Rectangular Maps Research Articles

End-to-End One-Shot Path-Planning Algorithm for an Autonomous Vehicle Based on a Convolutional Neural Network Considering Traversability Cost.

Path planning plays an important role in navigation and motion planning for robotics and automated driving applications. Most existing methods use iterative frameworks to calculate and plan the optimal path from the starting point to the endpoint. Iterative planning algorithms can be slow on large maps or long paths. This work introduces an end-to-end path-planning algorithm based on a fully convolutional neural network (FCNN) for grid maps with the concept of the traversability cost, and this trains a general path-planning model for 10 × 10 to 80 × 80 square and rectangular maps. The algorithm outputs the lowest-cost path while considering the cost and the shortest path without considering the cost. The FCNN model analyzes the grid map information and outputs two probability maps, which show the probability of each point in the lowest-cost path and the shortest path. Based on the probability maps, the actual optimal path is reconstructed by using the highest probability method. The proposed method has superior speed advantages over traditional algorithms. On test maps of different sizes and shapes, for the lowest-cost path and the shortest path, the average optimal rates were 72.7% and 78.2%, the average success rates were 95.1% and 92.5%, and the average length rates were 1.04 and 1.03, respectively.

Interpretation of a trained neural network based on genetic algorithms

Introduction: Artificial intelligence development strategy involves the use of deep machine learning algorithms in order to solve various problems. Neural network models trained on specific data sets are difficult to interpret, which is due to the “black box” approach when knowledge is formed as a set of interneuronal connection weights. Purpose: Development of a discrete knowledge model which explicitly represents information processing patterns encoded by connections between neurons. Methods: Adaptive quantization of a feature space using a genetic algorithm, and construction of a discrete model for a multidimensional OLAP cube with binary measures. Results: A genetic algorithm extracts a discrete knowledge carrier from a trained neural network. An individual's chromosome encodes a combination of values of all quantization levels for the measurable object properties. The head gene group defines the feature space structure, while the other genes are responsible for setting up the quantization of a multidimensional space, where each gene is responsible for one quantization threshold for a given variable. A discrete model of a multidimensional OLAP cube with binary measures explicitly represents the relationships between combinations of object feature values and classes. Practical relevance: For neural network prediction models based on a training sample, genetic algorithms make it possible to find the effective value of the feature space volume for the combinations of input feature values not represented in the training sample whose volume is usually limited. The proposed discrete model builds unique images of each class based on rectangular maps which use a mesh structure of gradations. The maps reflect the most significant integral indicators of classes that determine the location and size of a class in a multidimensional space. Based on a convolution of the constructed class images, a complete system of production decision rules is recorded for the preset feature gradations.

Sinkhorn–Knopp theorem for rectangular positive maps

ABSTRACTIn this work, we adapt Sinkhorn–Knopp theorem for rectangular positive maps. We extend their concepts of support and total support to these maps. We show that a positive map is equivalent to a doubly stochastic map if and only if is equivalent to a positive map with total support. Moreover, if k and m are coprime then support is sufficient for the equivalence with a doubly stochastic map. This result provides a necessary and sufficient condition for the filter normal form, which is commonly used in Quantum Information Theory. Let be a state and be the positive map . We show that A can be put in the filter normal form if and only if is equivalent to a positive map with total support. We prove that any state such that , if k=m, and , if , can be put in the filter normal form.

On Mathematical Optimization for the visualization of frequencies and adjacencies as rectangular maps

In this paper, we address the problem of visualizing a frequency distribution and an adjacency relation attached to a set of individuals. We represent this information using a rectangular map, i.e., a subdivision of a rectangle into rectangular portions so that each portion is associated with one individual, their areas reflect the frequencies, and the adjacencies between portions represent the adjacencies between the individuals. Due to the impossibility of satisfying both area and adjacency requirements, our aim is to fit as well as possible the areas, while representing as many adjacent individuals as adjacent rectangular portions as possible and adding as few false adjacencies, i.e., adjacencies between rectangular portions corresponding to non-adjacent individuals, as possible. We formulate this visualization problem as a Mixed Integer Linear Programming (MILP) model. We propose a matheuristic that has this MILP model at its heart. Our experimental results demonstrate that our matheuristic provides rectangular maps with a good fit in both the frequency distribution and the adjacency relation.

