Simplicity in vision - KU Leuvenu0084530/reprints/siv-lecture.pdf · Simplicity in vision: Topics Theoretical cycle Empirical cycle Tractability cycle The computability of simplicity

Simplicity in vision

Peter A. van der Helm

Laboratory of Experimental PsychologyKU Leuven – University of Leuven

Peter A. van der Helm Simplicity in vision

Simplicity in vision

Introduction

The veridicality of simplicity

Symmetry perception

Cognitive architecture


Human visual perceptual organization

The neuro-cognitive process that enables us to perceive scenes asstructured wholes consisting of objects arranged in space.

Yes No

YesNo



Visual perceptual organization may seem to occur effortlessly, butby all accounts, it must be both complex and flexible:

It singles out one hypothesis about the distal stimulus fromamong a myriad of hypotheses that fit the proximal stimulus.

To this end, multiple sets of features at multiple locations in astimulus have to be grouped in parallel.

This implies that the process must cope with a large numberof competing combinations simultaneously.

Hence, the combinatorial capacity of the process must be high,which, together with its high speed (it completes in the range of100–300 ms), reveals its truly impressive nature (Gray, 1999).



Visual perceptual organization may seem to occur effortlessly, butby all accounts, it must be both complex and flexible:

It singles out one hypothesis about the distal stimulus fromamong a myriad of hypotheses that fit the proximal stimulus.

To this end, multiple sets of features at multiple locations in astimulus have to be grouped in parallel.

This implies that the process must cope with a large numberof competing combinations simultaneously.

Hence, the combinatorial capacity of the process must be high,which, together with its high speed (it completes in the range of100–300 ms), reveals its truly impressive nature (Gray, 1999).


Why do things look as they do?

The likelihood principle (von Helmholtz, 1909)

Vision produces interpretations that are most likely to be true.

The Gestalt law of Pragnanz (as formulated by Koffka, 1935)

”Of several geometrically possible organizations that one willactually occur which possesses the best, the most stable shape.”

The minimum principle (Hochberg & McAlister, 1953)

”The less the amount of information needed to define a givenorganization as compared to the other alternatives, the more likelythat the figure will be so perceived.”

The simplicity principleVision produces simplest organizations.




















Keep it simple!

William of Occam (c. 1287–1347)

Occam’s razor:”Pluralitas non est ponenda sine necessitate.”(”Plurality is not to be posited without necessity.”)

”We consider it a good principle to explain phenomena by thesimplest hypothesis possible.” — Ptolemy (c. 90–168)

”We may assume the superiority, all things being equal, of thedemonstration which derives from fewer postulates or hypotheses.”

— Aristotle (c. 384–322 BC)


Keep it simple!

William of Occam (c. 1287–1347)

Occam’s razor:”Pluralitas non est ponenda sine necessitate.”(”Plurality is not to be posited without necessity.”)

”We consider it a good principle to explain phenomena by thesimplest hypothesis possible.” — Ptolemy (c. 90–168)

”We may assume the superiority, all things being equal, of thedemonstration which derives from fewer postulates or hypotheses.”

— Aristotle (c. 384–322 BC)


What if all things are not equal?

Observations on planet orbits

Simple theory Many unexplained observations

priorcomplexity

conditionalcomplexity

Before 1600, planet orbits were thought to be circular

priorcomplexity


Complex theory Few unexplained observations

Johannes Kepler (1571–1630): planet orbits are elliptical


What if all things are not equal?

It is not only a matter of how good a shape is in itself (the prior),but also of how well it fits the proximal stimulus (the conditional).


Could simplicity guide perceptual organization?

A competitive approach toperceptual organization

Leeuwenberg, E., & van der Helm, P. A. (2013).

Structural information theory.

Cambridge University Press.

Fundamental issues inperceptual organization

van der Helm, P. A. (2014).

Simplicity in vision.

Cambridge University Press.


Levels of description

Marr’s (1982) levels of description in vision:

The computational level — what is the logic that defines thenature of resulting mental representations of incoming stimuli?

The algorithmic level — how are the input and the outputrepresented and how is one transformed in the other?

The implementational level — how are those representationsand transformations neurally realized?

Epistemological pluralism: Complementary descriptions mayexplain how the goal is reached via a method allowed by the means.


Levels of description

Marr’s (1982) levels of description in vision:

The computational level — what is the logic that defines thenature of resulting mental representations of incoming stimuli?

The algorithmic level — how are the input and the outputrepresented and how is one transformed in the other?

The implementational level — how are those representationsand transformations neurally realized?

Epistemological pluralism: Complementary descriptions mayexplain how the goal is reached via a method allowed by the means.


Marr’s levels in the kitchen


Marr’s levels in biology

GOAL:Origin of species (Darwin, 1844)

Evolution theory

METHOD:Experiments in plant hybridization (Mendel, 1865)

Hereditary theory, classical genetics

MEANS:Molecular structure of DNA (Watson & Crick, 1953)

Molecular biology, modern genetics


Representations (”what”) versus processes (”how”)

Operating bases of the three major research paradigms:

Computational goal ← representational theory

l ”what”: molar – behavioral – competence – cognitive

Algorithmic method ← connectionist modeling

l ”how”: molecular – physiological – performance – neural

Implementational means ← dynamic systems theory

According to Marr, answers to both the ”what” and ”how”questions are needed, even though answering these questions maybe totally different endeavours using totally different tools.




















Levels of evaluation

Multidisciplinary research cycles:

The empirical cycle — has roots in physics; the idea is toconduct controlled experiments to test predictions inferredfrom theories and models (de Groot, 1961/1969).

The theoretical cycle — has roots in mathematics; the idea isto formalize ideas and assumptions in theories and models, tosee if they can be derived from first principles.

The tractability cycle — has roots in computer science; theidea is to assess if theories and models allow for feasibleimplementations in computers or brains (van Rooij, 2008).


