Computing the best grasp in a discrete point set

Addresses the issue of reconstruction of a 3D image from cone-beam projections measured for an arbitrary discrete set of vertex points. "Vertex point" refers to the position of the X-ray source in area CT or to the focal point of the CB collimator in SPECT imaging. The problem of defining a practical data sufficiency condition is investigated and a parameter /spl epsiv//sub 0/ is defined to describe how close a discrete vertex set comes to satisfying Tuy's condition. A reconstruction algorithm based on the Radon rebinning method is presented which is capable of near-optimal reconstructions from any discrete vertex set, whether or not it represents a sampled vertex path.

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This paper presents a general robust solution for the problem of affine object matching, whereby an object can be given as a discrete point set, a set of lines, or a closed region. Let be given two such objects which are related by a general affine transformation (up to noise and maybe some additional distortions of the object). Then we can determine the six parameters a/sub ik/ of the affine transformation using some new general moment invariants. These invariants are global, but assigned locally to any object point. With these invariants and using the Hungarian method or dynamic programming it can be computed a weighted point reference list. The affine parameters a/sub ik/ can be calculated from this list using the method of the least absolute differences (LAD) method. Our approach is very robust against noise and distortions. The algorithm can be used also for all subgroups of the affine group. Additionally, it is an unifying approach for all classes of objects: Discrete point sets, sets of lines, and closed regions. Many well known algorithms have problems with the case of symmetries of the objects, our approach is stable against symmetries. Experimental results both on simulated and real objects validate the robustness of the algorithm. In the case of closed regions our algorithm performs better than SQUID F. Mokhtarian et al. (1996).

In [24] the first named author developed the theory of singular fibers of generic differentiable maps between manifolds of negative codimension. Here, the codimension of a map f W M !N between manifolds is defined to be k D dimN dimM . For k 0, the fiber over a point in N is a discrete set of points, as long as the map is generic enough, and we can study the topology of such maps by using the well-developed theory of multi-jet spaces (see, for example, the article [13] by Mather). However, in the case where k < 0, the fiber over a point is no longer a discrete set, and is a complex of positive dimension k in general. This means that the theory of multi-jet spaces is not sufficient any more, and in [24] we have seen that the topology of singular fibers plays an essential role in such a study.

We construct a sequence of compact embedded minimal disks in a ball in R 3 , whose boundaries lie in the boundary of the ball, such that the curvature blows up only at a prescribed discrete (and hence, finite) set of points on the x 3 -axis. This extends a result of Colding and Minicozzi, who constructed a sequence for which the curvature blows up only at the center of the ball, and is a partial affirmative answer to the larger question of the existence of a sequence for which the curvature blows up precisely on a prescribed closed set on the x 3 -axis.

The article is devoted to the development of a method for aeroacoustic mapping of the state of the working blades of the flow part of turbomachines with the possibility of detecting a defect in a spatial format based on the results of fiber optic measurements in a distributed set of control points. It discusses the modeling of the gas flow cross-section nozzle for a turbomachine stationary and non-stationary operation modes, a mathematical model describing acoustic processes in the flow part of turbomachines and ways to improve them. The principles of creating complex systems of aeroacoustic cartography, which are accompanied by methods and means of measuring the parameters of gas dynamic flow in the flow part and in the cross-section of the turbomachine nozzle, are defined. The described principles make the control more informative and facilitate the development of an algorithm for a non-destructive method to monitor the state of the rotor blades. In general, it provides reliable data under conditions of parametric and structural uncertainty of the gas-air flow both on the controlled internal section of the flow path and on the output section of the turbomachine section. The problems of placement of control points and reconstruction of the spatial distribution of the field by dimensions in a discrete set of points are considered separately. The reconstruction of the acoustic field of the turbomachine is carried out based on measurements of the output of the fiber optic sensor only using a statistical approach.

When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, with a computational cost that is proportional to the number of these points. If that is fixed, the question arises, where should the points be placed in parameter space? The current literature advocates selecting the set of points (called a "template bank") whose Wigner-Seitz (also called Vorono\"i) cells have the smallest covering radius ($\equiv$ smallest maximal mismatch). Mathematically, such a template bank is said to have "minimum thickness". Here, for realistic populations of signal sources, we compute the fraction of potential detections which are "lost" because the template bank is discrete. We show that at fixed computational cost, the minimum thickness template bank does not maximize the expected number of detections. Instead, the most detections are obtained for a bank which minimizes a particular functional of the mismatch. For closely spaced templates, the fraction of lost detections is proportional to a scale-invariant "quantizer constant" G, which measures the average squared distance from the nearest template, i.e., the average expected mismatch. This provides a straightforward way to characterize and compare the effectiveness of different template banks. The template bank which minimizes G is mathematically called the "optimal quantizer", and maximizes the expected number of detections. We review optimal quantizer and minimum thickness template banks that are built as n-dimensional lattices, showing that even the best of these offer only a marginal advantage over template banks based on the humble cubic lattice.

