On a class of projective Ricci curvature of Finsler metrics

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we investigate the projective Ricci curvature of a class of general [Formula: see text]-metrics satisfying a certain condition, which is invariant under the change of volume form. Moreover, we construct a class of new nontrivial examples on such Finsler metrics.

A REVIEW OF THE ONE-PARAMETER DIVISION UNDISTORTION MODEL

The one-parameter division undistortion model by (Lenz, 1987) and (Fitzgibbon, 2001) is a simple radial distortion model with beneficial algebraic properties that allows to reason about some problems analytically that can only be handled numerically in other distortion models. One property of this distortion model is that straight lines in the undistorted image correspond to circles in the distorted image. These circles are fully described by their center point, as the radius can be calculated from the position of the center and the distortion parameter only. This publication collects the properties of this distortion model from several sources and reviews them. Moreover, we show in this publication that the space of this center is projectively isomorphic to the dual space of the undistorted image plane, i.e. its line space. Therefore, projective invariant measurements on the undistorted lines are possible by the according measurements on the centers of the distorted circles. As an example of application, we use this to find the metric distance of two parallel straight rails with known track gauge in a single uncalibrated camera image with significant radial distortion.

Fast Aerial Image Geolocalization Using the Projective-Invariant Contour Feature

We address the problem of aerial image geolocalization over an area as large as a whole city through road network matching, which is modeled as a 2D point set registration problem under the 2D projective transformation and solved in a two-stage manner. In the first stage, all the potential transformations aligning the query road point set to the reference road point set are found by local point feature matching. A local geometric feature, called the Projective-Invariant Contour Feature (PICF), which consists of a road intersection and the closest points to it in each direction, is specifically designed. We prove that the proposed PICF is equivariant under the 2D projective transformation group. We then encode the PICF with a projective-invariant descriptor to enable the fast search of potential correspondences. The bad correspondences are then removed by a geometric consistency check with the graph-cut algorithm effectively. In the second stage, a flexible strategy is developed to recover the homography transformation with all the PICF correspondences with the Random Sample Consensus (RANSAC) method or to recover the transformation with only one correspondence and then refine it with the local-to-global Iterative Closest Point (ICP) algorithm when only a few correspondences exist. The strategy makes our method efficient to deal with both scenes where roads are sparse and scenes where roads are dense. The refined transformations are then verified with alignment accuracy to determine whether they are accepted as correct. Experimental results show that our method runs faster and greatly improves the recall compared with the state-of-the-art methods.

New classes of projectively related Finsler metrics of constant flag curvature

We define a Weyl-type curvature tensor of [Formula: see text]-type to provide a characterization for Finsler metrics of constant flag curvature. This Weyl-type curvature tensor is projective invariant only to projective factors that are Hamel functions. Based on this aspect, we construct new families of projectively related Finsler metrics that have constant flag curvature.

Projective Ricci flat spherically symmetric Finsler metrics

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we characterize projective Ricci flat spherically symmetric Finsler metrics. Under a certain condition, we prove that a projective Ricci flat spherically symmetric Finsler metric must be a Douglas metric. Moreover, we construct a class of new nontrivial examples on projective Ricci flat Finsler metrics.