Multidisciplinary research cycles

Empiricalcycle

Tractabilitycyclecycle

Theoretical

Formalizations Implementations

Experiments

Theories & Models


Simplicity in vision: Topics

Theoreticalcycle

Empiricalcycle

Tractabilitycycle

The computabilityof simplicity

The nature ofvisual regularity in the brain

The visual hierarchy

The veridicalityof simplicity

Form classification, amodal completion,


symmetry perception


Simplicity in vision: Topics

Theoreticalcycle

Empiricalcycle

Tractabilitycycle

The computabilityof simplicity

The nature ofvisual regularity in the brain

The visual hierarchy

The veridicalityof simplicity


Form classification, amodal completion, symmetry perception


Part 1

The veridicality of simplicity


Vision and the world

Image properties

Perceived objects

Perceived objects

Image

Object properties

Likelihood principle: percepts are most likely to be true.→ external criterion — highly veridical by definition!

Simplicity principle: percepts have simplest representations.→ internal criterion — sufficiently veridical?


Bayes’ rule

Thomas Bayes (1702–1761)

Bayes’ rule: p(H|D) =p(H)∗p(D|H)

p(D)

The posterior probability p(H|D) that hypothesis H is true forgiven data D can be found by multiplying:

the prior probability p(H) that hypothesis H is true, and

the conditional probability p(D|H) that data D arises ifhypothesis H is true.

Note: p(D) is a normalization factor and is currently less relevant.


Bayes’ rule: An example

Imagine an HIV test for which 90% of all test results are correct:

90% of the HIV patients score positive.

90% of the others score negative.

Imagine further that you are one of 1000 arbitrarily chosenparticipants in a population test, and that you score positive... how worried should you be?

Bayes would say: That depends on how many people of the totalpopulation actually have HIV.

OK, say that 2% of the total population has HIV ... what then?



Imagine an HIV test for which 90% of all test results are correct:

90% of the HIV patients score positive.

90% of the others score negative.

Imagine further that you are one of 1000 arbitrarily chosenparticipants in a population test, and that you score positive... how worried should you be?

Bayes would say: That depends on how many people of the totalpopulation actually have HIV.

OK, say that 2% of the total population has HIV ... what then?



Bayesian prior probabilities (before the test starts):

About 20 (2%) of the 1000 participants has HIV.

About 980 (98%) of the 1000 participants does not have HIV.

Bayesian conditional probabilities (test results 90% correct):

Of the about 20 HIV patients, about 18 (90%) score positive.

Of the about 980 others, about 98 (10%) also score positive.

Hence, in total, about 116 positive scores are to be expected:

About 18 positive scores are correct (15.5% of 116).

About 98 positive scores are false alarms (84.5% of 116).

Bayes would say: Retest those 116 persons – prior then is 15.5%.

Note: If prior is 40%, then 85.7% correct and 14.3% false alarms.









































Bayes’ rule in vision

high conditional low conditionalprobability probability

But what are their prior probabilities?


Bayes’ rule in vision

high conditional low conditionalprobability probability

But what are their prior probabilities?


Helmholtzian likelihood principle in certainty terms

posteriorprobability

priorprobability

conditionalprobability

Likelihood principle: interpretation of data on the basis of the world

independencyviewpoint

dependencyviewpoint

maximize certainty p(H|D) = p(H) ∗ p(D|H)


Prior complexities: shapes as such

A descriptive code is a reconstruction recipe.

If y = 5 cm, z = 3 cm, a = 70◦, and b = 110◦, thenthe code 2 ∗ (yazb) is a recipe for a parallelogram.

If y = z = 3 cm and a = b = 90◦,then the parallelogram-code would yield a square.

But for a square, a shorter code is 4 ∗ (ya)... this square-code is simpler than theparallelogram-code, because it containsfewer structural information parameters (sip).

The prior complexity of a shape is given by the number of sip in itssimplest code – this is the minimum amount of structuralinformation needed to reconstruct the shape as such.
















Prior complexities: structural classes

Good patterns have few alternatives (Garner, 1970).


Conditional complexities: relative positions

The specific outcomes A and D are equally likely,but outcomes like A and outcomes like D are not.

Probabilities presuppose categories.



The specific outcomes A and D are equally likely

,but outcomes like A and outcomes like D are not.




The specific outcomes A and D are equally likely,but outcomes like A and outcomes like D are not.




From left to right:

In likelihood terms: increasing number of coincidences.

In simplicity terms: decreasing number of degrees of freedom.



The conditional complexity Iexternal reflects the effort to bring the twosticks in their proximal position starting from a general position.

van Lier et al. (1994)




















Occamian simplicity principle in information terms

A set of raw data as such explains nothing; it is better to searchfor a hypothesis (theory/model) by means of which the data canbe described more succinctly.

The best hypothesis H for data D then is the one that minimizesthe sum I (H|D) of:

the information I (H) needed to describe the hypothesis, and

the information I (D|H) needed to describe the data by meansof the hypothesis.

Analogous to Bayesian terminology:• the information I (H) is called the prior complexity;• the information I (D|H) is called the conditional complexity;• the sum I (H|D) is called the posterior complexity.



A set of raw data as such explains nothing; it is better to searchfor a hypothesis (theory/model) by means of which the data canbe described more succinctly.

The best hypothesis H for data D then is the one that minimizesthe sum I (H|D) of:

the information I (H) needed to describe the hypothesis, and

the information I (D|H) needed to describe the data by meansof the hypothesis.

Analogous to Bayesian terminology:• the information I (H) is called the prior complexity;• the information I (D|H) is called the conditional complexity;• the sum I (H|D) is called the posterior complexity.


Prior and conditional complexities

Observations on planet orbits

Simple theory Many unexplained observations

priorcomplexity


Before 1600, planet orbits were thought to be circular

priorcomplexity


Complex theory Few unexplained observations

Johannes Kepler (1571–1630): planet orbits are elliptical


Prior and conditional complexities

Yes No

YesNo



posteriorcomplexity

priorcomplexity


Simplicity principle: interpretation of the world on the basis of data


dependencyviewpoint

minimize information I(H|D) = I(H) + I(D|H)


The duality of simplicity and likelihood

posteriorprobability

posteriorcomplexity

priorprobability

priorcomplexity

conditionalprobability


Likelihood principle: interpretation of data on the basis of the world

Simplicity principle: interpretation of the world on the basis of data


dependencyviewpoint




From surprisals to precisals



likelihood: objective p

simplicity: descriptive I

precisal p = 2−I

modern IT

surprisal I = − log p

classical IT





likelihood: objective p

surprisal I = − log p simplicity: descriptive I

precisal p = 2−I

modern ITclassical IT





likelihood: objective p precisal p = 2−I

surprisal I = − log p simplicity: descriptive I

modern ITclassical IT


Classical information theory: From Morse to Shannon

Samuel Morse (1792–1872)

The Morse Code (1835)

More frequently used lettersget shorter codes:

V (1%) → dot-dot-dot-dashE (10%) → dot

Claude Shannon (1916–2001)

A mathematical theory ofcommunication (1948)

Proved that surprisalsI = − log p

are optimal code lengths

Problems: • Codes are just labels, not representations of content.• Often, the required probabilities p are unknown.


