Memoization method for storing of minimum-weight triangulation of a convex polygon

An analytical treatment of the problem of triangulation by stereophotogrammetry

This paper presents an algorithm to triangulate polygons optimally using order-k Delaunay triangulations, for a number of quality measures. The algorithm uses properties of higher order Delaunay triangulations to improve the O(n3) running time required for normal triangulations to O(k2n log k + kn log n) expected time, where n is the number of vertices of the polygon. An extension to polygons with points inside is also presented, allowing to compute an optimal triangulation of a polygon with h ≥ 1 components inside in O(kn log n) + O(k)h+2n expected time. Furthermore, through experimental results we show that, in practice, it can be used to triangulate point sets optimally for small values of k. This represents the first practical result on optimization of higher order Delaunay triangulations for k > 1.

We give a result that implies an improvement by a factor of log log n in the hypercube bounds for the geometric problems of batched planar point location, trapezoidal decomposition, and polygon triangulation. The improvements are achieved through a better solution to the multisearch problem on a hypercube, a parallel search problem where the elements in the data structure S to be searched are totally ordered, but where it is not possible to compare in constant time any two given queries q and q′. Whereas the previous best solution to this problem took O( log n( log log n)3) time on an n-processor hypercube, the solution given here takes O( log n( log log n)2) time on an n-processor hypercube. The hypercube model for which we claim our bounds is the standard one, SIMD, with O(1) memory registers per processor, and with one-port communication. Each register can store O( log n) bits, so that a processor knows its ID.

The High Bandwidth Memory (HBM) model is a theoretical computing model consisting of a logic circuit with a large external memory. Each address of the external memory can store <tex>$p$</tex> elements which can be read or written at the same time. Access to <tex>$p$</tex> elements stored at a given address in the external memory has a latency of <tex>$l$</tex> clock cycles. However, access to any <tex>$k$</tex> consecutive addresses can be done only in <tex>$(k+l-1)$</tex> clock cycles in a pipeline fashion by burst mode. A hardware algorithm is implemented in a logic circuit of the HBM to solve a particular problem. In this paper, we present an optimal implementation of the <tex>$O(n^{3})$</tex> -time dynamic programming algorithm for solving the optimal polygon triangulation (OPT) problem which is a problem to find a triangulation with minimum total weight of an input convex n-gon with weighted cords. We assume that the input weight matrix of a convex n-gon is stored in the external memory of the HBM model. Our hardware algorithm implemented in the logic circuit of size <tex>$O(s^{2})$</tex> operates on it and computes the optimal polygon triangulation of the input polygon in <tex>$O(\frac{n^{3}}{sp}+\frac{n^{3}}{s^{2}}+\frac{n^{3}}{s^{3}}l)$</tex> time. We also provide a theoretical proof showing that any hardware algorithm in a logic circuit of size <tex>$O(s^{2})$</tex> takes at least <tex>$\Omega(\frac{n^{3}}{sp}+\frac{n^{3}}{s^{2}})$</tex> time to solve the OPT problem. Thus, our implementation is optimal whenever <tex>$s^{2}\geq lp$</tex> or <tex>$s\geq l$</tex>, and this optimality condition is always satisfied from a practical point of view.

We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's Ω(n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n 2) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).

NEW RESULTS IN COMPUTATIONAL GEOMETRY RELEVANT TO PATTERN RECOGNITION IN PRACTICE

Optimal higher order Delaunay triangulations of polygons

We consider the following planar maximum weight triangulation (MAT) problem: given a set of n points in the plane, find a triangulation such that the total length of edges in triangulation is maximized. We prove an \(\Omega(\sqrt{n})\) lower bound on the approximation factor for several heuristics: maximum greedy triangulation, maximum greedy spanning tree triangulation and maximum spanning tree triangulation. We then propose the Spoke Triangulation algorithm, which approximates the maximum weight triangulation for points in general position within a factor of almost four in O(nlog n) time. The proof is simpler than the previous work. We also prove that Spoke Triangulation approximates the maximum weight triangulation of a convex polygon within a factor of two.

Unmanned aerial vehicles (UAVs) equipped with imaging and ranging sensors have become an effective remote sensing data acquisition tool for digital agriculture. Among potential products derived from UAVs, high-resolution orthophotos play an important role in several phenotyping activities, such as canopy cover estimation and flowering date identification. Current structure from motion (SfM) tools for image-based 3-D reconstruction and orthophoto generation cannot perform well when working with large-scale imagery over mechanized agricultural fields. This failure is mainly due to their inability to identify enough conjugate points among overlapping images captured at low altitudes. This study addresses such limitation through a new strategy that uses plant row segments as linear features in the triangulation process. The linear features are derived in two steps. First, an automated approach is implemented to extract plant row segments from the LiDAR data which are then back-projected to the imagery using available trajectory and system calibration parameters. In the second step, a machine-assisted strategy is used to adjust the line segments in image space for deriving accurate linear features. In the proposed framework, the triangulation process is conducted by investigating two mathematical models—referred to as object-space and image-space coplanarity constraints—for incorporating linear features in the bundle adjustment (BA). The orthophoto is generated using the refined trajectory and system calibration parameters derived from the BA process. Several experimental results over an agricultural filed show that the proposed framework outperforms commonly used SfM tools, e.g., Pix4D Mapper Pro and Agisoft Metashape in terms of generating orthophotos with high visual quality and geolocation accuracy. Also, results indicate that the object-space coplanarity constraint is more robust against potential noise in line measurements when compared to the image-space coplanarity model. However, both models lead to high absolute accuracy in the range of ±2–4 cm when the noise level in the image measurements of points along the line is reasonable, i.e., ~5–10 pixels.

Minimal Triangulations of Polygonal Domains

This paper is a review of some problems related to the triangulation of polygons or point sets in 2D and 3D space. It includes in particular a proof of the equiangularity properties of the Delaunay triangulation in 2D space and a short review on the different algorithms for the triangulation of polygons. The (still open) question of the intrinsic complexity of the triangulation problem for a simple polygon is also raised. As far as 3D space is concerned, the combinatory relations which provide bounds on the number of tetrahedra which appear in the triangulation of a set of points are given. A divide and conquer algorithm for triangulating arbitrary set of points is also presented. This algorithm is based on a splitting theorem which has been proved independently by Avis and ElGindy on one side and Edelsbrunner, Preparata and West on the other side.

Geometric classification of triangulations and their enumeration in a convex polygon

LetP andQ be two convex polygons withm andn vertices, respectively, which are specified by their cartesian coordinates in order. A simpleO(m+n) algorithm is presented for computing the intersection ofP andQ. Unlike previous algorithms, the new algorithm consists of a two-step combination of two simple algorithms for finding convex hulls and triangulations of polygons.

This paper considers the problem of finding an optimal order of the multiplication chain of matrices and the problem of finding an optimal triangulation of a convex polygon. For both these problems the best sequential algorithms run in ⊗(n log n) time. All parallel algorithms known use the dynamic programming paradigm and run in a polylogarithmic time using, in the best case, O(n6/logkn) processors for a constant k. We give a new algorithm which uses a different approach and reduces the problem to computing certain recurrence in a tree. We show that this recurrence can be optimally solved which enables us to improve the parallel bound by a few factors. Our algorithm runs in O(log3n) time using n2/log3n processors on a CREW PRAM.

A linear-time heuristic for minimum weight triangulation of convex polygons is presented. This heuristic produces a triangulation of length within a factor 1 + e from the optimum, where e is an arbitrarily small positive constant. This is the first subcubic algorithm that guarantees such an approximation factor, and it has interesting applications.