Geometry Imaginary Places

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Geometry of imaginary spaces

Jan J. Koenderink⇑

Delft University of Technology, Man Machine Interaction Group, EEMCS, P.O. Box 5031, 2600 GA Delft, The Netherlands

Katholieke Universiteit Leuven, Laboratorium voor Experimentele Psychologie, Tiensestraat 102, Bus 3711, 3000 Leuven, Belgium

The Flemish Academic Centre for Science and the Arts, Academy Palace, Hertogsstraat 1, 1000 Brussel, Belgium

a r t i c l e i n f o

Article history:

Available online 25 November 2011

Keywords:

Space

Imaginary space

Pictorial space

Local sign

External local sign

Cues

Natural perspective

Depth

a b s t r a c t

‘‘Imaginary space’’ is a three-dimensional visual awareness that feels different from what you experience

when you open your eyes in broad daylight. Imaginary spaces are experienced when you look ‘‘into’’ (as

distinct from ‘‘at’’) a picture for instance. Empirical research suggests that imaginary spaces have a tight,

coherent structure, that is very different from that of three-dimensional Euclidean space. This has to be

due to some constraints on psychogenesis, that is the development of awareness. I focus on the topic of

how, and where, the construction of such geometrical structures, that figure prominently in one’s aware-

ness, is implemented in the brain. My overall conclusion—with notable exceptions—is that present day

science has no clue. I indicate some possibly rewarding directions of research.

2011 Elsevier Ltd. All rights reserved.

1. Natural perspective

‘‘Natural perspective,’’ or perspectiva naturalis, is best knownfrom Euclid’s treatise (Burton, 1945) (Greek Optika; Latin: De

aspectibus). It should be sharply distinguished from ‘‘painter’s per-

spective,’’ or perspectiva artificialis, which plays no role in this

paper, but became generically known as ‘‘perspective.’’ The latter

involves ‘‘Alberti’s Window,’’ after Alberti’s (1435) Della pittura,

and deals with the representation of the visual field on planar sur-

faces. Unfortunately, these concepts are rarely distinguished. Here

I interpret natural perspective in its original sense of ‘‘optics,’’ a

proper subfield of physics. It deals with the potential of momentar-

ily seeing things with a single, punctate eye,1 and is thus to be con-

sidered a form of information theory.2 It has nothing to do with the

transport of radiant power, thus the frequent discussions on Euclid’s

use of the extramission theory are void (Koenderink, 1982).

I recapitulate the basics of natural perspective here. Considerthree-dimensional Euclidean space E3, augmented with a single

‘‘vantagepoint’’ O. Anypoint P 2 E3 O is seen ata uniquedirection,

called its ‘‘visual direction’’ with respect to the vantage point.

Consider the ‘‘optic array’’ (Gibson, 1950) at the vantage point,

whichis simplythe unit sphereS2, centeredon O. Then the directionof P is convenientlylabeled withits trace p 2 S

2, where p = (P O)/

kP Ok (see Fig. 1). One has P = O + .p, where . is the ‘‘range’’3 of P

with respect to the vantage point.

Points with the same traces are seen in the same direction.

When their ranges are different they are still distinct. I refer to such

points as mutually ‘‘parallel.’’4 Berkeley (MDCCIX) famously de-

clared that such parallel points cannot be distinguished on the basis

of optics proper, because ranges are, in modern jargon, not ‘‘optically

specified.’’

Slightly idealized, the human condition involves primarily day-

light, clear air, and opaque rigid objects. Daylight is solar radiation

in a narrow visual band centered at about 560 nm, often scattered

by cloud covers. Clear air optically acts much like the vacuum, a

perfect medium of the propagation of radiation. Opaque objectshave surfaces that scatter radiation into all directions. This leads

to the following basic laws of visual optics:

I. You can see any object given unobstructed range,

II. You cannot see objects that are occluded by other objects,

and

0928-4257/$ - see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.jphysparis.2011.11.002

⇑ Address: Delft University of Technology, Man Machine Interaction Group,

EEMCS, P.O. Box 5031, 2600 GA Delft, The Netherlands. Tel.: +31 152784145; fax:

+31 152787141.

E-mail address: [email protected] In computer vision this is known as the ‘‘pinhole camera model.’’ In geometrical

optics it is the center of the anterior nodal point of the optical system. In human

vision it is most natural to use the center of rotation of the eye-ball.2 Euclid uses the theory to account for visual acuity, by referring to a certain

sparsity and thickness of rays. This can be related to the minimum étendue of about a

wavelength squared of the wave theory of light.

3 ‘‘Range’’ is preferred over ‘‘distance,’’ because in the latter case one should not

omit to say ‘‘from the eye.’’4 This usage is natural in the singly isotropic plane introduced later, where one has

a perfect metrical duality between points and lines. Then it is natural to have both

‘‘parallel lines,’’ and ‘‘parallel points.’’

Journal of Physiology - Paris 106 (2012) 173–182

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III. If you can see it , then it can see you.

This is ‘‘physical optics’’ in a nutshell. What remains is informa-

tion theory, the formal tool essentially geometry.