Classical information theory: From Morse to Shannon

Samuel Morse (1792–1872)

The Morse Code (1835)

More frequently used lettersget shorter codes:

V (1%) → dot-dot-dot-dashE (10%) → dot

Claude Shannon (1916–2001)

A mathematical theory ofcommunication (1948)

Proved that surprisalsI = − log p

are optimal code lengths

Problems: • Codes are just labels, not representations of content.• Often, the required probabilities p are unknown.


Modern information theory in vision

Julian Hochberg (1923–)

The minimum principle (1953)

Wendell Garner (1921–2008)

Inferred subsets (1962)

Herbert Simon (1916–2001)

Language invariance (1972)


Modern information theory in mathematics

Andrey Kolmogorov (1903–1987):The complexity of an object is givenby the length of its shortestreconstruction recipe (1966).

Ray Solomonoff (1926–2009):Universal probabilities, based onKolmogorov complexities, might beused to make predictions (1964).

Both proved that it doesn’t matter much which coding language isused to describe objects (the Invariance Theorem).


The Fundamental Inequality

For the infinite number of enumerable probability distributionsP(x) over objects x holds, under some mild conditions, that:

K (x)− K (P) ≤ − log P(x) ≤ K (x)

where K is the Kolmogorov complexity.

In words: if a distribution P is simple, i.e., if K (P) is small,then K (x) ≈ − log P(x), and inversely, also P(x) ≈ 2−K(x).

Hence, if K (P) is small, one could use precisals 2−K(x) instead ofthe often unknown real probabilities P(x) to make predictions.

Note. An enumerable distribution is a rational-valued function of twononnegative integer arguments; examples are the uniform distribution,the normal distribution, and the Poisson distribution.

Li & Vitanyi (1997)


The Fundamental Inequality

For the infinite number of enumerable probability distributionsP(x) over objects x holds, under some mild conditions, that:

K (x)− K (P) ≤ − log P(x) ≤ K (x)

where K is the Kolmogorov complexity.

In words: if a distribution P is simple, i.e., if K (P) is small,then K (x) ≈ − log P(x), and inversely, also P(x) ≈ 2−K(x).

Hence, if K (P) is small, one could use precisals 2−K(x) instead ofthe often unknown real probabilities P(x) to make predictions.

Note. An enumerable distribution is a rational-valued function of twononnegative integer arguments; examples are the uniform distribution,the normal distribution, and the Poisson distribution.

Li & Vitanyi (1997)


The margin between simplicity and likelihood

Complexity K (P) of P is the length of the shortest descriptivecode that specifies the probabilities P(x) of things x .

The more categories, the more probabilities, the larger K (P).

Viewpoint-independent priors

The natural world exhibits an enormous shape diversity:K (P) probably large and prior precisals not very veridical.

Man-made environments exhibit a restricted shape diversity:K (P) probably small and prior precisals fairly veridical.

Viewpoint-dependent conditionals

An object generally gives rise to only a few view-categories:K (P) generally small and conditional precisals fairly veridical.

van der Helm (2000)










van der Helm (2000)










van der Helm (2000)


Priors versus conditionals

Simpler (i.e., more regular)objects belong to smallercategories.

Simpler (i.e., less coincidental)relative positions of objectsbelong to larger categories.


Everyday perception

You take a first glance

... and you make a first interpretation.

You move to take a second glance ... and you might see

... which will trigger you to update your interpretation.


Everyday perception

You take a first glance ... and you make a first interpretation.




Everyday perception


You move to take a second glance

... and you might see



Everyday perception





Everyday perception

A moving observer gets a growing sample of views of a distalscene, and can interpret it by recursively applying Bayes’ rule:

Sample D1: p1(H|D1) = p(D1|H) ∗ p(H)with p(H) as first prior.

Sample D2: p2(H|D2) = p(D2|H) ∗ p1(H|D1)with D1’s posterior as new prior.

etc ....

Well-chosen first priors speed up convergence, but their effect soonfades away, and the conditionals become decisive.

Prior precisals may not be veridical but conditional precisals are→ precisals and real probabilities give about the same predictivepower in everyday perception.


Everyday perception




etc ....




Everyday perception




etc ....




Everyday perception




etc ....




Summary (Part 1)

Helmholtzian likelihood principle:

Vision produces interpretations most likely to be true.

Priors cannot be quantified – conditionals probably can.

Special purpose principle:

highly veridical in – or adapted to – one environment.

Occamian simplicity principle:

Vision produces simplest organizations.

Priors perhaps not veridical – conditionals probably are.

General purpose principle:

fairly veridical in – or adaptive to – many environments.


Summary (Part 1)

Helmholtzian likelihood principle:

Vision produces interpretations most likely to be true.

Priors cannot be quantified – conditionals probably can.

Special purpose principle:

highly veridical in – or adapted to – one environment.

Occamian simplicity principle:

Vision produces simplest organizations.

Priors perhaps not veridical – conditionals probably are.

General purpose principle:

fairly veridical in – or adaptive to – many environments.


Part 2

Symmetry perception


William Blake (1794)

Tyger! Tyger! Burning bright,In the forests of the night,What immortal hand or eyeCould frame thy fearful symmetry?


Why symmetry perception?

Simplest organizations are obtained by capturing visual regularities,that is, regularities the visual system is sensitive to.

Fundamental questions

Which regularities?

What made them visual regularities:

evolutionary relevance of individual regularities?common underlying detection mechanism?

What is their perceptual nature?

What is their role in perceptual organization?


Why symmetry perception?

Simplest organizations are obtained by capturing visual regularities,that is, regularities the visual system is sensitive to.

Fundamental questions

Which regularities?

What made them visual regularities:

evolutionary relevance of individual regularities?common underlying detection mechanism?

