Related papers: Space-filling Curves for High-performance Data Min…
Different space-filling curves (SFCs) are briefly reviewed in this paper, and a new one is proposed. A century has passed between the inception of this kind of curves, since then they have been found useful in computer science, particularly…
This article describes sixteen different ways to traverse d-dimensional space recursively in a way that is well-defined for any number of dimensions. Each of these traversals has distinct properties that may be beneficial for certain…
Homogeneous Hilbert curves (HHC) in two dimensions are generalized by introducing the construction of the space filling curves from the same affine transformations but using an arbitrary kernel, we call such curves HHCK. The new curves are…
The 2x2 space-filling curve is a type of generalized space-filling curve characterized by a basic unit is in a "U-shape" that traverses a 2x2 grid. In this work, we propose a universal framework for constructing general 2x2 curves where…
We present Neural Space-filling Curves (SFCs), a data-driven approach to infer a context-based scan order for a set of images. Linear ordering of pixels forms the basis for many applications such as video scrambling, compression, and…
The order in which plane-filling curves visit points in the plane can be exploited to design efficient algorithms. Typically, the curves are useful because they preserve locality: points that are close to each other along the curve tend to…
This paper presents a general coding method where data in a Hilbert space are represented by finite dimensional coding vectors. The method is based on empirical risk minimization within a certain class of linear operators, which map the set…
This paper addresses the problem of exploring a region using the Hilbert's space-filling curve in the presence of obstacles. No prior knowledge of the region being explored is assumed. An online algorithm is proposed which can implement…
Space-filling curves can be used to organise points in the plane into bounding-box hierarchies (such as R-trees). We develop measures of the bounding-box quality of space-filling curves that express how effective different space-filling…
In order to be able to process the increasing amount of spatial data, efficient methods for their handling need to be developed. One major challenge for big spatial data is access. This can be achieved through space-filling curves, as they…
Membranes holomorphically embedded in flat noncompact space are constructed in terms of the degrees of freedom of an infinite collection of 0-branes. To each holomorphic curve we associate infinite-dimensional matrices which are static…
Kernel methods in machine learning use a kernel function that takes two data points as input and returns their inner product after mapping them to a Hilbert space, implicitly and without actually computing the mapping. For many kernel…
We describe a search for plane-filling curves traversing all edges of a grid once. The curves are given by Lindenmayer systems with only one non-constant letter. All such curves for small orders on three grids have been found. For all…
Finite plane geometry is associated with finite dimensional Hilbert space. The association allows mapping of q-number Hilbert space observables to the c-number formalism of quantum mechanics in phase space. The mapped entities reflect…
Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes.…
Hilbert order is widely applied in many areas. However, most of the algorithms are confined to low dimensional cases. In this paper, algorithms for encoding and decoding arbitrary dimensional Hilbert order are presented. Eight algorithms…
Semialgebraic splines are functions that are piecewise polynomial with respect to a cell decomposition into sets defined by polynomial inequalities. We study bivariate semialgebraic splines, formulating spaces of semialgebraic splines in…
We study kernel functions, and associated reproducing kernel Hilbert spaces $\mathscr{H}$ over infinite, discrete and countable sets $V$. Numerical analysis builds discrete models (e.g., finite element) for the purpose of finding…
One of the most startling mathematical discoveries of the nineteen century was the existence of plane-filling curves. As is well known, the first example of such a curve was given by the Italian mathematician Giuseppe Peano in 1890.…
A space-filling curve (SFC) maps points in a multi-dimensional space to one-dimensional points by discretizing the multi-dimensional space into cells and imposing a linear order on the cells. This way, an SFC enables the indexing of…