A Characterization of the Complement of the Hyperbolic Quadric in $\mathrm{PG}(3,q)$

Motivated by recent results by M. Mazalov about uniform approximation of functions by solutions of elliptic equations with constant complex coefficients we study two problems on approximation of functions by solutions of general homogeneous elliptic second-order systems of partial differential equations. The approximation is considered in spaces of continuous and -functions on compact sets in the complex plane.

If M M is a complete minimal surface in R n {R^n} , we denote by W W the set of points in R n {R^n} that do not lie on any tangent plane of M M . By taking a point in W W as origin, the position vector of M M determines a global unit normal vector field e e to M M . We prove that if e e is a minimal section, then M M is a plane. In particular, the set of tangent planes of a nonflat complete minimal surface in R 3 {R^3} covers all R 3 {R^3} . We also prove a similar result for a complete minimal surface M M in S 3 {S^3} , and deduce from it that if the spherical image of M M lies in a closed hemisphere, then M M is a great S 2 {S^2} .

Developable surface fitting to point clouds

The purpose of this paper is to study the structure of the sets of tangent vectors and tangent planes to a complex analytic variety, particularly in the neighborhood of singular points of the variety. We prove the existence of a stratification of the variety which has nice properties relative to tangent planes. Corresponding properties of real analytic varieties (which are the real parts of of complex ones) may be found by considering the corresponding complex analytic variety.

The reflected and refractive fields are calculated for a very rough surface which is assumed to be a set of tangent planes with small corrections due to the local finite radii of curvature. A perturbation technique similar to those used in the moderately rough surface calculations is developed to treat the solution for the tangent planes as the unperturbed solution with the curvature correction as the perturbation. Our results are expressed in terms of the slope distribution function of the tangent planes and the average radius of curvature. Numerical results for scattering of both polarizations are obtained. The depolarization calculation is shown to be consistent with the experimental data. Further, the relationship between the surface model and the conventional Gaussian model is discussed, and a comparison with previous works is given.

The matching of stereograms which contain periodic patterns suggests ways in which the stereo correspondence problem may be solved in human vision. The stereograms seem to be segmented by coarse-scale features. Within each segment a set of matches approximating a plane is chosen. In regions with periodic patterns there may be many such planar sets, and the disparity of coarse-scale features seems to guide the choice of a particular set. This emphasis on planarity may reflect the occurrence of correlation-like operations in cortical neurons. An attractive possibility is that segmentation effectively delimits areas of the visual field within which disparities are likely to be slow changing (eg local tangent planes to surfaces) so that the correlation sums evaluated in a segment can give the best estimate of depth. A mechanism of this kind cannot account for all of stereo matching, since not all visual objects are well described by ensembles of planes. But it is likely to be a component of the matching system which is particularly important where images are 'noisy' and averaging is needed to extract reliable disparities.

Geometric properties of the range of two-dimensional quasi-measures with respect to the Radon-Nikodym property

We prove in this paper that, given a nonempty open set G in the complex plane, a subset A of G which is not relatively compact and a holomorphic infinite order differential or antidiffeärential operator T, then there are holomorphic functions ƒ on G such that the image of A under T ƒ is dense in the complex plane. This extends a recent result about a property of boundary behaviour exhibited by the derivative operator.

Let π be an order- q -subplane of P G ( 2 , q 3 ) that is exterior to l ∞ . Then the exterior splash of π is the set of q 2 + q + 1 points on l ∞ that lie on an extended line of π . Exterior splashes are projectively equivalent to scattered linear sets of rank 3, covers of the circle geometry C G ( 3 , q ) , and hyper-reguli in P G ( 5 , q ) . We use the Bruck–Bose representation in P G ( 6 , q ) to investigate the structure of π , and the interaction between π and its exterior splash. We show that the point set of P G ( 6 , q ) corresponding to π is the intersection of nine quadrics, and that there is a unique tangent plane at each point, namely the intersection of the tangent spaces of the nine quadrics. In P G ( 6 , q ) , an exterior splash S has two sets of cover planes (which are hyper-reguli) and we show that each set has three unique transversal lines in the cubic extension P G ( 6 , q 3 ) . These transversal lines are used to characterise the carriers and the sublines of S .

