18 research outputs found
Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures using piecewise log-sum-exp inequalities
Information-theoretic measures such as the entropy, cross-entropy and the
Kullback-Leibler divergence between two mixture models is a core primitive in
many signal processing tasks. Since the Kullback-Leibler divergence of mixtures
provably does not admit a closed-form formula, it is in practice either
estimated using costly Monte-Carlo stochastic integration, approximated, or
bounded using various techniques. We present a fast and generic method that
builds algorithmically closed-form lower and upper bounds on the entropy, the
cross-entropy and the Kullback-Leibler divergence of mixtures. We illustrate
the versatile method by reporting on our experiments for approximating the
Kullback-Leibler divergence between univariate exponential mixtures, Gaussian
mixtures, Rayleigh mixtures, and Gamma mixtures.Comment: 20 pages, 3 figure
Voronoi diagrams in the max-norm: algorithms, implementation, and applications
Voronoi diagrams and their numerous variants are well-established objects in computational geometry. They have proven to be extremely useful to tackle geometric problems in various domains such as VLSI CAD, Computer Graphics, Pattern Recognition, Information Retrieval, etc. In this dissertation, we study generalized Voronoi diagram of line segments as motivated by applications in VLSI Computer Aided Design. Our work has three directions: algorithms, implementation, and applications of the line-segment Voronoi diagrams. Our results are as follows: (1) Algorithms for the farthest Voronoi diagram of line segments in the Lp metric, 1 ≤ p ≤ ∞. Our main interest is the L2 (Euclidean) and the L∞ metric. We first introduce the farthest line-segment hull and its Gaussian map to characterize the regions of the farthest line-segment Voronoi diagram at infinity. We then adapt well-known techniques for the construction of a convex hull to compute the farthest line-segment hull, and therefore, the farthest segment Voronoi diagram. Our approach unifies techniques to compute farthest Voronoi diagrams for points and line segments. (2) The implementation of the L∞ Voronoi diagram of line segments in the Computational Geometry Algorithms Library (CGAL). Our software (approximately 17K lines of C++ code) is built on top of the existing CGAL package on the L2 (Euclidean) Voronoi diagram of line segments. It is accepted and integrated in the upcoming version of the library CGAL-4.7 and will be released in september 2015. We performed the implementation in the L∞ metric because we target applications in VLSI design, where shapes are predominantly rectilinear, and the L∞ segment Voronoi diagram is computationally simpler. (3) The application of our Voronoi software to tackle proximity-related problems in VLSI pattern analysis. In particular, we use the Voronoi diagram to identify critical locations in patterns of VLSI layout, which can be faulty during the printing process of a VLSI chip. We present experiments involving layout pieces that were provided by IBM Research, Zurich. Our Voronoi-based method was able to find all problematic locations in the provided layout pieces, very fast, and without any manual intervention
Geometric aspects of the casting process
Manufacturing is the process of converting raw materials into useful products. Among the most important manufacturing processes, casting is a commonly used manufacturing process for plastic and metal objects. The industrial casting process consists of two stages. First, liquid is filled into a cavity formed by two cast parts. After the liquid has hardened, one cast part retracts, carrying the object with it. Afterwards, the object is ejected from the retracted cast part. In both retraction and ejection steps, the cast parts and the object should not be damaged, so that the quality of final object is guaranteed and the cast parts can be reused to produce another object. This mode of production is called ``mass production''.
Due to the geometric nature of the casting process, many geometric problems arise in the automation of casting. The problems we address here concern this aspect: Given a 3-dimensional object, is there a cast for it whose parts can be removed after the liquid has solidified? An object for which this is the case is called castable.
We consider the castability problem in three different casting models with a two-part cast. In the first casting model, two parts must be removed in opposite directions. There are two cases depending on whether the removal direction is specified in advance or not. The second model is identical to the first casting model, except that the cast machinery has a certain level of uncertainty in its directional movement. In the third model, the two cast parts are to be removed in two given directions and these directions need not be opposite. For all three casting models, we give complete characterizations of castability, and obtain algorithms to verify these conditions for polyhedral parts.
In manufacturing, features of an object imply manufacturing information, which facilitates the process of analyzing manufacturability and the automated design of a cast for the object. A small hole or a depression on the boundary of an object, for example, restricts the set of removal directions for which this object is castable, since the portion of the cast in the hole or in the depression must be removed from the object without breaking the object. We define a geometric feature, the cavity, which is related to the castability of objects, and provide algorithms to extract it from objects
On Geometric Range Searching, Approximate Counting and Depth Problems
In this thesis we deal with problems connected to range searching,
which is one of the central areas of computational geometry.
The dominant problems in this area are
halfspace range searching, simplex range searching and orthogonal range searching and
research into these problems has spanned decades.
For many range searching problems, the best possible
data structures cannot offer fast (i.e., polylogarithmic) query
times if we limit ourselves to near linear storage.
Even worse, it is conjectured (and proved in some cases)
that only very small improvements to these might be possible.