The most fundamental facts of natural perspective follow from

invariance properties. Consider the optical information available tothe human observer. There are many ways to specify it, e.g., you

might file a complete description of the environment. But this

would evidently be overkill. Following Berkeley (MDCCIX) and

Gibson (1950) I specify the available optical structure as the mere

simultaneous order of traces in the optic array.5 Any transformation

of the environment that leaves this structure invariant is undetect-

able, and thus irrelevant to vision. One such group of transforma-

tions are the rotations about the vantage point.6 Another group is

that of the homotheties (dilations or expansions) about the vantage

point.7

From such considerations I deduce that only ratios of ranges are

relevant, and that neither absolute range, nor range differences

matter. Thus the log-range dimension has the structure of the af-

fine line (Bennett, 1995) A1.The group of rotation-dilations about the vantage point gener-

ates an important family of curves, namely the planar logarithmic

spirals with center at the vantage point.8 These are shifted within

themselves by the transformations. In any sufficiently limited neigh-

borhood there is a unique arc connecting any two distinct points.

Thus they may be considered pre-geodesics (Buseman, 1955) (see

Fig. 2). They do not define a projective structure, for the nexus of

geodesic arcs that connect each vertex of a triangle to each point

of its opposite side fails to be a surface. It is a (non-convex) lens-like

volume (Berger, 2007; Koenderink et al., 2010a) (see Fig. 3).

A group of transformations that conserves the family of pre-

geodesics as a whole involves power-functions of the range (see

next paragraph). This group was empirically discovered by the Ger-

man sculptor Adolf Hildebrand at the close of the nineteenth cen-tury (von Hildebrand, 1901). It can be understood via the structure

of ‘‘Shape From X’’ algorithms studied in machine vision (Belhum-

eur et al., 1999).

The situation is much simpler in the tangent space of a point in

the optic array.9 Let {u, v } denote Cartesian coordinates in the tan-

gent plane of S2, and w = log.. This space is the Cayley–Klein space

(Cayley, 1859; Klein, 1893, 1872) that has two Euclidean and one

isotropic dimension (Yaglom, 1968, 1979), for the aforementioned

transformation group is

u0 ¼ h ðu cosu v sinuÞ þ t u; ð1Þ

v 0 ¼ h ðu sinuþ v cosuÞ þ t v ; ð2Þ

w0 ¼ g u u þ g v v þ kw þ t w; ð3Þ

with h, k > 0. It is an eight-parameter group, whereas the Euclidean

group of similarities is only a seven-parameter group. Reason is the

elliptical (periodic) angle measure of Euclidean space.10 The inter-

esting part is obtained by specializing to h = 1, u = 0, t u = t v = 0, a

subgroup that affects only the ranges. The parameter t w is a mere

shift in depth, that is a scaling of the range, and thus not interesting.

The relevant parameters are g = { g u, g v }, which denote isotropic rota-

tions, and k, which denotes the scaling of isotropic angles. The geom-

etry of this space is well known through Strubecker’s (1941, 1942,

1943, 1945) work, modern texts on its differential geometry are also

available (Yaglom, 1968, 1979; Sachs, 1990). Unfortunately, the best

texts are in German. However, the books by Yaglom (1968, 1979)have been translated into English, and make for an excellent intro-

duction, albeit only for the planar case.

This introduction merely mentions the more important aspects

of natural perspective. Unfortunately, there do not seem to exist

any textbooks on this important topic. Gibson’s (1950) work is

an informal attempt, but fails to arrive at the essential structure.

2. Imaginary space

When you look ‘‘into’’ a picture of an imaginary landscape, e.g.,

in a science fiction comic book, you experience spaces that do not

exist in a physical sense, and thus have to be considered ‘‘imagi-

nary’’ quite literally. I see no essential differences with looking into

the photograph of an actual scene, or your spatial awareness infront of an actual scene.11 Because too far removed from main-

stream thought I will not press the latter point. I focus on the topic

of whether such imaginary spaces can be granted geometrical struc-

tures, allow for formal descriptions, and perhaps some praxis of

geodesy.

Phenomenologically, these spaces are three-dimensional (or,

rather, ‘‘two plus one dimensional,’’ see below), and contain opa-

que, rigid objects. They differ from physical spaces in a number

of important aspects, due to the fact that their ontologies are

different.

Physical scenes are composed of physical objects that have phys-

ical properties. The formal description of physical spaces is in

terms of theoretical accounts (geometry, electromagnetic theory,

Fig. 1. The visual field S2 centered on the (black) vantage point, and a visual

direction d on which are two parallel points P and Q. Both points have the same

trace T 2 S2 .

5 The optic array is a mere formal device that has nothing to do with the structure

of the human eye. It would make absolutely no difference if the eye were cubical

instead of spherical.6 Any such rotation can be undone by a voluntary eye movement. Thus it cannot

generate exterospecific information.7 Thus Lilliput and Brobdignac are optically indistinguishable before the introduc-

tion of Gulliver.8 This is geometrically obvious. An algebraic proof may proceed from the metric

ds2 ¼ d x2 þ d y2 þ d z 2

x2 þ y2 þ z 2 ;

which is invariant with respect to rotation-dilations about the origin.

9 This case is more important than might be expected, for it describes the structure

of typical pictorial spaces.10 In the simpler case of the plane the transformations become

u0 ¼ hu þ t u;

w0 ¼ gu þ kw þ t w;

whereas the analogous group of Euclidean similarities is

u0 ¼ hðu cosu w sinuÞ þ t u;

w0 ¼ hðu sinuþ w cosuÞ þ t w:

In the latter case sizes are scaled by h, whereas the angles are not affected. In the for-

mer case both sizes and isotropic angles are scaled, the first by h, the latter by k. Both

angle and distance measure are parabolic, whereas although the distance measure is

also parabolic, the angle measure is elliptic in the latter case.11 At least not in the case of the stationary, monocular observer. Binocularity and

translational movement (not eye movements) make an essential difference since theygenerate exterospecific information.

