Carlos Velarde

A formal language approach for a 3D curve representation

Abstract A formal language approach for representing three-dimensional (3D) curves is presented. Based on the chain code for representing 3D curves defined in [1], we propose an approach for mapping 3D curves into strings. This mapping allows us to have a unique curve descriptor, which is invariant under translation and rotation. Also, it is possible to use inverse and mirroring operators and the use of formal language techniques for 3D curve generation and analysis. Finally, we present a result of this approach to represent and to generate polygonal sequences convergent to cube-filling Hilbert curves.

Entropies for severely contracted configuration space

We demonstrate that dual entropy expressions of the Tsallis type apply naturally to statistical–mechanical systems that experience an exceptional contraction of their configuration space. The entropic index α>1 describes the contraction process, while the dual index α′=2−α<1 defines the contraction dimension at which extensivity is restored. We study this circumstance along the three routes to chaos in low-dimensional nonlinear maps where the attractors at the transitions, between regular and chaotic behavior, drive phase-space contraction for ensembles of trajectories. We illustrate this circumstance for properties of systems that find descriptions in terms of nonlinear maps. These are size-rank functions, urbanization and similar processes, and settings where frequency locking takes place.

Mirror symmetry detection in curves represented by means of the Slope Chain Code

Abstract Symmetry is an important feature in natural and man-made objects; particularly, mirror symmetry is a relevant task in fields such as computer vision and pattern recognition. In the current work, we propose a new method to characterize mirror-symmetry in open and closed curves represented by means of the Slope Chain Code. This representation is invariant under scale, rotation, and translation, highly desirable properties for object recognition applications. The proposed method detects symmetries through simple inversion, concatenation and reflection operations on the chains, thus allowing the classification of symmetrical and asymmetrical contours. It also introduces a measure to quantify the degree of symmetry in quasi-mirror-symmetrical objects. Furthermore, it allows the identification of multiple symmetry axes and their location. Results show high performances in symmetrical/asymmetrical classification (0.9 recall, 0.9 accuracy, 0.97 precision) and axes’ detection (0.8 recall, 0.84 accuracy, 0.99 precision). Compared to other methods, the proposed algorithm provides properties such as: global, local, and multiple axes’ detection, as well as the capability to classify symmetrical objects, which makes it adequate for several practical applications, like the three exemplified in the paper.

Rank distributions: Frequency vs. magnitude

We examine the relationship between two different types of ranked data, frequencies and magnitudes. We consider data that can be sorted out either way, through numbers of occurrences or size of the measures, as it is the case, say, of moon craters, earthquakes, billionaires, etc. We indicate that these two types of distributions are functional inverses of each other, and specify this link, first in terms of the assumed parent probability distribution that generates the data samples, and then in terms of an analog (deterministic) nonlinear iterated map that reproduces them. For the particular case of hyperbolic decay with rank the distributions are identical, that is, the classical Zipf plot, a pure power law. But their difference is largest when one displays logarithmic decay and its counterpart shows the inverse exponential decay, as it is the case of Benford law, or viceversa. For all intermediate decay rates generic differences appear not only between the power-law exponents for the midway rank decline but also for small and large rank. We extend the theoretical framework to include thermodynamic and statistical-mechanical concepts, such as entropies and configuration.

Pascal (Yang Hui) triangles and power laws in the logistic map

We point out the joint occurrence of Pascal triangle patterns and power-law scaling in the standard logistic map, or more generally, in unimodal maps. It is known that these features are present in its two types of bifurcation cascades: period and chaotic- band doubling of attractors. Approximate Pascal triangles are exhibited by the sets of lengths of supercycle diameters and by the sets of widths of opening bands. Additionally, power-law scaling manifests along periodic attractor supercycle positions and chaotic band splitting points. Consequently, the attractor at the mutual accumulation point of the doubling cascades, the onset of chaos, displays both Gaussian and power-law distributions. Their combined existence implies both ordinary and exceptional statistical-mechanical descriptions of dynamical properties.

An Algorithm for Generating a Family of Alternating Knots

An algorithm for generating a family of alternating knots (which are described by means of a chain code) is presented. The family of alternating knots is represented on the cubic lattice, that is, each alternating knot is composed of constant orthogonal straight-line segments and is described by means of a chain code. This chain code is represented by a numerical string of finite length over a finite alphabet, allowing the usage of formal-language techniques for alternating-knot representation. When an alternating knot is described by a chain, it is possible to obtain its mirroring image in an easy way. Also, we have a compression efficiency for representing alternating knots, because chain codes preserve information and allow a considerable data reduction.