What is their perceptual nature?

What is their role in perceptual organization?


What is visual regularity?

Dots arranged equidistantly along an invisible rectangular spiral.



Dots arranged equidistantly along an invisible rectangular spiral.



Apparently, without the spiral, this is not a visual regularity.



Apparently, without the spiral, this is not a visual regularity.



Key phenomenon: Symmetry and Glass patterns are about equallydetectable, and generally better detectable than repetition.

Glass (1969)



Key phenomenon: Symmetry and Glass patterns are about equallydetectable, and generally better detectable than repetition.

Glass (1969)


Evolutionary considerations

PerceptionBiology

Growth

symmetry preferencein mate assessment

degree of body symmetryindicates genetic quality sensitivity to symmetry

high perceptual

in object recognitionsymmetry is useful cue

convergence on symmetricalforms in nature and art


Evolutionary considerations

PerceptionBiology

mental representationsnatural construction ofGrowth

symmetry preferencein mate assessment

degree of body symmetryindicates genetic quality sensitivity to symmetry

high perceptual

in object recognitionsymmetry is useful cue

convergence on symmetricalforms in nature and art


Transformational regularity: Invariance under motion

Regular patterns remain the same after rigid transformations.

Rotational symmetry Translational symmetry

Mathematically sound.

Suited for object classification.

Also suited for symmetry perception?




RepetitionSymmetry

Symmetry: symmetry halves identified by a 3D rotation.Repetition: repeats identified by a 2D translation.

Thus, both regularities are assigned a block structure.

Palmer (1983)




RepetitionSymmetry

Symmetry: symmetry halves identified by a 3D rotation.Repetition: repeats identified by a 2D translation.

Thus, both regularities are assigned a block structure.

Palmer (1983)


Regularity detection anchors

Virtual lines between corresponding elements are the anchors forthe regularity detection process.

Symmetry: parallel lines which are midpoint collinear.Repetition: parallel lines which are of constant length.

Thus, both regularities are assigned a point structure.

Jenkins (1983, 1985); Wagemans et al. (1993)


Regularity detection anchors

Virtual lines between corresponding elements are the anchors forthe regularity detection process.

Symmetry: parallel lines which are midpoint collinear.Repetition: parallel lines which are of constant length.

Thus, both regularities are assigned a point structure.

Jenkins (1983, 1985); Wagemans et al. (1993)


Why is symmetry more salient than repetition?

Bruce & Morgan (1975)

”It is important to realize that a repetition and a symmetry areequally redundant [in the transformational ”block” sense].

Attneave (1954) suggested that symmetric patterns contained anextra kind of perceptual redundancy, in that they could bedescribed by the relationship of each point in the figure to a singleaxis of symmetry.

But this is an intuitive rather than a mathematical notion: Arepetition pattern is equally constrained by the translation rule thatsimilar elements are all positioned the same distance apart.”


Holographic regularity: Invariance under growth

Regular patterns can be expanded preserving the regularity in them.

Symmetry

Pointwise body growth

Repetition

Blockwise queue growth

van der Helm & Leeuwenberg (1991)



Symmetry gets a point structure – repetition gets a block structure.

ModelW = E/n Wsym = 0.5 Wrep = 0.1

where n is the total number of elements in the pattern and E thenumber of holographic identities that constitute the regularity.

W = E/n is the weight of evidence for the regularity.


Suited for symmetry perception: it predicts a number effect inrepetition but not symmetry, and many other phenomena...

van der Helm & Leeuwenberg (1996, 1999, 2004); Csatho et al. (2003)




















Holographic weight of evidence W = E/n

Blobs strengthen repetition but weaken symmetry

Symmetry Repetition

Csatho et al. (2003)



Graceful degradation

R symmetry pairs, N noise elements: W = E/n = R/n = 1−N/n2

W = 0.4

W = 0.3

Barlow & Reeves’ (1979) data:

0

1

2

3

4

5

.1 .2 .3 .4 .5 .6 .7 .8 .90 1

Detectability(d

′ )

Noise proportion N/n

van der Helm (2010)




For a symmetry on R symmetry pairs or a Glass pattern on R dotdipoles, perturbed by N noise elements: E = R and n = 2R + N.

Then, W = E/n can be rewritten into W = 12+1/S with S = R/N.

On the basis of signal-detection considerations, Maloney et al.(1987) proposed the same formula for Glass patterns, and foundthat it fitted their empirical data on Glass patterns well.















Weber-Fechner’s law Holographic law

∆d ′ ∝ ∆SS d ′ ∝W = E/n

d ′ = k ln (S) + C d ′ = g 12+1/S

3

4

5

2

1

0

3

4

5

2

1

0

Barlow & Reeves’ (1979) data

for g = 7.64Best fit by g/(2 + 1/S)

Barlow & Reeves’ (1979) data

for k = 0.75, C = 2.34Best fit by k ∗ ln(S) + C

5 100.05 0.1 0.25 0.5 1 2.5 5 100.05 0.1 0.25 0.5 1 2.50.025 0.025

Regularity-to-noise ratio S = R/NRegularity-to-noise ratio S = R/N

Detectability(d

′ )

Detectability(d

′ )

0.01 0.01

van der Helm (2010)



Symmetry effect: overestimation of amounts of symmetry.Asymmetry effect: underestimation of amounts of symmetry.

The overall level of symmetry matters – but at every level, thedecisive factor is whether symmetry or noise is manipulated.

No incorrect estimates of amounts of symmetry or noise,but correct estimates of regularity-to-noise ratios.

Freyd & Tversky (1984); Csatho et al. (2004)














Symmetry perception and perceptual organization

On the one hand

Theoretically, in structural description approaches such as

RBC (Biederman, 1987), and

SIT (Leeuwenberg & van der Helm, 2013),

symmetry is taken to be a crucial component of how perceptionimposes view-independent, or object-centered, structure on stimuli.

Empirically, symmetry has been shown to play a role in

object recognition (Pashler, 1990; Vetter & Poggio, 1994);

figure-ground segregation (Driver et al., 1992; Leeuwenberg &

Buffart, 1984; Machilsen et al., 2009);

amodal completion (Kanizsa, 1985; van Lier et al., 1995).



