Quantifying Homology Classes

We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in $O(\beta^4 n^3 \log^2 n)$ time, where $n$ is the size of the simplicial complex and $\beta$ is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results

A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface

Given a graph $G$ cellularly embedded on a surface $\Sigma$ of genus $g$, a cut graph is a subgraph of $G$ such that cutting $\Sigma$ along $G$ yields a topological disk. We provide a fixed parameter tractable approximation scheme for the problem of computing the shortest cut graph, that is, for any $\varepsilon >0$, we show how to compute a $(1+ \varepsilon)$ approximation of the shortest cut graph in time $f(\varepsilon, g)n^3$. Our techniques first rely on the computation of a spanner for the problem using the technique of brick decompositions, to reduce the problem to the case of bounded tree-width. Then, to solve the bounded tree-width case, we introduce a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which may be of independent interest

A combined voxel and surface based method for topology correction of brain surfaces

Brain surfaces provide a reliable representation for cortical mapping. The construction of correct surfaces from magnetic resonance images (MRI) segmentation is a challenging task, especially when genus zero surfaces are required for further processing such as parameterization, partial inflation and registration. The generation of such surfaces has been approached either by correcting a binary image as part of the segmentation pipeline or by modifying the mesh representing the surface. During this task, the preservation of the structure may be compromised because of the convoluted nature of the brain and noisy/imperfect segmentations. In this paper, we propose a combined, voxel and surfacebased, topology correction method which preserves the structure of the brain while yielding genus zero surfaces. The topology of the binary segmentation is first corrected using a set of topology preserving operators applied sequentially. This results in a white matter/gray matter binary set with correct sulci delineation, homotopic to a filled sphere. Using the corrected segmentation, a marching cubes mesh is then generated and the tunnels and handles resulting from the meshing are finally removed with an algorithm based on the detection of nonseparating loops. The approach was validated using 20 young individuals MRI from the OASIS database, acquired at two different time-points. Reproducibility and robustness were evaluated using global and local criteria such as surface area, curvature and point to point distance. Results demonstrated the method capability to produce genus zero meshes while preserving geometry, two fundamental properties for reliable and accurate cortical mapping and further clinical studies

Randomly removing g handles at once

AbstractIndyk and Sidiropoulos (2007) proved that any orientable graph of genus g can be probabilistically embedded into a graph of genus g−1 with constant distortion. Viewing a graph of genus g as embedded on the surface of a sphere with g handles attached, Indyk and Sidiropoulos' method gives an embedding into a distribution over planar graphs with distortion 2O(g), by iteratively removing the handles. By removing all g handles at once, we present a probabilistic embedding with distortion O(g2) for both orientable and non-orientable graphs. Our result is obtained by showing that the minimum-cut graph of Erickson and Har-Peled (2004) has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma of Lee and Sidiropoulos (2009)

Schnyder woods for higher genus triangulated surfaces, with applications to encoding

Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into 3 spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus $g$ and compute a so-called $g$-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus $g$ and $n$ vertices in $4n+O(g \log(n))$ bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms presented here have execution time $O((n+g)g)$, hence are linear when the genus is fixed.Comment: 27 pages, to appear in a special issue of Discrete and Computational Geometr

Interactive Geometry Remeshing

We present a novel technique, both flexible and efficient, for interactive remeshing of irregular geometry. First, the original (arbitrary genus) mesh is substituted by a series of 2D maps in parameter space. Using these maps, our algorithm is then able to take advantage of established signal processing and halftoning tools that offer real-time interaction and intricate control. The user can easily combine these maps to create a control map – a map which controls the sampling density over the surface patch. This map is then sampled at interactive rates allowing the user to easily design a tailored resampling. Once this sampling is complete, a Delaunay triangulation and fast optimization are performed to perfect the final mesh. As a result, our remeshing technique is extremely versatile and general, being able to produce arbitrarily complex meshes with a variety of properties including: uniformity, regularity, semiregularity, curvature sensitive resampling, and feature preservation. We provide a high level of control over the sampling distribution allowing the user to interactively custom design the mesh based on their requirements thereby increasing their productivity in creating a wide variety of meshes