174 J.J. Koenderink/ Journal of Physiology - Paris 106 (2012) 173–182



statistical mechanics of the solid state, and so forth, see Feynman,1970), and ‘‘pointer readings’’ (Eddington, 1929). In objective, for-

mal accounts there figure no qualities, nor meanings. All there is, is

observed structure, and formal theory. The observer is of no special

importance (that would introduce subjectivity), but is yet another

physical structure that just so happens to be included in the

space.12

Imaginary spaces are composed of meaningful qualities in some

nexus of simultaneous relations. These spaces do not contain the

observer as just another object. The eye is not in the space at all.

Imaginary spaces are necessarily subjective, because only present

in personal awareness.13

This poses serious obstacles to the use of natural perspective in

the description of imaginary spaces. Most importantly, as the eye is

not in the space (Wittgenstein, 1992), there is no such a thing asthe range of a point.

A quality that might perhaps relate to range is that of ‘‘depth.’’

Depth is a feeling of ‘‘otherness,’’ or ‘‘remoteness.’’ Phenomenolog-

ically, depths of objects can be linearly ordered, and, to some ex-

tent, depth differences can be compared. There is no notion of an

absolute depth though. Thus the depth dimension has a structure

not unlike that of the affine line A1. If there were to be a relation

between depth and range, it would evidently have to be of a loga-

rithmic nature. However, there is no necessary, causal relation be-

tween depth and range. There cannot be any such relation between

the mental and the physical realms. If so, the physical would sub-

sume the mental.

3. Psychogenesis of imaginary space

The phenomenology of vision is that one is aware of an endless

sequence of ‘‘presentations.’’ Presentations just happen, nothing

you can do about them,14 except by closing or opening your

eyes, looking into various directions, and so forth. Such voluntary

actions are only a minor part of the vision-related actions that

occur involuntarily. Presentations are structured, composed of

qualities and meanings (Metzger, 1975). They are pre-reflective,

and proto-rational (Riedl, 1984). In cognition qualities have beenstripped off, and meanings formalized. You do not think presenta-

tions, they just happen. Ignoring cognition, I concentrate on immedi-

ate, optics related awareness.15 I focus mainly on simultaneous order,

that is spatial qualities.

This is the problem of ‘‘psychogenesis,’’ the genesis of the men-

tal. Unfortunately, the term ‘‘psychogenesis’’ is usually applied in a

different sense,16 the essential difference being time scale. I adapt

the perfectly descriptive term ‘‘psychogenesis’’ though. The relevant

time scale is that of the formation of a presentation, about a tenth of

a second.

No doubt the brain is involved in the generation of presenta-

tional awareness, but there consensus (in no way complete!)

stops. In the mainstream account (Marr, 1982; Palmer, 1999),

what happens when you stand in front of a scene and open youreyes, is roughly the following. The causal chain is cut at the level

of the absorption of radiant power in the retinal photoreceptors,

the layout of the receptor array being ‘‘given.’’ Thus one starts

from samples of a two-dimensional scalar field, the retinal ‘‘im-

age.’’ Then follows a sequence of image operations, yielding trans-

formed images galore. Finally, there is a magic step: the set of

derived images turns into a ‘‘representation of the scene in front

of you.’’

‘‘Magic,’’ because image transformations convert structures into

structures. Algorithms cannot convert mere structure into quality

and meaning, except by magic. In computer science this magic is

implemented through ‘‘formats’’ (Knuth, 1997). The same sequence

of keyboard presses may be interpreted as a password, a number, a

word in the English language, some code, an assembler command,gibberish, . . . depending, on the format applied by the currently ac-

tive algorithm. Input structure is not intrinsically meaningful,

meaning needs to be imposed (magically) by some arbitrary format.

Fig. 2. These figures illustrate a planar section through the eye. The figure at left shows a pencil of pre-geodesics through a fiducial point. The eye is indicated with the small

circle at bottom. At right the same configuration has been transformed (conformly) to log-range—angle space. The eye is not in this space, it is located at infinite distance

downwards from the (transformed) fiducial point, on the central mid-line. This central mid-line is a singular geodesic that passes through the eye, a visual ray. The pre-

geodesics are drawn at equal isotropic slope angle intervals. Note that the isotropic angle measure is parabolic (thus not periodic). The central mid-line is an isotropic

direction, it has infinite slope.

12 Thus verbal reports are to be considered nothing but the meaningless movement

of air molecules. Behaviorist psychology was consistent in this respect, though it is

‘‘non-invasive physiology,’’ rather than psychology proper.13 Verbal reports are potentially meaningful, first person reports are the only way to

come to know about other people’s awareness, otherwise there is only physiology.14 Much like sneezing.

