Classification results on surfaces in the isotropic 3-space

The isotropic 3-space I^3 which is one of the Cayley--Klein spaces is obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we give several classifications on the surfaces in I^3 with the constant relative curvature (analogue of the Gaussian curvature) and the constant isotropic mean curvature. In particular, we classify the helicoidal surfaces in I^3 with constant curvature and analyze some special curves on these.Comment: 12 pages, 2 figure

Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentzian geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of isotropic spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of pseudo-isotropic space, we also discuss on the distinct choices for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of Lorentz numbers (a complex-like system where the square of the imaginary unit is $+1$). Finally, we also show the possibility of obtaining an isotropic RM frame by rotation of the Frenet frame through the use of Galilean trigonometric functions and dual numbers (a complex-like system where the square of the imaginary unit vanishes).Comment: 2 figures. To appear in "Tamkang Journal of Mathematics

A Computational Model of Visual Anisotropy

Visual anisotropy has been demonstrated in multiple tasks where performance differs between vertical, horizontal, and oblique orientations of the stimuli. We explain some principles of visual anisotropy by anisotropic smoothing, which is based on a variation on Koenderink's approach in [1]. We tested the theory by presenting Gaussian elongated luminance profiles and measuring the perceived orientations by means of an adjustment task. Our framework is based on the smoothing of the image with elliptical Gaussian kernels and it correctly predicted an illusory orientation bias towards the vertical axis. We discuss the scope of the theory in the context of other anisotropies in perception

The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces

In this work, we are interested in the differential geometry of surfaces in simply isotropic $\mathbb{I}^3$ and pseudo-isotropic $\mathbb{I}_{\mathrm{p}}^3$ spaces, which consists of the study of $\mathbb{R}^3$ equipped with a degenerate metric such as $\mathrm{d}s^2=\mathrm{d}x^2\pm\mathrm{d}y^2$. The investigation is based on previous results in the simply isotropic space [B. Pavkovi\'c, Glas. Mat. Ser. III $\mathbf{15}$, 149 (1980); Rad JAZU $\mathbf{450}$, 129 (1990)], which point to the possibility of introducing an isotropic Gauss map taking values on a unit sphere of parabolic type and of defining a shape operator from it, whose determinant and trace give the known relative Gaussian and mean curvatures, respectively. Based on the isotropic Gauss map, a new notion of connection is also introduced, the \emph{relative connection} (\emph{r-connection}, for short). We show that the new curvature tensor in both $\mathbb{I}^3$ and $\mathbb{I}_{\mathrm{p}}^3$ does not vanish identically and is directly related to the relative Gaussian curvature. We also compute the Gauss and Codazzi-Mainardi equations for the $r$-connection and show that $r$-geodesics on spheres of parabolic type are obtained via intersections with planes passing through their center (focus). Finally, we show that admissible pseudo-isotropic surfaces are timelike and that their shape operator may fail to be diagonalizable, in analogy to Lorentzian geometry. We also prove that the only totally umbilical surfaces in $\mathbb{I}_{\mathrm{p}}^3$ are planes and spheres of parabolic type and that, in contrast to the $r$-connection, the curvature tensor associated with the isotropic Levi-Civita connection vanishes identically for $any$ pseudo-isotropic surface, as also happens in simply isotropic space.Comment: 18 pages in the published versio

The Atlas Structure of Images

Many operations of vision require image regions to be isolated and inter-related. This is challenging when they are different in detail and extent. Practical methods of Computer Vision approach this through the tools of downsampling, pyramids, cropping and patches. In this paper we develop an ideal geometric structure for this, compatible with the existing scale space model of image measurement. Its elements are apertures which view the image like fuzzy-edged portholes of frosted glass. We establish containment and cause/effect relations between apertures, and show that these link them into cross-scale atlases. Atlases formed of Gaussian apertures are shown to be a continuous version of the image pyramid used in Computer Vision, and allow various types of image description to naturally be expressed within their framework. We show that views through Gaussian apertures are approximately equivalent to the jets of derivative of Gaussian filter responses that form part of standard Scale Space theory. This supports a view of the simple cells of mammalian V1 as implementing a system of local views of the retinal image of varying extent and resolution. As a worked example we develop a keypoint descriptor scheme that outperforms previous schemes that do not make use of learning

Differential geometry of invariant surfaces in simply isotropic and pseudo-isotropic spaces

We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a three-dimensional space equipped with a rank 2 metric of index zero and one, respectively. We show that the one-parameter subgroups of isotropic rigid motions lead to seven types of invariant surfaces, which then generalizes the study of revolution and helicoidal surfaces in Euclidean and Lorentzian spaces to the context of singular metrics. After computing the two fundamental forms of these surfaces and their Gaussian and mean curvatures, we solve the corresponding problem of prescribed curvature for invariant surfaces whose generating curves lie on a plane containing the degenerated direction

Fast, Illumination Insensitive Face Detection Based on Multilinear Techniques and Curvature Features

Abstrac