Showing papers on "Point (geometry) published in 2004"

Principal manifolds and nonlinear dimensionality reduction via tangent space alignment

Abstract: We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized data points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approximation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimensional Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.

Review of some iterative root-finding methods from a dynamical point of view

Abstract: From a dynamical point of view applied to complex polynomials, we study a number of root–finding iterative methods. We consider Newton’s method, Newton’s method for multiple roots, Jarratt’s method, the super–Halley method, the convex as well as the double convex acceleration of Whittaker’s method, the methods of Chebyshev, Stirling, and Steffensen, among others. Since all of the iterative root–finding methods we study satisfy the Scaling Theorem, except for Stirling’s method and that of Steffensen, we obtain their conjugacy classes.

Uncertainty and variability in point cloud surface data

Abstract: We present a framework for analyzing shape uncertainty and variability in point-sampled geometry. Our representation is mainly targeted towards discrete surface data stemming from 3D acquisition devices, where a finite number of possibly noisy samples provides only incomplete information about the underlying surface. We capture this uncertainty by introducing a statistical representation that quantifies for each point in space the likelihood that a surface fitting the data passes through that point. This likelihood map is constructed by aggregating local linear extrapolators computed from weighted least squares fits. The quality of fit of these extrapolators is combined into a corresponding confidence map that measures the quality of local tangent estimates. We present an analysis of the effect of noise on these maps, show how to efficiently compute them, and extend the basic definition to a scale-space formulation. Various applications of our framework are discussed, including an adaptive re-sampling method, an algorithm for reconstructing surfaces in the presence of noise, and a technique for robustly merging a set of scans into a single point-based representation.

The 3D Line Motion Matrix and Alignment of Line Reconstructions

Abstract: We study the problem of aligning two 3D line reconstructions in projective, affine, metric or Euclidean space. We introduce the 6 × 6 3D line motion matrix that acts on Plucker coordinates. We characterize its algebraic properties and its relation to the usual 4 × 4 point motion matrix, and propose various methods for estimating 3D motion from line correspondences, based on cost functions defined in images or 3D space. We assess the quality of the different estimation methods using simulated data and real images.

Segmentation methods for smooth point regions of conventional engineering objects

Abstract: The purpose of reverse engineering is to convert a large point cloud into an accurate, fair and consistent CAD model. The class of conventional engineering objects comprises, by a priori assumption, objects bounded only by simple, analytic surfaces, linear extrusion surfaces and surfaces of revolution. In this case it is possible to create a CAD model from point clouds with a minimal amount of user assistance. The key issue is segmentation, i.e. how to subdivide the point cloud into smaller regions, where each of which can be approximated by a single surface. Segmentation is relatively simple, if regions are bounded by sharp edges and small blends; problems arise when smoothly connected regions need to be separated. The direct segmentation method described in this paper is based on a particular hierarchy of tests, by means of which a large point cloud can be robustly split into smaller and smaller subregions until no further subdivision is needed. Tests are given using different types of indicators. These are based on local estimates of various geometric and statistical quantities; several examples are given to show their effect. The structure of the smooth, multiple regions obtained provides the basis for constrained surface fitting, which completes the final model building phase.

Computing One-Dimensional Stable Manifolds and Stable Sets of Planar Maps without the Inverse

Abstract: We present an algorithm to compute the one-dimensional stable manifold of a saddle point for a planar map. In contrast to current standard techniques, here it is not necessary to know the inverse or approximate it, for example, by using Newton's method. Rather than using the inverse, the manifold is grown starting from the linear eigenspace near the saddle point by adding a point that maps back onto an earlier segment of the stable manifold. The performance of the algorithm is compared to other methods using an example in which the inverse map is known explicitly. The strength of our method is illustrated with examples of noninvertible maps, where the stable set may consist of many different pieces, and with a piecewise-smooth model of an interrupted cutting process. The algorithm has been implemented for use in the DsTool environment and is available for download with this paper.

The euclidean algorithm in algebraic number fields

Abstract: This article, which is an update of a version published 1995 in Expo. Math., intends to survey what is known about Euclidean number fields; we will do this from a number theoretical (and number geometrical) point of view. We have also tried to put some emphasis on the open problems in this field.

