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Classical geometry can be described either in terms of a metric tensor $g_{ab}(x)$ or in terms of the geodesic distance ${\sigma }^2(x,x^{\prime })$. Recent work, however, has shown that the geodesic distance is better suited to describe the quantum structure of spacetime. This is because one can incorporate some of the key quantum effects by replacing ${\sigma }^2$ by another function $S[{\sigma }^2]$ such that $S[0]=L_0^2$ is nonzero. This allows one to introduce a zero-point-length in the spacetime. I show that the geodesic distance can be an emergent construct, arising in the form of a correlator $S[{\sigma }^2(x,y)]=〈J(x)J(y)〉$, of a pregeometric variable $J(x)$, which can be interpreted as the quantum density of spacetime events. This approach also shows why null surfaces play a special role in the interface of quantum theory and gravity. I describe several technical and conceptual aspects of this construction and discuss some of its implications.

In this paper, the sequence of evolving networks is generated from some ‘dust-like’ cubes by applying the encoding methods in fractal and symbolic dynamical systems. Based on the self-similar structures of fractals, we study the mean clustering coefficient, the mean geodesic distance and the mean Fermat distance. The relevant results show the small-world effect of our evolving networks.

Road detection in aerial images is a crucial technique for visual navigation and scene understanding in relation to unmanned aerial vehicles (UAVs). A shape-aware road detection method for aerial images is proposed in this paper. It first employs the stroke width transform (SWT) and a geodesic distance based superpixel clustering to generate proposal regions. Then, a shape classification is responsible for selecting all potential road segments from the proposal regions which appear to be long and with consistent width. All road segments selected are clustered into several groups based on width and color features. A global graph based labeling model is then applied based on each group to remove potential background clutters, as well as to generate the final output. Experiments on two public datasets demonstrate that the proposed method can handle more diverse and challenging road scenes and needs less pre-training, leading to better performance compared to conventional methods.

Depth images, in particular depth maps estimated from stereo vision, may have a substantial amount of outliers and result in inaccurate 3D modelling and reconstruction. To address this challenging issue, in this paper, a graph-cut based multiple depth maps integration approach is proposed to obtain smooth and watertight surfaces. First, confidence maps for the depth images are estimated to suppress noise, based on which reliable patches covering the object surface are determined. These patches are then exploited to estimate the path weight for 3D geodesic distance computation, where an adaptive regional term is introduced to deal with the "shorter-cuts" problem caused by the effect of the minimal surface bias. Finally, the adaptive regional term and the boundary term constructed using patches are combined in the graph-cut framework for more accurate and smoother 3D modelling. We demonstrate the superior performance of our algorithm on the well-known Middlebury multi-view database and additionally on real-world multiple depth images captured by Kinect. The experimental results have shown that our method is able to preserve the object protrusions and details while maintaining surface smoothness.

A novel monitoring strategy is proposed for multimode process in which mode clustering and fault detection based on geodesic distance (GD) are integrated. To start with, the empowered adjacency matrix of normalized training dataset is obtained and improved Dijkstra algorithm (IDA) is utilized to calculate the geodesic distance between each sample data so as to characterize the shortest distance of the nonlinear data within local areas accurately. Next, GD matrix algorithm is presented as an optimal clustering solution for a multimode process dataset. Then, the GDS model is established in each operating mode. Monitoring statistics based on the power of geodesic distance are structured based on square sum of Euclidean distances. Once the test data is detected as fault data, mode location based on deviation coefficient is conducted to narrow the scope of the inspection fault. Finally, the validity and usefulness of the proposed GDMPM monitoring method are demonstrated through the Tennessee Eastman (TE) benchmark process.

We present a novel way to efficiently compute Riemannian geodesic distance over a two- or three-dimensional domain. It is based on a previously presented method for computation of geodesic distances on surface meshes. Our method is adapted for rectangular grids, equipped with a variable anisotropic metric tensor. Processing and visualization of such tensor fields is common in certain applications, for instance structure tensor fields in image analysis and diffusion tensor fields in medical imaging.

