Topological simplification of isosurfaces in volumetric data using octrees

Abstract

Volumetric data, such as output from CT scans or laser range scan processing methods, often have isosurfaces that contain topological noise—small handles and holes that are not present in the original model. Because this noise can significantly degrade the performance of other geometric processing algorithms, we present a volumetric method that removes the topological noise and patches holes in undefined regions for a given isovalue. We start with a surface completely inside the isosurface of interest and a surface completely outside the isosurface. These surfaces are expanded and contracted, respectively, on a voxel-by-voxel basis. Changes in topology of the surfaces are prevented at every step using a local topology test. The result is a pair of surfaces that accurately reflect the geometry of the model but have simple topology. We represent the volume in an octree format for improved performance in space and time.

Introduction

Three-dimensional volume data sets have become very popular in recent years, due to advances in 3D medical imaging and other scanning techniques. There are many methods of visualizing volumetric data, but a popular one is extracting a mesh that represents the boundary of the object (an isosurface or level set) using a method such as marching cubes [1]. The benefits of extracting a mesh include all the processing that can be done on meshes to enhance visualization, such as texture mapping and simplification. However, many algorithms for these purposes get bogged down in details in the mesh that are not important to visualization such as small handles and holes. We call these details topological noise because they artificially increase the genus of the model, often to a great extent, and they typically appear due to inaccuracies in scanning procedures.

We begin with a volumetric representation of a single component, three-dimensional object whose topology we wish to clean up. This representation consists of samples on a regular rectangular grid—for example, samples of the density of the object or of the distance to the object’s surface. Each sample is considered the center of a voxel—a closed solid cube. Each voxel is considered to be on the inside or on the outside of the object, depending on whether the sample value is above or below the isovalue. The voxel data are obtained directly from a process that gives volume data as output, such as medical imaging (CT and MRI scans) or the volumetric procedure for merging laser range scans described by Curless and Levoy [2]. After our procedure is finished, we still have a volumetric representation of the object, one that is very similar to the original input, except that the topological complexity of the object is under control. If the desired genus is known a priori, the user can specify the number of topological features the object is supposed to have. Otherwise a threshold on the size of the features can be specified. In either case, the topological noise is eliminated by cutting or patching over the small handles. This is accomplished by changing a small number of inside voxels to outside or a small number of outside voxels to inside. If there is any subvoxel information stored in the samples, for example the distance to the object surface, we retain this information for the output, except for the voxels that are changed. These samples are set to values that indicate a location slightly on the other side of the surface. The volume is now ready for mesh extraction.

Since volume data sets can be quite large, we implemented an octree representation and procedure to save memory. Some volume data sets contain unknown regions, but this is not a problem for our technique because it automatically constructs a well-defined sense of inside and outside for every voxel, so the extracted mesh will not only have controlled topology but will also be watertight.

Topologically simple models are important in many applications, such as texture mapping [3], simplification [4], compression, mapping between surfaces, and new techniques such as that of Zhang et al. [5]. For example, there are many methods of mesh simplification, but if the user wants the simplification to preserve the manifold structure of the mesh, the topology of the mesh must also be preserved. An example of this is Hoppe et al.’s [6] mesh simplification using edge contraction. The topology preservation can be desirable except when there is topological noise that the user wishes to be simplified away. Fortunately, our technique removes all the small handles but retains the meaningful topological features. Thus, when topology simplification is used as a preprocess to mesh extraction, there is no need to worry about getting bogged down in the topological noise during mesh simplification.