The Gestalt Computational Model by Persistent Homology

TL;DR

This paper utilizes persistent homology, a mathematical tool in computational topology, to develop a unified computational model for Gestalt principles, addressing the challenges of quantification and computation.

Abstract

Widely employed in cognitive psychology, Gestalt theory elucidates basic principles in visual perception. However, the Gestalt principles are validated mainly by psychological experiments, lacking quantitative research supports and theoretical coherence. In this paper, we utilize persistent homology, a mathematical tool in computational topology, to develop a unified computational model for Gestalt principles, addressing the challenges of quantification and computation. On the one hand, the Gestalt computational model presents quantitative supports for Gestalt theory. On the other hand, it shows that the Gestalt principles can be uniformly calculated using persistent homology, thus developing a coherent theory for Gestalt principles in computation. Moreover, it is anticipated that the Gestalt computational model can serve as a significant computational model in the field of computational psychology, and help the understanding of human being visual perception.

Figures (9)

• Figure 1: Calculation of Gestalt similarity principle. a, d: Examples for similarity principle. b, e: The clustering result of the corresponding 0-PD with highlighted zero-dimensional significant points. c, f: The VR complex $VR(\varepsilon_g)$ that coincides with the similarity principle.
• Figure 2: Calculation of the principle of proximity, closure and good continuation. a, d, g: Examples of Gestalt proximity, closure and good continuation principles. b: The clustering result of the corresponding 0-PD with highlighted zero-dimensional significant points. c: The VR complex $VR(\varepsilon_g)$ that coincides with the Gestalt proximity principle. e, h: The clustering result of the corresponding 1-PD with highlighted one-dimensional significant points. f: The hidden loop in the point cloud. i: The corresponding 1-skeleton when all one-dimensional topological features (loops) have formed. j: Two circles are successfully identified. Here edges whose two terminal nodes are the starting and ending points are drawn in yellow.
• Figure 3: Calculation of Gestalt pragnanz principle. a: The shape of the five Olympic rings. b: It is unlikely that the shapes in Fig. \ref{['fig:fig3']} a will be seen as nine parts. c: An example of the pragnanz principle. The point cloud represents the shape of the five Olympic rings. d: The clustering result of the corresponding 1-PD with highlighted one-dimensional significant points. e: Five circles that represent five significant topological features correspond to the five significant points in the 1-PD.
• Figure 4: Illustration of conflicts. a: An example of a conflict between different attributes of the similarity principle itself. b: The clustering result of the corresponding 0-PD with highlighted zero-dimensional significant points. Here two situations lead to the same 0-PD. c: The computation result when the shape feature is dominant. d: The computation result when the color feature is dominant.
• Figure 5: Point sets with clutter points. a, c, e, g, i: Point sets we have used in previous sections, with additional random clutter points adhering to a Gaussian distribution and random colors. b, d, f, h, j: Computational results of our model. To improve clarity, simplices formed by clutter points are omitted. Results aligning with Gestalt theory can still be reconstructed.