Maximizing the Number of Visible Labels on a Rotating Map

For a map that can be rotated, we consider the following problem. There are a number of feature points on the map, each having a geometric object as a label. The goal is to find the largest subset of these labels such that when the map is rotated and the labels remain vertical, no two labels in the subset intersect. We show that, even if the labels are vertical bars of zero width, this problem remains NP-hard, and present a polynomial approximation scheme for solving it. We also introduce a new variant of the problem for vertical labels of zero width, in which any label that does not appear in the output must be coalesced with a label that does. Coalescing a subset of the labels means to choose a representative among them and set its label height to the sum of the individual label heights.

Kinser Inequalities and Related Matroids

<p>Kinser developed a hierarchy of inequalities dealing with the dimensions of certain spaces constructed from a given quantity of subspaces. These inequalities can be applied to the rank function of a matroid, a geometric object concerned with dependencies of subsets of a ground set. A matroid which is representable by a matrix with entries from some finite field must satisfy each of the Kinser inequalities. We provide results on the matroids which satisfy each inequality and the structure of the hierarchy of such matroids.</p>

Kinser Inequalities and Related Matroids

<p>Kinser developed a hierarchy of inequalities dealing with the dimensions of certain spaces constructed from a given quantity of subspaces. These inequalities can be applied to the rank function of a matroid, a geometric object concerned with dependencies of subsets of a ground set. A matroid which is representable by a matrix with entries from some finite field must satisfy each of the Kinser inequalities. We provide results on the matroids which satisfy each inequality and the structure of the hierarchy of such matroids.</p>

Image classification and cognition using contour curvature statistics

Although the higher order mechanisms behind object representation and classification in the visual system are still not well understood, there are hints that simple shape primitives such as “curviness” might activate neural activation and guide this process. Drawing on elementary invariance principles, we propose that a statistical geometric object, the probability distribution of the normalized contour curvatures (NCC) in the intensity field of a planar image, has the potential to represent and classify categories of objects. We show that NCC is sufficient for discriminating between cognitive categories such as animacy, size and type, and demonstrate the robustness of this metric to variation in illumination and viewpoint, consistent with neurobiological constraints and psychological experiments. A generative model for producing artificial images with the observed NCC distributions highlights the key features that our metric captures and just as importantly, those that it does not. More broadly, our study points to the need for statistical geometric approaches to cognition that build in both the statistics and the natural invariances of the sensory world.

An Accelerated Slicing Algorithm for Frep Models

Complex 3D objects with microstructures can be modelled using the function representation (FRep) approach and then manufactured. The task of modelling a geometric object with a sophisticated microstructure based on unit cell repetition is often too computationally expensive for CAD systems. FRep provides efficient tools to solve this problem. However, even for FRep the slicing step required for manufacturing can take a significant amount of time. An accelerated slicing algorithm for FRep 3D objects is proposed in this paper. This algorithm allows the preparation of FRep models for 3D printing without surface generation stage. The spatial index is employed to accelerate the slicing process. A novel compound adaptive criterion and a novel acceleration criterion are proposed to speed up the evaluation of the defining function of an FRep object. The use of these criteria is significantly reducing the computational time for contour construction during the slicing process. The k-d tree and R-tree data structures are used as spatial indexes. The performance of the accelerated slicing algorithm was tested. The contouring time was reduced 100-fold due to using the novel compound adaptive criterion with the novel acceleration criterion.

RESEARCH OF THE METHOD OF INCREASING THE OBJECT DETERMINATION ACCURACY ON THE LOW-RESOLUTION VIDEO STREAM

Study subject. The article proposes and investigates a method for increasing the accuracy of determination of the distance and the obstacle geometric parameters based on object contours determination using a computer vision system that uses low-resolution sensors. The goal is the effectiveness evaluation of the proposed method. Tasks: to conduct experimental researches of the quality indicators of the method of increasing the object contours determination accuracy; evaluate the effectiveness of this method. Used methods: statistical modeling, laboratory scale tests. The obtained results: the analysis of the proposed method efficiency was carried out and the influence of this method on the determination accuracy of the distance and object geometric parameters was evaluated. Conclusions: the considered method made it possible to achieve the increasing the determination accuracy of the distance and geometric object parameters by compensating for image blur using the Lucy-Richardson deconvolution algorithm. The obtained data showed a decrease in the maximum error in determining the distance from 8% to 4% and the error in the geometric object parameters from 7.7% to 5.8%. The implementation of this approach was carried out in the Python programming language.

Finite Element Analysis and Its Applications in Dentistry

Finite Element Analysis or Finite Element Method is based on the principle of dividing a structure into a finite number of small elements. It is a sophisticated engineering tool, which has been used extensively in design optimization and structural analysis first originated in the aerospace industry to study stress in complex airframe structures. This method is a way of getting a numerical solution to a specific problem, used to analyze stresses and strains in complex mechanical systems. It enables the mathematical conversion and analysis of mechanical properties of a geometric object with wide range of applications in dental and oral health science. It is useful for specifying predominantly the mechanical aspects of biomaterials and human tissues that cannot be measured in vivo. It has various advantages, can be compared with studies on real models, and the tests are repeatable, with accuracy and without ethical concerns.

Robust coding with spiking networks: a geometric perspective

The interactions of large groups of spiking neurons have been difficult to understand or visualise. Using simple geometric pictures, we here illustrate the spike-by-spike dynamics of networks based on efficient spike coding, and we highlight the conditions under which they can preserve their function against various perturbations. We show that their dynamics are confined to a small geometric object, a ‘convex polytope’, in an abstract error space. Changes in network parameters (such as number of neurons, dimensionality of the inputs, firing thresholds, synaptic weights, or transmission delays) can all be understood as deformations of this polytope. Using these insights, we show that the core functionality of these network models, just like their biological counterparts, is preserved as long as perturbations do not destroy the shape of the geometric object. We suggest that this single principle—efficient spike coding—may be key to understanding the robustness of neural systems at the circuit level.

Squares and associative representations of two-dimensional evolution algebras

We associate a square to any two-dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behavior of the algebra. We determine the identities of degrees at most four, as well as derivations and automorphisms. We look at the group of automorphisms as an algebraic group, getting in this form a new algebraic invariant. The study of associative representations of evolution algebras is also started and we get faithful representations for most two-dimensional evolution algebras. In some cases, we prove that faithful commutative and associative representations do not exist, giving rise to the class of what could be termed as “exceptional” evolution algebras (in the sense of not admitting a monomorphism to an associative algebra with deformed product).