MeshPipe: a Python-based tool for easy automation and demonstration of geometry processing pipelines

The popularization of inexpensive 3D scanning, 3D printing, 3D publishing and AR/VR display technologies have renewed the interest in open-source tools providing the geometry processing algorithms required to clean, repair, enrich, optimize and modify point-based and polygonal-based models. Nowadays, there is a large variety of such open-source tools whose user community includes 3D experts but also 3D enthusiasts and professionals from other disciplines. In this paper we present a Python-based tool that addresses two major caveats of current solutions: the lack of easy-to-use methods for the creation of custom geometry processing pipelines (automation), and the lack of a suitable visual interface for quickly testing, comparing and sharing different pipelines, supporting rapid iterations and providing dynamic feedback to the user (demonstration). From the user's point of view, the tool is a 3D viewer with an integrated Python console from which internal or external Python code can be executed. We provide an easy-to-use but powerful API for element selection and geometry processing. Key algorithms are provided by a high-level C library exposed to the viewer via Python-C bindings. Unlike competing open-source alternatives, our tool has a minimal learning curve and typical pipelines can be written in a few lines of Python code.Peer ReviewedPostprint (published version

Concrete Computation of Global Illumination Using Structured Sampling

A new methodology is presented for the computation of global illumination using structured sampling. Analytical/numerical solutions for illumination are developed for simple lighting configurations. These solutions are subsequently used to generate accurate reference images. The structured sampling solution for global illumination is then discussed, comprising sample placement for illumination calculation, reconstruction for light transfer and finally resampling and filtering of illumination samples for display. A first approximation to this technique is presented using a priori placement of samples, irregular polygon reflectors, grid resampling and a conical filter for display. The new algorithm is evaluated for image quality, and compared to the traditional radiosity-based approach. These first results show that the structured sampling solution yields significant computational savings while maintaining high image quality. 1. Goals of the Approach The calculation of global illumination is inherently complex, even for environments that are simpl

Exploring Periodic Orbit Expansions and Renormalisation with the Quantum Triangular Billiard

A study of the quantum triangular billiard requires consideration of a boundary value problem for the Green's function of the Laplacian on a trianglar domain. Our main result is a reformulation of this problem in terms of coupled non--singular integral equations. A non--singular formulation, via Fredholm's theory, guarantees uniqueness and provides a mathematically firm foundation for both numerical and analytic studies. We compare and contrast our reformulation, based on the exact solution for the wedge, with the standard singular integral equations using numerical discretisation techniques. We consider in detail the (integrable) equilateral triangle and the Pythagorean 3-4-5 triangle. Our non--singular formulation produces results which are well behaved mathematically. In contrast, while resolving the eigenvalues very well, the standard approach displays various behaviours demonstrating the need for some sort of ``renormalisation''. The non-singular formulation provides a mathematically firm basis for the generation and analysis of periodic orbit expansions. We discuss their convergence paying particular emphasis to the computational effort required in comparision with Einstein--Brillouin--Keller quantisation and the standard discretisation, which is analogous to the method of Bogomolny. We also discuss the generalisation of our technique to smooth, chaotic billiards.Comment: 50 pages LaTeX2e. Uses graphicx, amsmath, amsfonts, psfrag and subfigure. 17 figures. To appear Annals of Physics, southern sprin

Recursive tilings and space-filling curves with little fragmentation

This paper defines the Arrwwid number of a recursive tiling (or space-filling curve) as the smallest number w such that any ball Q can be covered by w tiles (or curve sections) with total volume O(vol(Q)). Recursive tilings and space-filling curves with low Arrwwid numbers can be applied to optimise disk, memory or server access patterns when processing sets of points in d-dimensional space. This paper presents recursive tilings and space-filling curves with optimal Arrwwid numbers. For d >= 3, we see that regular cube tilings and space-filling curves cannot have optimal Arrwwid number, and we see how to construct alternatives with better Arrwwid numbers.Comment: Manuscript accompanying abstract in EuroCG 2010, including full proofs, 20 figures, references, discussion et

Solid NURBS Conforming Scaffolding for Isogeometric Analysis

This work introduces a scaffolding framework to compactly parametrise solid structures with conforming NURBS elements for isogeometric analysis. A novel formulation introduces a topological, geometrical and parametric subdivision of the space in a minimal plurality of conforming vectorial elements. These determine a multi-compartmental scaffolding for arbitrary branching patterns. A solid smoothing paradigm is devised for the conforming scaffolding achieving higher than positional geometrical and parametric continuity. Results are shown for synthetic shapes of varying complexity, for modular CAD geometries, for branching structures from tessellated meshes and for organic biological structures from imaging data. Representative simulations demonstrate the validity of the introduced scaffolding framework with scalable performance and groundbreaking applications for isogeometric analysis

Efficient Decimation of Polygonal Models Using Normal Field Deviation

A simple and robust greedy algorithm has been proposed for efficient and quality decimation of polygonal models. The performance of a simplification algorithm depends on how the local geometric deviation caused by a local decimation operation is measured. As normal field of a surface plays key role in its visual appearance, exploiting the local normal field deviation in a novel way, a new measure of geometric fidelity has been introduced. This measure has the potential to identify and preserve the salient features of a surface model automatically. The resulting algorithm is simple to implement, produces approximations of better quality and is efficient in running time. Subjective and objective comparisons validate the assertion. It is suitable for applications where the focus is better speed-quality trade-off, and simplification is used as a processing step in other algorithms

Polygonal path simplification with angle constraints

We present efficient geometric algorithms for simplifying polygonal paths in R2 and R3 that have angle constraints, improving by nearly a linear factor over the graph-theoretic solutions based on known techniques. The algorithms we present match the time bounds for their unconstrained counterparts. As a key step in our solutions, we formulate and solve an off-line ball exclusion search problem, which may be of interest in its own right

Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

We consider preprocessing a set $S$ of $n$ points in convex position in the plane into a data structure supporting queries of the following form: given a point $q$ and a directed line $\ell$ in the plane, report the point of $S$ that is farthest from (or, alternatively, nearest to) the point $q$ among all points to the left of line $\ell$. We present two data structures for this problem. The first data structure uses $O(n^{1+\varepsilon})$ space and preprocessing time, and answers queries in $O(2^{1/\varepsilon} \log n)$ time, for any $0 < \varepsilon < 1$. The second data structure uses $O(n \log^3 n)$ space and polynomial preprocessing time, and answers queries in $O(\log n)$ time. These are the first solutions to the problem with $O(\log n)$ query time and $o(n^2)$ space. The second data structure uses a new representation of nearest- and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only $O(\log n)$ amortized pointer changes, in addition to $O(\log n)$-time point-location queries, even though every such update may make $\Theta(n)$ combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in $o(n)$ amortized pointer changes per operation while keeping $O(\log n)$-time point-location queries.Comment: 17 pages, 6 figures. Various small improvements. To appear in Algorithmic