The included benchmark study shows that our method provides significantly better results in anisotropic regions in 2-D and 3-D and is faster than current stat-of-the-art solvers in 2-D grids. Additionally, our method is straightforward to code; the test implementation is less than 150 lines of C++ code. The paper is an extension of a previously presented conference paper and includes new sections on 3-D grids in particular.

Measuring the distance is an important task in many clustering and image-segmentation algorithms. The value of the distance decides whether two image points belong to a single or, respectively, to two different image segments. The Euclidean distance is used quite often. In more complicated cases, measuring the distances along the surface that is defined by the image function may be more appropriate. The geodesic distance, i.e. the shortest path in the corresponding graph, has become popular in this context. The problem is that it is determined on the basis of only one path that can be viewed as infinitely thin and that can arise accidentally as a result of imperfections in the image. Considering the k shortest paths can be regarded as an effort towards the measurement of the distance that is more reliable. The drawback remains that measuring the distance along several paths is burdened with the same problems as the original geodesic distance. Therefore, it does not guarantee significantly better results. In addition to this, the approach is computationally demanding. This paper introduces the resistance-geodesic distance with the goal to reduce the possibility of using a false accidental path for measurement. The approach can be briefly characterised in such a way that the path of a certain chosen width is sought for, which is in contrast to the geodesic distance. Firstly, the effective conductance is computed for each pair of the neighbouring nodes to determine the local width of the path that could possibly run through the arc connecting them. The width computed in this way is then used for determining the costs of arcs; the arcs whose use would lead to a small width of the final path are penalised. The usual methods for computing the shortest path in a graph are then used to compute the final distances. The corresponding theory and the experimental results are presented in this paper.

We obtain the average geodesic distance on the Sierpinski carpet in terms of the integral of geodesic distance on self-similar measure. We find out the finite pattern phenomenon of integral inspired by the notion of finite type on self-similar sets with overlaps.

It is of great interest to analyze geodesics in fractals. We investigate the structure of geodesics in $n$-dimensional Sierpinski gasket $F_n$ for $n\ge 3$, and prove that there are at most eight geodesics between any pair of points in $F_n$. Moreover, we obtain that there exists a unique geodesic for almost every pair of points in $F_n$.

This paper concerns the average distances of evolving networks modeled by Sierpinski tetrahedron. We express the limit of average distances on reorganized networks as an integral of geodesic distance on Sierpinski tetrahedron. Based on the self-similarity and renewal theorem, we obtain the asymptotic formula on the average distance of our evolving networks.

For any Lalley–Gatzouras self-affine carpet, we find out that the length of any curves from $(x_1,x_2)$ to $(y_1,y_2)$ in the carpet is no less than $|x_1-y_1|+|x_2-y_2|$.

In this paper, we investigate the equivalence of connectedness for the Sierpinski-like sponge and skeleton networks, and find out the relation between the geodesic distance on the sponge and renormalized shortest path distance on the skeleton networks. Furthermore, under some assumption on the IFS, we obtain the comparability of the Manhattan distance and the geodesic distance on the sponge.

We consider a two-spin system of $XXX$ Heisenberg type submitted to an external magnetic field. Using the associated $ℂP^3$ geometry, we investigate the dynamics of the system. We explicitly give the corresponding Fubini–Study metric. We show that for arbitrary pure initial states, the dynamics occurs on a torus. We compute the geometric phase, the dynamic phase and the topological phase. We investigate the interplay between the torus geometry and the entanglement of the two spins. In this respect, we provide a detailed analysis of the geometric phase, the dynamics velocity and the geodesic distance measured by the Fubini–Study metric in terms of the degree of entanglement between the two spins.

We study the shape of inflated surfaces introduced in [3] and [12]. More precisely, we analyze profiles of surfaces obtained by inflating a convex polyhedron, or more generally an almost everywhere flat surface, with a symmetry plane. We show that such profiles are in a one-parameter family of curves which we describe explicitly as the solutions of a certain differential equation.