On the Oval Shapes of Beach Stones

This article introduces a new stochastic non-isotropic frictional abrasion model, in the form of a single short partial integro-differential equation, to show how frictional abrasion alone of a stone on a planar beach might lead to the oval shapes observed empirically. The underlying idea in this theory is the intuitive observation that the rate of ablation at a point on the surface of the stone is proportional to the product of the curvature of the stone at that point and the likelihood the stone is in contact with the beach at that point. Specifically, key roles in this new model are played by both the random wave process and the global (non-local) shape of the stone, i.e., its shape away from the point of contact with the beach. The underlying physical mechanism for this process is the conversion of energy from the wave process into the potential energy of the stone. No closed-form or even asymptotic solution is known for the basic equation, which is both non-linear and non-local. On the other hand, preliminary numerical experiments are presented in both the deterministic continuous-time setting using standard curve-shortening algorithms and a stochastic discrete-time polyhedral-slicing setting using Monte Carlo simulation.

Computational Methods for Light Scattering by Metallic Nanoparticles

<p>The unifying theme of this thesis is that of light scattering by particles, using computational approaches. This contributions here are separated into two main areas. The first consists of examining the behaviour of the extended boundary-condition method and T-matrix method, and providing a modified set of equations to use to calculate the relevant integrals. From this, some linear relations between integrals were found, which hint at the possibility of a more efficient means of performing these calculations. As well as this, the severe numerical problems associated with this method were investigated, and the primary source of these problems was identified in the case of two commonly-used shapes, spheroids and offset spheres. The cause of these numerical problems is that dominant, leading terms in the power series expansion of the integrands integrate identically to zero, but in practice, numerical calculations have insufficient precision to compute this exactly, and the overwhelming errors from this lead to drastically incorrect results. Following this identification, a new formulation of the integrals for spheroids is presented, which allows the much easier treatment of spheroids, approaching the level of ease of calculations for spheres in Mie theory. This formulation replaces some terms in the integrands with modified terms, that do not contain the parts of the power series that cause problems. As these should integrate to zero, we are able to remove them from the integrand without affecting the correct result. The second area of this thesis is concerned with calculations of the near-field for systems of interest in plasmonics, and specifically in surface-enhanced Raman spectroscopy. Here, the enhancement of the electric field in the vicinity of a metallic surface has a large effect on measured signals. The contribution of this thesis is to study the geometric parameters that influence the distribution of the field enhancement at the particle's resonance, specifically focusing on different effects caused by the overall shape of the particle, as opposed to those effects due to the local shape of the particle in regions of high enhancement. It is shown that the overall shape determines the location of the resonance, while the local shape determines how strongly the enhancement is localised. Understanding the factors that influence the enhancement localisation will help in guiding the design of suitable plasmonic substrates.</p>

Computational Methods for Light Scattering by Metallic Nanoparticles

<p>The unifying theme of this thesis is that of light scattering by particles, using computational approaches. This contributions here are separated into two main areas. The first consists of examining the behaviour of the extended boundary-condition method and T-matrix method, and providing a modified set of equations to use to calculate the relevant integrals. From this, some linear relations between integrals were found, which hint at the possibility of a more efficient means of performing these calculations. As well as this, the severe numerical problems associated with this method were investigated, and the primary source of these problems was identified in the case of two commonly-used shapes, spheroids and offset spheres. The cause of these numerical problems is that dominant, leading terms in the power series expansion of the integrands integrate identically to zero, but in practice, numerical calculations have insufficient precision to compute this exactly, and the overwhelming errors from this lead to drastically incorrect results. Following this identification, a new formulation of the integrals for spheroids is presented, which allows the much easier treatment of spheroids, approaching the level of ease of calculations for spheres in Mie theory. This formulation replaces some terms in the integrands with modified terms, that do not contain the parts of the power series that cause problems. As these should integrate to zero, we are able to remove them from the integrand without affecting the correct result. The second area of this thesis is concerned with calculations of the near-field for systems of interest in plasmonics, and specifically in surface-enhanced Raman spectroscopy. Here, the enhancement of the electric field in the vicinity of a metallic surface has a large effect on measured signals. The contribution of this thesis is to study the geometric parameters that influence the distribution of the field enhancement at the particle's resonance, specifically focusing on different effects caused by the overall shape of the particle, as opposed to those effects due to the local shape of the particle in regions of high enhancement. It is shown that the overall shape determines the location of the resonance, while the local shape determines how strongly the enhancement is localised. Understanding the factors that influence the enhancement localisation will help in guiding the design of suitable plasmonic substrates.</p>

