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Grouping by Proximity and Grouping by Proximity and Multistability in Dot LatticesMultistability in Dot Lattices

A Quantitative Gestalt TheoryA Quantitative Gestalt Theory

Grouping by Proximity and Grouping by Proximity and Multistability in Dot LatticesMultistability in Dot Lattices

A Quantitative Gestalt TheoryA Quantitative Gestalt Theory

A Paper by Michael Kubovy and Johan WagemansA Paper by Michael Kubovy and Johan Wagemans

Presentation by Adrian IliePresentation by Adrian Ilie

A lot of the material of these slides is taken from Dr. Kubovy’s website with his consent.A lot of the material of these slides is taken from Dr. Kubovy’s website with his consent.

http://faculty.virginia.edu/kubovylab/kubovy/

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OutlineOutline

• What is the Gestalt Theory?

• Contributions

• A two-parameter space of dot lattices

• A one parameter model of grouping by proximity

• Experiment

• Discussion

• Conclusion

• What is the Gestalt Theory?

• Contributions

• A two-parameter space of dot lattices

• A one parameter model of grouping by proximity

• Experiment

• Discussion

• Conclusion

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The Gestalt TheoryThe Gestalt Theory

• Gestalt theory is a broadly interdisciplinary general theory which provides a framework for a wide variety of psychological phenomena, processes, and applications.

• Human beings are viewed as open systems in active interaction with their environment. It is especially suited for the understanding of order and structure in psychological events.

( ( http://www.enabling.org/ia/gestalt/gtax1.html#kap2http://www.enabling.org/ia/gestalt/gtax1.html#kap2 ) )

• Gestalt theory is a broadly interdisciplinary general theory which provides a framework for a wide variety of psychological phenomena, processes, and applications.

• Human beings are viewed as open systems in active interaction with their environment. It is especially suited for the understanding of order and structure in psychological events.

( ( http://www.enabling.org/ia/gestalt/gtax1.html#kap2http://www.enabling.org/ia/gestalt/gtax1.html#kap2 ) )

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ContributionsContributions

• Enlarge the domain of stimuli:Enlarge the domain of stimuli: systematic systematic variation of parameters using a geometric analysis of variation of parameters using a geometric analysis of lattices.lattices.

• Broaden the notion of ambiguity:Broaden the notion of ambiguity: dot dot lattices are at list quadristable (ambiguous figures lattices are at list quadristable (ambiguous figures with 4 aspects).with 4 aspects).

• Information-theoretic model of grouping Information-theoretic model of grouping and multistability.and multistability.

• Enlarge the domain of stimuli:Enlarge the domain of stimuli: systematic systematic variation of parameters using a geometric analysis of variation of parameters using a geometric analysis of lattices.lattices.

• Broaden the notion of ambiguity:Broaden the notion of ambiguity: dot dot lattices are at list quadristable (ambiguous figures lattices are at list quadristable (ambiguous figures with 4 aspects).with 4 aspects).

• Information-theoretic model of grouping Information-theoretic model of grouping and multistability.and multistability.

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Dot LatticesDot Lattices

• Are collections of dots in the plane, invariant under Are collections of dots in the plane, invariant under two independent translations.two independent translations.

• Properties:Properties:

They are They are discretediscrete (dots are not too close from each (dots are not too close from each other).other).

The dots are spread over the entire plane (dots are The dots are spread over the entire plane (dots are not too far from each other), thus lattices are not too far from each other), thus lattices are infiniteinfinite..

• They are predominantly organized by proximity, and They are predominantly organized by proximity, and they are somewhat ambiguous.they are somewhat ambiguous.

• Are collections of dots in the plane, invariant under Are collections of dots in the plane, invariant under two independent translations.two independent translations.

• Properties:Properties:

They are They are discretediscrete (dots are not too close from each (dots are not too close from each other).other).

The dots are spread over the entire plane (dots are The dots are spread over the entire plane (dots are not too far from each other), thus lattices are not too far from each other), thus lattices are infiniteinfinite..

• They are predominantly organized by proximity, and They are predominantly organized by proximity, and they are somewhat ambiguous.they are somewhat ambiguous.

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Example LatticesExample Lattices

• Shape: rectangular (1), square (2), hexagonal (3).Shape: rectangular (1), square (2), hexagonal (3).

• Ambiguity: ambiguous (1), more ambiguous (2), the Ambiguity: ambiguous (1), more ambiguous (2), the most ambiguous (3).most ambiguous (3).

• Shape: rectangular (1), square (2), hexagonal (3).Shape: rectangular (1), square (2), hexagonal (3).

