A technique for adding range restrictions to generalized searching problems

In a generalized searching problem, a set $S$ of $n$ colored geometric objects has to be stored in a data structure, such that for any given query object $q$, the distinct colors of the objects of $S$ intersected by $q$ can be reported efficiently. In this paper, a general technique is presented for adding a range restriction to such a problem. The technique is applied to the problem of querying a set of colored points (resp.\ fat triangles) with a fat triangle (resp.\ point). For both problems, a data structure is obtained having size $O(n^{1+\epsilon})$ and query time $O((\log n)^2 + C)$. Here, $C$ denotes the number of colors reported by the query, and $\epsilon$ is an arbitrarily small positive constant

Further results on generalized intersection searching problems: counting, reporting, and dynamization

In a generalized intersection searching problem, a set, $S$, of colored geometric objects is to be preprocessed so that given some query object, $q$, the distinct colors of the objects intersected by $q$ can be reported efficiently or the number of such colors can be counted efficiently. In the dynamic setting, colored objects can be inserted into or deleted from $S$. These problems generalize the well-studied standard intersection searching problems and are rich in applications. Unfortunately, the techniques known for the standard problems do not yield efficient solutions for the generalized problems. Moreover, previous work on generalized problems applies only to the static reporting problems. In this paper, a uniform framework is presented to solve efficiently the counting/reporting/dynamic versions of a variety of generalized intersection searching problems, including: 1-, 2-, and 3-dimensional range searching, quadrant searching, interval intersection searching, 1- and 2-dimensional point enclosure searching, and orthogonal segment intersection searching

Optimal placement of a limited number of observations for period searches

Robotic telescopes present the opportunity for the sparse temporal placement of observations when period searching. We address the best way to place a limited number of observations to cover the dynamic range of frequencies required by an observer. We show that an observation distribution geometrically spaced in time can minimise aliasing effects arising from sparse sampling, substantially improving signal detection quality. The base of the geometric series is however a critical factor in the overall success of this strategy. Further, we show that for such an optimal distribution observations may be reordered, as long as the distribution of spacings is preserved, with almost no loss of quality. This implies that optimal observing strategies can retain significant flexibility in the face of scheduling constraints, by providing scope for on-the-fly adaptation. Finally, we present optimal geometric samplings for a wide range of common observing scenarios, with an emphasis on practical application by the observer at the telescope. Such a sampling represents the best practical empirical solution to the undersampling problem that we are aware of. The technique has applications to robotic telescope and satellite observing strategies, where target acquisition overheads mean that a greater total target exposure time (and hence signal-to-noise) can often in practice be achieved by limiting the number of observations.Comment: 8 pages with 16 figure

Effizient algorithms for generalized intersection searching on non-iso-oriented objects

In a generalized intersection searching problem, a set $S$ of colored geometric objects is to be preprocessed so that, given a query object $q$, the distinct colors of the objects of $S$ that are intersected by $q$ can be reported or counted efficiently. These problems generalize the well-studied standard intersection searching problems and are rich in applications. Unfortunately, the solutions known for the standard problems do not yield efficient solutions to the generalized problems. Recently, efficient solutions have been given for generalized problems where the input and query objects are iso-oriented, i.e., axes-parallel, or where the color classes satisfy additional properties, e.g., connectedness. In this paper, efficient algorithms are given for several generalized problems involving non-iso-oriented objects. These problems include: generalized halfspace range searching in ${\cal R}^d$, for any fixed $d \geq 2$, segment intersection searching, triangle stabbing, and triangle range searching in ${\cal R}^2$. The techniques used include: computing suitable sparse representations of the input, persistent data structures, and filtering search