Two Extensions of the Quadratic Nonuniform B-Spline Curve with Local Shape Parameter Series

Two extensions of the quadratic nonuniform B-spline curve with local shape parameter series, called the W3D3C1P2 spline curve and the W3D4C2P1 spline curve, are introduced in the paper. The new extensions not only inherit most excellent properties of the quadratic nonuniform B-spline curve but also can move locally toward or against the fixed control polygon by varying the shape parameter series. They are C1 and C2 continuous separately. Furthermore, the W3D3C1P2 spline curve includes the quadratic nonuniform B-spline curve as a special case. Two applications, the interpolation of the position and the corresponding tangent direction and the interpolation of a line segment, are discussed without solving a system of linear functions. Several numerical examples indicated that the new extensions are valid and can easily be applied.

Local Shape of the Vapor–Liquid Critical Point on the Thermodynamic Surface and the van der Waals Equation of State

Differential geometry is a powerful tool to analyze the vapor–liquid critical point on the surface of the thermodynamic equation of state. The existence of usual condition of the critical point (∂p/∂V)T=0 requires the isothermal process, but the universality of the critical point is its independence of whatever process is taken, and so we can assume (∂p/∂T)V=0. The distinction between the critical point and other points on the surface leads us to further assume that the critical point is geometrically represented by zero Gaussian curvature. A slight extension of the van der Waals equation of state is to letting the two parameters a and b in it vary with temperature, which then satisfies both assumptions and reproduces its usual form when the temperature is approximately the critical one.

Extracting Multiple Features for Dynamic Hand Gesture Recognition

In this work a framework based on histogram of orientation of optical flow (HOOF) and local binary pattern from three orthogonal planes (LBP_TOP) is proposed for recognizing dynamic hand gestures. HOOF algorithm extracts local shape and dynamic motion information of gestures from image sequences and local descriptor LBP is extended to three orthogonal planes to create an efficient motion descriptor. These features are invariant to scale, translation, illumination and direction of motion. The performance of the new framework is tested in two different ways. The first one is by fusing the global and local features as one descriptor and the other is using features separately to train the multi class support vector machine. Performance analysis shows that the proposed approach produces better results for recognizing dynamic hand gestures when compared with state of the art methods

Local Shape Descriptors for Neuron Segmentation

We present a simple, yet effective, auxiliary learning task for the problem of neuron segmentation in electron microscopy volumes. The auxiliary task consists of the prediction of Local Shape Descriptors (LSDs), which we combine with conventional voxel-wise direct neighbor affinities for neuron boundary detection. The shape descriptors are designed to capture local statistics about the neuron to be segmented, such as diameter, elongation, and direction. On a large study comparing several existing methods across various specimen, imaging techniques, and resolutions, we find that auxiliary learning of LSDs consistently increases segmentation accuracy of affinity-based methods over a range of metrics. Furthermore, the addition of LSDs promotes affinitybased segmentation methods to be on par with the current state of the art for neuron segmentation (Flood-Filling Networks, FFN), while being two orders of magnitudes more efficient—a critical requirement for the processing of future petabyte-sized datasets. Implementations of the new auxiliary learning task, network architectures, training, prediction, and evaluation code, as well as the datasets used in this study are publicly available as a benchmark for future method contributions.

A Framework for Curvature-Based CAD Mesh Partitioning

In ISO Geometrical Product Specifications and Verification Standards (GPS) [1], partition is one of the fundamental operations used to obtain ideal or non-ideal features of a product. The operation of partition produces independent geometrical features by decomposing the object. A curvature-based CAD mesh partitioning framework is proposed in this paper. The framework combines several key steps including curvature-based attribute calculation, local shape type refinement, region growing, slippage analysis and statistical modeling. The partitioned features are classified into seven invariance classes of surface in the context of ISO GPS. A case study shows that not only appropriate partitioning but also accurate invariance class recognition for GPS are achieved by the proposed framework.