How many three-dimensional Hilbert curves are there

Hilbert's space-filling curve is a continuous function that maps the unit interval onto the unit square. The construction of such curves in the 1890s surprised mathematicians of the time and led, in part, to the development of dimension theory. In this note, we discuss how modem notions of self-similarity illuminate the structure of this curve. In particular, we show that Hilbert's curve has a basic self-similar structure and can be generated using what is called an iterated function system, or IFS. Furthermore, its coordinate functions display a generalized type of self-similarity called digraph self-affinity and may be described using an appropriately generalized iterated function system. The notions of self-similarity used here are described in the text by Edgar [2, Ch. 4]. The definition of a digraph IFS was originally formulated in a research paper by Mauldin and Williams [4], although similar ideas have appeared elsewhere. The author has published a Mathematica package implementing the digraph IFS scheme [6]. A broad introduction to space-filling curves may be found in the book by Sagan [7].

To analyze multidimensional images we need a mapping of feature vectors from a multidimensional space to a lower dimensional space. In general, these are performed using linear transformation methods, such as principal component analysis, etc. Linear transformation requires many rotations of data from several points of view because the mapping is not one‐to‐one. Here, a new interactive method for classifying multispectral images using a Hilbert curve is presented. The Hilbert curve is a one‐to‐one mapping from N‐dimensional space to one‐dimensional space and preserves the neighborhood as much as possible. Hilbert curve is a kind of space filling curves, and provides a continuous scan. The merit of the system presented is that the user can extract category clusters without computing any distance in N‐dimensional space easily. The method presented here is explained in brief. Clusters are extracted from 1‐D data mapped by a Hilbert curve interactively, i.e., a pixel is classified as a category. The user can analyze multidimensional images hierarchically from gross data distribution to fine data distribution. To realize the real time response from the system, data tables storing the addresses and the occurrences of data are used. Here, the address is defined by using the coordinates in N‐dimensional space, and a part of mapping which cannot preserve the neighborhood is utilized. In the experiments ex‐extracting categories from LANDSAT data, it is confirmed that the user can obtain the real time response from the system after once making the data tables.

Advances in metamaterials, additive manufacturing, and computational design have enabled significant progress in airborne sound absorption. However, most absorbers still rely on two conventional mechanisms, i.e., friction-induced viscous loss and pressure-fluctuation-induced thermal loss, leaving their thickness constrained by the Rozanov bound. Here, we investigate an acoustic metamaterial based on a single three-dimensional Hilbert curve, designed to achieve sound absorption beyond this theoretical limit. The Hilbert curve forms continuous, tightly spaced winding channels whose geometry enables laminar flow-like shear dissipation. Experimental, numerical, and theoretical results demonstrate that with an optimal slit width close to the viscous boundary layer thickness, the actual thickness can be approximately 86% of the calculated Rozanov bound. This finding suggests a dissipation mechanism distinct from classical porous and resonator-based absorbers, opening different avenues for ultra-thin broadband sound absorbers.

Horizontal and vertical lines hold significant aesthetic and psychological importance, providing a sense of order, stability, and security. This paper presents an image stylization method that quickly generates non‐self‐intersecting and regular continuous lines based on the Hilbert curve, a well‐known space‐filling curve consisting of only horizontal and vertical segments. We first calculate the grayscale threshold based on gray quantization for the original image and recursively subdivide the cells according to the density in each cell. To avoid generating new feature curves due to limited gray quantization, a recursive subdivision with probability is designed to smooth the density. Then, we utilize the rule of Hilbert curve to generate continuous lines connecting all the cells. Between different degrees of Hilbert curves, bridge curves composed of horizontal and vertical lines are constructed, which are also intersection‐free, instead of a straight line linking them directly. There are two parameters provided for feasibly adjusting variate effects. The image stylization framework could be generalized to other space‐filling curves like the Peano curve. Compared to existing methods, our approach can generate pleasing results quickly and is fully automated. Many results show our method is robust and effective.

A Space Filling Curve (SFC) has been proposed, which allows for exchange among departments and offers increased flexibility to the optimization process, as a facility layout technique which can treat any departments of unequal area. However little attention has been paid to the department shape in spite of the fact that SFC determines the department shape. The Hilbert Curve (HC) is used as the SFC, but it gives poor estimation of department shape and is not applicable for use in some rectangular buildings. A new SFC is thus developed, which offers better estimation of department shape than SFC based on HC, and applicability for any rectangular building. Numerical results are given to demonstrate the effectiveness of the presented SFC.