On Geometric Range Searching, Approximate Counting and Depth Problems

In this thesis we deal with problems connected to range searching, which is one of the central areas of computational geometry. The dominant problems in this area are halfspace range searching, simplex range searching and orthogonal range searching and research into these problems has spanned decades. For many range searching problems, the best possible data structures cannot offer fast (i.e., polylogarithmic) query times if we limit ourselves to near linear storage. Even worse, it is conjectured (and proved in some cases) that only very small improvements to these might be possible. This inefficiency has encouraged many researchers to seek alternatives through approximations. In this thesis we continue this line of research and focus on relative approximation of range counting problems. One important problem where it is possible to achieve significant speedup through approximation is halfspace range counting in 3D. Here we continue the previous research done and obtain the first optimal data structure for approximate halfspace range counting in 3D. Our data structure has the slight advantage of being Las Vegas (the result is always correct) in contrast to the previous methods that were Monte Carlo (the correctness holds with high probability). Another series of problems where approximation can provide us with substantial speedup comes from robust statistics. We recognize three problems here: approximate Tukey depth, regression depth and simplicial depth queries. In 2D, we obtain an optimal data structure capable of approximating the regression depth of a query hyperplane. We also offer a linear space data structure which can answer approximate Tukey depth queries efficiently in 3D. These data structures are obtained by applying our ideas for the approximate halfspace counting problem. Approximating the simplicial depth turns out to be much more difficult, however. Computing the simplicial depth of a given point is more computationally challenging than most other definitions of data depth. In 2D we obtain the first data structure which uses near linear space and can answer approximate simplicial depth queries in polylogarithmic time. As applications of this result, we provide two non-trivial methods to approximate the simplicial depth of a given point in higher dimension. Along the way, we establish a tight combinatorial relationship between the Tukey depth of any given point and its simplicial depth. Another problem investigated in this thesis is the dominance reporting problem, an important special case of orthogonal range reporting. In three dimensions, we solve this problem in the pointer machine model and the external memory model by offering the first optimal data structures in these models of computation. Also, in the RAM model and for points from an integer grid we reduce the space complexity of the fastest known data structure to optimal. Using known techniques in the literature, we can use our results to obtain solutions for the orthogonal range searching problem as well. The query complexity offered by our orthogonal range reporting data structures match the most efficient query complexities known in the literature but our space bounds are lower than the previous methods in the external memory model and RAM model where the input is a subset of an integer grid. The results also yield improved orthogonal range searching in higher dimensions (which shows the significance of the dominance reporting problem). Intersection searching is a generalization of range searching where we deal with more complicated geometric objects instead of points. We investigate the rectilinear disjoint polygon counting problem which is a specialized intersection counting problem. We provide a linear-size data structure capable of counting the number of disjoint rectilinear polygons intersecting any rectilinear polygon of constant size. The query time (as well as some other properties of our data structure) resembles the classical simplex range searching data structures

Probabilistic Search for Object Segmentation and Recognition

The problem of searching for a model-based scene interpretation is analyzed within a probabilistic framework. Object models are formulated as generative models for range data of the scene. A new statistical criterion, the truncated object probability, is introduced to infer an optimal sequence of object hypotheses to be evaluated for their match to the data. The truncated probability is partly determined by prior knowledge of the objects and partly learned from data. Some experiments on sequence quality and object segmentation and recognition from stereo data are presented. The article recovers classic concepts from object recognition (grouping, geometric hashing, alignment) from the probabilistic perspective and adds insight into the optimal ordering of object hypotheses for evaluation. Moreover, it introduces point-relation densities, a key component of the truncated probability, as statistical models of local surface shape

Probabilistic Search for Object Segmentation and Recognition

The problem of searching for a model-based scene interpretation is analyzed within a probabilistic framework. Object models are formulated as generative models for range data of the scene. A new statistical criterion, the truncated object probability, is introduced to infer an optimal sequence of object hypotheses to be evaluated for their match to the data. The truncated probability is partly determined by prior knowledge of the objects and partly learned from data. Some experiments on sequence quality and object segmentation and recognition from stereo data are presented. The article recovers classic concepts from object recognition (grouping, geometric hashing, alignment) from the probabilistic perspective and adds insight into the optimal ordering of object hypotheses for evaluation. Moreover, it introduces point-relation densities, a key component of the truncated probability, as statistical models of local surface shape.Comment: 18 pages, 5 figure

Searching the Sky with CONFIGR-STARS

SyNAPSE program of the Defense Advanced Projects Research Agency (HRL Laboratories LLC, subcontract #801881-BS under DARPA prime contract HR0011-09-C-0001); CELEST, a National Science Foundation Science of Learning Center (SBE-0354378)CONFIGR-STARS, a new methodology based on a model of the human visual system, is developed for registration of star images. The algorithm first applies CONFIGR, a neural model that connects sparse and noisy image components. CONFIGR produces a web of connections between stars in a reference starmap or in a test patch of unknown location. CONFIGR-STARS splits the resulting, typically highly connected, web into clusters, or "constellations." Cluster geometry is encoded as a signature vector that records edge lengths and angles relative to the cluster’s baseline edge. The location of a test patch cluster is identified by comparing its signature to signatures in the codebook of a reference starmap, where cluster locations are known. Simulations demonstrate robust performance in spite of image perturbations and omissions, and across starmaps from different sources and seasons. Further studies would test CONFIGR-STARS and algorithm variations applied to very large starmaps and to other technologies that may employ geometric signatures. Open-source code, data, and demos are available from http://techlab.bu.edu/STARS/

Scaffold searching: automated identification of similar ring systems for the design of combinatorial libraries

Rigid ring systems can be used to position receptor-binding functional groups in 3D space and they thus play an increasingly important role in the design of combinatorial libraries. This paper discusses the use of shape-similarity methods to identify ring systems that are structurally similar to, and aligned with, a user-defined target ring system. These systems can be used as alternative scaffolds for the construction of a combinatorial library