Convex Boundary Research Articles

OPTIMAL INTEGRABILITY CONDITION FOR THE LOG-SOBOLEV INEQUALITY

AbstractLet M denote a connected complete Riemannian manifold (possibly with a convex boundary), ρ the Riemannian distance function from a fixed point and V ∈ C 2 (M) such that dμ V ≔ e V d x is a probability measure. For any K ≥ 0, we prove that K/2 is the infimum over all λ > 0 such that Ric M − Hess V ≥ −K and imply the log-Sobolev inequality for the Dirichlet form μ V (|∇ f | 2 ).

Proper groupoids and momentum maps: Linearization, affinity, and convexity

We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural integral affine structure, which makes it into an affine orbifold with locally convex polyhedral boundary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps.

Recovering convex boundaries from blurred and noisy observations

We consider the problem of estimating convex boundaries from blurred and noisy observations. In our model, the convolution of an intensity function f is observed with additive Gaussian white noise. The function f is assumed to have convex support G whose boundary is to be recovered. Rather than directly estimating the intensity function, we develop a procedure which is based on estimating the support function of the set G. This approach is closely related to the method of geometric hyperplane probing, a well-known technique in computer vision applications. We establish bounds that reveal how the estimation accuracy depends on the ill-posedness of the convolution operator and the behavior of the intensity function near the boundary.

Global Detection of Salient Convex Boundaries

As an important geometric property of many structures or structural components, convexity plays an important role in computer vision and image understanding. In this paper, we describe a general approach that can force various edge-grouping algorithms to detect only convex structures from a set of boundary fragments. The basic idea is to remove some fragments and fragment connections so that, on the remaining ones, a prototype edge-grouping algorithm that detects closed boundaries without the convexity constraint can only produce convex closed boundaries. We show that this approach takes polynomial time and preserves the grouping optimality by not excluding any valid convex boundary from the search space. Choosing the recently developed ratio-contour algorithm as the prototype grouping algorithm, we develop a new convex-grouping algorithm, which can detect convex salient boundaries with good continuity and proximity in a globally optimal fashion. To facilitate the application of this convex-grouping algorithm, we develop a new fragment-connection method based on four-point Bezier curves. We demonstrate the performance of this convex-grouping algorithm by conducting experiments on both synthetic and real images. In addition, we provide a comparison with some prior edge-grouping algorithms. Finally, we show that the proposed convex-grouping algorithm can be further extended to detect convex open boundaries, derive region-based image hierarchies, and incorporate some simple human-computer interactions.

Data‐Driven Classification Using Boundary Observations

ABSTRACTClassification is often a critical task for business managers in their decision‐making processes. It is generally more difficult for a classification scheme to produce accurate results when the input domains of the various output classes coincide, to some degree, with one another. In an attempt to address this issue, this article discusses a data‐driven algorithm that identifies the region of coincidence, or overlap, for two‐group classification problems by empirically determining the convex boundary for each group. The results are extendable to multigroup classification. The class membership of a new observation is then determined by its relative position with respect to each of these boundaries. Due to minimal data storage requirements, this boundary‐point classification technique can adapt to changing conditions far more easily than other approaches. Test results demonstrate that the new classification technique has similar performance to a back‐propagation neural network under static conditions and significantly outperforms a back‐propagation neural network under dynamic conditions.

Multiple contour extraction from graylevel images using an artificial neural network

For active contour modeling (ACM), we propose a novel self-organizing map (SOM)-based approach, called the batch-SOM (BSOM), that attempts to integrate the advantages of SOM- and snake-based ACMs in order to extract the desired contours from images. We employ feature points, in the form of an edge-map (as obtained from a standard edge-detection operation), to guide the contour (as in the case of SOM-based ACMs) along with the gradient and intensity variations in a local region to ensure that the contour does not "leak" into the object boundary in case of faulty feature points (weak or broken edges). In contrast with the snake-based ACMs, however, we do not use an explicit energy functional (based on gradient or intensity) for controlling the contour movement. We extend the BSOM to handle extraction of contours of multiple objects, by splitting a single contour into as many subcontours as the objects in the image. The BSOM and its extended version are tested on synthetic binary and gray-level images with both single and multiple objects. We also demonstrate the efficacy of the BSOM on images of objects having both convex and nonconvex boundaries. The results demonstrate the superiority of the BSOM over others. Finally, we analyze the limitations of the BSOM.

