Topology preserving data simplification with error bounds

Abstract

Many approaches to simplification of triangulated terrains and surfaces have been proposed which permit bounds on the error introduced. A few algorithms additionally bound errors in auxiliary functions defined over the triangulation. We present an approach to simplification of scalar fields over unstructured grids which preserves the topology of functions defined over the triangulation, in addition to bounding of the errors. The topology of a 2D scalar field is defined by critical points (local maxima, local minima, saddle points), in addition to integral curves between them, which together segment the field into regions which vary monotonically. By preserving this shape description, we guarantee that isocontours of the scalar function maintain the correct topology in the simplified model. Methods for topology preserving simplification by both point-insertion (refinement) and point-deletion (coarsening) are presented and compared.

Introduction

Scientific data is often sampled or computed over a dense mesh in order to capture high-frequency components or achieve a desired error bound. Interactive display and navigation of such large meshes is impeded by the sheer number of triangles required to sufficiently model highly complex data. A number of simplification techniques have been developed which reduce the number of triangles to a particular desired triangle count or until a particular error threshold is met. Given an initial triangulation M of a domain $\text{D}$ and a function $\text{F}$(x) defined over the triangulation, the simplified mesh can be called M′ and the resulting function $\text{F'}$(x). The measure of error in a simplified mesh M′ is usually represented as

$$\text{ϵ(M')=}\text{max}{}_{\mathrm{<mtext>x\in </mtext><mtext>D</mtext>}}\text{(|}\text{F}\text{(x)−}\text{F'}\text{(x)|).}$$

The ability to bound the error ϵ(M′) is very important, but the error definition (1) is inherently a local measure, neglecting to consider global features of the data. We introduce new criteria for the simplification of sampled functions which preserves scalar field features in addition to bounding local errors. Two-dimensional scalar field topology is described by the critical points and arcs between them. Preserving the scalar field criticalities maintains an invariant of the connectivity and combinatorial structure (topological genus) of successively simplified isocontours.

In Section 2we discuss related work in mesh simplification and feature detection. Section 3introduces the definition of 2D scalar topology as it will be used in our simplification strategy. In Section 4we introduce two algorithms for simplification with topology preserving characteristics. The first is an extension to existing coarsening techniques which iteratively delete vertices or contract edges in the mesh. The second algorithm adopts an inverse approach, iteratively introducing detail (refinement) to an initially sparse mesh, preserving the scalar topology of the fine mesh.