An image encryption scheme based on an improved Hilbert curve and a novel 4D hyper-chaotic system

The 2-dimensional Hilbert scan (HS) is a one-to-one mapping between 2-dimensional (2-D) space and one-dimensional (1-D) space along the 2-D Hilbert curve. Because Hilbert curve can preserve the spatial relationships of the patterns effectively, 2-D HS has been studied in digital image processing actively, such as compressing image data, pattern recognition, clustering an image, etc. However, the existing HS algorithms have some strict restrictions when they are implemented. For example, the most algorithms use recursive function to generate the Hilbert curve, which makes the algorithms complex and takes time to compute the one-to-one correspondence. And some even request the sides of the scanned rectangle region must be a power of two, that limits the application scope of HS greatly. Thus, in order to improve HS to be proper to real-time processing and general application, we proposed a Pseudo-Hilbert scan (PHS) based on the look-up table method for arbitrarily-sized arrays in this paper. Experimental results for both HS and PHS indicate that the proposed generalized Hilbert scan algorithm also reserves the good property of HS that the curve preserves point neighborhoods as much as possible, and gives competitive performance in comparison with Raster scan.

We have devised and implemented a tessellation algorithm with two typical space filling curves, Hilbert curve and Sierpinski curve, and applied the tessellation algorithm into the construction of 3D fractal models. In this paper we shall describe our tessellation algorithm with Hilbert curve and Sierpinski curve, and provide the method how to map the two tiled planes on the surface of models. Some result of verification experiments will be provided also.

DHT networks offer a scalable structure for use in massively multi-user virtual environments (MMVEs). However, an issue with DHT structures is their use of uniform location-independent ID assignment. This conflicts with the locality-sensitive non-uniform ID assignment needed to achieve efficient latency-aware routing in MMVE applications. Our proposed solution is to use a modified version of the Hilbert space-filling curve in order to map users¿ locations from three dimensional VE space into single dimensional DHT ID space in the best locality preserving way. The need for such modified Hilbert curve is due to the fact that users are not typically homogeneously distributed in MMVEs while Hilbert curves fill space in a uniform manner. Generation of, and mapping to, an entire Hilbert curve can be very expensive in terms of computational complexity. Thus we present a fast dynamic mapping of MMVE locations to the modified Hilbert curve to reduce the mapping complexity.

An efficient trajectory design for mobile sink in a wireless sensor network

We present the design of pService system for P2P Web services discovery supporting similarity search with its locality-preserving feature based on modified Skip Graph. In pService, the most important property (e.g. service name) of a service is viewed as the indexing key of a peer node. The other properties are extracted as multiattributes which would be mapped into one-dimensional indices using HSFC (Hilbert Space Filling Curve), and assigned to membership vector of the peer node. These two features help to support similarity search in two aspects: peer nodes with similar indexing keys would be distributed neighborly, peer nodes with similar indices will connect to each other through routing table. Our evaluation showed that the pService performs considerable Web services discovery efficiency.

Efficiently querying large trajectory datasets is a challenge of growing importance. Abstracting trajectory segments with minimum bounding boxes and indexing them in R-Trees results in a high false positive rate due to high dead space. Space filling curves (SFCs), which have excellent locality preserving and dimensionality reduction properties, have been shown to be effective for indexing points in space. However, they can yield a high false positive count and slow query times if used to index trajectory segments. Our work shows how to use SFCs to index trajectory polylines. In our experiments, the proposed method runs 2--15 times faster than other state-of-the-art approaches.

In the novel area of membrane computing, the bio-inspired computing models with the generic name of P systems have turned out to be a convenient framework for handling different kinds of problems and for developing suitable solutions. One such problem area is generation of geometric patterns of approximations of space-filling curves encoded as words over appropriate chain code symbols. Parallel chain code P systems with rewriting in parallel of words in the regions using context-free rules have been shown to generate the languages of chain code words representing the finite approximation patterns of the well-known space-filling Peano and Hilbert curves. Here we consider the approximating polygons that converge to the space-filling curves of Hilbert and construct a parallel chain code P system generating these approximating polygons. We also construct a parallel chain code P system for generating the approximating polygons corresponding to another space-filling curve, known as Lebesgue’s curve, which is almost everywhere differentiable unlike any other space-filling curve.

An efficient algorithm for the generation of Hilbert's space-filling curve is given. The algorithm implements a recursive procedure that involves simple integer operations and quickly converges to the set of points that make the Hilbert curve. The algorithm is elegant, short, and considerably easier to implement than previous recursive and nonrecursive algorithms and can be efficiently implemented in all programming languages that have integer operations and allow recursion. The fundamental Hilbert shape (a line joining the four corners of a square) is represented by two variables with values of either 0 or 1. This coding technique could be successfully applied to the generation of other regular space-filling curves, such as the Peano curve.

An algorithm for generating Hilbert's space-filling curve in a byte-oriented manner is presented. In the context of one application of space-filling curves, the algorithm may be modified so that the results are correct for continua rather than for quantized spaces.