Accurate Synthesis of Multi‐Class Disk Distributions

Abstract

While analysing and synthesising 2D distributions of points has been applied both to the generation of textures with discrete elements and for populating virtual worlds with 3D objects, the results are often inaccurate since the spatial extent of objects cannot be expressed. We introduce three improvements enabling the synthesis of more general distributions of elements. First, we extend continuous pair correlation function (PCF) algorithms to multi‐class distributions using a dependency graph, thereby capturing interrelationships between distinct categories of objects. Second, we introduce a new normalised metric for disks, which makes the method applicable to both point and possibly overlapping disk distributions. The metric is specifically designed to distinguish perceptually salient features, such as disjoint, tangent, overlapping, or nested disks. Finally, we pay particular attention to convergence of the mean PCF as well as the validity of individual PCFs, by taking into consideration the variance of the input. Our results demonstrate that this framework can capture and reproduce real‐life distributions of elements representing a variety of complex semi‐structured patterns, from the interaction between trees and the understorey in a forest to droplets of water. More generally, it applies to any category of 2D object whose shape is better represented by bounding circles than points.

Data-driven authoring of large-scale ecosystems

“…In this category techniques range from relatively straightforward but computationally efficient half-toning of density images [Deussen et al 1998], dart throwing [Andújar et al 2014], and dominant-species placement using greedy cluster merging [Li et al 2018], to more complex field-of-neighbourhood plant distributions placed on Wang tiles [Alsweis and Deussen 2006], probability density maps [Lane and Prusinkiewicz 2002] and interand intra-species distribution histograms [Emilien et al 2015;. There has been a recent move from point-based to discbased synthesis [Ecormier-Nocca et al 2019], since this represents both trunk position and canopy extent and better captures biotic interactions, such as the shadowing of shade-tolerant plants.…”

“…Instead, we turn to simulation as a source. Recognizing that simulation on its own is several orders of magnitude too slow for interactive authoring we use it as a basis for disk-based statistical run-time synthesis of undergrowth [Ecormier-Nocca et al 2019;].…”

Image‐based authoring of herd animations

“…Firstly, most of them only handle nonoverlapping shapes, while animals within herds are often in close contact. The only method explicitly handling contacts and overlaps is limited to disks and is not suited to our data, where animals are anisotropic and oriented. Secondly, all current methods tackled the analysis of a supposedly uniform distribution of elements.…”

“…PCFs are smooth functions that represent the average density of objects in a distribution with respect to an increasing distance between those objects. They are known to have many interesting properties such as being density invariant and characterizing well the visual aspect of distributions (see the work of Ecormier‐Nocca for details).…”

Virtual Forestry Generation: Evaluating Models for Tree Placement in Games

“…There have been a number of papers which show their use in the procedural placement of objects, including trees and forestry [28,29]. A recent example of this is by Ecormier et al [30], in which a variance-aware disk-based distribution algorithm is presented. In particular, the authors highlight its usage in synthesising virtual forest scenes.…”