Spatial clustering and its effect on perceived clustering, numerosity, and dispersion

Human observers are able to estimate the numerosity of large sets of visual elements. The occupancy model of perceived numerosity in intermediate numerical ranges is based on overlapping regions of influence. The key idea is that items within a certain range count for less than their actual numerical value and more so the closer they are to their neighbours. Therefore occupancy is sensitive to the grouping of elements, but there are other spatial properties of  configurations that could also influence perceived numerosity, such as: area of convex hull, occupancy area, total degree of connectivity, and local clustering For all indices apart from convex hull, we varied the radius of the area that defined neighbours. We tested perceived numerosity using a fixed number of elements placed at random within a circular region. Observers compared two patterns (presented in two intervals) and chose the one that appeared more numerous. The same observers performed two other separate tasks in which they judged which pattern appeared more dispersed or more clustered. In each pair of images, the number was always the same (22, 28, 34, or 40 items), because we were interested in which "appeared" more numerous on the basis of spatial configuration. The results suggest that estimates of numerosity, dispersion, and clustering are based on different spatial information, that there are alternative approaches to quantifying clustering, and that in all cases clustering is linked to a decrease in perceived numerosity. The alternative measures have different properties and different practical and computational advantages.