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

1. 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. Kubovy’s website with his consent. http://faculty.virginia.edu/kubovylab/kubovy/

2. 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

3. 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.html#kap2 )

4. 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.

5. 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.

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

7. 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. The Basic Parallelogram

8. The Space of Lattices • If a is held constant, all lattices are defined by b and g. • |b|≥|a| • 60°<g<90°

9. Hexagonal

10. Rhombic

11. Rhombic

12. Rhombic

13. Square

14. Rectangular

15. Rectangular

16. Rectangular

17. Rectangular

18. Centered Rectangular

19. Centered Rectangular

20. Centered Rectangular

21. Centered Rectangular

22. Oblique

23. 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.

24. Formal Model (1) • Let V={a,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 functionf(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.

25. 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 u=f(a)+f(b)+f(c)+f(d), the estimated entropy.

26. 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.

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

28. Parameters for 16 Lattices

29. 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.

30. 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.

31. 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).

32. 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).

33. 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).

34. 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.

35. The End