In this paper we present an automatic algorithm to detect basic shapes in unorganized point clouds. The algorithm decomposes the point cloud into a concise, hybrid structure of inherent shapes and a set of remaining points. Each detected shape serves as a proxy for a set of corresponding points. Our method is based on random sampling and detects planes, spheres, cylinders, cones and tori. For models with surfaces composed of these basic shapes only, for example, CAD models, we automatically obtain a representation solely consisting of shape proxies. We demonstrate that the algorithm is robust even in the presence of many outliers and a high degree of noise. The proposed method scales well with respect to the size of the input point cloud and the number and size of the shapes within the data. Even point sets with several millions of samples are robustly decomposed within less than a minute. Moreover, the algorithm is conceptually simple and easy to implement. Application areas include measurement of physical parameters, scan registration, surface compression, hybrid rendering, shape classification, meshing, simplification, approximation and reverse engineering.

Efficient RANSAC for Point-Cloud Shape Detection

The theory of Markov Decision Processes is the theory of controlled Markov chains. Its origins can be traced back to R. Bellman and L. Shapley in the 1950’s. During the decades of the last century this theory has grown dramatically. It has found applications in various areas like e.g. computer science, engineering, operations research, biology and economics. In this article we give a short introduction to parts of this theory. We treat Markov Decision Processes with finite and infinite time horizon where we will restrict the presentation to the so-called (generalized) negative case. Solution algorithms like Howard’s policy improvement and linear programming are also explained. Various examples show the application of the theory. We treat stochastic linear-quadratic control problems, bandit problems and dividend pay-out problems.

/pdf/markov-decision-processes-3cwgsogfja.pdf

Markov Decision Processes

This paper provides a tutorial and survey of methods for parameterizing surfaces with a view to applications in geometric modelling and computer graphics. We gather various concepts from differential geometry which are relevant to surface mapping and use them to understand the strengths and weaknesses of the many methods for parameterizing piecewise linear surfaces and their relationship to one another.

/pdf/surface-parameterization-a-tutorial-and-survey-3u3e0o47hm.pdf

Surface Parameterization: a Tutorial and Survey

We present the essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem. We emphasize its computer implementation, which will enable interested readers to readily implement the method.

http://arturo.imati.cnr.it/brezzi/papers/hitchhikers-preprint.pdf

The Hitchhiker's Guide to the Virtual Element Method

Multi-armed bandits is a rich, multi-disciplinary area that has been studied since 1933, with a surge of activity in the past 10-15 years. This is the first monograph to provide a textbook like treatment of the subject. The work on multi-armed bandits can be partitioned into a dozen or so directions. Each chapter tackles one line of work, providing a self-contained introduction and pointers for further reading. Introduction to Multi-Armed Bandits concentrates on fundamental ideas and elementary, teachable proofs over the strongest possible results. It emphasizes accessibility of the material; while exposure to machine learning and probability/statistics would certainly help, a standard undergraduate course on algorithms should suffice for background. The first four chapters are devoted to IID rewards with adversarial rewards being covered in the next 3 chapters. Contextual bandits are discussed in a separate chapter before the monograph concludes with connections to economics. Each chapter contains a section on bibliographic notes and further directions. Many of the chapters conclude with some exercises. Introduction to Multi-Armed Bandits provides an accessible treatment for students of a topic that has gained importance in the last decade. Lecturers can use it as a text for an introductory course on the subject.

/pdf/introduction-to-multi-armed-bandits-4xr7k57ndi.pdf

Introduction to Multi-Armed Bandits

We consider the problem of minimizing the uniform norm on $[0, 1]$ over non-zero polynomials $p$ of the form $$p(x) = \sum_{j=0}^n a_jx^j \quad\text{with } |a_j| \le 1,\, a_j \in {\Bbb C},$$ where the modulus of the first non-zero coefficient is at least $\delta > 0$. Essentially sharp bounds are given for this problem. An interesting related result states that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$
 \exp (-c_1 \sqrt{n} ) \le \inf_{0 \ne p \in {\cal F}_n }\|p\|_{[0, 1]} \le \exp (-c_2 \sqrt{n}),$$ for every $n \ge 2$, where ${\cal F}_n$ denotes the set of polynomials of degree at most $n$ with coefficients from $\{-1, 0, 1 \}$. This Chebyshev-type problem is closely related to the question of how many zeros a polynomial from the above classes can have at $1$. We also give essentially sharp bounds for this problem. {\em Inter alia} we prove that there is an absolute constant $c > 0$ such
 that every polynomial $p$ of the form $$p(x) = \sum_{j=0}^n a_j x^j, \quad\text{with } |a_j| \le 1,\,|a_0|=|a_n|=1,\, a_j \in {\Bbb C},$$ has at most $c\sqrt{n}$ real zeros. This improves the old bound $c\sqrt{n\log n}$ given by Schur in 1933, as well as more recent related bounds of Bombieri and Vaaler, and, up to the constant $c$, this is the best possible result. All the analysis rests critically on the key estimate stating that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$|f(0)|^{c_1/a} \le \exp (c_2/a)\|f\|_{[1-a,1]},$$ for every $f \in {\cal S}$ and $a \in (0, 1]$, where $\cal S$ denotes the collection of all analytic functions $f$ on the open unit disk $D := \{ z \in {\Bbb C}: |z| $$|f(z)| \le \frac{1}{1-|z|}\quad \text{for } z \in D.$$

Littlewood-type problems on [0,1]

Methods to recover constant radius rolling ball blends in reverse engineering

Segmenting point clouds and fitting surfaces onto subsets of measured data are crucial elements of reverse engineering algorithms. Due to the variety in data acquisition procedures and the requirements concerning the representation of the model to be created there are several methods to approach this problem. Our current interest is to construct exact, 'watertight' boundary representation models of regular objects, based on laser scanned point clouds with relatively high density and high accuracy. After defining the term 'regular object', a non-iterative algorithm for direct segmentation is presented, where well-known techniques from computer vision are combined with new procedures for processing point and normal vector data. These include the separation of primary surfaces and transition elements, special filtering methods according to planarity and dimensionality and the detection of simple analytic surface types.

Reverse engineering regular objects: simple segmentation and surface fitting procedures

Reverse engineering regular objects: simple segmentation and surface fitting procedures. (Research report of Geometric Modelling Laboratory, GML 1997/3.)

Reconstructing surfaces from a set of unorganised sample points in the 3D space is a very important problem in reverse engineering. Most algorithms first build a triangular mesh to obtain an approximate surface representation. In this paper we describe an algorithm which works by creating and merging local triangular complexes to obtain an unambiguous 2D-manifold triangulation. We use all the given sample points as vertices, which is a natural requirement. Our method is able to handle open boundaries and holes, different geni (for example tori) and unoriented surfaces in a computationally efficient way.

An Algorithm to Triangulate Surfaces in 3D Using Unorganised Point Clouds

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