Proximity, Curvature, and Feldman's Eureka! - University …people.virginia.edu/~mk9y/documents/papers/20041121.pdf · Phenomenological Psychophysics & Grouping Curved vs. Collinear

Phenomenological Psychophysics & GroupingCurved vs. Collinear Organizations

Feldman’s “Eureka!” to the Rescue?

Proximity, Curvature, and Feldman’s Eureka!

Michael Kubovy and Lars Strother

Sunday, November 10, 2004, 9:40–9:5545th Annual Meeting of the Psychonomic Society

Hyatt Regency, Minneapolis

Kubovy & Strother Proximity, Curvature, & Eureka!



Outline

1 Phenomenological Psychophysics & the Strength of GroupingGrouping by proximityGrouping by similarity

2 Curved vs. Collinear OrganizationsStandard early-vision resultsCurves are more complexDot-sampled grids

3 Feldman’s “Eureka!” to the Rescue?




Outline







Outline







Two take-home questions

1 Particular: Are collinear organizations more salient thancurvilinear ones?

2 General: Can middle-level vision be reduced to early vision?




Two take-home questions

1 Particular: Are collinear organizations more salient thancurvilinear ones?

2 General: Can middle-level vision be reduced to early vision?




The difference between early vs. middle-level vision

Middle-level vision is a.k.a. perceptual organization

Like respiration, it is bottom-up most of time but top-downsome of the timeOne goal of research in early vision: to account for perceptualorganizationDifferences in method

Early: studies responses at threshold (by and large)Middle-level: studies responses to easily-perceived stimuli





Middle-level vision is a.k.a. perceptual organizationLike respiration, it is bottom-up most of time but top-downsome of the time

One goal of research in early vision: to account for perceptualorganizationDifferences in method






Middle-level vision is a.k.a. perceptual organizationLike respiration, it is bottom-up most of time but top-downsome of the timeOne goal of research in early vision: to account for perceptualorganization

Differences in method






Middle-level vision is a.k.a. perceptual organizationLike respiration, it is bottom-up most of time but top-downsome of the timeOne goal of research in early vision: to account for perceptualorganizationDifferences in method







Early: studies responses at threshold (by and large)

Middle-level: studies responses to easily-perceived stimuli










Grouping by proximityGrouping by similarity

How we study middle-level vision

Use ambiguous stimuli: dot lattices

Near equilibriumDisplace from equilibriumRecord how the stimulus is perceivedNo correct responsePhenomenological psychophysics






Use ambiguous stimuli: dot latticesNear equilibrium

Displace from equilibriumRecord how the stimulus is perceivedNo correct responsePhenomenological psychophysics






Use ambiguous stimuli: dot latticesNear equilibriumDisplace from equilibrium

Record how the stimulus is perceivedNo correct responsePhenomenological psychophysics






Use ambiguous stimuli: dot latticesNear equilibriumDisplace from equilibriumRecord how the stimulus is perceived

No correct responsePhenomenological psychophysics






Use ambiguous stimuli: dot latticesNear equilibriumDisplace from equilibriumRecord how the stimulus is perceivedNo correct response

Phenomenological psychophysics






Use ambiguous stimuli: dot latticesNear equilibriumDisplace from equilibriumRecord how the stimulus is perceivedNo correct responsePhenomenological psychophysics





Dot Lattices

Hexagonal Lattice: three-way equilibrium





Dot Lattices

Hexagonal Lattice: three-way equilibrium





Dot Lattices

Square Lattice: two-way equilibrium





Dot Lattices

Square Lattice: two-way equilibrium





Dot Lattices

Rectangular Lattice: ambiguous, but not at equilibrium





Dot Lattices

Rectangular Lattice: ambiguous, but not at equilibrium





Dot Lattices

They vary continuously





Dot Lattices

The dots are at the intersections of a regular grid





Dot Lattices

Unit: the basic parallelogram

a → short sideb → long sidec → short diagonald → long diagonal





Dot Lattices







Dot Lattices


a

a → short side

b → long sidec → short diagonald → long diagonal





Dot Lattices


a

b

a → short sideb → long side

c → short diagonald → long diagonal





Dot Lattices


a

bc

a → short sideb → long sidec → short diagonal

d → long diagonal





Dot Lattices


a

bc

d






What a trial might look like

Task: which organization did you see?

Four possible responses: a,b,c,d . No correct response







Four possible responses: a,b,c,d .