Total positive curvature of hypersurfaces with convex boundary

We prove that if Σ is a compact hypersurface in Euclidean space Rn, its boundary lies on the boundary of a convex body C, and meets C orthogonally from the outside, then the total positive curvature of Σ is bigger than or equal to half the area of the sphere Sn-1. Also, we obtain necessary and sufficient conditions for the equality to hold.

Discussion on: “An Adaptive Gradient Law with Projection for Non-smooth Convex Boundaries”

Discussion on: “An Adaptive Gradient Law with Projection for Non-smooth Convex Boundaries”

Periodic orbits in outer billiard

It is shown that the set of 4-period orbits in outer billiard in the Euclidean plane with piecewise smooth convex boundary has an empty interior, provided that no four corners of the boundary form a parallelogram.

An Adaptive Gradient Law with Projection for Non-smooth Convex Boundaries

The parameter projection method, which has been designed for solving on-line parameter identification problems under smooth convex restrictions, is extended here to arbitrary convex parameter domains. Such problems arise naturally in cases where an admissible solution set results from the intersection of several (possibly smooth) convex constraints. The corresponding adaptive gradient law has the form of a special evolution projected dynamical system with a discontinuous right-hand side. The paper develops an alternative formulation of this projected dynamical system based on the multidimensional stop operator. The advantage of this approach is that the new right-hand side is continuous and the problem is thus accessible to conventional analysis methods, which easily give results on existence, uniqueness, and convergence properties of the corresponding solution trajectories. The method is tested on the parameter identification problem for complex hysteresis nonlinearities.

Hyperbolic manifolds with convex boundary

Let (M,∂M) be a 3-manifold, which carries a hyperbolic metric with convex boundary. We consider the hyperbolic metrics on M such that the boundary is smooth and strictly convex. We show that the induced metrics on the boundary are exactly the metrics with curvature K>-1, and that the third fundamental forms of ∂M are exactly the metrics with curvature K<1, for which the closed geodesics which are contractible in M have length L>2π. Each is obtained exactly once. Other related results describe existence and uniqueness properties for other boundary conditions, when the metric which is achieved on ∂M is a linear combination of the first, second and third fundamental forms.

Discrete one-forms on meshes and applications to 3D mesh parameterization

We describe how some simple properties of discrete one-forms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte's celebrated “spring-embedding” theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result generalizes the first, dealing with the case where the mesh contains multiple boundaries, which are free to be non-convex in the embedding. We characterize when it is still possible to achieve an embedding, despite these boundaries being non-convex. The third result is an analogous embedding theorem for meshes with genus 1 (topologically equivalent to the torus). Applications of these results to the parameterization of meshes with disk and toroidal topologies are demonstrated. Extensions to higher genus meshes are discussed.

Scattering of E polarized whispering-gallery mode from concave boundary

In this work we present numerical results for the 2D problem of scattering E polarised whispering-gallery mode from concave convex perfectly conducting boundary. The results were obtained by applying the developed method of currants integral equations (CIE) [6,7] for high frequency domain when the size of the scatterer match is greater than the wave length. We have applied the described procedure in order to find numerical solutions of scattering whispering-gallery mode by concave finite convex boundary as a part of a circular cylinder or part of parabolic cylinder. As incident wave we have considered cylindrical waves from line source and Gauss beam [6] with different effective width. It is shown that we have a complicated process of focusing and oscillating of the beam’s reflected field, both cylindrical and Gauss beam incident fields. The distortions of reflected field depend on shape of the boundary and parameters of the incident fields.

On the optimality of spectral compression of mesh data

Spectral compression of the geometry of triangle meshes achieves good results in practice, but there has been little or no theoretical support for the optimality of this compression. We show that, for certain classes of geometric mesh models, spectral decomposition using the eigenvectors of the symmetric Laplacian of the connectivity graph is equivalent to principal component analysis on that class, when equipped with a natural probability distribution. Our proof treats connected one-and two-dimensional meshes with fixed convex boundaries, and is based on an asymptotic approximation of the probability distribution in the two-dimensional case. The key component of the proof is that the Laplacian is identical, up to a constant factor, to the inverse covariance matrix of the distribution of valid mesh geometries. Hence, spectral compression is optimal, in the mean square error sense, for these classes of meshes under some natural assumptions on their distribution.

