A Parallel Method for Open Hole Filling in Large-Scale 3D Automatic Modeling Based on Oblique Photography

Common methods of filling open holes first reaggregate them into closed holes and then use a closed hole filling method to repair them. These methods have problems such as long calculation times, high memory consumption, and difficulties in filling large-area open holes. Hence, this paper proposes a parallel method for open hole filling in large-scale 3D automatic modeling. First, open holes are automatically identified and divided into two categories (internal and external). Second, the hierarchical relationships between the open holes are calculated in accordance with the adjacency relationships between partitioning cells, and the open holes are filled through propagation from the outer level to the inner level with topological closure and height projection transformation. Finally, the common boundaries between adjacent open holes are smoothed based on the Laplacian algorithm to achieve natural transitions between partitioning cells. Oblique photography data from an area of 28 km2 in Dongying, Shandong, were used for validation. The experimental results reveal the following: (i) Compared to the Han method, the proposed approach has a 12.4% higher filling success rate for internal open holes and increases the filling success rate for external open holes from 0% to 100%. (ii) Concerning filling efficiency, the Han method can achieve hole filling only in a small area, whereas with the proposed method, the size of the reconstruction area is not restricted. The time and memory consumption are improved by factors of approximately 4–5 and 7–21, respectively. (iii) In terms of filling accuracy, the two methods are basically the same.

Closure System and Its Semantics

It is well known that topological spaces are axiomatically characterized by the topological closure operator satisfying the Kuratowski Closure Axioms. Equivalently, they can be axiomatized by other set operators encoding primitive semantics of topology, such as interior operator, exterior operator, boundary operator, or derived-set operator (or dually, co-derived-set operator). It is also known that a topological closure operator (and dually, a topological interior operator) can be weakened into generalized closure (interior) systems. What about boundary operator, exterior operator, and derived-set (and co-derived-set) operator in the weakened systems? Our paper completely answers this question by showing that the above six set operators can all be weakened (from their topological counterparts) in an appropriate way such that their inter-relationships remain essentially the same as in topological systems. Moreover, we show that the semantics of an interior point, an exterior point, a boundary point, an accumulation point, a co-accumulation point, an isolated point, a repelling point, etc. with respect to a given set, can be extended to an arbitrary subset system simply by treating the subset system as a base of a generalized interior system (and hence its dual, a generalized closure system). This allows us to extend topological semantics, namely the characterization of points with respect to an arbitrary set, in terms of both its spatial relations (interior, exterior, or boundary) and its dynamic convergence of any sequence (accumulation, co-accumulation, and isolation), to much weakened systems and hence with wider applicability. Examples from the theory of matroid and of Knowledge/Learning Spaces are used as an illustration.

closure-complement-frontier problem in saturated polytopological spaces

Let $X$ be a space equipped with $n$ topologies $\tau_1,\ldots,\tau_n$ which are pairwise comparable and saturated, and for each $1\leq i\leq n$ let $k_i$ and $f_i$ be the associated topological closure and frontier operators, respectively. Inspired by the closure-complement theorem of Kuratowski, we prove that the monoid of set operators $\mathcal{KF}_n$ generated by $\{k_i,f_i:1\leq i\leq n\}\cup\{c\}$ (where $c$ denotes the set complement operator) has cardinality no more than $2p(n)$ where $p(n)=\frac{5}{24}n^4+\frac{37}{12}n^3+\frac{79}{24}n^2+\frac{101}{12}n+2$. The bound is sharp in the following sense: for each $n$ there exists a saturated polytopological space $(X,\tau_1,...,\tau_n)$ and a subset $A\subseteq X$ such that repeated application of the operators $k_i, f_i, c$ to $A$ will yield exactly $2p(n)$ distinct sets. In particular, following the tradition for Kuratowski-type problems, we exhibit an explicit initial set in $\mathbb{R}$, equipped with the usual and Sorgenfrey topologies, which yields $2p(2)=120$ distinct sets under the action of the monoid $\mathcal{KF}_2$.