System and method of curve fitting

Abstract: The invention relates to fitting a curve to a plurality of data points. A “seed curve” is determined from a first set of data points selected from the plurality of data points. From the remaining data points, data points are individually selected and a determination is made for each selected data point as to whether the data point is acceptable to be included with the first set of data points. When a data point is determined to be acceptable, the data point is included with the first set of data points to form another set of data points. After each of the other data points are evaluated for inclusion with the first set of data points, a best fit curve is determined from a final set of data points.

A mesh reconstruction algorithm driven by an intrinsic property of a point cloud

Abstract: This paper presents an algorithm for reconstructing a triangle mesh surface from a given point cloud. Starting with a seed triangle, the algorithm grows a partially reconstructed triangle mesh by selecting a new point based on an intrinsic property of the point cloud, namely, the sampling uniformity degree. The reconstructed mesh is essentially an approximate minimum-weight triangulation to the point cloud constrained to be on a two-dimensional manifold. Thus, the reconstructed surface has only small topological difference from the surface of the sampled object. Topological correct reconstruction can be guaranteed by adding a post-processing step. q 2003 Elsevier Ltd. All rights reserved.

Intrinsic point cloud simplification

Abstract: Modelling and visualisation methods working directly with point-sampled geometry have developed into attractive alternatives to more traditional mesh-based surface processing. In this paper, we consider a vital step in any point-based surface processing pipeline, point cloud simplification. Building upon the intrinsic point cloud simplification idea put forward in [14], we obtain a simplification algorithm allowing for intuitive density control and satisfying a set of important requirements unsupported by existing simplification techniques. The algorithm operates efficiently and gives a point set density guarantee. It supports both sub- and resampling of the input point set and allows for uniform and user-controlled feature-sensitive simplification. It can further deal with non-uniformly distributed point sets and point-sampled geometry featuring illegitimate holes of simple complexity. The algorithm is inherently progressive and supports the generation of multiresolution representations in the form of levels of detail. We are primarily concerned with describing the conceptual framework of our intrinsic approach and show its viability by giving a number of application examples using massive data sets.

Boundary labeling: models and efficient algorithms for rectangular maps

Abstract: In this paper, we present boundary labeling, a new approach for labeling point sets with large labels. We first place disjoint labels around an axis-parallel rectangle that contains the points. Then we connect each label to its point such that no two connections intersect. Such an approach is common e.g. in technical drawings and medical atlases, but so far the problem has not been studied in the literature. The new problem is interesting in that it is a mixture of a label-placement and a graph-drawing problem.

A near-linear constant-factor approximation for euclidean bipartite matching?

Abstract: In the Euclidean bipartite matching problem, we are given a set R of "red" points and a set B of "blue" points in R3 where |R| = |B| = n, and we want to pair up each red point with a distinct blue point so that the sum of distances between the paired points is minimized. We present an approximation algorithm that given any parameter 0

ISA and IBFVS: image space-based visualization of flow on surfaces

Abstract: We present a side-by-side analysis of two recent image space approaches for the visualization of vector fields on surfaces. The two methods, image space advection (ISA) and image-based flow visualization for curved surfaces (IBFVS) generate dense representations of time-dependent vector fields with high spatio-temporal correlation. While the 3D vector fields are associated with arbitrary surfaces represented by triangular meshes, the generation and advection of texture properties is confined to image space. Fast frame rates are achieved by exploiting frame-to-frame coherency and graphics hardware. In our comparison of ISA and IBFVS, we point out the strengths and weaknesses of each approach and give recommendations as to when and where they are best applied.

Color gamut mapping

Abstract: Mapping of color gamut in a three-dimensional color space is performed by converting coordinates of source reference points to coordinates of destination reference points, and converting coordinates of an initial point to coordinates of a target point based on a first relationship among a first set of volumes and a second relationship among a second set of volumes. The first set of volumes are defined at least in part by the initial point and at least a subset of the source reference points, and the second set of volumes are defined at least in part by the target point and at least a subset of the destination reference points.

Insertion support system

Abstract: According to an insertion support system of the present invention, when a biopsy area is specified at a periphery of the bronchi, the barycenter of the biopsy area is extracted. A circle centering on the barycenter is determined as a search area. The search area is expanded until the bronchi are located within the search area. A point in the search area to which the bronchi first reach is determined as an end point. A first route choice connecting the end point and a start point is determined. If the first route choice has not been registered yet, the first route choice is registered as a first registered route. Accordingly, a location of interest can be specified as an arbitrary region, and navigation leading to the specified region is appropriately set.

