Figure 5.17: Distance function propagation on a surfel
Illustration of the distance function overlaid on some surfels (left). Same portion with adjacencies of the Reeb graph (contours) added (right).
5.3.1.5 Discussion
One of the benefits of this approach is the low memory overhead for the Reeb graph representation. In the case of anO(n3)volume the storage requirement for propagating the distance function isO(n2), as it depends on the size of the surface. Once again the computation time for the Reeb graph will beO(nlogn) as the Reeb graph construction is equivalent to running a Dijkstra’s algorithm over the mesh to propagate distances and then anO(n)time step to collect surfels into contours. The only other data that we need to store for generation of the coarse mesh is dependent on the contours in the Reeb graph and isO(n). Memory overhead for contours is minimized by keeping only, (i) the contours selected to be part of the coarse mesh;
(ii) the last contour constructed and (iii) the current contour, which is being evaluated for possible selection.
Conclusion Once we have a coarse mesh, we use quadrisection to create a semi-regular mesh. We use a force based solver to position the vertices to best match the desired isosurface. This work presents a novel algorithm to extract isosurfaces in the form of hierarchical, adaptive semi-regular meshes. It relies on a novel approach to construct a coarsest mesh with the same global topology as the isosurface. The resulting meshes have a natural multi-resolution structure since they are semi-regular, making them suitable for a variety of powerful digital geometry processing algorithms.
Chapter 6
Conclusion
This thesis has presented computational topology algorithm for acquired 2-manifolds used in computer graph- ics. These algorithms are tailored for the discrete representation of 2-manifolds as triangle meshes or scalar volumes. We have presented algorithms for the identifying and isolating handles within a surface, algorithms for measuring the length of two locally minimal-length non-separating loops for each handle and methods to re-sample the geometry of the surface to retain or simplify handles. We have demonstrated the use of these algorithms through applications which simplify the topology of triangle meshes and scalar volumes. In these applications, we introduced methods for automatically removing handles from isosurfaces and triangle meshes through direct processing of the original volume data or mesh. We have demonstrated the effective- ness of these methods on several complex models. We have also demonstrated that removing topological artifacts is important for many subsequent modeling operations.
6.1 Contributions
The contributions of this thesis are the following:
A robust and efficient method to localize and isolate handles for discrete 2-manifolds. We propose a method where handles are efficiently identified through methods tuned to the discrete setting. The handle identification traversal of the surface is varied for efficiency while guaranteeing that all handles are located.
We present a traversal method with a complexity ofO(nlogn), with proof that our traversal methods will detect all handles during the traversal. Handles can subsequently be efficiently identified during the traversal of the surface as cycles in the augmented Reeb graph as it is incrementally constructed (see next paragraph).
Section 4.2.1.1 of Chapter 4 presented these methods in detail and the associated combinatorial proof.
A method to robustly represent the topology of the surface with an augmented Reeb graph. We present a method to construct an augmented Reeb graph which stores additional geometric information about the surface to facilitate isolating handles. We present a method to construct the augmented Reeb graph, which
guarantees that for each interval of the traversal, each ribbon has genus equal to zero. In addition, we guarantee that for each interval, the number of cycles in the augmented Reeb graph matches the genus of the surface traversed thus far. Geometric properties of the surface are encoded in the augmented Reeb graph which allows geometrically succinct handles to be isolated within the original surface. Section 4.2.2.2 of Chapter 4 introduced the augmented Reeb graph and the methods used to guarantee consistency between the number of handles and the number of cycles in the graph.
A method to find two locally minimal-length non-separating cycles for each handle. This thesis introduces a simple measure of handle size to be the length of two transverse non-separating cycles. The locally minimal-length non-separating cycles are detected efficiently for handles of the surface. We present a proof that we find two discrete locally minimal-length non-separating cycles with a complexity comparable to related approaches. See Section 4.3.2 in Chapter 4 for more details.
A simple method to simplify the topology for volume data and triangle meshes which preserves the local geometry as much as possible. Cutting the surface along the locally minimal-length non-separating cycle will reduce the genus of the model while retaining as much of the fine geometrical detail as possible. By using the smaller of the two non-separating cycles for each handle, the topology of the surface is only modified in a small local region. This targeted approach to modifying the topology preserves the fine geometry of the surface as much as possible. We propose a simple method to simplify the topology of triangle meshes and isosurfaces. Refer to Section 4.4.1 in Chapter 4 for more details. In particular, for isosurface topology simplification, to remove a handle, we alter the scalar values of the volume, thus indirectly modifying the isosurface. Since isosurfaces are always manifold, operating on the volume is robust. Also, by operating on the volume directly, we avoid computing an expensive triangle mesh and never compute or store floating point values to represent the geometric position of the vertices of the surface. Since our algorithm creates a topologically clean volume, this volume can then be used for surface extraction or other applications [48]
that depend on a topologically accurate volumetric representation.
An out-of-core method for topology simplification for volume data. Complex 3D models are rep- resented by large volumes that may not fit entirely in main memory. The model in Figure 1.7 is from a 885×709×736grid, and much larger models now exist [59]. The isosurface topology simplification algo- rithm is applied to such volumes using out-of-core methods. The algorithm employs a sweep method to read the volume in planar slices, so the data access pattern is highly regular. We encode surface topology as the sweep progresses using an augmented Reeb graph, requiring only a few slices in memory at any given time.
For some large handles, previous slices may need to be reloaded to perform simplification. However, simpli- fication can be performed on small segments of the volume one at a time, resulting in a purely out-of-core algorithm. The details of this out-of-core method are presented in Section 5.2.1 of Chapter 5.