15 Thus the Gestalt school of psychology was interested in perception (immediate

awareness) per se, whereas cognitive psychology deals only with the thought

processes that follow presentations. This is a crucial ontological difference.16 A common definition of ‘‘psychogenesis’’ is: 1. The origin and development of

psychological processes, personality, or behavior. 2. Development of a physical disorder or

illness resulting from psychic, rather than physiological, factors. This is evidently not my

intended meaning. The interested reader should pursue (e.g., in Google search)

‘‘microgenesis’’ instead. In short, microgenesis is the pre-conscious process that

purportedly presents you with immediate visual awareness. Its study was initiated by

the psychologists of the early Gestalt schools. A modern account is given by Brown(1996).

J.J. Koenderink/ Journal of Physiology - Paris 106 (2012) 173–182 175



The same input structure may thus give rise to multifarious, per-

haps mutually incompatible, meanings.

This immediately applies to psychogeny, the process that makesyou ‘‘see the scene in front of you’’ when you open your eyes in

broad daylight. This is Berkeley’s (MDCCIX) original argument. It

seems impossible to refute. The mainstream account bridges the

ontological gap via spooky Deus ex machina mechanisms.

3.1. Probing

Main reason that one is forced to refer to such spooky mecha-

nisms is that visual perception is understood as ‘‘inverse optics’’

(Poggio, 1984). Because optical structure is just structure, that is

meaningless, one is forced to postulate mysterious mechanisms

for intentions, qualities and meanings. The only way out of this di-

lemma is to deny the existence of the latter, which is to deny the

existence of the mental realm (Dennett, 1992). Preferred by some(Dennett, 1992), this is perhaps less desirable to most of us.

Alternatives to the mainstream account have to invert the chain

of events, that is to say, replace inverse optics with ‘‘controlled hal-

lucination.’’ Such accounts have been proposed by Bergson (1907),

Schrödinger (1992), among more. A modern account is by Brown

(1996). The mainstream tends to ignore these as ‘‘unscientific.’’

Yet it is actually the mainstream account itself that is incoherent,

and has to rely on magic.

Organisms have a natural urge to grow, and expand their realm.

Such tendencies are observed from the simplest organisms to man.

Organisms poke their environment, partly randomly, partly inten-

tionally. When poking meets resistance it becomes probing. When

probing meets resistance, it is informative. Organisms learn about

their world through informative probings. In Schrödinger’s(1992) view the world lights up to organisms when resistance to

a probing is met, as a spark of enlightenment, a germ of

awareness.17

In formal terms, probing may be understood as ‘‘questioning.’’

Probing is intentional, probing for something. The intention, that is

the question, presupposes possible answers. Thus the meaning is in

the question, not in the answer. In that sense questions are like

formats. The difference is that questions are intentional (Brentano,

1874) to start with. Formats proper are merely reactive, questions(probings) are intentional, world directed, active, and therefore

meaningful by their very nature.

This account is seamlessly in line with biological thoughts (the

ethology of Lorenz (1973), Tinbergen (1951), Riedl (1984), etc.),

differently from the mainstream account of vision which is in

many respects unduly anthropocentric.18

3.2. The Sherlock model

The mechanism of optics related awareness (seeing) is perfectly

illustrated by the time honored methods of forensic investigation,

which may be labeled with the name of that prototypical detective

Sherlock Holmes (see Conan Doyle, 1887).

As the investigator is confronted with the scene of the crime,

what is he to do? Compare the dumb village policeman to the

superior detective.

The clueless village policeman will proceed to collect and pho-

tograph anything even mildly out of the ordinary. This results in a

file of mutually unrelated facts. The size of this file is potentially

limitless, for the world is infinitely structured. There is no end to

which fact, perhaps even on the molecular scale (think of DNA

traces), might eventually prove to be important. Facts are not ‘‘evi-

dence,’’ they are simply facts. Facts yield no account of what hap-

pened at the time of the crime. Record of a headless corpse is only

suggestive. Speculation goes beyond the facts. This ‘‘bottom up’’

modus operandi of the village policeman is analogous to the ‘‘in-

verse optics’’ account of visual perception.

Sherlock Holmes’ method is different. He conceives of likely

‘‘plots,’’ and on the basis of these hunts for evidence. In doing this,he ignores the bulk of facts. Any mere fact may become evidence in

the context of some plot, whereas it will be mere structure that

may be ignored in the context of other plots. Different facts (say

a discarded cigarette butt and a broken flower pot) become mean-

ingfully related in the context of some plots, but are totally unre-

lated in other contexts. Some plots ‘‘work’’ (perhaps to different

extents), others do not. Sherlock Holmes keeps generating plots

until one fits a variety of otherwise mutually unrelated facts so

well, that the odds are overwhelmingly in its favor. Since the prob-

abilities of unrelated rare facts combine multiplicatively, this pro-

cess is almost bound to yield virtual certainty (Pearl, 2000).

Generating plots is not that hard either, at least when the investi-

gator understands the environment he is working in. It is like play-

ing the game of twenty questions with nature, his chances to winare substantial. Some very successful optimization algorithms

work like this, starting from mere random guesses, for instance

‘‘harmony finding’’ (Geem et al., 2001).

The analogy to vision is immediate. Facts in vision are optical

structures. They are overwhelmingly abundant, but meaningless.

Evidence is fact considered in the context of a plot. In vision such

evidence is known as ‘‘cue.’’ Facts in themselves are not cues.

Fig. 3. A geodesic triangle defined by the points {1, 0, 0}, {0, 2, 0}, and {0, 0, 4}

(vantage point at {0, 0, 0}), with some geodesic arcs that connect a vertex to its

opposite side. Notice that the nexus of geodesic arcs fails to ‘‘mesh,’’ although only

by a little margin. The triangle is not a patch of surface, but is ‘‘thick,’’ volumetric.

