Title: The role of visual clustering in approximate numerosity perception

Date: Tuesday, April 15, 2025

Time: 12:30pm - 2:30pm

Location: Hybrid

  Online (Zoom): https://gatech.zoom.us/j/94723168281?pwd=h0r5Byi9RPYptLYHyQACQ4b6dnq3zn.1

  In-person: IC Cafe (TSRB 204), 85 5th St NW, Atlanta, GA 30332

Vijay Marupudi

Ph.D. Candidate, Human-Centered Computing School of Interactive Computing College of Computing Georgia Institute of Technology

Committee:

Dr. Sashank Varma (Advisor), School of Interactive Computing, Georgia Institute of Technology Dr. Cindy Xiong Bearfield, School of Interactive Computing, Georgia Institute of Technology Dr. Chris MacLellan, School of Interactive Computing, Georgia Institute of Technology Dr. Elizabeth Brannon, Department of Psychology, University of Pennsylvania Dr. Priti Shah, Department of Psychology, University of Michigan

Summary:

The ability to visually perceive distinct objects as groups that belong together is a core part of human perception. This ability, called visual clustering, is important for a wide range of human behaviors such as ensemble perception, visual working memory, spatial problem-solving, and understanding information visualizations. However, surprisingly little is known about this ability, which is also known as Gestalt proximity grouping. It is complex. From a computational perspective, for any given criterion --- for example minimizing the variance of the distances between objects within clusters and maximizing the variance of the distances between clusters --- we are not currently aware of an algorithm that can find an optimal solution efficiently, i.e., in polynomial time.

However, people cluster visual input regularly. How are they doing so? In this thesis, I conduct a series of human-subjects experiments to determine how people perceive clusters and how these clusters may be used as a basis for other cognitive abilities. In one experiment focused on understanding how stable people's clusters are, my results indicate that people's clusterings of the same stimulus at two different time points are remarkably similar to each other. Participants displayed greater reliability for clustered (clumped) compared to dispersed (spread out) stimuli. This reliability partially diminished with increasing number of points. I found that people's clusterings were consistent with a Gaussian distribution and were also highly similar to each other, suggesting a shared cognitive process.

Given its stability, visual clustering is likely implicated in how people solve problems they encounter in the world. To demonstrate this, I investigated whether people use clustering to solve the traveling salesperson problem (TSP). People are unexpectedly performant at this task, staying within 10% of optimal solutions and solving problems in time linear to the number of points. I asked participants to solve TSP problems at two different time points. I found that the reliability patterns of the TSP for clustered and dispersed stimuli followed the same patterns as visual clustering, implying that clustering may be used to solve the TSP. In a second experiment, I asked people to cluster and solve the TSP on the same stimuli and found that people's TSP solutions were remarkably consistent with their clusterings, with 52% of TSP solutions being perfectly congruent with their clusterings. Even when their solutions weren't perfectly congruent, they displayed high amounts of congruency in general, providing further support to the theory that people use the clusters they perceive to solve the TSP.

Finally, I investigated whether people use visual clustering for numerosity perception. Babies as young as 6 months old are sensitive to the number of objects they see in the world. People can also estimate and compare numerosities displayed for only 16 milliseconds! It is not clear how people can determine the numerosity of objects so quickly and automatically. This ability falls under the purview of the approximate number system (ANS), a cognitive system responsible for the abstract representation of number. Research on the ANS has uncovered several illusions and effects where properties associated with clustering result in people underestimating them. Some of these include the number of clusters perceived, whether objects are connected to each other, and how points are distributed in space (cluster structure). These findings raise the possibility that people may be visually clustering points to determine the numerosity of visual input. In the final chapter of this thesis, I test this hypothesis by conducting two experiments where participants compare and estimate the numerosities of sets of points. In the first experiment, participants judged the numerosities of the same stimuli used to investigate clustering reliability, which varied in cluster structure and the number of points present. From that study, we knew how reliably people clustered the stimuli. This allowed us to evaluate the influence of these properties on numerosity perception. In Experiment 2, participants were shown stimuli with increased numerosities and were provided limited time to view the sets of points. Additionally, I varied how the sets of points were presented on comparison trials by presenting them either simultaneously or sequentially to alleviate concerns that clustering effects may only appear when stimuli appear next to each other.

In both experiments, I found a robust effect of cluster structure, i.e., whether stimuli are clustered or dispersed, on the magnitude comparison task. When people were asked to determine which of two sets of points are more numerous, they were more likely to choose the dispersed stimulus. I found that this effect increased with if participants were given more time to view the stimulus. On the magnitude estimation task, where participants were asked to report the number of points they see, I only observed a small effect of cluster structure. Other clustering properties, such as the number of clusters, clustering reliability, and how points were assigned to clusters, were not reliable predictors of participants' performance. In fact, I observed a negative effect of the number of clusters on the magnitude comparisons, contrary to what is often assumed in the literature. These findings shed light on the cognitive processes underlying the ANS. They suggest that visual clustering, one prevailing explanation for why people underestimate clustered stimuli, is unlikely to be responsible for the effect. Instead, it is possible that the effect occurs due to how attention is allocated and information is encoded. The work presented here can help our understanding of numerical behavior across a wide variety of fields and has important implications for decision-making and information visualization research.