Alexandroff topologies and monoid actions

Given a monoid S acting (on the left) on a set X, all the subsets of X which are invariant with respect to such an action constitute the family of the closed subsets of an Alexandroff topology on X. Conversely, we prove that any Alexandroff topology may be obtained through a monoid action. Based on such a link between monoid actions and Alexandroff topologies, we firstly establish several topological properties for Alexandroff spaces bearing in mind specific examples of monoid actions. Secondly, given an Alexandroff space X with associated topological closure operator σ, we introduce a specific notion of dependence on union of subsets. Then, in relation to such a dependence, we study the family {\mathcal{A}_{\sigma,X}} of the closed subsets Y of X such that, for any {y_{1},y_{2}\in Y}, there exists a third element {y\in Y} whose closure contains both {y_{1}} and {y_{2}}. More in detail, relying on some specific properties of the maximal members of the family {\mathcal{A}_{\sigma,X}}, we provide a decomposition theorem regarding an Alexandroff space as the union (not necessarily disjoint) of a pair of closed subsets characterized by such a dependence. Finally, we refine the study of the aforementioned decomposition through a descending chain of closed subsets of X of which we give some examples taken from specific monoid actions.

CHARACTERIZING O-MINIMAL GROUPS IN TAME EXPANSIONS OF O-MINIMAL STRUCTURES

We establish the first global results for groups definable in tame expansions of o-minimal structures. Let ${\mathcal{N}}$ be an expansion of an o-minimal structure ${\mathcal{M}}$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of ${\mathcal{M}}$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil’s group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if ${\mathcal{N}}$ expands ${\mathcal{M}}$ by a dense independent set, then every definable group is o-minimal.

On gaps in the closures of images of divisor functions

Given a complex number [Formula: see text], define the divisor function [Formula: see text] by [Formula: see text]. In this paper, we look at [Formula: see text], the topological closure of the image of [Formula: see text], when [Formula: see text]. We exhibit new lower bounds on the number of connected components of [Formula: see text], bringing this bound from linear in [Formula: see text] to exponential. We also discuss the general structure of gaps of [Formula: see text] in order to work toward a possible monotonicity result.

Algebraic flows on abelian varieties

Let A be an abelian variety. The abelian Ax–Lindemann theorem shows that the Zariski closure of an algebraic flow in A is a translate of an abelian subvariety of A. The paper discusses some conjectures on the usual topological closure of an algebraic flow in A. The main result is a proof of these conjectures when the algebraic flow is given by an algebraic curve.

On the Closure of the Image of the Generalized Divisor Function

For any real number s, let σs be the generalized divisor function, i.e., the arithmetic function defined by σs(n) := ∑d|n ds, for all positive integers n. We prove that for any r > 1 the topological closure of σ−r(N+) is the union of a finite number of pairwise disjoint closed intervals I1, . . . , Iℓ. Moreover, for k = 1, . . . , ℓ, we show that the set of positive integers n such that σ−r(n) ∈ Ik has a positive rational asymptotic density dk. In fact, we provide a method to give exact closed form expressions for I1, . . . , Iℓ and d1, . . . , dℓ, assuming to know r with sufficient precision. As an example, we show that for r = 2 it results ℓ = 3, I1 = [1, π2/9], I2 = [10/9, π2/8], I3 = [5/4, π2/6], d1 = 1/3, d2 = 1/6, and d3 = 1/2.

A DEFINABLE -ADIC ANALOGUE OF KIRSZBRAUN’S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$-adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

Jordan-Wigner formalism for arbitrary 2-input 2-output matchgates and their classical simulation

In Valiant's matchgate theory, 2-input 2-output matchgates are $4\times 4$ matrices that satisfy ten so-called matchgate identities. We prove that the set of all such matchgates (including non-unitary and non-invertible ones) coincides with the topological closure of the set of all matrices obtained as exponentials of linear combinations of the 2-qubit Jordan-Wigner (JW) operators and their quadratic products, extending a previous result of Knill. In Valiant's theory, outputs of matchgate circuits can be classically computed in poly-time. Via the JW formalism, Terhal \& DiVincenzo and Knill established a relation of a unitary class of these circuits to the efficient simulation of non-interacting fermions. We describe how the JW formalism may be used to give an efficient simulation for all cases in Valiant's simulation theorem, which in particular includes the case of non-interacting fermions generalised to allow arbitrary 1-qubit gates on the first line at any stage in the circuit. Finally we give an exposition of how these simulation results can be alternatively understood from some basic Lie algebra theory, in terms of a formalism introduced by Somma et al.