• Ambiguity: ambiguous (1), more ambiguous (2), the Ambiguity: ambiguous (1), more ambiguous (2), the most ambiguous (3).most ambiguous (3).

11 22 33

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The Basic ParallelogramThe Basic Parallelogram

• aa and and bb are the sides are the sides

• is the angle is the angle between thembetween them

• cc and and dd are the are the diagonalsdiagonals

• aa and and bb are the sides are the sides

• is the angle is the angle between thembetween them

• cc and and dd are the are the diagonalsdiagonals

• Note that Note that aa, , bb and and are enough to define the are enough to define the parallelogram.parallelogram.

• If If aa is held constant, all lattices are defined by is held constant, all lattices are defined by bb and and ..

• Note that Note that aa, , bb and and are enough to define the are enough to define the parallelogram.parallelogram.

• If If aa is held constant, all lattices are defined by is held constant, all lattices are defined by bb and and ..

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The Space of LatticesThe Space of Lattices

• If If aa is held constant, all lattices are is held constant, all lattices are defined by defined by bb and and ..

• ||bb||≥|≥|aa||

• 60°<60°<<90°<90°

• If If aa is held constant, all lattices are is held constant, all lattices are defined by defined by bb and and ..

• ||bb||≥|≥|aa||

• 60°<60°<<90°<90°

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HexagonalHexagonal

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RhombicRhombic

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RhombicRhombic

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RhombicRhombic

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SquareSquare

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RectangularRectangular

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RectangularRectangular

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RectangularRectangular

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RectangularRectangular

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Centered RectangularCentered Rectangular

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Centered RectangularCentered Rectangular

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Centered RectangularCentered Rectangular

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Centered RectangularCentered Rectangular

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ObliqueOblique

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Grouping by ProximityGrouping by Proximity

• Koffka (1935): “we must think of group information Koffka (1935): “we must think of group information as due to actual forces of attraction between the as due to actual forces of attraction between the members of the group”.members of the group”.

• Grouping by proximity means that a lattice will be Grouping by proximity means that a lattice will be seen as parallel strips of dots along the shortest seen as parallel strips of dots along the shortest vector vector aa..

• As As ||b|b| approaches approaches ||a|a|, the perceived organization may , the perceived organization may change to parallel strips of dots along vector change to parallel strips of dots along vector bb..

• Different distributions of Different distributions of aa, , bb, , cc and and dd lead to lead to different degrees of multistability.different degrees of multistability.

• Koffka (1935): “we must think of group information Koffka (1935): “we must think of group information as due to actual forces of attraction between the as due to actual forces of attraction between the members of the group”.members of the group”.

• Grouping by proximity means that a lattice will be Grouping by proximity means that a lattice will be seen as parallel strips of dots along the shortest seen as parallel strips of dots along the shortest vector vector aa..

• As As ||b|b| approaches approaches ||a|a|, the perceived organization may , the perceived organization may change to parallel strips of dots along vector change to parallel strips of dots along vector bb..

• Different distributions of Different distributions of aa, , bb, , cc and and dd lead to lead to different degrees of multistability.different degrees of multistability.

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Formal Model (1)Formal Model (1)

• Let Let V={V={aa,,bb,,cc,,dd}} be the set of vector magnitudes be the set of vector magnitudes ||a|a|, , ||bb||, , ||cc||, , ||dd||..

• Grouping by proximity: the probability of Grouping by proximity: the probability of organizing a lattice in a direction organizing a lattice in a direction vvVV is an is an attraction functionattraction function f(f(vv)=e)=e--((((vv//aa)-1))-1), where , where is the is the attraction constant.attraction constant.

• The attraction constant is directly related to the The attraction constant is directly related to the grouping tendency: the larger grouping tendency: the larger , the stronger , the stronger the proximity grouping tendency.the proximity grouping tendency.

• Let Let V={V={aa,,bb,,cc,,dd}} be the set of vector magnitudes be the set of vector magnitudes ||a|a|, , ||bb||, , ||cc||, , ||dd||..

• Grouping by proximity: the probability of Grouping by proximity: the probability of organizing a lattice in a direction organizing a lattice in a direction vvVV is an is an attraction functionattraction function f(f(vv)=e)=e--((((vv//aa)-1))-1), where , where is the is the attraction constant.attraction constant.

• The attraction constant is directly related to the The attraction constant is directly related to the grouping tendency: the larger grouping tendency: the larger , the stronger , the stronger the proximity grouping tendency.the proximity grouping tendency.

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Formal Model (2)Formal Model (2)

• Let Let p(p(vv)=f()=f(vv)/(f()/(f(aa)+f()+f(bb)+f()+f(cc)+f()+f(dd)))) be the be the probability of grouping in direction probability of grouping in direction vv. Note that . Note that p(p(vv)=1 )=1 for for vvVV..

