, 1979.

### Optics/units (Knoblauch)

From: Ken Knoblauch <Ken.Knoblauch@cismibm.univ-lyon1.fr>

Has anybody suggested that 1.13 mm pupil diameter (artificial
probably) should give you 1 troland = 1 cd/m^2?

### Field of view (Grigsby)

From: Scott Grigsby <SGRIGSBY@FALCON.AAMRL.WPAFB.AF.MIL>

total extent of human field-of-view under optimal conditions: ~200 deg
extent of overlap of the two eye fields: ~120 deg
extent of FOV that does binocular processing: ~40 deg
(see Grigsby and Tsou (1994), Vision  Research 34, 2841-2848)

extent of eye movement before a compensating head movement: 20 deg
(Robinson (1979), Human Factors 21, 343-352)

1 microcandle in the plane of the pupil = 1 troland

From: mike@monkeybiz.Stanford.EDU (Mike Shadlen)

I don't know; but here are some other numbers
in a cubic mm of cortex there are
10^5 neurons
10^9 synapses
2 miles of axons

These and several other fun numbers can be found in Stevens CF, 'What
form should a cortical theory take' in Large-Scale Neuronal Theories
of the Brain, Koch C and Davis JL eds., MIT press, Cambridge, Mass.,
1994, pp.239-255.

From: groh@monkeybiz.Stanford.EDU (Jennifer M. Groh)

Peak saccade velocities in humans can reach 700+ deg/sec, though
in my experience tenured faculty are slower by ~250 deg/sec.  You
can cite

RHS Carpenter, Movements of the Eyes, Pion, London, 1988
p. 70

Bahill AT, Clark MR, Stark L, 1975, Glissades--eye movements generated
by mismatched components of the saccadic motoneuronal control signal,
Mathematical Biosciences 26:303-318

as a specific reference on the "main sequence", which is the
amplitude-duration-velocity relationship of saccadic eye movements.

Another rule of thumb for you:

duration of saccades > 5 deg = 20-30 ms + 2 ms per degree.

This also comes from Carpenter.

### Rod circuit sensitivity (Robson)

From: From: John RobsonI have a good number for you.  It is 1 and is the number of
photoisomerisations per rod required to saturate the retinal rod circuit.

### Number of Colors

From: Mark Fairchild> From Wandell to Fairchild:
>
> How many discriminable colors
> are there, and how did people count them?  No important reason for asking,
> a vague answer I don't like.
>

I get asked that question alot also and I also give vague answers. I
guess I usually get away with something like "it depends on the viewing
conditions."  I did a little digging and found on page 21 of Hunt's
Measuring Colour 2nd Ed. a line saying that 10 million was one reliable
estimate and he referenced Judd and Wyszecki.  I went to Judd &
Wyszecki, Color in Business Science and Industry, 3rd Ed. and found them
saying " since about 10 million surface colors can be distinguished by
the normal human eye ..." on page 388.  They however gave no direct
reference.  That section was talking about the ISCC-NBS method of color
naming so there might be something in the paper that describes that
(Judd and Kelly, Method of Designating Colors, J. Res. Netl. Bur. Std.,
23 355 (1939).).  I'm not going to look that one up.  I guess if 10
million is a good enough estimate for Judd, Wyszecki, and Hunt, then
it's good enough for me.
From shev@midway.uchicago.edu Fri Oct 25 08:15:55 1996

Mark and Brian, Thanks for replying to my inquiry.  The following FYI.

A conservatively low estimate of the number of discriminable colors,
ignoring luminance, is 6,000.  I arrived at this using two different
approaches.

APPROACH #1

1.  10,000,000 colors with luminance included (Hunt, Judd, Wyszecki)
2.  1% Weber fraction from 1 to 1,000,000 trolands (implying ~1620
discriminable luminance steps). This assumption is the conservative
part, leading to a low-end estimate.
So, 10,000,000/1620 = 6,173.

The number is 10,800 if the 1% Weber fraction is applied over 1-10,000 td.
The number is 21,500 if the Weber fraction is 2% over 1-10,000 td.

APPROACH #2

This approach takes into consideration that the 10,000,000 colors are
*surface* colors but ignores overall level of illumination.

1.  10,000,000 surface colors (Hunt, Judd, Wyszecki), ignoring luminance.
2.  Assume incremental and decremental contrast are *each* discriminable
with a 1% Weber fraction over a 100:1 contrast range (implying
2x463=926 discriminable contrast steps).  Again, this is conservative.
So, 10,000,000/(926) = 10,799.

Another approach is (1) to take discrimination around the spectral locus
(using wavelength discrim data) and along the non-spectral purple line,
and then (2) to count the number of JNDs from each discrminable point on
the locus to EE white.  This requires some pretty heavy assumptions in the
model of JNDs and will lead to an inflated estimate, because discriminable
colors at the spectrum locus may not be discriminable when desaturated.
I may try it as an exercise.

..Steve

### Location of Optic Nerve Head (Blindspot)

From wsimpson@io.uwinnipeg.ca Fri Jan 10 13:02:15 1997

Dear Brian,

I just looked at your list, which includes

Retina
5. ...size of optic nerve head (optic disc/blind spot)

BUT I always need to look up its location whenever I use a stimulus
outside the fovea.  Maybe you could add the location?  15 deg nasal.

Best wishes,
Bill

### Visual angle of a pencil line

 From: "Robert A. (Bob) Morris"

I just worked this out for a course I'm teaching on art and vision: A
line drawn by a number two pencil subtends about 1/5 degree when
viewed at 18", and so falls approximately at the peak of adult human
grating acuity.

I wonder if crayon line widths fall at the peak of 3-year old acuity?

Bob Morris