A Discrete Probabilistic Approach to Dense Flow Visualization

TL;DR

This paper introduces a novel discrete probabilistic framework for dense flow visualization, utilizing spectral embeddings derived from a similarity matrix to reveal flow mixing processes across scales.

Contribution

It presents a new discrete formulation based on probability theory, leading to spectral embedding methods for flow visualization, differing from traditional continuous models.

Findings

Spectral embeddings effectively visualize flow mixing at multiple scales.

The method applies to both 2D and 3D flows, demonstrating versatility.

Connections to Fourier analysis enhance interpretability of the embeddings.

Abstract

Dense flow visualization is a popular visualization paradigm. Traditionally, the various models and methods in this area use a continuous formulation, resting upon the solid foundation of functional analysis. In this work, we examine a discrete formulation of dense flow visualization. From probability theory, we derive a similarity matrix that measures the similarity between different points in the flow domain, leading to the discovery of a whole new class of visualization models. Using this matrix, we propose a novel visualization approach consisting of the computation of spectral embeddings, i.e., characteristic domain maps, defined by particle mixture probabilities. These embeddings are scalar fields that give insight into the mixing processes of the flow on different scales. The approach of spectral embeddings is already well studied in image segmentation, and we see that spectral embeddings are connected to Fourier expansions and frequencies. We showcase the utility of our method using different 2D and 3D flows.