An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing $\ensuremath{\hbar}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ by the operator $\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}}\ensuremath{-}\frac{e\stackrel{\ensuremath{\rightarrow}}{A}}{c}$. The resultant Schr\"odinger equation becomes a finite-difference equation whose eigenvalues can be computed by a matrix method. The magnetic flux which passes through a lattice cell, divided by a flux quantum, yields a dimensionless parameter whose rationality or irrationality highly influences the nature of the computed spectrum. The graph of the spectrum over a wide range of "rational" fields is plotted. A recursive structure is discovered in the graph, which enables a number of theorems to be proven, bearing particularly on the question of continuity. The recursive structure is not unlike that predicted by Azbel', using a continued fraction for the dimensionless parameter. An iterative algorithm for deriving the clustering pattern of the magnetic subbands is given, which follows from the recursive structure. From this algorithm, the nature of the spectrum at an "irrational" field can be deduced; it is seen to be an uncountable but measure-zero set of points (a Cantor set). Despite these-features, it is shown that the graph is continuous as the magnetic field varies. It is also shown how a spectrum with simplified properties can be derived from the rigorously derived spectrum, by introducing a spread in the field values. This spectrum satisfies all the intuitively desirable properties of a spectrum. The spectrum here presented is shown to agree with that predicted by A. Rauh in a completely different model for crystal electrons in a magnetic field. A new type of magnetic "superlattice" is introduced, constructed so that its unit cell intercepts precisely one quantum of flux. It is shown that this cell represents the periodicity of solutions of the difference equation. It is also shown how this superlattice allows the determination of the wave function at nonlattice sites. Evidence is offered that the wave functions belonging to irrational fields are everywhere defined and are continuous in this model, whereas those belonging to rational fields are only defined on a discrete set of points. A method for investigating these predictions experimentally is sketched.

Most band structure calculations approximate the integral over the Brillouin zone of momentum (i.e., wave vector) dependent properties with an appropriately weighted sum over a discrete set of points in the Brillouin zone. The best choice for such a set of points has long been a point of discussion in crystalline band structure calculations. For one-dimensionally periodic systems, however, the usual choice of points has been evenly spaced points in the one-dimensional Brillouin zone with equal weights. We have analyzed the exact error for the integral over the π band of a tight-binding model of trans-polyacetylene as a function of bond alternation. We find that the error in π band energy decreases in magnitude as q−2, where q is the total number of points treated in the Brillouin zone, for the metallic polyacetylene system with equal bond lengths. As bond alternation increases, however, we find that the error in π band energy decreases in magnitude roughly exponentially as a function of bond alternation for any given value of q. We find that this systematic change in error as a function of bond alternation can lead to either apparent overestimation or underestimation of the equilibrium dimerization and stabilization energy of Peierls distorted systems using first-principles total energy calculations.

Maxwell’s equations are usually formulated as differential equations (1.1)–(1.4), where we have derivatives with respect to space (represented by the curl and divergence operators) and derivatives with respect to time. The sought electromagnetic field is a function of space and time that must satisfy Maxwell’s equations. A rather simple way to pursue a numerical solution is to represent the electromagnetic field by its function values at a (finite) set of discrete grid points, where we would normally use a uniform Cartesian grid with respect to space and time. Then, it is quite natural to approximate the derivatives of the electromagnetic field in Maxwell’s equations by finite-difference approximations that involve the fields at neighboring points on the grid. This type of approach is referred to as a finite-difference method.KeywordsGrid PointStagger GridFinite Difference ApproximationComplex ExponentialDifference FormulaThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The aim of this study was to explore a method for automatically reconstructing the occlusal surface by CAD technology. A 3D graphic database of standard tooth crown (STC) was established from physical models. Referring to the contact rule, the occlusal surface of STC was partitioned into several functional regions, and the regions related with occlusal adjustment were further defined as some sets of discrete points. Additionally, geometrical marks were added to the database to embody dimensional contours and orientation of STC. Then an adaptive redesign methodology was developed to form occlusal surface: firstly, a STC similar to patient's anatomical characteristics was scaled and registered to the abutment teeth; secondly, the distances between the point sets and the antagonist were calculated to quantify the occluding relation; finally, the occlusal surface was reconstructed by local deformation. The program provided a novel method to automatically adjust the size and location of the restoration, avoid interpenetrations and create necessary contacts. The final occlusal surface fulfills the esthetic and morphologic criteria in clinical dentistry and meets the physiologic demands of occlusal and proximal contacts.

TheT-hull of both discrete point sets and continua is a generalization of the convex hull with respect to a reference objectT. VariousT-hulls serve as tools for shape analysis and as a basis for the introduction of a family of shape similarity measures. Some potential applications of these measures to shape problems arising in the natural sciences and a specific chemical example are discussed. The method ofT-hulls is applicable for the description of molecular shape effects in solvent-solute interactions, in external electromagnetic fields and within enzyme cavities.

Simple quasipenodic tilings with 8-fold and 12-fold symmetry are presented that possess local de-linflation symmetry and perfect matching rules.The special feature of these tilings is that the fuit information is already derivable from the set of vertex sites atone.This means that the latter is a valid representative of the corresponding equivalence class of mutual local derivability.l.

Kernel density estimates are a robust way to reconstruct a continuous distribution from a discrete point set. Typically their effectiveness is measured either in L1 or L2 error. In this paper we investigate the challenges in using L∞ (or worst case) error, a stronger measure than L1 or L2. We present efficient solutions to two linked challenges: how to evaluate the L∞ error between two kernel density estimates and how to choose the bandwidth parameter for a kernel density estimate built on a subsample of a large data set.

Let K be a bounded open convex set in euclidean n-space Rn symmetric in the origin 0. Further let L be a discrete point set in Rn containing 0 and at least n linearly independent points of Rn. Put mi = inf ui extended over all positive real numbers ui for which uiK contains i linearly independent points of L, i = 1, 2, …, n.