17 That the only learning is by mistakes will readily be accepted. The ‘‘magic’’ in

Schrödinger’s account is in the (micro-) enlightenment. It cannot be accounted for by

the exact sciences. It is very intuitive in a phenomenological sense though. You vividly

experience the results of your mistakes, whereas much sensorimotor behavior (e.g.,

walking under optical control) goes by unnoticed.18 For instance, the mainstream account stresses the veridicality of perception. But

evolution drives fitness, not veridicality. It develops idiosyncratic user interfaces, notrepresentations of basic physics. Your vision is different from that of your cat or dog.




The observer selects structure, and promotes it to cue status. A cue

is like the answer to a question (or probing), and therefore mean-

ingful. It may prove misleading though. Wrong question, wrong

answer! In the psychological literature plots are known as ‘‘situa-

tional awareness.’’ The general knowledge of the investigator is

known as ‘‘background,’’ (Searle, 1983), ‘‘frames,’’ (Minsky, 1974),

‘‘mental models,’’ (Lakoff, 1987; Johnson-Laird, 1983), and so forth.

It is crucial.Since plots are freely invented, they are technically speaking

hallucinations.19 Holmes’ method of generating plots, discarding

them when they do not fit the evidence, preferring one over another

if the odds are in its favor, is much like the process of biological evo-

lution. Only the fittest plots surface into awareness. Thus presenta-

tions (momentary visual awarenesses) are generated by endless

diversification and merciless pruning. This renders ‘‘controlled hallu-

cination’’ a powerful information generating mechanism (Dawkins,

1986). It is the only such mechanism known—or even imaginable.

It is not essentially different from ‘‘the scientific method.’’

Notice that the ‘‘Sherlock model’’ is not in need of a special

‘‘attention mechanism’’ (Treisman, 1969), for the very method is

attention at work. Nor is it in need of some special mechanism to

‘‘solve the binding problem’’ (Revonsuo and Newman, 1999). There

is no such a thing as a binding problem. Any disjunct structures

that figure as evidence in some plot are thereby automatically

‘‘bound.’’

3.3. Neural mechanisms

In the mainstream account one often refers to the optical struc-

ture as ‘‘data,’’ or ‘‘information.’’ This is thoroughly misleading

because to be understood in the Shannon (1948) sense of utterly

meaningless information. As the brain structures transform the

optical structure into a variety of structured neural activities, main-

stream often uses semantic terms to describe them. This confuses

facts with evidence. In the case of an ‘‘edge detector’’ (Canny,

1986) the very name suggests that the edge exists before being de-tected. Thisis nonsensical, the so-callededge detectoris really noth-

ing but a ‘‘first order directional derivative operator’’ (Koenderink

and van Doorn, 1992). The latter term is to be preferred because it

describes the transformation of structure into structure, whereas

the former suggest some spooky operation.

When vision is understood as inverse optics, it is natural to think

of the early transformations as to take the brunt of the effort. The

primary visual cortex is supposed to play a key role in constructing

a ‘‘representation’’ of reality. From a biological perspective this is

highly unlikely. Throughout the evolution of brains layer after layer

was added, the earlier ones always remaining (Striedter, 2005). The

later structures are built on top of the earlier ones and originally

served to refine already existing processes. This is the way evolu-

tion proceeds (Dawkins, 1986). One should look for the origin of presentations in the early structures, rather than the recent ones.

An alternative way to understand the role of primary visual cor-

tex, and related structures, focusses on vision as ‘‘optics related

awareness.’’ There is also vision as ‘‘optically guided behavior,’’

which is for the larger part irrelevant to momentary awareness,

since it applies equally to zombies. The latter part is important to

survival, and largely co-determined the evolution of the brain. I

ignore it here, since it is unrelated to awareness, e.g., you are not

aware of how you walk, you merely do it.

To become aware of something is due to the promotion of cer-

tain optical structures to the status of evidence. It is not the thing

itself, but some ‘‘sign’’ of it. Usually many layers of indirection can

be distinguished. For example, consider how you may become

visually aware of ‘‘human presence,’’ as may happen in the context

of a hide and seek game. Seeing a person works, as does spotting

part of a person (e.g., a foot sticking out from behind an occluder),

as does finding a footprint. The ‘‘footprint’’ is really a depression inthe sand that might have resulted from a dust devil, or lightning

struck, but is taken by your vision to be an impression of a human

foot.

Of course ‘‘seeing a person’’ also involves the presence of a cer-

tain optical structure at the eye, a certain cortical activity, and so

forth. Thus ‘‘seeing human presence’’ necessarily involves a long

chain of indirection. The footprint, the field of radiant power, and

the cortical activity are on the same ontological level, being all

physical structures. The cortex is no more a ‘‘footprint detector’’

than the sand of the beach is. The cortical activity per se is just

as meaningless as the depression in the sand. In this sense you

use your cortex much as you use your muscles.

Wet sand is much better than dry sand if you are interested in

foot prints. Likewise, the cortex has developed into a highly func-

tional substrate for the representation of optical structure. It may

be understood as a volatile buffer of readily available, conveniently

packaged facts. In order for this to be possible it is structured as a

‘‘geometry engine’’ (Koenderink, 1990; Petitot, 2008), implement-

ing differential geometric operators that allow invariant, frugal

description of optical facts. Analogous reasoning applies to various

later, increasingly dedicated cortices.

