Given the birds, where is the flock? Visual estimation of the location of collections of points

Abstract

Birds in flight form a flock, and pedestrians form a crowd. Humans readily perceive the center of such clouds, even though no individual bird or pedestrian marks that center. Developing accurate models of human perception of such phenomena is an important goal in vision research. Here, we reframed this central problem of perceptual organization in statistical terms: observers saw a sample drawn from an invisible probability density function (pdf) and attempted to estimate the location of that pdf, with better performance corresponding to lower estimation variance. Optimal minimum variance estimators differ across distribution families. We evaluated human performance in estimating the locations of pdfs from samples drawn from Gaussian, Laplacian, and Uniform location families. Observers approximated the family-specific optimal estimators for the Laplacian and Uniform distributions but deviated systematically from the optimum for the Gaussian distribution. How could the visual system flexibly adapt its estimates across different location families? We propose that observers adopt a single estimator based on clusters in the sample. According to this Visual Cluster Model, perceptual organization first aggregates individual "parts" into clusters and only then combines these clusters into a representation of the "whole", with the contribution of each cluster determined by the likelihood of the underlying distribution location. The model accounted for performance across all three distribution families, providing a general computational mechanism for estimating the location of a cloud from its parts across different statistical environments.