Grouping by Proximity and Multistability in Dot Lattices

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• Grouping by Proximity and Multistability in Dot
  Lattices
• A Quantitative Gestalt Theory

A Paper by Michael Kubovy and Johan
Wagemans Presentation by Adrian Ilie A lot of
the material of these slides is taken from Dr.
Kubovys website with his consent. http//faculty
.virginia.edu/kubovylab/kubovy/
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Outline

• 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 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.htmlka
  p2 )

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Contributions

• Enlarge the domain of stimuli systematic
  variation of parameters using a geometric
  analysis of lattices.
• Broaden the notion of ambiguity dot lattices are
  at list quadristable (ambiguous figures with 4
  aspects).
• Information-theoretic model of grouping and
  multistability.

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Dot Lattices

• Are collections of dots in the plane, invariant
  under two independent translations.
• Properties
• They are discrete (dots are not too close from
  each other).
• The dots are spread over the entire plane (dots
  are not too far from each other), thus lattices
  are infinite.
• They are predominantly organized by proximity,
  and they are somewhat ambiguous.

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Example Lattices
1
2
3

• Shape rectangular (1), square (2), hexagonal
  (3).
• Ambiguity ambiguous (1), more ambiguous (2), the
  most ambiguous (3).

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The Basic Parallelogram

• a and b are the sides
• g is the angle between them
• c and d are the diagonals

• Note that a, b and g are enough to define the
  parallelogram.
• If a is held constant, all lattices are defined
  by b and g.

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The Space of Lattices

• If a is held constant, all lattices are defined
  by b and g.
• ba
• 60ltglt90

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Hexagonal
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Rhombic
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Rhombic
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Rhombic
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Square
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Rectangular
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Rectangular
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Rectangular
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Rectangular
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Centered Rectangular
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Centered Rectangular
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Centered Rectangular
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Centered Rectangular
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Oblique
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Grouping by Proximity

• Koffka (1935) we must think of group
  information as due to actual forces of attraction
  between the members of the group.
• Grouping by proximity means that a lattice will
  be seen as parallel strips of dots along the
  shortest vector a.
• As b approaches a, the perceived organization
  may change to parallel strips of dots along
  vector b.
• Different distributions of a, b, c and d lead to
  different degrees of multistability.

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Formal Model (1)

• Let Va,b,c,d be the set of vector magnitudes
  a, b, c, d.
• Grouping by proximity the probability of
  organizing a lattice in a direction vÎV is an
  attraction function f(v)e-a((v/a)-1), where a is
  the attraction constant.
• The attraction constant is directly related to
  the grouping tendency the larger a, the stronger
  the proximity grouping tendency.

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Formal Model (2)

• Let p(v)f(v)/(f(a)f(b)f(c)f(d)) be the
  probability of grouping in direction v. Note that
  Sp(v)1 for vÎV.
• Let H-Sp(v)log2p(v) be the entropy (average
  uncertainty, Shannon and Wiener, 1948).
• We have H Sp(v)log2p(v)/u log2u, where
  uf(a)f(b)f(c)f(d), the estimated entropy.

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Experiment

• Designed to test grouping by proximity, so other
  grouping principles were minimized (similarity,
  reference frames, orientation biases).
• 7 subjects the 2nd author, 3 graduate students,
  3 undergraduate students.
• Apparatus computer screen with simulated
  aperture.

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Stimuli and Responses

• Stimuliyellow dots of 5-pixel radius, separated
  by fixed distance a60 pixels (1.5 visual
  angle).
• Responsesproposed orientations.

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Parameters for 16 Lattices
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Procedure

• 3 sessions.
• 1600 trials (100 of each of the 16 lattices) in
  each session.
• Each session took 1 hour, with breaks every 400
  trials.
• Sessions were separated by at least 1 hour.
• Subjects were told lattices are collections of
  strips of dots, and could have more than one
  organization.
• Used the mouse to indicate perceived
  organization.

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Results

• Computed p(a), p(b), p(c) and p(d), and used them
  to compute H for each of the 16 lattices and each
  subject.
• Regressed the values of H on H while also
  determining a (varied a until the regression
  coefficients were maximized).
• The model slightly but systematically
  underestimates the values of H.

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Discussion (1)

• Proposed a negatively-accelerated attraction
  function not unique, other functions work well
  too unknown under which condition a is unique.
• Estimated choice probabilities the choice axiom
  (Luce, 1959 independence from irrelevant
  alternatives) may not hold in this case (choice
  may be random in absence of preferred
  alternative).

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Discussion (2)

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

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Limitations

• The model cannot describe perceptual clustering
  of random dots in the plane. Models that do
  unfortunately fail with dot lattices instead
  they can only predict proximity grouping, not
  multistability.
• The model cannot distinguish between small
  fragments of lattices and extensive lattices.
  This is due to the phenomenon of cooperativity
  (Julesz 1971) lattice fragments are not seen as
  organized in strips, and lattices do not undergo
  piecemeal organization (organization of local
  sets of dots in different parts of the lattice is
  linked).

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Conclusion

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

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The End