The plots must derive from the earliest structures of the brain,

and go through processes of diversification and pruning as they

branch out towards the newer structures (Brown, 1996). These hal-

lucinations must pass through various dreamlike phases as they

evolve, their initial qualities being emotional. The generation in-

volves the immediately preceding states, as well as much earlier

states (often denoted memories, background, etc.). Presentational

awareness involves the single surviving plot from an evolution thatinvolved numerous alternatives. As it occurs it immediately makes

place for the next presentation, which is likely to be very similar,

but may occasionally differ greatly. A small number of presenta-

tions (at most a few seconds worth) make up a ‘‘specious moment.’’

As presentations enter cognition they lose their immediate, vivid

qualities and meaning. Thoughts are different from presentations

in that they lack ‘‘mental paint.’’ You cannot know presentations,

they just happen.

Thus presentations are the final stage of an evolution, where

further development is no longer possible (perhaps awaiting the

overgrowth of more intricate brain structures). They are like the

outer, rigidified crust of an ever changing, tremendously flexible

process.

The entities of awareness are the objects of your environment.Perversely, mainstream considers them to be the ‘‘causes’’ of your

perceptions. This is mistaken, because the objects of your percep-

tion (e.g., a ‘‘chair’’) have no equivalents in physics (some odd, ill

defined collection of elementary particles perhaps?).

Here the behaviorists (Skinner, 1938) were more consistent

than present day cognitive scientists, their immediate successors

and heirs, in holding that a verbal utterance is nothing but the

movement of air molecules. Although perhaps not entirely wrong,

it certainly is not right. If anything, it is inhuman.

4. Iconogenesis

On opening your eyes in broad daylight, presentations of thescene in front of you happen to you, at least if you are not blind

19 ‘‘Hallucination’’ has a bad ring to it. But notice that even a scientific theory is

nothing but a hallucination until there is sufficient empirical evidence in its favor.

How do you know you are not hallucinating? By noticing the coherence or mismatch

of your (visual) presentations with those from other modalities (hearing, touch, etc.),

with your situational awareness, and with your observed behavior of others. For anyof these it is easy to point out spectacular failures of course.




or daydreaming. You experience objects, relations, causal histories

and futures. You experience in terms of ‘‘mental paint’’ (qualities),

and meaning (intentionality). Here I focus on spatial qualities only.

I refer to this process of constructing the visual field as iconogen-

esis,20 a subprocess of psychogenesis.

4.1. The geometry of the ‘‘visual field’’

The ‘‘visual field’’ is the simultaneous order of presentationswhen the depth quality is ignored. You become aware of it when

looking at , instead of into, a painting, attending to the simultaneous

order of pigments.

For most human observers the topology of the visual field is

well developed. Ill developed topology is known as ‘‘tarachopia,’’

scrambled visual field (Hess, 1982). One speaks of ‘‘local sign,’’ an

address associated with each optic nerve fiber (Lotze, 1852). An

idea by von Helmholtz (1977) provides a possible neural imple-

mentation. Correlation of nerve activities signals spatial overlap

of receptive fields. As I showed (Koenderink, 1984) this can be

developed into a mechanism that generates a C ech cohomology

(see Fig. 4).

A metric is probably calibrated via eye movements,21 the rele-

vant observations and theory are due to von Helmholtz (1856,1867). In this paper I simply treat the visual field as the Euclidean

plane E2, ignoring the various (important) conceptual problems with

this notion.

4.2. The geometry of imaginary spaces

You are aware of entities at various degrees of remoteness.22

This quality of remoteness is called ‘‘depth.’’ When a depth label is

assigned to each point of the visual field, ‘‘visual space’’ becomes a

fiber bundle E2D, where D denotes the depth domain. Each point

in the visual field has its own copy of the depth domain. In establish-

ing a spatial configuration the psychogenetic process shifts depth

values along the depth fibers, much like one shifts beads along the

wires of an abacus (see Fig. 5). The structure of visual space resultsfrom this ‘‘Glasperlenspiel.’’ The resulting configuration is consistent

with the depth cues identified by the iconogenetic process.

The fibers of visual space are ‘‘visual rays,’’ these are loci in vi-

sual space that are composed of parallel points. Visual rays should

not be confused with the rays of geometrical optics, or with the

propagation of radiant power. They are closer to the notion of

‘‘rays’’ as used by Euclid (see Burton, 1945) and the other ancient

authors. The rays of geometrical optics fan out from the anterior

nodal point of the eye,23 which, for relatively distant objects, is

not that different from the bundle of concurrent rays at the centerof rotation of the eye ball. Whether the human observer has estab-

lished a correlation between the visual rays, and the rays of geomet-

rical optics, is something on which the literature is silent. We have

recently investigated this problem (Koenderink et al., 2009,

2010b), which may be referred to as the issue of ‘‘external local

sign,’’ so as to distinguish it from Lotze’s local sign (Lotze, 1852;

von Helmholtz, 1977; Koenderink, 1984). We find quite a bit of in-

ter-observer variability. Most people treat their visual rays roughly

as if it were a bundle of parallel geometrical optics rays (Fig. 6), as

is evident from the perhaps surprisingly huge (tens of degrees of vi-

sual angle) errors they make in the judgment of angular relations.