On the other hand

Is symmetry really a cue for the presence of a single object, andis repetition really a cue for the presence of multiple objects?

The proximal features of a regularity vary with viewpoint, so, howcan it be an effective grouping factor if viewed non-orthofrontally?

Symmetry Repetition

The Hoffding step, or the problem of viewpoint generalization:

How does vision arrive at a view-independent representationof a 3D scene, starting from a 2D view of the scene?

Corballis & Roldan (1974); Treder & van der Helm (2007);Szlyk et al. (1995); van der Vloed et al. (2005);Hoffding (1891); Wagemans (1993); Schmidt & Schmidt (2013)



On the other hand



Symmetry Repetition






On the other hand



Symmetry Repetition






Sawada, Li, & Pizlo (2011): Any pair of 2D curves is consistentwith a 3D symmetric interpretation (but is not always seen as such).

http://www.tadamasasawada.com/demos/sym2011


Multiple symmetry perception

Two-fold versus three-fold symmetry (bootstrapping)

Detection of symmetry propagates via trapezoids:

One-fold symmetry has ”1-way” trapezoids.

Two-fold symmetry has accelerating ”2-way” trapezoids.

Three-fold symmetry only has ”1-way” trapezoids.

Wagemans et al. (1993)



Two-fold versus three-fold symmetry (descriptive coding)

abbaabbaabba 3*(abba)

S[(a)(b)(c)(c)(b)(a)]abccbaabccba

abbaabbaabba S[(a)(b)(b)(a)(a)(b)]

The regularity in two-fold symmetry can be captured completely— that in three-fold symmetry only partly.

Three-fold symmetry contains ”hidden order”, which is known totrigger curiosity, interest, and aesthetical feelings.

Boselie & Leeuwenberg (1985)



Two-fold versus three-fold symmetry (descriptive coding)

abbaabbaabba 3*(abba)

S[(a)(b)(c)(c)(b)(a)]abccbaabccba

abbaabbaabba S[(a)(b)(b)(a)(a)(b)]

The regularity in two-fold symmetry can be captured completely— that in three-fold symmetry only partly.

Three-fold symmetry contains ”hidden order”, which is known totrigger curiosity, interest, and aesthetical feelings.

Boselie & Leeuwenberg (1985)



Predicted detectability

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7 8

Predicted

goodness

Number of symmetry axes

Transformational: 1− 1/(2A) with A is #axes

Holographic: E/n with E is #identities in code

The detectability of multiple symmetry is not as might be expectedon the basis of the number of global symmetry axes alone.

van der Helm (2011); Wenderoth & Welsh (1998); Treder et al. (2011)



Predicted detectability

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4 5 6 7 8

Predicted

goodness

Number of symmetry axes

Transformational: 1− 1/(2A) with A is #axes

Holographic: E/n with E is #identities in code

The detectability of multiple symmetry is not as might be expectedon the basis of the number of global symmetry axes alone.

van der Helm (2011); Wenderoth & Welsh (1998); Treder et al. (2011)


Multiple symmetry in nature and art

Skewed distribution in flowers

monocotyledons dicotyledons

80% 70%

Perhaps, 3-fold and 5-fold symmetrical flowers have a procreationadvantage over others, because their visual appearance attractsmore pollinators.

Note. Insect vision exists 400 million years — flowers 150 million years.

Heywood (1993), van der Helm (2011)



Skewed distribution in flowers

monocotyledons dicotyledons

80% 70%

Perhaps, 3-fold and 5-fold symmetrical flowers have a procreationadvantage over others, because their visual appearance attractsmore pollinators.

Note. Insect vision exists 400 million years — flowers 150 million years.

Heywood (1993), van der Helm (2011)



Skewed distribution in designs

10

20

30

40

50

60

87654321

Per

cen

tag

e o

f d

eco

rati

ve b

and

s

Number of symmetry axes in motifs

Decorative art Mystical art

Throughout history, humans have seen 3-fold and 5-foldsymmetrical motifs as being more appropriate for mystical art.

Hardonk (1999); van der Helm (2011)



Skewed distribution in designs

10

20

30

40

50

60

87654321

Per

cen

tag

e o

f d

eco

rati

ve b

and

s

Number of symmetry axes in motifs

Decorative art Mystical art

Throughout history, humans have seen 3-fold and 5-foldsymmetrical motifs as being more appropriate for mystical art.

Hardonk (1999); van der Helm (2011)


Summary (Part 2)

Symmetry perception

Evolution did not select individual visual regularities, but itselected a detection mechanism with sufficient survival value.

Many known regularity-detection phenomena can be explainedin terms of representation and/or process properties.

It is clear that symmetry plays a role in perceptualorganization, but its exact role is still unclear.


Part 3




In artificial intelligence research, a cognitive architecture is acomputational model of not only resulting behaviour but alsostructural properties of intelligent systems.

These structural properties can be abstract as well as physicalproperties implemented in such systems.

In cognitive neuroscience, cognitive architectures

are blueprints for systems that act like the human system;

should have an eye for neural plausibility;

unify processes and representations.

Anderson (1983); Newell (1990); Thagard (2012)



In artificial intelligence research, a cognitive architecture is acomputational model of not only resulting behaviour but alsostructural properties of intelligent systems.

These structural properties can be abstract as well as physicalproperties implemented in such systems.

In cognitive neuroscience, cognitive architectures

are blueprints for systems that act like the human system;

should have an eye for neural plausibility;

unify processes and representations.

Anderson (1983); Newell (1990); Thagard (2012)


Visual pathways in the brain

Visualcortex

(b)(a)

LGN

OC

Retina Object perception

Spatial perception

Ungerleider & Mishkin (1982)


PATVISH: Combined action of perception and attention

Attention

Binding of similar features

Extraction of visual features

Global structures

Unorganized parts

Organized wholes

RepresentationsSubprocesses


Perception

Selection of different features

Local features

Perception comprises three neurally intertwined subprocesses(Lamme, Super, & Spekreijse, 1998; van der Helm, 2012)

Resulting percepts reflect hierarchical stimulus organizations(Leeuwenberg & van der Helm, 1991)

Attention subserves top-down scrutiny of established percepts(Hochstein & Ahissar, 2002)


Hierarchical organizations: Global vs local

Stimulus

Local features

Compatible with perceived global structure

Incompatible with perceived global structure

Perceived global organization


The reverse hierarchy theory of perceptual learning

Attention can be deployed in a top-down fashion to any level in thevisual hierarchy (Hochstein & Ahissar, 2002; see also Wolfe, 2007)

Thus, it first captures global structures coded in higher areas and– if required by task and allowed by time – it may descend alongrecurrent connections to capture local features coded in lower areas


Wholes versus parts

Local features are the first ones processed by perception,but global structures are the first ones encountered by attention.