Gas reservoir evaluation and assessment tool method and apparatus and program storage device

Abstract: A Gas Reservoir Evaluation and Assessment Tool utilizes an Analytical Engine to produce predictions of pressure values and other production data at any point in space and at any point in time in a reservoir. A computer system, such as a workstation, stores a Gas Reservoir Evaluation and Assessment software which includes the Analytical Engine and responds to input data (which includes a reservoir description and fluid properties) by generating an output record which represents a prediction of the pressure values and other data at ‘any point in space’ and at ‘any point in time’ in a reservoir. The Analytical Engine will first calculate a pressure value in 1D for a single layer of a reservoir at a single point in space and time; it will then calculate a pressure value in 1D for multiple layers in the reservoir at the single point in space and time; it will then calculate a pressure value in 2D for the multiple layers at the single point in space and time; it will then calculate a pressure value in 3D for the multiple layers at the single point in space and time; and it will then calculate a pressure values in 3D for multiple layers not only at a single point in space but also at any future point in time.

A General Framework for Trajectory Triangulation

Abstract: The multiple view geometry of static scenes is now well understood. Recently attention was turned to dynamic scenes where scene points may move while the cameras move. The triangulation of linear trajectories is now well handled. The case of quadratic trajectories also received some attention. We present a complete generalization and address the problem of general trajectory triangulation of moving points from non-synchronized cameras. Two cases are considered: (i) the motion is captured in the images by tracking the moving point itself, (ii) the tangents of the motion only are extracted from the images. The first case is based on a new representation (to computer vision) of curves (trajectories) where a curve is represented by a family of hypersurfaces in the projective space \Bbb P5. The second case is handled by considering the dual curve of the curve generated by the trajectory. In both cases these representations of curves allow: (i) the triangulation of the trajectory of a moving point from non-synchronized sequences, (ii) the recovery of more standard representation of the whole trajectory, (iii) the computations of the set of positions of the moving point at each time instant an image was made. Furthermore, theoretical considerations lead to a general theorem stipulating how many independent constraints a camera provides on the motion of the point. This number of constraint is a function of the camera motion. On the computation front, in both cases the triangulation leads to equations where the unknowns appear linearly. Therefore the problem reduces to estimate a high-dimensional parameter in presence of heteroscedastic noise. Several method are tested.

Fluorescence-enhanced optical imaging of large phantoms using single and simultaneous dual point illumination geometries.

Abstract: Fluorescence-enhanced optical tomography is typically performed using single point illumination and multiple point collection measurement geometry. Single point illumination is often insufficient to illuminate greater volumes of large phantoms and results in an inadequate fluorescent signal to noise ratio ~SNR! for the majority of measurements. In this work, the use of simultaneous multiple point illumination geometry is proposed for acquiring a large number of fluorescent measurements with a sufficiently high SNR. As a feasibility study, dual point excitation sources, which are in-phase, were used in order to acquire surface measurements and perform three-dimensional reconstructions on phantoms of large volume and/or significant penetration depth. Measurements were acquired in the frequency‐domain using a modulated intensified CCD imaging system under different experimental conditions of target depth ~1.4 ‐2.8 cm deep! with a perfect uptake optical contrast. Three-dimensional reconstructions of the fluorescence absorption from the dual point illumination geometry compare well with the reconstructions from the single point illumination geometry. Targets located up to 2 cm deep were located successfully, establishing the feasibility of reconstructions from simultaneous multiple point excitation sources. With improved excitation light rejection, multiple point illumination geometry may prove useful in reconstructing more challenging domains containing deeply embedded targets. Image quality assessment tools are required to determine the optimal measurement geometry for the largest set off imaging tasks. © 2004 American Association of Physicists in Medicine. @DOI: 10.1118/1.1639321#

Classification of Six-Point Metrics

Abstract: There are $339$ combinatorial types of generic metrics on six points. They correspond to the $339$ regular triangulations of the second hypersimplex $\Delta(6,2)$, which also has $14$ non-regular triangulations.

Finite elements on point based surfaces

Abstract: framework efficiently and effectively brings well-known PDE-based processing techniques to the field of point-based surfaces. Our method is based on the construction of local tangent planes and a local Delaunay triangulation of adjacent points projected onto this plane. The definition of tangent spaces relies on moment-based computation with proven scaling and stability properties. Once local couplings are obtained, we are able to easily assemble PDE-specific mass and stiffness matrices and solve corresponding linear systems by standard iterative solvers. We demonstrate our framework by different classes of PDE-based surface processing applications, such as texture synthesis and processing, geometric fairing, and segmentation.

