Decimation of triangle meshes

TL;DR: An application independent algorithm that uses local operations on geometry and topology to reduce the number of triangles in a triangle mesh and results from two different geometric modeling applications illustrate the strengths of the algorithm.

Abstract: The polygon remains a popular graphics primitive for computer graphics application. Besides having a simple representation, computer rendering of polygons is widely supported by commercial graphics hardware and software. However, because the polygon is linear, often thousands or millions of primitives are required to capture the details of complex geometry. Models of this size are generally not practical since rendering speeds and memory requirements are proportional to the number of polygons. Consequently applications that generate large polygonal meshes often use domain-specific knowledge to reduce model size. There remain algorithms, however, where domainspecific reduction techniques are not generally available or appropriate. One algorithm that generates many polygons is marching cubes. Marching cubes is a brute force surface construction algorithm that extracts isodensity surfaces from volume data, producing from one to five triangles within voxels that contain the surface. Although originally developed for medical applications, marching cubes has found more frequent use in scientific visualization where the size of the volume data sets are much smaller than those found in medical applications. A large computational fluid dynamics volume could have a finite difference grid size of order 100 by 100 by 100, while a typical medical computed tomography or magnetic resonance scanner produces over 100 slices at a resolution of 256 by 256 or 512 by 512 pixels each. Industrial computed tomography, used for inspection and analysis, has even greater resolution, varying from 512 by 512 to 1024 by 1024 pixels. For these sampled data sets, isosurface extraction using marching cubes can produce from 500k to 2,000k triangles. Even today’s graphics workstations have trouble storing and rendering models of this size. Other sampling devices can produce large polygonal models: range cameras, digital elevation data, and satellite data. The sampling resolution of these devices is also improving, resulting in model sizes that rival those obtained from medical scanners. This paper describes an application independent algorithm that uses local operations on geometry and topology to reduce the number of triangles in a triangle mesh. Although our implementation is for the triangle mesh, it can be directly applied to the more general polygon mesh. After describing other work related to model creation from sampled data, we describe the triangle decimation process and its implementation. Results from two different geometric modeling applications illustrate the strengths of the algorithm.

Summary

Introduction

• Besides having a simple representation, computer rendering of polygons is widely supported by commercial graphics hardware and software.
• Marching cubes is a brute force surface construction algorithm that extracts isodensity surfaces from volume data, producing from one to five triangles within voxels that contain the surface.
• Industrial computed tomography, used for inspection and analysis, has even greater resolution, varying from 512 by 512 to 1024 by 1024 pixels.
• The goal of the decimation algorithm is to reduce the total number of triangles in a triangle mesh, preserving the original topology and a good approximation to the original geometry.

2.1 OVERVIEW

• Multiple passes are made over all vertices in the mesh.
• During a pass, each vertex is a candidate for removal and, if it meets the specified decimation criteria, the vertex and all triangles that use the vertex are deleted.
• The vertex removal process repeats, with possible adjustment of the decimation criteria, until some termination condition is met.
• Usually the termination criterion is specified as a percent reduction of the original mesh (or equivalent), or as some maximum decimation value.
• The three steps of the algorithm are: 1. characterize the local vertex geometry and topology, 2. evaluate the decimation criteria, and 3. triangulate the resulting hole.

2.2 CHARACTERIZING LOCAL

• The first step of the decimation algorithm characterizes the local geometry and topology for a given vertex.
• Each vertex may be assigned one of five possible classifications: simple, complex, boundary, interior edge, or corner vertex.
• Examples of each type are shown in the figure below.
• A simple vertex is surrounded by a complete cycle of Simple Complex Boundary Interior Edge Corner triangles, and each edge that uses the vertex is used by exactly two triangles.
• All other vertices become candidates for deletion.

2.3 EVALUATING THE DECIMATION CRITERIA

• The characterization step produces an ordered loop of vertices and triangles that use the candidate vertex.
• The evaluation step determines whether the triangles forming the loop can be deleted and replaced by another triangulation exclusive of the original vertex.
• Boundary and interior edge vertices use the distance to edge criterion .
• Meshes may contain areas of relatively small triangles with large feature angles, contributing relatively little to the geometric approximation.
• The authors call this edge preservation, a user specifiable parameter.