This agrees with the Gestalt idea that, behaviourally, wholesdominate parts — which also has been framed in terms of

global precedence (Navon, 1977);

configural superiority (Pomerantz et al., 1977);

superstructure dominance (Leeuwenberg & van der Helm, 1991);

primacy of holistic properties (Kimchi, 2003).

Note. Participants in behavioural experiments respond on the basis ofwhat they have perceived, so, responses are indicative of properties ofpercepts rather than of properties of the perceptual process itself.


Wholes versus parts

Local features are the first ones processed by perception,but global structures are the first ones encountered by attention.

This agrees with the Gestalt idea that, behaviourally, wholesdominate parts — which also has been framed in terms of

global precedence (Navon, 1977);

configural superiority (Pomerantz et al., 1977);

superstructure dominance (Leeuwenberg & van der Helm, 1991);

primacy of holistic properties (Kimchi, 2003).

Note. Participants in behavioural experiments respond on the basis ofwhat they have perceived, so, responses are indicative of properties ofpercepts rather than of properties of the perceptual process itself.


Global structures mask incompatible local features

Stimulus

Local features

Compatible with perceived global structure

Incompatible with perceived global structure

Perceived global organization


PATVISH: Combined action of perception and attention

Attention



Global structures

Unorganized parts

Organized wholes

RepresentationsSubprocesses


Perception


Local features

Perception comprises three neurally intertwined subprocesses(Lamme, Super, & Spekreijse, 1998; van der Helm, 2012)

Resulting percepts reflect hierarchical stimulus organizations(Leeuwenberg & van der Helm, 1991)

Attention subserves top-down scrutiny of established percepts(Hochstein & Ahissar, 2002)


Neurally intertwined subprocesses of perception





Integrated percepts (perceptual organizations) are yielded by:

feedforward extraction of – or tuning to – visual features

horizontal binding of similar features

— synchronization

recurrent selection of different features








feedforward extraction of – or tuning to – visual features

horizontal binding of similar features — synchronization



Neuronal synchronization

Neuronal synchronization

The phenomenon that, depending on the input, transientassemblies of neurons temporarily synchronize their activity.

How those transient assemblies of neurons physically go in and outof existence is typically a topic in dynamic systems theory.(e.g., van Leeuwen et al., 1997; Campbell et al., 1999; Harris et al., 2003)

Neuronal synchronization has been associated with corticalintegration, and more general, with cognitive processing.(Milner, 1974; von der Malsburg, 1981)

Synchronization in the gamma band (30–70 Hz), in particular, hasbeen associated with ”horizontal” feature binding in vision.(Eckhorn et al., 1988; Gray & Singer, 1989; Gilbert, 1992)


Perceptual integration capability

In PATVISH, gamma synchronization is taken to subserve bindingof similar features – thereby, it also has an effect on selection ofdifferent features, so, in total, it underlies perceptual integration.(or incremental grouping, as Roelfsema, 2006, called it)

What happens if gamma synchronization is impaired?(as in autism and schizophrenia; Sun et al., 2012; Uhlhaas et al., 2005)

Then, by PATVISH, one gets reduced perceptual integration,that is, reduced construction of global structures ...so that top-down atention has better access to local features.(as in autism and Williams syndrome; Frith, 1989; Bernardino et al., 2012)












The local advantage phenomenon in autism

The local advantage phenomenon: Autistics perform betterthan typical on tasks in which local features are to be discerned.(Shah & Frith, 1983, 1993; Jolliffe & Baron-Cohen, 1997)

Embedded figures taskBlock design task

Perceived global structures mask incompatible local features— in autism, this masking is weaker than typical, due to reducedperceptual integration caused by impaired gamma synchronization.


The local advantage phenomenon in autism

The local advantage phenomenon: Autistics perform betterthan typical on tasks in which local features are to be discerned.(Shah & Frith, 1983, 1993; Jolliffe & Baron-Cohen, 1997)

Embedded figures taskBlock design task

Perceived global structures mask incompatible local features— in autism, this masking is weaker than typical, due to reducedperceptual integration caused by impaired gamma synchronization.








feedforward extraction of visual features

horizontal binding of similar features



The minimal-coding algorithm PISA

PISA computes – for symbol strings – simplest hierarchical codesby capturing a maximum amount of visual regularity.

Brain PISA





All-pairs shortest path method

All-substrings identification

Hyperstrings

Hyperstrings

The correspondence between the three intertwined subprocesses isstriking, but does this correspondence go any deeper?

van der Helm (2004, 2012, 2014)


The minimal-coding algorithm PISA


Brain PISA







Hyperstrings

Hyperstrings

The correspondence between the three intertwined subprocesses isstriking, but does this correspondence go any deeper?

van der Helm (2004, 2012, 2014)


Structural information theory (initiated by Leeuwenberg, 1968)

Structural information theory (SIT)

A general theory of human visual perceptual organization.

It adopts the simplicity principle, which holds that thesimplest organization of a stimulus is the one perceived.

To make predictions, it proposes a coding language todetermine simplest codes of symbol strings (which mayrepresent visual patterns) by capturing a maximumamount of repetitions, symmetries, and alternations.

Note: Repetitions, symmetries, and alternations are the only transparentholographic regularities – this mathematical notion explains much of humansymmetry perception (van der Helm & Leeuwenberg, 1991, 1996, 1999, 2004).