No correct response







Four possible responses: a,b,c,d . No correct response





The data

Variables

four IVs: the lengths |a|, |b|, |c|, |d|four DVs: the probabilities p(a), p(b), p(c), p(d)

p(a) + p(b) + p(c) + p(d) = 1 ⇒ three dfSo, new variables:

three IVs: the relative distances |b|/|a|, |c|/|a|, |d|/|a|three DVs: the odds p(b)/p(a), p(c)/p(a), p(d)/p(a)

We plot the log-oddslog[p(b)/p(a)], log[p(c)/p(a)], log[p(d)/p(a)]





The data

Variablesfour IVs: the lengths |a|, |b|, |c|, |d|

four DVs: the probabilities p(a), p(b), p(c), p(d)








The data

Variablesfour IVs: the lengths |a|, |b|, |c|, |d|four DVs: the probabilities p(a), p(b), p(c), p(d)








The data


p(a) + p(b) + p(c) + p(d) = 1 ⇒ three df

So, new variables:







The data









The data



three IVs: the relative distances |b|/|a|, |c|/|a|, |d|/|a|

three DVs: the odds p(b)/p(a), p(c)/p(a), p(d)/p(a)






The data









The data









The data: proximity

Kubovy, Holcombe, & Wagemans (1998)

basic parallelogram

a

b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity


basic parallelograma

b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0

c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0

c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0

c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0

c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: proximity



b

|b||a|

1 1.1 1.2 1.3

log p(b)p(a)

0c

|c||a|

log p(c)p(a)

attraction function





The data: similarity

van den Berg & Kubovy (in prep.): Similarity along b


b

|b||a|1 1.1 1.2 1.3

log p(b)p(a)

0






van den Berg & Kubovy (in prep.): Similarity along b


b

|b||a|1 1.1 1.2 1.3

log p(b)p(a)

0






van den Berg & Kubovy (in prep.): Similarity along a


b

|b||a|1 1.1 1.2 1.3

log p(b)p(a)

0






van den Berg & Kubovy (in prep.): Similarity along a


b

|b||a|1 1.1 1.2 1.3

log p(b)p(a)

0




Standard early-vision resultsCurves are more complexDot-sampled grids

It’s harder to detect curves than straight lines

Pizlo, Salach-Golyska, & Rosenfeld (1997): stimulus






Pizlo, Salach-Golyska, & Rosenfeld (1997): stimulus






Pizlo, Salach-Golyska, & Rosenfeld (1997): data

4

3

2

1

0none low high

d’

curvature





Feldman & Singh (2004), Psychological Review

Curve sampled every ∆s; α = ∆φ = angle between successivetangents φ





Complexity of a curve

The expected change in tangent direction α is distributed as a vonMises distribution centered on 0 (straight).





At a point where the angle is α, the surprisal is:

µ(α) = − log[p(α)]

If for p(α) we substitute the value given by the von Misesdistribution, we get:

µ(α) = − log A− b cos (α)





Complexity of a curve

Contour information (surprisal) on the boundary of a shape





Dot-sampled grids

The stimulus





Dot-sampled grids

The grid





Possible organizations

The grid





The response screen

The response screen





Prediction

Strother & Kubovy (in prep.)

nomenclature

σκ

|κ||σ|

1.2−1 1.1−1 1 1.1 1.2

log p(κ)p(σ)

0

attraction functionwe expect collinearityto group more strongly





Prediction


nomenclature

σκ

|κ||σ|

1.2−1 1.1−1 1 1.1 1.2

log p(κ)p(σ)

0






Prediction


nomenclature

σκ

|κ||σ|

1.2−1 1.1−1 1 1.1 1.2

log p(κ)p(σ)

0

attraction function

we expect collinearityto group more strongly





Prediction


nomenclature

σκ

|κ||σ|

1.2−1 1.1−1 1 1.1 1.2

log p(κ)p(σ)

0






Prediction


nomenclature

σκ

|κ||σ|

1.2−1 1.1−1 1 1.1 1.2

log p(κ)p(σ)

0

attraction function

we expect collinearityto group more strongly





The Experiment

Method10 observers50 trials/conditionfactorsaspect ratio: |κ|/|σ| = 1.3−1, 1.2−1, 1.1−1, 1, 1.1, 1.2, 1.3

curvature: none, moderate, highdensity: high, moderate, low

duration: 200ms, 300 msaperture: circular, square





The Experiment

The Data

nomenclature

σκ

|κ||σ|

1

log p(κ)p(σ)

0

no curvature

moderate curvature

high curvature





The Experiment

The Data

nomenclature

σκ

|κ||σ|

1

log p(κ)p(σ)

0

no curvature

moderate curvature

high curvature





The Experiment

The Data

nomenclature

σκ

|κ||σ|

1

log p(κ)p(σ)

0

no curvature

moderate curvature

high curvature





The Experiment

The Data

nomenclature

σκ

|κ||σ|

1

log p(κ)p(σ)

0

no curvature

moderate curvature

high curvature





Prediction falsified!