For a single space curve (that is, a smooth curve embedded in ℝ3) much geometrical information is contained in the dual and the focal set of the curve. These are both (singular) surfaces in ℝ3, the dual being a model of the set of all tangent planes to the curve, and the focal set being the locus of centres of spheres having at least 3-point contact with the curve. The local structures of the dual and the focal set are (for a generic curve) determined by viewing them as (respectively) the discriminant of a family derived from the height functions on the curve, and the bifurcation set of the family of distance-squared functions on the curve. For details of this see for example [6, pp. 123–8].

360° cameras can capture complete environments in a single shot, which makes 360° imagery alluring in many computer vision tasks. However, monocular depth estimation remains a challenge for 360° data, particularly for high resolutions like 2K (2048 × 1 024) and beyond that are important for novel-view synthesis and virtual reality applications. Current CNN-based methods do not support such high resolutions due to limited GPU memory. In this work, we propose aflexible framework for monocular depth estimation from high-resolution 360° images using tangent images. We project the 360° input image onto a set of tangent planes that produce perspective views, which are suitable for the latest, most accurate state-of-the-art perspective monocular depth estimators. To achieve globally consistent disparity estimates, we recombine the individual depth estimates using deformable multi-scale alignment followed by gradient-domain blending. The result is a dense, high-resolution 360° depth map with a high level of detail, also for outdoor scenes which are not supported by existing methods. Our source code and data are available at https://manurare.github.io/360monodepth/.

A β-accurate linearization method of Euclidean distance for the facility layout problem with heterogeneous distance metrics

This paper presents a novel method for alignment of geometrically similar 3D models. It is based on the prior that both models have at least one common tangent plane on which both can stand stably and when standing on it the models are partially aligned. To determine the final rotation around the stable plane's normal, needed for a complete alignment, we adapt an image alignment technique based on the log-polar transformation. Because the set of stable planes of a model is small enough, alignment is efficiently approached as a global optimization problem that finds the common stable plane providing the best alignment according to a similarity measure. As the method does not rely on any kind of global symmetry features, we show it can be used to register incomplete stereo point clouds of objects located on a stable plane (table, ground, etc.) with the corresponding similar 3D models. We evaluate the 3D-alignment method by comparing it to the well-known CPCA and show a significant improvement when aligning 120 models belonging to 12 different classes.

If $M$ is a complete minimal surface in ${R^n}$, we denote by $W$ the set of points in ${R^n}$ that do not lie on any tangent plane of $M$. By taking a point in $W$ as origin, the position vector of $M$ determines a global unit normal vector field $e$ to $M$. We prove that if $e$ is a minimal section, then $M$ is a plane. In particular, the set of tangent planes of a nonflat complete minimal surface in ${R^3}$ covers all ${R^3}$. We also prove a similar result for a complete minimal surface $M$ in ${S^3}$, and deduce from it that if the spherical image of $M$ lies in a closed hemisphere, then $M$ is a great ${S^2}$.

Shape approximation algorithms aim at computing simple geometric descriptions of dense surface meshes. Many such algorithms are based on mesh decimation techniques, generating coarse triangulations while optimizing for a particular metric which models the distance to the original shape. This approximation scheme is very efficient when enough polygons are allowed for the simplified model. However, as coarser approximations are reached, the intrinsic piecewise linear point interpolation which defines the decimated geometry fails at capturing even simple structures. We claim that when reaching such extreme simplification levels, highly instrumental in shape analysis, the approximating representation should explicitly and progressively model the volumetric extent of the original shape. In this paper, we propose Sphere-Meshes , a new shape representation designed for extreme approximations and substituting a sphere interpolation for the classic point interpolation of surface meshes. From a technical point-of-view, we propose a new shape approximation algorithm, generating a sphere-mesh at a prescribed level of detail from a classical polygon mesh. We also introduce a new metric to guide this approximation, the Spherical Quadric Error Metric in R 4 , whose minimizer finds the sphere that best approximates a set of tangent planes in the input and which is sensitive to surface orientation, thus distinguishing naturally between the inside and the outside of an object. We evaluate the performance of our algorithm on a collection of models covering a wide range of topological and geometric structures and compare it against alternate methods. Lastly, we propose an application to deformation control where a sphere-mesh hierarchy is used as a convenient rig for altering the input shape interactively.