Towards analysis and predicting maps of ultraviolet index from experimental astronomical parameters (solar elevation, total ozone level, aerosol index, reflectivity). Artificial neural networks global scale approach

UV radiation is an important problem in climatology, ecology but also has direct effect on human health. A novel method for analysis and prediction of global erythemal UV for clear-sky at noon at any localisation, expressed as the UV index, has been proposed. The supervised artificial neural networks (ANN) were trained using purely experimental astronomical parameters (input: solar elevation, total ozone level, aerosol index, reflectivity and required output: erythemal local noon UV irradiance expressed as the UV index) for all dates from a 3-year representative period (2001–2003) collected by Total Ozone Mapping Spectrometer (TOMS). The input data from the 3-year period provide three sets of 1095 grids, each consisting of 288×180 i.e. 56,764,800 training vectors for total ozone level, aerosol index, reflectivity, while the output data provide only one set of data of the same resolution. The trained network delivers a good long-term representation of the physical problem as it is able to predict clear-sky global UV index maps (UVI for any location and date) with an excellent accuracy close to the detection error (3%, 0.5 unit of UVI). The omission of data on aerosol index (slightly) and reflectivity (highly) deteriorates the quality of UVI prediction (MSPE error 6.4%, 1 unit of UVI) but also confirms the importance of total ozone level for the UVI prediction. The neglect of data on aerosol index and reflectivity evidently removes the inhomogeneities and results in smoother and less reliable UVI maps. Reflectivity plays a very important role in variation of UV radiation level. The results are presented in the form of 2D rectangular maps (WGS-84 projection) of UV index. The neural network approach to UVI forecasting and analysis yields reasonable results and can be considered as an alternative to traditional approaches mainly based on radiative-transfer or regression models.

Raman microspectroscopy of organic inclusions in spodumenes from Nilaw (Nuristan, Afghanistan)

The color varieties of spodumene (green spodumene, kunzite) from Nilaw mine (Nuristan, Afghanistan) have been investigated by microthermometry and Raman spectroscopy analyses. These minerals are rich in primary and secondary fluid inclusions. Measured values of temperature homogenization ( T h) and pressure ( P) for selected fluid-inclusion assemblages (I–IV) FIA in green spodumene and (I–II) FIA in kunzite ranges from 370 to 430 °C, 1.16 to 1.44 kbar and 300 to 334 °C, 0.81 to 1.12 kbar, respectively. The brine content and concentration varies from 4.3 to 6.6 wt.% eq. NaCl. Numerous and diverse mineral phases (quartz, feldspars, mica, beryl, zirconium, apatite, calcite, gypsum) present in this mineral as solid inclusions were studied by Raman microspectroscopy. Raman spectra of selected fluid, organic and solid inclusions were collected as line or rectangular maps and also depth profiles to study their size and contents. There appeared very interesting calcite (156, 283, 711and 1085 cm −1), beryl (324, 397, 686, 1068 and 3610 cm −1), topaz (231, 285, 707, 780 and 910 cm −1) and spodumene (355, 707 and 1073 cm −1) inclusions accompanied by fluid and/or organic inclusions (liquid and gas hydrocarbons) with bands at 2350 cm −1 (CO 2, N 2), 2550 cm −1 (H 2S) and 2900 cm −1 (C 2H 6–CH 3). Some solid inclusions contain carbonaceous matter (D-band at ca. 1320 cm −1 and/or G-band at ca. 1600 cm −1).

Synchrotron infrared microspectroscopy of retinal layers

Viewing a microscopic frozen section of a rat retina with differential interference contrast (DIC) reveals the layered structure. An infrared microspectrometer equipped with all reflecting infinity corrected optics provides such viewing capability just prior to probing individual layers or performing mapping experiments. Three-dimensional functional group images reveal the localized chemical microstructure of the layered tissue. This family of images is constructed from spectra scanned at each location in the area mapped. The excellent detail and sharpness of the transition between layers is achieved using a small projected image plane mask. Experimentation done with enhanced spatial resolution, achievable with the very bright synchrotron source that has no thermal noise and is non-divergent, provides high density (100%) rectangular maps from step sizes of 3 or 5 μm between points in the grid. In addition to the perspective provided by the high spatially resolved chemical images, mapping across layers revealed variation in layer contour and chemical composition complimenting the spectral information obtained by previous microspectroscopy investigations with a thermal (globar) source using a narrow but long mask to obtain spectra for each retinal layer. The mapping also showed consistent composition along layers providing baseline data for future investigation that may involve diseased states. Excellent spectra with 4 cm −1 resolution were extracted from any pixel obtained in the mapping process.

Equal Area World Maps: A Case Study

Maps that show the area of all regions of the earth's surface in their correct proportions, appropriately called equal area maps, are often used by cartographers to display area-based data, such as the extent of rain forests, the range of butterfly migrations, or the access of people in various regions to medical facilities. In this article, we will examine the step-by-step construction of an equal area map first announced by Karl Brandan Mollweide (1774-1825) and commonly used in atlases today. Aesthetically, Mollweide's map, which represents the whole world in an ellipse whose axes are in a 2:1 ratio, reflects the essentially round character of the earth better than rectangular maps. The mathematics involved in the construction requires mainly high school algebra and trigonometry with only a bit of calculus (which can plausibly be avoided if one so desires). All of this material, except for the calculus, was included in a first-year liberal arts math/geography course taught at Villanova University.

The case against rectangular world maps

The case against rectangular world maps

The case against rectangular world maps

The case against rectangular world maps