TY - JOUR AB - We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in double-struck R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on double-struck S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard. AU - Attali, Dominique AU - Bauer, Ulrich AU - Devillers, Olivier AU - Glisse, Marc AU - Lieutier, André ID - 1805 IS - 8 JF - Computational Geometry: Theory and Applications TI - Homological reconstruction and simplification in R3 VL - 48 ER - TY - JOUR AB - We present a software platform for reconstructing and analyzing the growth of a plant root system from a time-series of 3D voxelized shapes. It aligns the shapes with each other, constructs a geometric graph representation together with the function that records the time of growth, and organizes the branches into a hierarchy that reflects the order of creation. The software includes the automatic computation of structural and dynamic traits for each root in the system enabling the quantification of growth on fine-scale. These are important advances in plant phenotyping with applications to the study of genetic and environmental influences on growth. AU - Symonova, Olga AU - Topp, Christopher AU - Edelsbrunner, Herbert ID - 1793 IS - 6 JF - PLoS One TI - DynamicRoots: A software platform for the reconstruction and analysis of growing plant roots VL - 10 ER - TY - GEN AU - Symonova, Olga AU - Topp, Christopher AU - Edelsbrunner, Herbert ID - 9737 TI - Root traits computed by DynamicRoots for the maize root shown in fig 2 ER - TY - JOUR AB - Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology. AU - Pausinger, Florian AU - Svane, Anne ID - 1792 IS - 6 JF - Journal of Complexity TI - A Koksma-Hlawka inequality for general discrepancy systems VL - 31 ER - TY - THES AB - This thesis is concerned with the computation and approximation of intrinsic volumes. Given a smooth body M and a certain digital approximation of it, we develop algorithms to approximate various intrinsic volumes of M using only measurements taken from its digital approximations. The crucial idea behind our novel algorithms is to link the recent theory of persistent homology to the theory of intrinsic volumes via the Crofton formula from integral geometry and, in particular, via Euler characteristic computations. Our main contributions are a multigrid convergent digital algorithm to compute the first intrinsic volume of a solid body in R^n as well as an appropriate integration pipeline to approximate integral-geometric integrals defined over the Grassmannian manifold. AU - Pausinger, Florian ID - 1399 SN - 2663-337X TI - On the approximation of intrinsic volumes ER -