Analysis and synthesis of point distributions based on pair correlation

Abstract

Analyzing and synthesizing point distributions are of central importance for a wide range of problems in computer graphics. Existing synthesis algorithms can only generate white or blue-noise distributions with characteristics dictated by the underlying processes used, and analysis tools have not been focused on exploring relations among distributions. We propose a unified analysis and general synthesis algorithms for point distributions. We employ the pair correlation function as the basis of our methods and design synthesis algorithms that can generate distributions with given target characteristics, possibly extracted from an example point set, and introduce a unified characterization of distributions by mapping them to a space implied by pair correlations. The algorithms accept example and output point sets of different sizes and dimensions, are applicable to multi-class distributions and non-Euclidean domains, simple to implement and run in O ( n ) time. We illustrate applications of our method to real world distributions.

Topology-Driven Trajectory Synthesis with an Example on Retinal Cell Motions

“…We first test the performance of the synthesis algorithm using PCF only [13]. In this case, the distance between two point sets measures the difference between their pairwise correlations in (2).…”

“…To define closeness, we combine two high-level distance measures: firstly, we employ the pair correlation function (PCF) which extracts pairwise correlations in the point cloud data by measuring how density varies as a function of distance from a reference point. The PCF is widely accepted as an informative statistical measure for point set analysis, and has been used for trajectory synthesis in previous work [13]. As major novelty, we propose to combine the PCF with a topological distance measure: we compare persistence diagrams of alpha-shape filtrations which capture the evolution of holes that arise when the points are thickened to disks with increasing radius.…”

“…It has been shown that such a simple linear interpolation can generate valid PCFs from which distributions can be synthesized [13]. Thus, we can use the synthesis algorithm to generate data at any time t 0 ≤ t ≤ t M .…”

An adaptive point sampler on a regular lattice

“…This approach is well-suited for adaptive sampling, and aims at substantially higher spectral qualities than Wang tiles; but this comes at a considerable cost in memory, since the whole tiling structure has to be stored. Through two subsequent steps of development [Ostromoukhov 2007;Wachtel et al 2014], this approach reached a quality that enables almost full control over the spectral properties of the conveyed point sets, using sophisticated optimization techniques [Heck et al 2013;Öztireli and Gross 2012;Zhou et al 2012] that were developed concurrently. Unfortunately, to that end the required memory footprint grows beyond the practical limits of many applications: gigabyte-sized lookup tables for a single spectral profile.…”

“…11 Spectral Control. Target matching algorithms [Heck et al 2013;Öztireli and Gross 2012;Zhou et al 2012] can easily be adapted to our framework thanks to the toroidal domain optimization environment we have. The only change needed is to average the displacements of the points holding the same ID, as advocated by all of [Ahmed et al 2015;Ostromoukhov 2007;Ostromoukhov et al 2004;Wachtel et al 2014].…”