Results 1  10
of
13
Homological Illusions of Persistence and Stability
, 2008
"... In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of points in the extended plane called persistence diagram. We start with the question of ridding the fu ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of points in the extended plane called persistence diagram. We start with the question of ridding the function of topological noise as suggested by its persistence diagram. We give an algorithm for hierarchically finding such εsimplifications on 2manifolds as well as answer the question of when it is impossible to simplify a function in higher dimensions. We continue by examining timevarying functions. The original algorithm computes the persistence pairing from an ordering of the simplices in a triangulation and takes worstcase time cubic in the number of simplices. We describe how to maintain the pairing in linear time per transposition of consecutive simplices. A side effect of the update algorithm is an elementary proof of the stability of persistence diagrams. We introduce a parametrized family of persistence diagrams called persistence vineyards and illustrate the concept with a vineyard describing a folding of a small peptide. We also base a simple algorithm to compute the rank invariant of a collection of functions on the update procedure.
TerraStream: From elevation data to watershed hierarchies
 Proc. ACM Sympos. on Advances in Geographic Information Systems
"... We consider the problem of extracting a river network and a watershed hierarchy from a terrain given as a set of irregularly spaced points. We describe TerraStream, a “pipelined ” solution that consists of four main stages: construction of a digital elevation model (DEM), hydrological conditioning, ..."
Abstract

Cited by 11 (7 self)
 Add to MetaCart
We consider the problem of extracting a river network and a watershed hierarchy from a terrain given as a set of irregularly spaced points. We describe TerraStream, a “pipelined ” solution that consists of four main stages: construction of a digital elevation model (DEM), hydrological conditioning, extraction of river networks, and construction of a watershed hierarchy. Our approach has several advantages over existing methods. First, we design and implement the pipeline so each stage is scalable to massive data sets; a single nonscalable stage would create a bottleneck and limit overall scalability. Second, we develop the algorithms in a general framework so that they work for both TIN and grid DEMs. TerraStream is flexible and allows users to choose from various models and parameters, yet our pipeline is designed to reduce (or eliminate) the need for manual intervention between stages. We have implemented TerraStream and present experimental results on real elevation point sets that show that our approach handles massive multigigabyte terrain data sets. For example, we can process a data set containing over 300 million points—over 20GB of raw data—in under 26 hours, where most of the time (76%) is spent in the initial CPUintensive DEM construction stage. 1
Flooding countries and destroying dams
 WADS 2007. LNCS
, 2007
"... In many applications of terrain analysis, pits or local minima are considered artifacts that must be removed before the terrain can be used. Most of the existing methods for local minima removal work only for raster terrains. In this paper we consider algorithms to remove local minima from polyhedr ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
In many applications of terrain analysis, pits or local minima are considered artifacts that must be removed before the terrain can be used. Most of the existing methods for local minima removal work only for raster terrains. In this paper we consider algorithms to remove local minima from polyhedral terrains, by modifying the heights of the vertices. To limit the changes introduced to the terrain, we try to minimize the total displacement of the vertices. Two approaches to remove local minima are analyzed: lifting vertices and lowering vertices. For the former we show that all local minima in a terrain with n vertices can be removedintheoptimalwayinO(n log n) time. For the latter we prove that the problem is NPhard, and present an approximation algorithm with factor 2 ln k, wherek is the number of local minima in the terrain.
Lipschitz unimodal and isotonic regression on paths and trees
, 2008
"... Let M = (V, A) be a planar graph, let γ ≥ 0 be a real parameter, and t: V → R a height function. A γLipschitz unimodal regression (γLUR) of t is a function s: V → R such that s has a unique local minimum, s(u) − s(v)  ≤ γ for each {u, v} ∈ A, and ‖s − t‖2 = ∑ v∈V (s(v) − t(v))2 is minimized. ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Let M = (V, A) be a planar graph, let γ ≥ 0 be a real parameter, and t: V → R a height function. A γLipschitz unimodal regression (γLUR) of t is a function s: V → R such that s has a unique local minimum, s(u) − s(v)  ≤ γ for each {u, v} ∈ A, and ‖s − t‖2 = ∑ v∈V (s(v) − t(v))2 is minimized. Here, a local minimum of s is a vertex v such that s(u)> s(v) for any neighbor u of v. For a directed planar graph, s: V → R is the γLipschitz isotonic regression (γLIR) of t if s(u) ≤ s(v) ≤ s(u)+γ for each directed edge (u, v) and ‖s − t‖2 is minimized. These problems arise, for example, in topological simplification of a height function. We present nearlineartime algorithms for LUR and LIR problems for two special cases where M is a path or a tree.
