An absolute interval scale of order for point patterns

Abstract

Human observers readily make judgements about the degree of order in planar arrangements of points (point patterns). Here, based on pairwise ranking of 20 point patterns by degree of order, we have been able to show that judgements of order are highly consistent across individuals and the dimension of order has an interval scale structure spanning roughly 10 justnotable-differences ( jnd) between disorder and order. We describe a geometric algorithm that estimates order to an accuracy of half a jnd by quantifying the variability of the size and shape of spaces between points. The algorithm is 70% more accurate than the best available measures. By anchoring the output of the algorithm so that Poisson point processes score on average 0, perfect lattices score 10 and unit steps correspond closely to jnds, we construct an absolute interval scale of order. We demonstrate its utility in biology by using this scale to quantify order during the development of the pattern of bristles on the dorsal thorax of the fruit fly.

Difference magnitude is not measured by discrimination steps for order of point patterns

“…The ML estimated lapse rate parameter was 0.3%. We note that 16.5 jnds is larger than the 10 jnds we found between order and disorder in our previous work (Protonotarios et al, 2014). We attribute this difference to the greater variation in pattern types in the previous experiment.…”

“…Providing a mathematical scale of order as an objective surrogate of human perception seems a promising alternative. In previous work (Protonotarios, Baum, Johnston, Hunter, & Griffin, 2014) we have shown, using pairwise comparisons of point patterns from a diverse set, that observers are highly consistent in their judgments of order, and that these are compatible with an interval scale structure. This means that the preference frequencies with respect to order for pairs of point patterns of the whole set can be predicted based on the distances between the attribute values on this scale.…”

“…The space of approximately ordered point patterns is vast. In our previous work (Protonotarios et al, 2014) we used a multistep process to synthesize diverse examples from this space. Starting from perfect lattices of points, which could be triangular, rectangular, or hexagonal, we independently perturbed the position of the points, randomly deleted and added varying proportions of them, and finally applied a smooth nonlinear positional warp.…”

“…This means that a large number of trials simply validated the ordinal structure of the scale. Although we have established in previous work (Protonotarios et al, 2014), the interval structure of the responses for diverse point patterns, for the sake of completeness, we preferred again a balanced design to check for that for the new set. We could not exclude beforehand that patterns of high jitter may appear more ordered than patterns of intermediate jitter (e.g., because of random pairs or clusters of points).…”

Coordinated control of Notch-Delta signalling and cell cycle progression drives lateral inhibition mediated tissue patterning

“…During notum development, all ECs divide once ( Bosveld et al, 2012 ) ( ), before undergoing terminal differentiation. At the same time, an initially disordered array of cells expressing proneural genes is refined to generate an ordered pattern of bristles in adults ( Cohen et al, 2010 ; Protonotarios et al, 2014 ) ( Fig. 1 A).…”