More resultsWhen density is low, curvature is less salient and the effect isreduced.When duration is only 200 ms the effect is reduced.When aperture is square the effect is preserved, hence thesalience of curvilinearity is not a frame of reference effect




Feldman’s “Eureka!”

Feldman (2004), CognitionQuantifies the principle of nonaccidentality

Given the properties of a set of objects, how unlikely wouldthe pattern be if they were generated at random?In statistics we answer this question by reference to a thedistribution under H0 (such as the t-distribution).The paper gives the answer in the realm of concept learning.





Feldman (2004), CognitionQuantifies the principle of nonaccidentalityGiven the properties of a set of objects, how unlikely wouldthe pattern be if they were generated at random?

In statistics we answer this question by reference to a thedistribution under H0 (such as the t-distribution).The paper gives the answer in the realm of concept learning.





Feldman (2004), CognitionQuantifies the principle of nonaccidentalityGiven the properties of a set of objects, how unlikely wouldthe pattern be if they were generated at random?In statistics we answer this question by reference to a thedistribution under H0 (such as the t-distribution).

The paper gives the answer in the realm of concept learning.





Feldman (2004), CognitionQuantifies the principle of nonaccidentalityGiven the properties of a set of objects, how unlikely wouldthe pattern be if they were generated at random?In statistics we answer this question by reference to a thedistribution under H0 (such as the t-distribution).The paper gives the answer in the realm of concept learning.





Application to our resultFeldman & Singh: each curve has a certain probability, thelikelihood under the smooth hypothesis.

Curvilinearity is less likely than collinearityWhat happens when you have a second curve?When a complex, low-probability event is precisely repeated,this is stronger evidence of a relationship then when a simple,high-probability even is repeated.The significance level (probability under H0) is lower, H0 canbe rejected more readily, and the alternative hypothesis of“pattern” can be accepted more readily.





Application to our resultFeldman & Singh: each curve has a certain probability, thelikelihood under the smooth hypothesis.Curvilinearity is less likely than collinearity

What happens when you have a second curve?When a complex, low-probability event is precisely repeated,this is stronger evidence of a relationship then when a simple,high-probability even is repeated.The significance level (probability under H0) is lower, H0 canbe rejected more readily, and the alternative hypothesis of“pattern” can be accepted more readily.





Application to our resultFeldman & Singh: each curve has a certain probability, thelikelihood under the smooth hypothesis.Curvilinearity is less likely than collinearityWhat happens when you have a second curve?

When a complex, low-probability event is precisely repeated,this is stronger evidence of a relationship then when a simple,high-probability even is repeated.The significance level (probability under H0) is lower, H0 canbe rejected more readily, and the alternative hypothesis of“pattern” can be accepted more readily.





Application to our resultFeldman & Singh: each curve has a certain probability, thelikelihood under the smooth hypothesis.Curvilinearity is less likely than collinearityWhat happens when you have a second curve?When a complex, low-probability event is precisely repeated,this is stronger evidence of a relationship then when a simple,high-probability even is repeated.

The significance level (probability under H0) is lower, H0 canbe rejected more readily, and the alternative hypothesis of“pattern” can be accepted more readily.





Application to our resultFeldman & Singh: each curve has a certain probability, thelikelihood under the smooth hypothesis.Curvilinearity is less likely than collinearityWhat happens when you have a second curve?When a complex, low-probability event is precisely repeated,this is stronger evidence of a relationship then when a simple,high-probability even is repeated.The significance level (probability under H0) is lower, H0 canbe rejected more readily, and the alternative hypothesis of“pattern” can be accepted more readily.




The take-home answers

1 Particular: Collinear organizations moresalient than curvilinear ones? No.

2 General: Middle-level vision reducible toearly vision? #1 suggests that it cannot.

1 Particular: No.2 General: No.

























ThanksDaniel Dunbar for programmingNIE and NIDCD for grantingThe developers of LATEX and BEAMER for making possiblePDF presentations created with free software.

Download this presentation from MK’s home page:http://www.people.virginia.edu/~mk9y/.


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Proximity, Curvature, and Feldman's Eureka! - University …people.virginia.edu/~mk9y/documents/papers/20041121.pdf · Phenomenological Psychophysics & Grouping Curved vs. Collinear