The Complexity of Flow on Fat Terrains and its I/OEfficient Computation
"... We study the complexity and the I/Oefficient computation of flow on triangulated terrains. We present an acyclic graph, the descent graph, that enables us to trace flow paths in triangulations i/oefficiently. We use the descent graph to obtain i/oefficient algorithms for computing river networks ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We study the complexity and the I/Oefficient computation of flow on triangulated terrains. We present an acyclic graph, the descent graph, that enables us to trace flow paths in triangulations i/oefficiently. We use the descent graph to obtain i/oefficient algorithms for computing river networks and watershedarea maps in O(Sort(d + r)) i/o’s, where r is the complexity of the river network and d of the descent graph. Furthermore we describe a data structure based on the subdivision of the terrain induced by the edges of the triangulation and paths of steepest ascent and descent from its vertices. This data structure can be used to report the boundary of the watershed of a query point q or the flow path from q in O(l(s) + Scan(k)) i/o’s, where s is the complexity of the subdivision underlying the data structure, l(s) is the number of i/o’s used for planar point location in this subdivision, and k is the size of the reported output. On αfat terrains, that is, triangulated terrains where the minimum angle of any triangle is bounded from below by α, we show that the worstcase complexity of the descent graph and of any path of steepest descent is O(n/α 2), where n is the number of triangles in the terrain. The worstcase complexity of the river network and the abovementioned data structure on such terrains is O(n 2 /α 2). When α is a positive constant this improves the corresponding bounds for arbitrary terrains by a linear factor. We prove that similar bounds cannot be proven for Delaunay triangulations: these can have river networks of complexity Θ(n 3). 1
I/OEfficient Algorithms for Computing Contours on a Terrain
 In SCG ’08: Proceedings of the twentyfourth annual symposium on Computational geometry
, 2008
"... A terrain M is the graph of a bivariate function. We assume that M is represented as a triangulated surface with N vertices. A contour (or isoline) of M is a connected component of a level set of M. Generically, each contour is a closed polygonal curve; at “critical ” levels these curves may touch e ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
A terrain M is the graph of a bivariate function. We assume that M is represented as a triangulated surface with N vertices. A contour (or isoline) of M is a connected component of a level set of M. Generically, each contour is a closed polygonal curve; at “critical ” levels these curves may touch each other or collapse to a point. We present I/Oefficient algorithms for the following two problems related to computing contours of M: (i) Given a sequence ℓ1 < · · · < ℓs of real numbers, we present an I/Ooptimal algorithm that reports all contours of M at heights ℓ1,..., ℓs using O(sort(N) + T/B) I/Os, where T is the total number edges in the output contours, B is the “block size, ” and sort(N) is the number of I/Os needed to sort N elements. The
Removing Local Extrema from Imprecise Terrains
"... In this paper we consider imprecise terrains, that is, triangulated terrains with a vertical error interval in the vertices. In particular, we study the problem of removing as many local extrema (minima and maxima) as possible from the terrain; that is, finding an assignment of one height to each ve ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
In this paper we consider imprecise terrains, that is, triangulated terrains with a vertical error interval in the vertices. In particular, we study the problem of removing as many local extrema (minima and maxima) as possible from the terrain; that is, finding an assignment of one height to each vertex, within its error interval, so that the resulting terrain has minimum number of local extrema. We show that removing only minima or only maxima can be done optimally in O(n log n) time, for a terrain with n vertices. Interestingly, however, the problem of finding a height assignment that minimizes the total number of local extrema (minima as well as maxima) is NPhard, and is even hard to approximate within a factor of O(log log n) unless P = NP. Moreover, we show that even a simplified version of the problem where we can have only three different types of intervals for the vertices is already NPhard, a result we obtain by proving hardness of a special case of 2Disjoint Connected Subgraphs, a problem that has lately received considerable attention from the graphalgorithms community. 1
I/OEfficient Computation of Water Flow Across a Terrain
"... Consider rain falling at a uniform rate onto a terrain T represented as a triangular irregular network. Over time, water collects in the basins of T, forming lakes that spill into adjacent basins. Our goal is to compute, for each terrain vertex, the time this vertex is flooded (covered by water). We ..."
Abstract
 Add to MetaCart
Consider rain falling at a uniform rate onto a terrain T represented as a triangular irregular network. Over time, water collects in the basins of T, forming lakes that spill into adjacent basins. Our goal is to compute, for each terrain vertex, the time this vertex is flooded (covered by water). We present an I/Oefficient algorithm that solves this problem using O(sort(X) log(X/M) + sort(N)) I/Os, where N is the number of terrain vertices, X is the number of pits of the terrain, sort(N) is the cost of sorting N data items, and M is the size of the computer’s main memory. Our algorithm assumes that the volumes and watersheds of the basins of T have been precomputed using existing methods.
I/OEfficient Contour Queries on Terrains
"... A terrain M can be represented as a triangulation of the plane along with a height function associated with the vertices (and linearly interpolated within the edges and triangles) of M. We investigate the problem of answering contour queries on M: Given a height ℓ and a triangle f of M that intersec ..."
Abstract
 Add to MetaCart
A terrain M can be represented as a triangulation of the plane along with a height function associated with the vertices (and linearly interpolated within the edges and triangles) of M. We investigate the problem of answering contour queries on M: Given a height ℓ and a triangle f of M that intersects the level set of M at height ℓ, report the list of the edges of the connected component of this level set that intersect f, sorted in clockwise or counterclockwise order. Contour queries are different from levelset queries in that only one contour (connected component of the level set) out of all those that may exist is expected to be reported. We present an I/Oefficient data structure of linear size that answers a contour query in O(log B N + T/B) I/Os, where N is the number of triangles in the terrain and T is the number of edges in the output contour. The data structure can be constructed using O(Sort(N)) I/Os.
Cleaning Massive Sonar Point Clouds ABSTRACT
"... We consider the problem of automatically cleaning massive sonar data point clouds, that is, the problem of automatically removing noisy points that for example appear as a result of scans of (shoals of) fish, multiple reflections, scanner selfreflections, refraction in gas bubbles, and so on. We de ..."
Abstract
 Add to MetaCart
We consider the problem of automatically cleaning massive sonar data point clouds, that is, the problem of automatically removing noisy points that for example appear as a result of scans of (shoals of) fish, multiple reflections, scanner selfreflections, refraction in gas bubbles, and so on. We describe a new algorithm that avoids the problems of previous localneighbourhood based algorithms. Our algorithm is theoretically I/Oefficient, that is, it is capable of efficiently processing massive sonar point clouds that do not fit in internal memory but must reside on disk. The algorithm is also relatively simple and thus practically efficient, partly due to the development of a new simple algorithm for computing the connected components of a graph embedded in the plane. A version of our cleaning algorithm has already been incorporated in a commercial product.