TY - CHAP
AB - Alpha shapes have been conceived in 1981 as an attempt to define the shape of a finite set of point in the plane. Since then, connections to diverse areas in the sciences and engineering have developed, including to pattern recognition, digital shape sampling and processing, and structural molecular biology. This survey begins with a historical account and discusses geometric, algorithmic, topological, and combinatorial aspects of alpha shapes in this sequence.
AU - Herbert Edelsbrunner
ID - 3311
T2 - Tessellations in the Sciences
TI - Alpha shapes - a survey
ER -
TY - GEN
AB - We study the 3D reconstruction of plant roots from multiple 2D images. To meet the challenge caused by the delicate nature of thin branches, we make three innovations to cope with the sensitivity to image quality and calibration. First, we model the background as a harmonic function to improve the segmentation of the root in each 2D image. Second, we develop the concept of the regularized visual hull which reduces the effect of jittering and refraction by ensuring consistency with one 2D image. Third, we guarantee connectedness through adjustments to the 3D reconstruction that minimize global error. Our software is part of a biological phenotype/genotype study of agricultural root systems. It has been tested on more than 40 plant roots and results are promising in terms of reconstruction quality and efficiency.
AU - Zheng, Ying
AU - Gu, Steve
AU - Edelsbrunner, Herbert
AU - Tomasi, Carlo
AU - Benfey, Philip
ID - 3312
T2 - Proceedings of the IEEE International Conference on Computer Vision
TI - Detailed reconstruction of 3D plant root shape
ER -
TY - CONF
AB - Interpreting an image as a function on a compact sub- set of the Euclidean plane, we get its scale-space by diffu- sion, spreading the image over the entire plane. This gener- ates a 1-parameter family of functions alternatively defined as convolutions with a progressively wider Gaussian ker- nel. We prove that the corresponding 1-parameter family of persistence diagrams have norms that go rapidly to zero as time goes to infinity. This result rationalizes experimental observations about scale-space. We hope this will lead to targeted improvements of related computer vision methods.
AU - Chen, Chao
AU - Edelsbrunner, Herbert
ID - 3313
T2 - Proceedings of the IEEE International Conference on Computer Vision
TI - Diffusion runs low on persistence fast
ER -
TY - CONF
AB - We report on a generic uni- and bivariate algebraic kernel that is publicly available with CGAL 3.7. It comprises complete, correct, though efficient state-of-the-art implementations on polynomials, roots of polynomial systems, and the support to analyze algebraic curves defined by bivariate polynomials. The kernel design is generic, that is, various number types and substeps can be exchanged. It is accompanied with a ready-to-use interface to enable arrangements induced by algebraic curves, that have already been used as basis for various geometric applications, as arrangements on Dupin cyclides or the triangulation of algebraic surfaces. We present two novel applications: arrangements of rotated algebraic curves and Boolean set operations on polygons bounded by segments of algebraic curves. We also provide experiments showing that our general implementation is competitive and even often clearly outperforms existing implementations that are explicitly tailored for specific types of non-linear curves that are available in CGAL.
AU - Berberich, Eric
AU - Hemmer, Michael
AU - Kerber, Michael
ID - 3328
TI - A generic algebraic kernel for non linear geometric applications
ER -
TY - CONF
AB - We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance µ in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution shape P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(n log n)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. An alternative algorithm, based purely on rational arithmetic, answers the same deconstruction problem, up to an uncertainty parameter, and its running time depends on the parameter δ (in addition to the other input parameters: n, δ and the radius of the disk). If the input shape is found to be approximable, the rational-arithmetic algorithm also computes an approximate solution shape for the problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution shape P with at most one more vertex than a vertex-minimal one. Our study is motivated by applications from two different domains. However, since the offset operation has numerous uses, we anticipate that the reverse question that we study here will be still more broadly applicable. We present results obtained with our implementation of the rational-arithmetic algorithm.
AU - Berberich, Eric
AU - Halperin, Dan
AU - Kerber, Michael
AU - Pogalnikova, Roza
ID - 3329
T2 - Proceedings of the twenty-seventh annual symposium on Computational geometry
TI - Deconstructing approximate offsets
ER -