Leeuwenberg & van der Helm (2013); van der Helm (2014)


Computing simplest codes of strings: The basic idea

String:

a b a b a b q c d e p d e c f p q k p q l m p q u

Substrings:

a b a b a b → 3 ∗ (ab)

c d e p d e c → S [(c)(de), (p)]

p q k p q l m p q u → 〈(pq)〉/〈(k)(lm)(u)〉

Once every substring has thus been encoded in the simplest way,there are O(2N) candidate codes for the entire string, but thenDijkstra’s (1959) O(N2) shortest path method can be applied toselect a simplest one – which, in this example, is:

3 ∗ (ab) q S [(c)(de), (p)] f 〈(pq)〉/〈(hk)(lm)(u)〉



Computing simplest codes of strings: The basic idea

String:

a b a b a b q c d e p d e c f p q k p q l m p q u

Substrings:

a b a b a b → 3 ∗ (ab)

c d e p d e c → S [(c)(de), (p)]

p q k p q l m p q u → 〈(pq)〉/〈(k)(lm)(u)〉

Once every substring has thus been encoded in the simplest way,there are O(2N) candidate codes for the entire string, but thenDijkstra’s (1959) O(N2) shortest path method can be applied toselect a simplest one – which, in this example, is:

3 ∗ (ab) q S [(c)(de), (p)] f 〈(pq)〉/〈(hk)(lm)(u)〉



Distributed representations of SIT codes

Assume that, for the string ababfabab, a simplest ISA-form is known foreach of the O(N2) substrings (a few are shown):

1 2 3 4 5 6 7 8 9 10

S[(a),(b)] <(a)>/<(bf) 2*((b))>

b b f

S[(2*(ab)), (f)]

2*(ab)2*(ab) S[(b),(fa)]

S[(ab),(f)]

b ba a aa

Then, O(2N) codes for the entire string are possible – for instance, thepath along nodes 1, 4, 5, 6, 10 yields code S [(a), (b)] b f 2 ∗ (ab)

How to select the simplest code?




1 2 3 4 5 6 7 8 9 10

<(a)>/<(bf) 2*((b))>

b

S[(2*(ab)), (f)]

2*(ab) S[(b),(fa)]

S[(ab),(f)]

b b

S[(a),(b)]

b f

2*(ab)

a a aa






1 2 3 4 5 6 7 8 9 10

<(a)>/<(bf) 2*((b))>

b

S[(2*(ab)), (f)]

2*(ab) S[(b),(fa)]

S[(ab),(f)]

b b

S[(a),(b)]

b f

2*(ab)

a a aa




Serial distributed processing to select simplest codes

Take the complexities of the ISA-forms as the lengths of the edges ...

1 2 3 4 5 6 7 8 9 10

4

3 4

4

3

6

1 11 1 1

2

1 11 1

and apply Dijkstra’s (1959) shortest path method which yields theminimal distance dmin(1,N) from node 1 to node N, by determining

dmin(1, k) = MINp<k{dmin(1, p) + d(p, k)} for k = 2, 3, ...,N.

This is a ”smart” O(N2) method to evaluate O(2N) paths.

Dijkstra (1959)




1 2 3 4 5 6 7 8 9 10

4

3 4

4

3

6

1 11 1 1

2

1 11 1




Dijkstra (1959)




1 2 3 4 5 6 7 8 9 10

4

3 4

4

3

6

1 11 1 1

2

1 11 1




Dijkstra (1959)


Parallel distributed processing to select shortest paths

”Smart” hilly tube system

T = 0

1

3

4

5

2

0

Fluid takes one time unit to ”exite” one straight tube segment

-




T = 1

0

1

2

3

4

5

Fluid, arriving at a node, ”inhibits” other incoming tubes

-




T = 2

0

2

3

4

5

1

Fluid flows on, also in blocked tubes

-




T = 3

0

1

2

3

4

5

Fluid in blocked tubes hardens in one time unit

-




T = 4

0

1

3

4

5

2

Fluid exits through the shortest path

-




T = 5

0

1

2

3

4

5

All other paths harden, leaving a flow in only the shortest path

-


Computing simplest codes of strings: The problem

Say, for some string, substring ababpbaba has been encoded into

S [(a)(b)(a)(b), (p)]

then, argument (a)(b)(a)(b) can be recoded hierarchically into

2 ∗ ((a)(b))

yielding the simpler hierarchical code S [2 ∗ ((a)(b)), (p)]

In principle, a substring can be encoded into an exponentialnumber of symmetries – or, likewise, alternations – which, each,have to be recoded hierarchically in order to find a simplest one.

Hierarchical recoding of each symmetry and alternation separatelywould require an intractable, superexponential, amount of work.


Computing simplest codes of strings: The problem

Say, for some string, substring ababpbaba has been encoded into

S [(a)(b)(a)(b), (p)]

then, argument (a)(b)(a)(b) can be recoded hierarchically into

2 ∗ ((a)(b))

yielding the simpler hierarchical code S [2 ∗ ((a)(b)), (p)]

In principle, a substring can be encoded into an exponentialnumber of symmetries – or, likewise, alternations – which, each,have to be recoded hierarchically in order to find a simplest one.

Hierarchical recoding of each symmetry and alternation separatelywould require an intractable, superexponential, amount of work.


Selection of longest pencil (1)

Measure the pencil lengths serially or in parallel ..... not smart!



Measure the pencil lengths serially or in parallel

..... not smart!



Measure the pencil lengths serially or in parallel ..... not smart!



Is this serial or parallel processing? No, it is transparallel processing!






Is this serial or parallel processing?

No, it is transparallel processing!





Computing simplest codes of strings: The solution

Gather, in O(N2) time, the arguments of the O(2N) symmetriesinto which a substring can be encoded in a directed acyclic graph.

Substring: a b a b f a b a b g b a b a f b a b a

6

7

8

9

101

2

3

4

5

(a)

(b)

(bab)(bab)

(a)

(b)

(f)

(b)

(a)

(b)

(a)

(aba) (aba)

S [(aba)(b)(f )(a)(bab), (g)]

The graph is provably a hyperstring, which implies that thoseO(2N) arguments can be recoded in a transparallel fashion, thatis, simultaneously as if only one argument were concerned.

van der Helm (2004, 2014)





6

7

8

9

101

2

3

4

5

(a)

(b)

(bab)

(a)(b)

(a)

(b)

(aba)(aba)

(b)

(f)

(a)

(bab)

S [(aba)(b)(f )(a)(bab), (g)]







6

7

8

9

101

2

3

4

5

(a)

(b)

(bab)

(a)(b)

(a)

(b)

(aba)(aba)

(b)

(f)

(a)

(bab)

S [(aba)(b)(f )(a)(bab), (g)]