2.4 TRIANGULATION

• Deleting a vertex and its associated triangles creates one (simple or boundary vertex) or two loops (interior edge vertex).
• In addition, there are two important characteristics of the loop that can be used to advantage.
• Second, since every loop is star-shaped, triangulation schemes based on recursive loop splitting are effective.
• The next section describes one such scheme.
• Once the triangulation is complete, the original vertex and its cycle of triangles are deleted.

3.1 DATA STRUCTURES

• The data structure must contain at least two pieces of information: the geometry, or coordinates, of each vertex, and the definition of each triangle in terms of its three vertices.
• Their implementation uses a space-efficient vertex-triangle hierarchical ring structure.
• Taken together these pointers form a ring relationship.
• The authors implementation uses three lists: a list of vertex coordinates, a list of triangle definitions, and another list of lists of triangles using each vertex.
• Edge definitions are not explicit, instead edges are implicitly defined as ordered vertex pairs in the triangle definition.

3.2 TRIANGULATION

• The authors chose a recursive loop splitting procedure.
• In order to determine whether the split forms two non-overlapping loops, the split plane is used for a half-space comparison.
• Certain special cases may occur during the triangulation process.
• Eliminating a vertex in this case would modify the topology of the mesh.
• Two different applications illustrate the triangle decimation algorithm.

4.1 VOLUME MODELING

• The first application applies the decimation algorithm to isosurfaces created from medical and industrial computed tomography scanners.
• Earlier work reported a triangle reduction strategy that used averaging to reduce the number of triangles on this same data set.
• The first set of figures shows the resulting bone isosurfaces for 0%, 75%, and 90% decimation, using a decimation threshold of 1/5 the voxel dimension.
• The isosurface created from the original blade data contains 1.7 million triangles.
• In fact, the authors could not render the original model because they exceeded the swap space on their graphics hardware.

4.2 TERRAIN MODELING

• The authors applied the decimation algorithm to two digital elevation data sets: Honolulu, Hawaii and the Mariner Valley on Mars.
• First the authors applied a decimation threshold of zero, eliminating over 30% of the co-planar triangles.
• The data represents the western end of the Mariner Valley and is about 1000km by 500km on a side.
• The last set of figures compares the shaded and wireframe models obtained via sub-sampling and decimation.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Decimation of triangle meshes" ?

This paper describes an application independent algorithm that uses local operations on geometry and topology to reduce the number of triangles in a triangle mesh. After describing other work related to model creation from sampled data, the authors describe the triangle decimation process and its implementation. 

Q2. What is the way to split a loop?

A loop of three ver-daverage planed boundarytices forms a triangle, that may be added to the mesh, and terminates the recursion process. 

Q3. What is the termination criterion for a vertex?

Usually the termination criterion is specified as a percent reduction of the original mesh (or equivalent), or as some maximum decimation value. 

Q4. What is the decimation algorithm for simple vertex?

A simple vertex is surrounded by a complete cycle ofSimple Complex Boundary Interior EdgeCornertriangles, and each edge that uses the vertex is used by exactly two triangles. 

Q5. What is the first application of the decimation algorithm?

The first application applies the decimation algorithm to isosurfaces created from medical and industrial computed tomography scanners. 

Q6. What is the way to determine whether a loop is split?

That is, if every point in a candidate loop is on one side of the split plane, then the two loops do not overlap and the split plane is acceptable. 

Q7. What is the definition of a vertex?

If the edge is not used by two triangles, or if the vertex is used by a triangle not in the cycle of triangles, then the vertex is complex. 

Q8. Why is the Mars example an appropriate test?

The Mars example is an appropriate test because the authors had access to sub-sampled resolution data that could be compared with the decimated models. 

Q9. How did the authors generate the initial mesh?

In both examples the authors generated an initial mesh by creating two triangles for each uniform quadrilateral element in the sampled data. 

Q10. What is the simplest way to reduce the mesh?

From the Euler relation it follows that removal of a simple, corner, or interior edge vertex reduces the mesh by precisely two triangles. 

Q11. What are the steps of the decimation algorithm?

The three steps of the algorithm are: 1. characterize the local vertex geometry and topology, 2. evaluate the decimation criteria, and 3. triangulate the resulting hole. 

Q12. What is the process of removing a vertex?

During a pass, each vertex is a candidate for removal and, if it meets the specified decimation criteria, the vertex and all triangles that use the vertex are deleted.