Depth is a serial order, as can be shown as follows (van Doorn

et al., 2011). When I indicate two locations in a painting, of a real-

istic landscape say, you can generally tell which location is ‘‘clo-ser.’’ Doing this for all pairs taken out of a set of N points, I

collect P ¼ 12

N ðN 1Þ pairwise rankings. A linear depth order of N

items has only Q = N 1 degrees of freedom, thus it is a priori un-

likely that it will account for the data. The observations have far

too many degrees of freedom. For N = 50, a realistic number, one

may account only for Q = 49 of the P = 1225 degrees of freedom.

Thus it is non-trivial that I find that, empirically, such a set of

observations can always be accounted for in terms of a linear order

within the variance found in repeated sessions (see Fig. 7). Since

P Q , this is a strong indication for the existence of a coherent

one-dimensional realm. In the illustrated result the rankings were

almost perfectly accounted for. Human observers easily resolved

dozens of depth layers in a painting.

Such methods work because a mark put on the picture planewill travel into depth in your visual awareness, until it attaches

to the closest surface of an object. This property is crucial in the

psychophysics, though its neural basis is fully in the dark. The

reader may easily try this on a portrait by using a marker to put

a freckle or beauty spot ‘‘on the cheek.’’ One sees the principle

put in practice on many poster boards where politicians or

super-models have acquired moustaches or black teeth.

It is not hard to introduce a metric either (van Doorn et al.,

2011). Instead of merely indicating two locations I put a circular

disc at each location, and grant the observer control over their rel-

ative sizes (see Fig. 8). I instruct them to set these relative sizes

such that the disks look like two ‘‘equally large’’ pictorial objects.

Fig. 4. The start of a C ech cohomology. At left two receptive fields overlap, leading

to correlation. Inversely, correlation indicates overlap. At right the notion of

inclusion: B is included in A iff for any C that overlaps with B, it is the case that C

also overlaps with A. In a similar way one defines simplices and their boundaries.

This enables one to boot up a C ech cohomology on the basis of a correlation

structure.

Fig. 5. The fiber bundle E1 D (in reality the base space is E2). All visual rays, like

RR 0 , are mutually independent. The psychogenetic process assigns depths as if by

sliding beads on strings, like the white bead on RR 0 . The resulting cross section is not

necessarily smooth. If it is (like here) one speaks of a ‘‘pictorial relief.’’

20 ‘‘Iconogenesis’’ is an apt term for the subprocess of psychogenesis that is aimed at

the genesis of visual awareness.21 The crucial observation made by Helmholtz is that ‘‘Listing’s Law’’ of eye

movement constrains the group of rotations of the eye ball to an abelian group. A

study of the orbits of the group then leads to a useful geometrical structure of the

visual field.22 Remote ‘‘from the ego’’ if you want. The eye as a physical object is not involved.

23 It is natural to let them ‘‘fan out’’ instead of ‘‘fan in,’’ because they are related toprobing. This in no way implies an extramission theory of radiant power.




Fig. 6. At left geometrical rays fanning out from the eye. They roughly fill a half-space. At right the visual rays as the mind knows them. There is no eye in this picture and the

rays do not fan out. Many observers have a bit of fanning out, about a ninety degree cone being typical.

Fig. 7. At the top left a frontal view of pictorial space, its base space. The base space is simply the picture plane. The picture is a copy after a wash drawing by Francesco

Guardi, due to Anne-Sophie Bonno (http://www.atelier-bonno.fr/). The colored dots are points whose depths were psychophysically compared, one pair at a time, in randomorder. This yields the ranking order shown in the 3D plot at top right, and in the plan and elevation at bottom. The lines are the depth fibers of pictorial space.


http://www.atelier-bonno.fr/

http://www.atelier-bonno.fr/



I consider the logarithm of the size ratio as a measure of the depth

difference, using the heuristic that depth differences should com-

bine linearly. Given the depth differences between the

P ¼ 12

N ðN 1Þ pairs taken out of N locations, I find the best fitting

N depth values that explain the differences. Since absolute depth is

undefined I arbitrarily set the average to zero. Perhaps surprisingly,

this works really well, within the spread from repeated sessions.

Since Q = N 1 P , this is again a strong indication for a one-

dimensional realm, this time with a metric (see Fig. 9). The depths

for different observers are related as w0

= g uu + g v v + k w + t w, withapparently idiosyncratic g and k. Notice that t w is determined by

the constraint put on the average. Thus the depth domain D appar-

ently has the structure of the affine line A1.

How are these depths related to the coordinates of the picture

plane? One way to study this is as follows (van Doorn et al.,

2011). I superimpose the pictures of a pointer and of a target on

the picture plane. The picture of the pointer can be put in various

(pictorial) spatial attitudes, and is put under the observer’s control.

The task is to let the pointer apparently point to the target in pic-

torial space (see Fig. 8). Since the pointing can be done either way

one obtains N (N 1) directions. The two-way directions, combinedwith the distance in the picture plane define a unique parabolic arc,

Fig. 8. In the left column a pointer and target, in the right column the relative size cue. In the top row the left side is closer, in the bottom row the right size. These probes

would be superimposed over a picture in an actual experiment.

