Similarity and symmetry measures for convex sets based on Minkowski addition

This paper discusses similarity and symmetry measures for convex shapes. Their definition is based on Minkowski addition and the Brunn-Minkowski inequality. All measures considered are invariant under translations; furthermore, they may also be invariant under rotations, multiplications, reflections, or the class of all affine transformations. The examples discussed in this paper allow efficient algorithms if one restricts oneselves to convex polygons. Although it deals exclusively with the 2-dimensional case, many of the theoretical results carry over almost directly to higher-dimensional spaces. Some results obtained in this paper are illustrated by experimental data

Associahedra via spines

An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the associahedron with relevant properties related to the symmetric group and the classical permutahedron. We introduce the spine of a triangulation as its dual tree together with a labeling and an orientation. This notion extends the classical understanding of the associahedron via binary trees, introduces a new perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's original approach, and sheds light upon the combinatorial and geometric properties of the resulting realizations of the associahedron. It also leads to noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction

VLSI Routing for Advanced Technology

Routing is a major step in VLSI design, the design process of complex integrated circuits (commonly known as chips). The basic task in routing is to connect predetermined locations on a chip (pins) with wires which serve as electrical connections. One main challenge in routing for advanced chip technology is the increasing complexity of design rules which reflect manufacturing requirements. In this thesis we investigate various aspects of this challenge. First, we consider polygon decomposition problems in the context of VLSI design rules. We introduce different width notions for polygons which are important for width-dependent design rules in VLSI routing, and we present efficient algorithms for computing width-preserving decompositions of rectilinear polygons into rectangles. Such decompositions are used in routing to allow for fast design rule checking. A main contribution of this thesis is an O(n) time algorithm for computing a decomposition of a simple rectilinear polygon with n vertices into O(n) rectangles, preseverving two-dimensional width. Here the two-dimensional width at a point of the polygon is defined as the edge length of a largest square that contains the point and is contained in the polygon. In order to obtain these results we establish a connection between such decompositions and Voronoi diagrams. Furthermore, we consider implications of multiple patterning and other advanced design rules for VLSI routing. The main contribution in this context is the detailed description of a routing approach which is able to manage such advanced design rules. As a main algorithmic concept we use multi-label shortest paths where certain path properties (which model design rules) can be enforced by defining labels assigned to path vertices and allowing only certain label transitions. The described approach has been implemented in BonnRoute, a VLSI routing tool developed at the Research Institute for Discrete Mathematics, University of Bonn, in cooperation with IBM. We present experimental results confirming that a flow combining BonnRoute and an external cleanup step produces far superior results compared to an industry standard router. In particular, our proposed flow runs more than twice as fast, reduces the via count by more than 20%, the wiring length by more than 10%, and the number of remaining design rule errors by more than 60%. These results obtained by applying our multiple patterning approach to real-world chip instances provided by IBM are another main contribution of this thesis. We note that IBM uses our proposed combined BonnRoute flow as the default tool for signal routing

Calculating the EHZ Capacity of Polytopes

In this thesis we are concerned with symplectic geometry. This rather new field of research gained much interest due to some famous results in the late 20th century, such as the celebrated non-squeezing theorem by Gromov. An intriguing feature of this theorem is, that it is true if and only if there are global symplectic invariants that satisfy certain properties. These global invariants are called symplectic capacities and their existence is a nontrivial fact. By now, several constructions are known but the computation of symplectic capacities has not received much attention. In this thesis, we address this aspect for a certain symplectic capacity, namely the Ekeland-Hofer-Zehnder capacity, which maps a nonnegative number or infinity to every convex body in $\mathbb{R}^{2n}$. To compute the Ekeland-Hofer-Zehnder capacity of sets of the form $K\times T$, where both $K$ and $T$ are convex sets, we use an approach based on Minkowski Billiards. More precisely, we build on an algorithm by Alkoumi and Schlenk, that is formulated for the case where $K$ is a two-dimensional convex set and $T$ is the Euclidean unit ball. On the one hand, we adapt this algorithm to allow convex sets $K$ with arbitrary dimension. On the other hand, we generalize the approach by Alkoumi and Schlenk from the Euclidean setting (where $T$ is the Euclidean unit ball) to the Minkowski setting (where $T$ is an arbitrary convex set). In particular, we consider the case where both $K$ and $T$ are polytopes. Aside from this approach we consider a formulation of the Ekeland-Hofer-Zehnder capacity as a maximization problem, which is due to Abbondandolo and Majer. This maximization problem is a hybrid of a quadratic assignment problem and a quadratic program with non-convex objective function. We employ different optimization techniques and obtain upper and lower bounds on the Ekeland-Hofer-Zehnder capacity of polytopes. Amongst others, we obtain a very good upper bound that is equal to the exact value for many small problem instances at the cost of a rapidly increasing running time

Leaf recognition for accurate plant classification.

Doctor of Philosophy in Computer Science, University of KwaZulu-Natal, Durban 2017.Plants are the most important living organisms on our planet because they are sources of energy and protect our planet against global warming. Botanists were the first scientist to design techniques for plant species recognition using leaves. Although many techniques for plant recognition using leaf images have been proposed in the literature, the precision and the quality of feature descriptors for shape, texture, and color remain the major challenges. This thesis investigates the precision of geometric shape features extraction and improved the determination of the Minimum Bounding Rectangle (MBR). The comparison of the proposed improved MBR determination method to Chaudhuri's method is performed using Mean Absolute Error (MAE) generated by each method on each edge point of the MBR. On the top left point of the determined MBR, Chaudhuri's method has the MAE value of 26.37 and the proposed method has the MAE value of 8.14. This thesis also investigates the use of the Convexity Measure of Polygons for the characterization of the degree of convexity of a given leaf shape. Promising results are obtained when using the Convexity Measure of Polygons combined with other geometric features to characterize leave images, and a classification rate of 92% was obtained with a Multilayer Perceptron Neural Network classifier. After observing the limitations of the Convexity Measure of Polygons, a new shape feature called Convexity Moments of Polygons is presented in this thesis. This new feature has the invariant properties of the Convexity Measure of Polygons, but is more precise because it uses more than one value to characterize the degree of convexity of a given shape. Promising results are obtained when using the Convexity Moments of Polygons combined with other geometric features to characterize the leaf images and a classification rate of 95% was obtained with the Multilayer Perceptron Neural Network classifier. Leaf boundaries carry valuable information that can be used to distinguish between plant species. In this thesis, a new boundary-based shape characterization method called Sinuosity Coefficients is proposed. This method has been used in many fields of science like Geography to describe rivers meandering. The Sinuosity Coefficients is scale and translation invariant. Promising results are obtained when using Sinuosity Coefficients combined with other geometric features to characterize the leaf images, a classification rate of 80% was obtained with the Multilayer Perceptron Neural Network classifier. Finally, this thesis implements a model for plant classification using leaf images, where an input leaf image is described using the Convexity Moments, the Sinuosity Coefficients and the geometric features to generate a feature vector for the recognition of plant species using a Radial Basis Neural Network. With the model designed and implemented the overall classification rate of 97% was obtained