• Let Let H=-H=-p(p(vv)log)log22p(v)p(v) be the entropy (average be the entropy (average uncertainty, Shannon and Wiener, 1948).uncertainty, Shannon and Wiener, 1948).

• We have We have H’= H’= p(p(vv)log)log22p(p(vv)/)/uu log log22uu, where , where uu=f(=f(aa)+f()+f(bb)+f()+f(cc)+f()+f(dd)), the estimated entropy., the estimated entropy.

• Let Let p(p(vv)=f()=f(vv)/(f()/(f(aa)+f()+f(bb)+f()+f(cc)+f()+f(dd)))) be the be the probability of grouping in direction probability of grouping in direction vv. Note that . Note that p(p(vv)=1 )=1 for for vvVV..

• Let Let H=-H=-p(p(vv)log)log22p(v)p(v) be the entropy (average be the entropy (average uncertainty, Shannon and Wiener, 1948).uncertainty, Shannon and Wiener, 1948).

• We have We have H’= H’= p(p(vv)log)log22p(p(vv)/)/uu log log22uu, where , where uu=f(=f(aa)+f()+f(bb)+f()+f(cc)+f()+f(dd)), the estimated entropy., the estimated entropy.

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ExperimentExperiment

• Designed to test grouping by proximity, so Designed to test grouping by proximity, so other grouping principles were minimized other grouping principles were minimized (similarity, reference frames, orientation (similarity, reference frames, orientation biases).biases).

• 7 subjects: the 27 subjects: the 2ndnd author, 3 graduate students, author, 3 graduate students, 3 undergraduate students.3 undergraduate students.

• Apparatus: computer screen with simulated Apparatus: computer screen with simulated aperture.aperture.

• Designed to test grouping by proximity, so Designed to test grouping by proximity, so other grouping principles were minimized other grouping principles were minimized (similarity, reference frames, orientation (similarity, reference frames, orientation biases).biases).

• 7 subjects: the 27 subjects: the 2ndnd author, 3 graduate students, author, 3 graduate students, 3 undergraduate students.3 undergraduate students.

• Apparatus: computer screen with simulated Apparatus: computer screen with simulated aperture.aperture.

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Stimuli and ResponsesStimuli and Responses

• Stimuli=yellow dots of Stimuli=yellow dots of 5-pixel radius, separated 5-pixel radius, separated by fixed distance a=60 by fixed distance a=60 pixels (1.5° visual pixels (1.5° visual angle).angle).

• Responses=proposed Responses=proposed orientations.orientations.

• Stimuli=yellow dots of Stimuli=yellow dots of 5-pixel radius, separated 5-pixel radius, separated by fixed distance a=60 by fixed distance a=60 pixels (1.5° visual pixels (1.5° visual angle).angle).

• Responses=proposed Responses=proposed orientations.orientations.

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Parameters for 16 LatticesParameters for 16 Lattices

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ProcedureProcedure

• 3 sessions.3 sessions.

• 1600 trials (100 of each of the 16 lattices) in each 1600 trials (100 of each of the 16 lattices) in each session.session.

• Each session took 1 hour, with breaks every 400 Each session took 1 hour, with breaks every 400 trials.trials.

• Sessions were separated by at least 1 hour.Sessions were separated by at least 1 hour.

• Subjects were told lattices are collections of strips of Subjects were told lattices are collections of strips of dots, and could have more than one organization.dots, and could have more than one organization.

• Used the mouse to indicate perceived organization.Used the mouse to indicate perceived organization.

• 3 sessions.3 sessions.

• 1600 trials (100 of each of the 16 lattices) in each 1600 trials (100 of each of the 16 lattices) in each session.session.

• Each session took 1 hour, with breaks every 400 Each session took 1 hour, with breaks every 400 trials.trials.

• Sessions were separated by at least 1 hour.Sessions were separated by at least 1 hour.

• Subjects were told lattices are collections of strips of Subjects were told lattices are collections of strips of dots, and could have more than one organization.dots, and could have more than one organization.

• Used the mouse to indicate perceived organization.Used the mouse to indicate perceived organization.

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ResultsResults

• Computed Computed p(p(aa)), , p(p(bb)), , p(p(cc)) and and p(p(dd)), and used , and used them to compute them to compute HH for each of the 16 lattices for each of the 16 lattices and each subject.and each subject.

• Regressed the values of Regressed the values of HH on on H’H’ while also while also determining determining (varied (varied until the regression until the regression coefficients were maximized).coefficients were maximized).

• The model slightly but systematically The model slightly but systematically underestimates the values of underestimates the values of HH..

• Computed Computed p(p(aa)), , p(p(bb)), , p(p(cc)) and and p(p(dd)), and used , and used them to compute them to compute HH for each of the 16 lattices for each of the 16 lattices and each subject.and each subject.

• Regressed the values of Regressed the values of HH on on H’H’ while also while also determining determining (varied (varied until the regression until the regression coefficients were maximized).coefficients were maximized).

• The model slightly but systematically The model slightly but systematically underestimates the values of underestimates the values of HH..

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Discussion (1)Discussion (1)

• Proposed a negatively-accelerated Proposed a negatively-accelerated attraction function:attraction function: not unique, other functions not unique, other functions work well too; unknown under which condition work well too; unknown under which condition is is uniqueunique..

• Estimated choice probabilities: Estimated choice probabilities: the choice the choice axiom (Luce, 1959: independence from irrelevant axiom (Luce, 1959: independence from irrelevant alternatives) may not hold in this case (choice may alternatives) may not hold in this case (choice may be random in absence of preferred alternative).be random in absence of preferred alternative).

• Proposed a negatively-accelerated Proposed a negatively-accelerated attraction function:attraction function: not unique, other functions not unique, other functions work well too; unknown under which condition work well too; unknown under which condition is is uniqueunique..

• Estimated choice probabilities: Estimated choice probabilities: the choice the choice axiom (Luce, 1959: independence from irrelevant axiom (Luce, 1959: independence from irrelevant alternatives) may not hold in this case (choice may alternatives) may not hold in this case (choice may be random in absence of preferred alternative).be random in absence of preferred alternative).

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Discussion (2)Discussion (2)

• Used entropy to estimate ambiguity:Used entropy to estimate ambiguity: unusual, others count the time it takes to switch unusual, others count the time it takes to switch alternatives, or the number of switches per unit of alternatives, or the number of switches per unit of time; may benefit others (orientation of triangles).time; may benefit others (orientation of triangles).

• Used entropy to estimate ambiguity:Used entropy to estimate ambiguity: unusual, others count the time it takes to switch unusual, others count the time it takes to switch alternatives, or the number of switches per unit of alternatives, or the number of switches per unit of time; may benefit others (orientation of triangles).time; may benefit others (orientation of triangles).

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LimitationsLimitations

• The model cannot describe perceptual The model cannot describe perceptual clustering of random dots in the plane. clustering of random dots in the plane. Models Models that do unfortunately fail with dot lattices instead: they can that do unfortunately fail with dot lattices instead: they can only predict proximity grouping, not multistability.only predict proximity grouping, not multistability.

• The model cannot distinguish between small The model cannot distinguish between small fragments of lattices and extensive lattices. fragments of lattices and extensive lattices. This is due to the phenomenon of cooperativity This is due to the phenomenon of cooperativity (Julesz 1971): (Julesz 1971): lattice fragments are not seen as organized in lattice fragments are not seen as organized in strips, and lattices do not undergo piecemeal organization strips, and lattices do not undergo piecemeal organization (organization of local sets of dots in different parts of the (organization of local sets of dots in different parts of the lattice is linked).lattice is linked).

• The model cannot describe perceptual The model cannot describe perceptual clustering of random dots in the plane. clustering of random dots in the plane. Models Models that do unfortunately fail with dot lattices instead: they can that do unfortunately fail with dot lattices instead: they can only predict proximity grouping, not multistability.only predict proximity grouping, not multistability.

• The model cannot distinguish between small The model cannot distinguish between small fragments of lattices and extensive lattices. fragments of lattices and extensive lattices. This is due to the phenomenon of cooperativity This is due to the phenomenon of cooperativity (Julesz 1971): (Julesz 1971): lattice fragments are not seen as organized in lattice fragments are not seen as organized in strips, and lattices do not undergo piecemeal organization strips, and lattices do not undergo piecemeal organization (organization of local sets of dots in different parts of the (organization of local sets of dots in different parts of the lattice is linked).lattice is linked).

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ConclusionConclusion

• Proposed a quantitative Gestalt model Proposed a quantitative Gestalt model according to which dots in a lattice are according to which dots in a lattice are attracted to each other as a decreasing attracted to each other as a decreasing exponential function of the distance between exponential function of the distance between them, independently of the lattice geometry.them, independently of the lattice geometry.

• Proposed a quantitative Gestalt model Proposed a quantitative Gestalt model according to which dots in a lattice are according to which dots in a lattice are attracted to each other as a decreasing attracted to each other as a decreasing exponential function of the distance between exponential function of the distance between them, independently of the lattice geometry.them, independently of the lattice geometry.

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The EndThe End