Hyperstrings

Manystrings

h1 h8h7h6h5h4h3h2One string

representationDistributed

xcv

abcfabcg

abcv

xcfabcg

ayv

abcfw

abcfayg

xcfw

xcfxcg

xcfayg

ayfxcg

ayfayg

ayfabcg

abcfxcg

ayfw

1 2 3 4 5 6 7 8 9

v

w

gf

x

a b c

y

a

x

b

y

c


Hyperstrings


Manystrings


xcv

abcfabcg

abcv

xcfabcg

ayv

abcfw

abcfayg

xcfw

xcfxcg

xcfayg

ayfxcg

ayfayg

ayfabcg

abcfxcg

ayfw

1 2 3 4 5 6 7 8 9

v

w

gf

x

a b c

y

a

x

b

y

c


Hyperstrings


Manystrings


xcv

abcfabcg

abcv

xcfabcg

ayv

abcfw

abcfayg

xcfw

xcfxcg

xcfayg

ayfxcg

ayfayg

ayfabcg

abcfxcg

ayfw

1 2 3 4 5 6 7 8 9

x

v

x

y y

w

gcbafcba


Hyperstrings


Manystrings

h1 h8h7h6h5h4h3h2

xcv

abcfabcg

abcv

xcfabcg

ayv

abcfw

abcfayg

xcfw

xcfxcg

xcfayg

ayfxcg

ayfayg

ayfabcg

abcfxcg

ayfw

One string

1 2 3 4 5 6 7 8 9

x

v

x

y y

w

gcbafcba


Transparallel processing by hyperstrings

h1 h8h7h6h5h4h3h2

1 2 3 4 5 6 7 8 9

x

v

x

y y

w

gcbafcba

Substrings h1h2h3 and h5h6h7 are identical, so that the string canbe encoded into the alternation 〈(h1h2h3)〉/〈(h4)(h8)〉 which, inone go, represents alternations in three different strings, namely:

〈(abc)〉/〈(f )(g)〉 in the string abcfabcg〈(xc)〉/〈(f )(g)〉 in the string xcfxcg〈(ay)〉/〈(f )(g)〉 in the string ayfayg

Hence, the O(2N) strings in a hyperstring can be encoded as ifonly one string were concerned – an exponential reduction in work.



h1 h8h7h6h5h4h3h2

1 2 3 4 5 6 7 8 9

x

v

x

y y

w

gcbafcba

Substrings h1h2h3 and h5h6h7 are identical, so that the string canbe encoded into the alternation 〈(h1h2h3)〉/〈(h4)(h8)〉 which, inone go, represents alternations in three different strings, namely:

〈(abc)〉/〈(f )(g)〉 in the string abcfabcg〈(xc)〉/〈(f )(g)〉 in the string xcfxcg〈(ay)〉/〈(f )(g)〉 in the string ayfayg

Hence, the O(2N) strings in a hyperstring can be encoded as ifonly one string were concerned – an exponential reduction in work.




Brain PISA







Hyperstrings

Hyperstrings

Proposal: Hyperstrings correspond to transient neural assemblies,whose synchronization manifests transparallel feature processing.





Brain PISA







Hyperstrings

Hyperstrings

Proposal: Hyperstrings correspond to transient neural assemblies,whose synchronization manifests transparallel feature processing.



Hypotheses about the mind

Inspired by Feynman’s (1982) idea of quantum computers – which,for some applications, promise an exponential reduction in the timeneeded to complete a computing job – Penrose (1989) proposed:

The quantum mind hypothesis: Quantum mechanicalphenomena, such as quantum entanglement andsuperposition, are the basis of neuronal synchronization.

However, quantum phenomena do not seem to last long enough tobe useful for neuro-cognitive processing (e.g., Tegmark, 2000).

Transparallel processing can be done on classical computers andimplies, for some applications, an exponential reduction too.

The transparallel mind hypothesis: Flexible cognitivearchitecture is implemented in the brain by synchronizedneural assemblies mediating transparallel feature processing.

Koffka, 1935; Hebb, 1949; Kelso, 1995; Lehar, 2003; Buzsaki, 2006


Hypotheses about the mind

Inspired by Feynman’s (1982) idea of quantum computers – which,for some applications, promise an exponential reduction in the timeneeded to complete a computing job – Penrose (1989) proposed:

The quantum mind hypothesis: Quantum mechanicalphenomena, such as quantum entanglement andsuperposition, are the basis of neuronal synchronization.

However, quantum phenomena do not seem to last long enough tobe useful for neuro-cognitive processing (e.g., Tegmark, 2000).

Transparallel processing can be done on classical computers andimplies, for some applications, an exponential reduction too.

The transparallel mind hypothesis: Flexible cognitivearchitecture is implemented in the brain by synchronizedneural assemblies mediating transparallel feature processing.

Koffka, 1935; Hebb, 1949; Kelso, 1995; Lehar, 2003; Buzsaki, 2006


Summary (Part 3)

Cognitive architectures call for specifications of ingredients neededto build unified theories of cognition — thereby, they stimulateresearchers to think about metatheoretical aspects such as

representational theory — connectionism — DST;

forms of neuro-cognitive processing;

metaphors of cognition;

and more specifically, about things such as

tractability of proposed processes;

unification of processes and representations;

neural building blocks of cognition;

different than typical cognitive processing.


Conclusion


Well, there is still much to do before cognitive neuroscience mayarrive at a ”grand unified theory” of perceptual organization, but:

The high combinatorial capacity and speed of the perceptualorganization process might be enabled by a flexible cognitivearchitecture, constituted by transient neural assemblies exhibitingsynchronization as manifestation of transparallel feature processing.

The resulting mental representation of a scene can, at the neurallevel, be described as a relatively stable physical state, and at thecognitive level, as a state which is informationally simplest due tomaximal extraction of visual regularities.

A perceptual organization process yielding simplest organizationscan be conceived of as a form of unconscious inference which is

an efficient user of internal resources;a fairly reliable source of knowledge about the external world.


Conclusion








Conclusion








Conclusion








Documents

Simplicity in vision - KU Leuvenu0084530/reprints/siv-lecture.pdf · Simplicity in vision: Topics Theoretical cycle Empirical cycle Tractability cycle The computability of simplicity