Fig. 9. At left a frontal view of pictorial space, which is simply the picture plane. The colored dots are points whose depths were psychophysically compared by way of the size

cue probe illustrated in Fig. 8 right. The points labeled with square marks are in the far field. This yields a metrical order shown in the 3D plot at right.

Fig. 10. At left a frontal view of pictorial space, which is simply the picture plane. The colored dots are points whose depths were psychophysically compared by means of

two-way pointing, using the probes illustrated in Fig. 8 left. Pointing yields a metrical order shown in the 3D plot at right.




and thus another depth difference, a total of P ¼ 12

N ðN 1Þ of

them. Once again I obtain a metrical depth order (see Fig. 10).

The procedure turns out to be consistent too. Moreover, the three

methods mutually agree up to transformations of the type

w0 = g uu + g v v + kw + t w.

An example of a more general ‘‘gauge transformation’’ is shown

in Fig. 11. Observers had to adjust a gauge figure superimposed onthe picture so as to sample the spatial attitude of the tangent

planes to a pictorial relief at 422 barycentra of the faces of a regular

hexagonal triangulation. From this one finds the depths at the ver-

tices that best explain these spatial attitude observations. The tri-

angulation covers the larger part of the torso of the frontmost

figure in the drawing. The gauge figure was a small wireframe el-

lipse that had to be adjusted so as to appear as ‘‘a circle painted

upon the surface.’’ Notice that the pictorial relief covers a white

area in the picture, thus it has to be due to contour information.

In this case the stimulus is so ambiguous that the observers

come up with somewhat different presentations. For the case of

more realistic drawings, or photographs, we find that the simple

affine transformation invariably succeeds, implying that the picto-

rial space is homogeneous. In the present case an overall affine

transformation does not suffice though. One needs a transforma-

tion that varies slightly from place to place in the picture plane. Lo-

cally, this transformation is again of the type w0(u, v , w) = g uu +

g v v + kw + t w, but the parameters { g u(u, v ), g v (u, v ), k(u, v )} (the

depth shift is irrelevant) change smoothly from place to place.

With such a transformation the coefficient of variation for the

regression of depths from two observers at corresponding points

increases from 0.49 (straight regression) to 0.97 (regression withthe best transformation). As ‘‘best’’ transformation we applied

the optimal quartics (leading to the highest coefficient of variation)

for the parameters { g u(u, v ), g v (u, v ), k(u, v )}. In the figure we pres-

ent the surfaces w0(u, v , 0) and w0(u, v , 1). These can be interpreted

geometrically as frontoparallel planes at depths w = 0,1 for one ob-

server, mapped into the pictorial space of another observer. Obser-

vations such as these suggest that it may be of interest to consider

more general ‘‘gauge transformations’’ than the simple overall

affinities.

Apparently the idea of a consistent ‘‘imaginary space’’ is viable

in the sense that it yields an economical description of a large body

of otherwise unrelated data. Formally, it is consistent with a Cay-

ley-Klein space (Cayley, 1859; Klein, 1893, 1872; Yaglom, 1979)

of the type described formally by Strubecker (1941, 1942, 1943,

1945) and Sachs (1990).

Fig. 11. At top left a drawing by Picasso that appears to combine multiple viewpoints. We measured the spatial attitudes of tangent planes of the pictorial relief of the body of

the frontmost person at 422 points. The scatterplot at bottom left compares the depths at corresponding points for two different observers. The coefficient of variation is only

0.49. A nonlinear gauge transformation (illustrated at top right) brings the coefficient of variation on a much higher level, namely 0.97 (scatter plot at bottom right).

Apparently the observers use different ‘‘mental viewpoints,’’ and, moreover, assume different mental viewpoints for different parts of the picture. The surfaces at top right are

the loci of the zero and unit points on the depth fibers of one observer that correspond to the canonical loci of the other observer.




5. Final remarks

Human ‘‘imaginary space’’ is a fiber bundle E2 A1, where the

base space is the ‘‘visual field,’’ and the fibers the ‘‘depth’’ domain.

The psychogenetic process shifts depth values along visual rays

like beads on their strings. It does this on the basis of ‘‘depth cues’’

that are identified as such by the process itself. The result is ambig-

uous by its very nature, and the observer’s optical awareness con-sists of a sequence of ‘‘presentations’’ that are often quite similar to

each other, though occasionally very different, just think of the

familiar ‘‘flips’’ of a Necker cube.

A large part of the ambiguity can be formalized as the group of

isotropic rotations, angular scalings, and depth translations in a

singly isotropic (otherwise Euclidean) Cayley–Klein space. This

can be understood as resulting from the fundamental invariance

properties of natural perspective.

The psychogenetic process constrains its articulations through

probing the visual front end. This part of the brain is readily avail-

able for formal descriptions that are close to the neural hardware.

The implementation of the group of isotropic similarities, a geo-

metrical object that can easily be probed through psychophysical

means, remains fully in the dark though. Processes that seem read-

ily amenable to neurophysiological study are the implementation

of ‘‘probings’’ of the primary visual regions by the deep structures

where the intentional processes are launched.

Acknowledgments

The empirical work on pictorial space (van Doorn et al., 2011)

was done with Johan Wagemans (Laboratorium voor Experimen-

tele Psychologie, Katholieke Universiteit Leuven) and Andrea van

Doorn (Industrial Design, Delft University of Technology). This

work was supported by the Methusalem program by the Flemish

Government (METH/08/02), awarded to Johan Wagemans.

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