On the separation of cohomology groups of increasing unions of $(1, 1)$ convex-concave manifolds

We construct a complex manifold $X$, $dimX \geq 3$, which is an increasing union of (1, 1) convex-concave open subsets having the same fixed convex boundary, and a holomorphic line bundle $L$ on $X$, such that the cohomology group $H^{1}(X,L)$ is not separated.The manifold $X$ is constructed as a proper modification of the (1, 1) convex-concave manifold $\mathbb{C}^{k} \backslash \{0\}$ at a discrete subset. It is also remarked that an increasing union of 1-concave manifolds has always separated cohomology (for locally free sheaves).

On solving large strain hyperelastic problems with the natural element method

AbstractIn this paper, an extension of the natural element method (NEM) is presented to solve finite deformation problems. Since NEM is a meshless method, its implementation does not require an explicit connectivity definition. Consequently, it is quite adequate to simulate large strain problems with important mesh distortions, reducing the need for remeshing and projection of results (extremely important in three‐dimensional problems). NEM has important advantages over other meshless methods, such as the interpolant character of its shape functions and the ability of exactly reproducing essential boundary conditions along convex boundaries. The α‐NEM extension generalizes this behaviour to non‐convex boundaries. A total Lagrangian formulation has been employed to solve different problems with large strains, considering hyperelastic behaviour. Several examples are presented in two and three dimensions, comparing the results with the ones of the finite element method. NEM performs better showing its important capabilities in this kind of applications. Copyright © 2004 John Wiley &amp; Sons, Ltd.

The 1999 Taiwan Earthquake: A Proposed Stress-Focusing, Heel-Shaped Model

Stress focusing in a proposed heel-shaped model reasonably explains the occurrence of the largest onland earthquake ( M W 7.6) in Taiwan9s history. The epicenter of the mainshock was located in a low-seismicity block, where, historically, large earthquakes have not been documented for the past century. Spatial variations in crustal deformation reported in Global Positioning System surveys suggest that the crustal strength in the low-seismicity block was stronger than in the surrounding area. Since the low-seismicity block, which roughly coincides with the northern part of the Peikang High, is defined by a convex curve, stress partitioning or reorientation might well have taken place along the convex boundary; it then likely stored a lot of strain energy before eventually failing. The low-seismicity block with strong crust characteristics was finally overcome by stress focusing, resulting in the generation of the 921 Chi-Chi earthquake near the focus of the low-seismicity block, where a cluster of earthquakes had occasionally occurred during the past decade. Significant surface ruptures with vertical offset ranging from 2 to 8 m, primarily along the Chelungpu fault, were restricted to within the low-seismicity block. In short, when a low-seismicity block with small velocity fields of crustal deformation is identified in an active foreland belt, there should be a greater risk of a larger earthquake to release energy that accumulates in the strong crust. This finding provides some useful insight with which to evaluate potential earthquakes in Taiwan and other convergent zones.

Whispering Gallery Modes Inside Asymmetric Resonant Cavities

Two dimensional resonators with a smooth strictly convex boundary are known to possess a whispering gallery region supporting modes concentrated near the boundary. A new class of asymmetric resonant cavities is introduced, where a whispering gallery-like region is found deep inside the resonator. The construction of such resonators is a novel application of the geometric control methods. The results of numerical simulations and experiments are presented.

The role of environmental factors in growth accounting

AbstractThis paper explores a relatively new methodology, the directional distance function method, to analyse productivity growth. The method allows us to explicitly evaluate the role that undesirable outputs of the economy, such as carbon dioxide and other greenhouse gases, have on the frontier production process which we specify as a piecewise linear and convex boundary function. We decompose productivity growth into efficiency change (catching up) and technology change (innovation). We test the statistical significance of the estimates using recently developed bootstrap methods. We also explore implications for growth of total factor productivity in the OECD and Asian economies. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Local absorbing boundaries of elliptical shape for scalar waves

We discuss the performance of a family of local and weakly-non-local in space and time absorbing boundary conditions, prescribed on truncation boundaries of elliptical shape for the solution of the two-dimensional wave equation in both the time- and frequency-domains. The elliptical artificial boundaries are derived as particular cases of general convex boundaries for which the absorbing conditions have been developed. The conditions, via an operator-splitting scheme, are shown to lend themselves to easy incorporation in a variational form that, in turn, leads to a standard Galerkin finite element approach. The resulting wave absorbing finite elements are shown to preserve the sparsity and symmetry of standard finite element schemes in both the time- and frequency-domains. Numerical experiments for transient and time-harmonic cases attest to the computational savings realized when elongated scatterers are surrounded by elliptically-shaped boundaries, as opposed to the more commonly used circular truncation geometries.