Transformers excel when dealing with sequential data. Generalizing transformer models to geometric domains, such as manifolds, we encounter the problem of not having a well-defined global order. We propose a solution with attention heads following a space-filling curve. As a first experimental example, we present the Spiroformer, a transformer that follows a polar spiral on the $2$-sphere.

Identifying driving maneuvers plays an essential role on-board vehicles to monitor driving and driver states, as well as off-board to train and evaluate machine learning algorithms for automated driving for example. Maneuvers can be characterized by vehicle kinematics or data from its surroundings including other traffic participants. Extracting relevant maneuvers therefore requires analyzing time-series of (i) structured, multi-dimensional kinematic data, and (ii) unstructured, large data samples for video, radar, or LiDAR sensors. However, such data analysis requires scalable and computationally efficient approaches, especially for non-annotated data. In this paper, we are presenting a maneuver detection approach based on two variants of space-filling curves (Z-order and Hilbert) to detect maneuvers when passing roundabouts that do not use GPS data. We systematically evaluate their respective performance by including permutations of selections of kinematic signals at varying frequencies and compare them with two alternative baselines: All manually identified roundabouts, and roundabouts that are marked by geofences. We find that encoding just longitudinal and lateral accelerations sampled at 10Hz using a Hilbert space-filling curve is already successfully identifying roundabout maneuvers, which allows to avoid the use of potentially sensitive signals such as GPS locations to comply with data protection and privacy regulations like GDPR.

The paper presents a strategy for robotic exploration problems using Space-Filling curves (SFC). The region of interest is first tessellated, and the tiles/cells are connected using some SFC. A robot follows the SFC to explore the entire area. However, there could be obstacles that block the systematic movement of the robot. We overcome this problem by providing an evading technique that avoids the blocked tiles while ensuring all the free ones are visited at least once. The proposed strategy is online, implying that prior knowledge of the obstacles is not mandatory. It works for all SFCs, but for the sake of demonstration, we use Hilbert curve. We present the completeness of the algorithm and discuss its desirable properties with examples. We also address the non-uniform coverage problem using our strategy.

This paper deals with the aerial survey of a closed region using the Space-Filling curve, particularly Sierpinski curve. The specified region is triangulated, and the Sierpinski curve is used to explore each smaller triangular region. The entire region may have one or more obstacles. An algorithm is presented which suggests evasive manoeuvre (detour) if an obstacle is detected. The algorithm is online; that is, it does not require prior knowledge of the location of obstacles and can be applied while the robotic system is traversing the designated path. The fractal nature of the Sierpinski curve and simple geometric observations were used to formulate and validate the algorithm. The non-uniform coverage and multiple obstacle problems are also dealt with towards the end.

This paper presents the first classical Convolutional Neural Network (CNN) that can be applied directly to data from unstructured finite element meshes or control volume grids. CNNs have been hugely influential in the areas of image classification and image compression, both of which typically deal with data on structured grids. Unstructured meshes are frequently used to solve partial differential equations and are particularly suitable for problems that require the mesh to conform to complex geometries or for problems that require variable mesh resolution. Central to the approach are space-filling curves, which traverse the nodes or cells of a mesh tracing out a path that is as short as possible (in terms of numbers of edges) and that visits each node or cell exactly once. The space-filling curves (SFCs) are used to find an ordering of the nodes or cells that can transform multi-dimensional solutions on unstructured meshes into a one-dimensional (1D) representation, to which 1D convolutional layers can then be applied. Although developed in two dimensions, the approach is applicable to higher dimensional problems. To demonstrate the approach, the network we choose is a convolutional autoencoder (CAE) although other types of CNN could be used. The approach is tested by applying CAEs to data sets that have been reordered with an SFC. Sparse layers are used at the input and output of the autoencoder, and the use of multiple SFCs is explored. We compare the accuracy of the SFC-based CAE with that of a classical CAE applied to two idealised problems on structured meshes, and then apply the approach to solutions of flow past a cylinder obtained using the finite-element method and an unstructured mesh.

Space-filling curves like the Hilbert-curve, Peano-curve and Z-order map natural or real numbers from a two or higher dimensional space to a one dimensional space preserving locality. They have numerous applications like search structures, computer graphics, numerical simulation, cryptographics and can be used to make various algorithms cache-oblivious. In this paper, we describe some details of the Hilbert-curve. We define the Hilbert-curve in terms of a finite automaton of Mealy-type which determines from the two-dimensional coordinate space the Hilbert order value and vice versa in a logarithmic number of steps. And we define a context-free grammar to generate the whole curve in a time which is linear in the number of generated coordinate/order value pairs, i.e. a constant time per coordinate pair or order value. We also review two different strategies which enable the generation of curves without the usual restriction to square-like grids where the side-length is a power of two. Finally, we elaborate on a few applications, namely matrix multiplication, Cholesky decomposition, the Floyd-Warshall algorithm, k-Means clustering, and the similarity join.

DNA is perceived as a sequence over the letters {A, C, G, T }, the alphabet of nucleotides. This sequence constitutes the code that acts as a blueprint for all processes taking place in a cell. But beyond merely reflecting primary sequence, DNA is a molecule, which implies that DNA assumes spatial structure and shape. The spatial organization of DNA is achieved by integrating ("recruiting") other molecules, the histone proteins, that help to assume the correct spatial configuration. The combination of DNA and helper molecules is called chromatin; the spatial configuration of the chromatin, finally, defines the functional properties of local areas of the DNA [de Graaf and van Steensel, 2013]. Chromatin can assume several function-defining epigenetic states, where states vary along the genome [Ernst et al., 2011]. The key determinant for spatial configuration is the underlying primary DNA sequence: sequential patterns are responsible for recruiting histone proteins and their chemical modifications, which in turn give rise to or even define the chromatin states. The exact configuration of the chromatin and its interplay with the underlying raw DNA sequence are under active research.