This inefficiency has encouraged many researchers to seek alternatives through approximations.
In this thesis we continue this line of research and focus on
relative approximation of range counting problems.
One important problem where it is possible to achieve significant speedup
through approximation is halfspace range counting in 3D.
Here we continue the previous research done
and obtain the first optimal data structure for approximate halfspace range counting in 3D.
Our data structure has the slight advantage of being Las Vegas (the result is always correct) in contrast
to the previous methods that were Monte Carlo (the correctness holds with high probability).
Another series of problems where approximation can provide us with
substantial speedup comes from robust statistics.
We recognize three problems here:
approximate Tukey depth, regression depth and simplicial depth queries.
In 2D, we obtain an optimal data structure capable of approximating
the regression depth of a query hyperplane.
We also offer a linear space data structure which can answer approximate
Tukey depth queries efficiently in 3D.
These data structures are obtained by applying our ideas for the
approximate halfspace counting problem.
Approximating the simplicial depth turns out to be much more
difficult, however.
Computing the simplicial depth of a given point is more computationally
challenging than most other definitions of data depth.
In 2D we obtain the first data structure which uses near linear space
and can answer approximate simplicial depth queries in polylogarithmic time.
As applications of this result, we provide two non-trivial methods to
approximate the simplicial depth of a given point in higher dimension.
Along the way, we establish a tight combinatorial relationship between
the Tukey depth of any given point and its simplicial depth.
Another problem investigated in this thesis is the dominance reporting problem,
an important special case of orthogonal range reporting.
In three dimensions, we solve this
problem in the pointer machine model and the external memory model
by offering the first optimal data structures in these models of computation.
Also, in the RAM model and for points from
an integer grid we reduce the space complexity of the fastest
known data structure to optimal.
Using known techniques in the literature, we can use our
results to obtain solutions for the orthogonal range searching problem as well.
The query complexity offered by our orthogonal range reporting data structures
match the most efficient query complexities
known in the literature but our space bounds are lower than the previous methods in the external
memory model and RAM model where the input is a subset of an integer grid.
The results also yield improved orthogonal range searching in
higher dimensions (which shows the significance
of the dominance reporting problem).
Intersection searching is a generalization of range searching where
we deal with more complicated geometric objects instead of points.
We investigate the rectilinear disjoint polygon counting problem
which is a specialized intersection counting problem.
We provide a linear-size data structure capable of counting
the number of disjoint rectilinear polygons
intersecting any rectilinear polygon of constant size.
The query time (as well as some other properties of our data structure) resembles
the classical simplex range searching data structures
Abstracts for the twentyfirst European workshop on Computational geometry, Technische Universiteit Eindhoven, The Netherlands, March 9-11, 2005
This volume contains abstracts of the papers presented at the 21st European Workshop on Computational Geometry, held at TU Eindhoven (the Netherlands) on March 9–11, 2005. There were 53 papers presented at the Workshop, covering a wide range of topics. This record number shows that the field of computational geometry is very much alive in Europe. We wish to thank all the authors who submitted papers and presented their work at the workshop. We believe that this has lead to a collection of very interesting abstracts that are both enjoyable and informative for the reader. Finally, we are grateful to TU Eindhoven for their support in organizing the workshop and to the Netherlands Organisation for Scientific Research (NWO) for sponsoring the workshop
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by Erd˝os
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
Detection and identification of elliptical structure arrangements in images: theory and algorithms
Cette thèse porte sur différentes problématiques liées à la détection, l'ajustement et l'identification de structures elliptiques en images. Nous plaçons la détection de primitives géométriques dans le cadre statistique des méthodes a contrario afin d'obtenir un détecteur de segments de droites et d'arcs circulaires/elliptiques sans paramètres et capable de contrôler le nombre de fausses détections. Pour améliorer la précision des primitives détectées, une technique analytique simple d'ajustement de coniques est proposée ; elle combine la distance algébrique et l'orientation du gradient. L'identification d'une configuration de cercles coplanaires en images par une signature discriminante demande normalement la rectification Euclidienne du plan contenant les cercles. Nous proposons une technique efficace de calcul de la signature qui s'affranchit de l'étape de rectification ; elle est fondée exclusivement sur des propriétés invariantes du plan projectif, devenant elle même projectivement invariante. ABSTRACT : This thesis deals with different aspects concerning the detection, fitting, and identification of elliptical features in digital images. We put the geometric feature detection in the a contrario statistical framework in order to obtain a combined parameter-free line segment, circular/elliptical arc detector, which controls the number of false detections. To improve the accuracy of the detected features, especially in cases of occluded circles/ellipses, a simple closed-form technique for conic fitting is introduced, which merges efficiently the algebraic distance with the gradient orientation. Identifying a configuration of coplanar circles in images through a discriminant signature usually requires the Euclidean reconstruction of the plane containing the circles. We propose an efficient signature computation method that bypasses the Euclidean reconstruction; it relies exclusively on invariant properties of the projective plane, being thus itself invariant under perspective
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum