From 3D Models to 3D Prints: an Overview of the Processing Pipeline

Due to the wide diffusion of 3D printing technologies, geometric algorithms for Additive Manufacturing are being invented at an impressive speed. Each single step, in particular along the Process Planning pipeline, can now count on dozens of methods that prepare the 3D model for fabrication, while analysing and optimizing geometry and machine instructions for various objectives. This report provides a classification of this huge state of the art, and elicits the relation between each single algorithm and a list of desirable objectives during Process Planning. The objectives themselves are listed and discussed, along with possible needs for tradeoffs. Additive Manufacturing technologies are broadly categorized to explicitly relate classes of devices and supported features. Finally, this report offers an analysis of the state of the art while discussing open and challenging problems from both an academic and an industrial perspective.Comment: European Union (EU); Horizon 2020; H2020-FoF-2015; RIA - Research and Innovation action; Grant agreement N. 68044

On Triangular Splines:CAD and Quadrature

The standard representation of CAD (computer aided design) models is based on the boundary representation (B-reps) with trimmed and (topologically) stitched tensor-product NURBS patches. Due to trimming, this leads to gaps and overlaps in the models. While these can be made arbitrarily small for visualisation and manufacturing purposes, they still pose problems in downstream applications such as (isogeometric) analysis and 3D printing. It is therefore worthwhile to investigate conversion methods which (necessarily approximately) convert these models into water-tight or even smooth representations. After briefly surveying existing conversion methods, we will focus on techniques that convert CAD models into triangular spline surfaces of various levels of continuity. In the second part, we will investigate efficient quadrature rules for triangular spline space

Principal arc analysis on direct product manifolds

We propose a new approach to analyze data that naturally lie on manifolds. We focus on a special class of manifolds, called direct product manifolds, whose intrinsic dimension could be very high. Our method finds a low-dimensional representation of the manifold that can be used to find and visualize the principal modes of variation of the data, as Principal Component Analysis (PCA) does in linear spaces. The proposed method improves upon earlier manifold extensions of PCA by more concisely capturing important nonlinear modes. For the special case of data on a sphere, variation following nongeodesic arcs is captured in a single mode, compared to the two modes needed by previous methods. Several computational and statistical challenges are resolved. The development on spheres forms the basis of principal arc analysis on more complicated manifolds. The benefits of the method are illustrated by a data example using medial representations in image analysis.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS370 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Continuous Medial Models in Two-Sample Statistics of Shape

In questions of statistical shape analysis, the foremost is how such shapes should be represented. The number of parameters required for a given accuracy and the types of deformation they can express directly influence the quality and type of statistical inferences one can make. One example is a medial model, which represents a solid object using a skeleton of a lower dimension and naturally expresses intuitive changes such as "bending", "twisting", and "thickening". In this dissertation I develop a new three-dimensional medial model that allows continuous interpolation of the medial surface and provides a map back and forth between the boundary and its medial axis. It is the first such model to support branching, allowing the representation of a much wider class of objects than previously possible using continuous medial methods. A measure defined on the medial surface then allows one to write integrals over the boundary and the object interior in medial coordinates, enabling the expression of important object properties in an object-relative coordinate system. I show how these properties can be used to optimize correspondence during model construction. This improved correspondence reduces variability due to how the model is parameterized which could potentially mask a true shape change effect. Finally, I develop a method for performing global and local hypothesis testing between two groups of shapes. This method is capable of handling the nonlinear spaces the shapes live in and is well defined even in the high-dimension, low-sample size case. It naturally reduces to several well-known statistical tests in the linear and univariate cases