Imposing and recovering correlithm objects in conjunction with table lookup

Abstract: Encoding bits includes receiving a bit set to encode. An encoding lookup table associates correlithm objects of a space with bit sets. The space refers to an N-dimensional space, a correlithm object refers to a point of the space. The correlithm object corresponding to the received bit set is identified. The received bit set is encoded as the identified correlithm object. The identified correlithm object is imposed to encode the received bit set and subsequently decoded with table lookup using the reverse process.

Three-dimensional shape measuring method and apparatus for the same

Abstract: A three-dimensional shape measuring apparatus comprising a line laser light source (1-1) and an imaging unit (1-2). A three-dimensional shape measuring method and apparatus characterized by having a process (2-1) for irradiating a line laser beam to an object (1-3) to be measured by using a laser unit (1-1), a process (2-2) for imaging by the imaging unit (1-2) a point (1-6) irradiated with the laser beam, a process (2-3) for detecting a point on an image as a cross point (1-8), the point being detected from the taken image arrays and being detected as a common point where line laser beams pass, a process (2-4) for deriving equations including the position of a laser plane and the three-dimensional position of the cross point as unknowns based on the cross point, a process (2-5) for determining the three-dimensional position of the cross point and the position of a laser plane by solving simultaneous equations consisting of the plurality of equations, a process (2-6) for calculating, according to the principle of triangulation, the three-dimensional position of a point irradiated with the laser beam other than the cross point by using the position of a laser plane, and a process for obtaining the three-dimensional shape of the object by repeating the series of process.

Systems and methods for multi-objective portfolio analysis using dominance filtering

Abstract: The systems and methods of the invention are directed to portfolio optimization and related techniques. For example, the invention provides a method for multi-objective portfolio optimization for use in investment decisions based on competing objectives and a plurality of constraints constituting a portfolio problem in a space, the method comprising: generating a set of solutions of portfolio allocations in the space, the space having a plurality of dimensions; selecting a first dimension from the plurality of dimensions, the first dimension being a dimension under consideration; dividing the space into bins based on each dimension in the plurality of dimensions other than the dimension under consideration; determining a respective point in each bin with the most extreme value in the dimension under consideration; determining, based on the point in each bin with the most extreme value, whether other points in the space are dominant or dominated; and removing the dominated points from further consideration, so as to result in a reduced set of solutions, the reduced set of solutions being used in investment decisioning.

Dispersive estimates for Schroedinger operators in dimension two

Abstract: We prove dispersive estimates for linear Schroedinger equations in two space dimensions. The potential is assumed to be real-valued with some polynomial decay (faster than a negative third power), and zero energy is assumed to be a regular point for the perturbed resolvent.

Image processing device

Abstract: PROBLEM TO BE SOLVED: To provide an image processing device allowing a view point to be arbitrarily moved in a virtual space defined by dimensionality, and capable of providing a suitable operation environment. SOLUTION: In a simulation game device for playing a game based on an image formed by viewing a landscape or a character composed in a virtual space in three dimensions, a view point is allowed to be three-dimensionally moved in overlooking respective stages to increase amusingness of the game. In entering into a topographic map data check mode (ST1), the view point is three-dimensionally moved based on a predetermined path while panning or zooming like, for instance, in a movie (ST2), and a message is displayed at a message display point formed in the route of the path (ST4). COPYRIGHT: (C)2004,JPO&NCIPI

Determining sets of n-dimensional colorant control signals

Abstract: A method for determining a set of n-dimensional colorant control signals that map to a point P in an m-dimensional color space where n>m, including determining a forward device model that maps n-dimensional colorant control signals to m-dimensional color space signals; determining a set of n-dimensional colorant control signal simplexes that span a domain of the forward device model; and determining a set of range space simplexes in the m-dimensional color space corresponding to the set of n-dimensional colorant control space signal simplexes by mapping the vertices of the n-dimensional colorant control signal simplexes into the m-dimensional color space using the forward device model. The method also includes specifying a point P in the m-dimensional color space; and determining the set of n-dimensional colorant control signals that map to the point P in the m-dimensional color space, by performing m sequential slicing processes through the set of range space simplexes.