Invariant representation and matching of space curves

Space curves are highly descriptive features for 3-D objects. Two invariant representations for space curves are discussed in this paper. One represents space curves by complex waveforms. The other represents space curves using the 3-D moment invariants of the data points on the curves. Space curve matching using invariant global features is discussed. An algorithm for matching partially occluded 3-D curves is also presented, in which rigidity constraints on pairwise curve segments are used to determine the globally consistent matching. An association graph can be constructed from the local matches. The maximal cliques of the graph will determine the visible part of the model curves in the scene. Experimental results using 3-D curves obtained from stereo matching and edges detected from the range data are also presented

New Directions in Geometric and Applied Knot Theory

The aim of this book is to present recent results in both theoretical and applied knot theory—which are at the same time stimulating for leading researchers in the field as well as accessible to non-experts. The book comprises recent research results while covering a wide range of different sub-disciplines, such as the young field of geometric knot theory, combinatorial knot theory, as well as applications in microbiology and theoretical physics

Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores

We provide the explicit solutions of linear, left-invariant, (convection)-diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2). These diffusion equations are forward Kolmogorov equations for stochastic processes for contour enhancement and completion. The solutions are group-convolutions with the corresponding Green's function, which we derive in explicit form. We mainly focus on the Kolmogorov equations for contour enhancement processes which, in contrast to the Kolmogorov equations for contour completion, do not include convection. The Green's functions of these left-invariant partial differential equations coincide with the heat-kernels on SE(2), which we explicitly derive. Then we compute completion distributions on SE(2) which are the product of a forward and a backward resolvent evolved from resp. source and sink distribution on SE(2). On the one hand, the modes of Mumford's direction process for contour completion coincide with elastica curves minimizing $\int \kappa^{2} + \epsilon ds$, related to zero-crossings of 2 left-invariant derivatives of the completion distribution. On the other hand, the completion measure for the contour enhancement concentrates on geodesics minimizing $\int \sqrt{\kappa^{2} + \epsilon} ds$. This motivates a comparison between geodesics and elastica, which are quite similar. However, we derive more practical analytic solutions for the geodesics. The theory is motivated by medical image analysis applications where enhancement of elongated structures in noisy images is required. We use left-invariant (non)-linear evolution processes for automated contour enhancement on invertible orientation scores, obtained from an image by means of a special type of unitary wavelet transform

Multi-scale active shape description in medical imaging

Shape description in medical imaging has become an increasingly important research field in recent years. Fast and high-resolution image acquisition methods like Magnetic Resonance (MR) imaging produce very detailed cross-sectional images of the human body - shape description is then a post-processing operation which abstracts quantitative descriptions of anatomically relevant object shapes. This task is usually performed by clinicians and other experts by first segmenting the shapes of interest, and then making volumetric and other quantitative measurements. High demand on expert time and inter- and intra-observer variability impose a clinical need of automating this process. Furthermore, recent studies in clinical neurology on the correspondence between disease status and degree of shape deformations necessitate the use of more sophisticated, higher-level shape description techniques. In this work a new hierarchical tool for shape description has been developed, combining two recently developed and powerful techniques in image processing: differential invariants in scale-space, and active contour models. This tool enables quantitative and qualitative shape studies at multiple levels of image detail, exploring the extra image scale degree of freedom. Using scale-space continuity, the global object shape can be detected at a coarse level of image detail, and finer shape characteristics can be found at higher levels of detail or scales. New methods for active shape evolution and focusing have been developed for the extraction of shapes at a large set of scales using an active contour model whose energy function is regularized with respect to scale and geometric differential image invariants. The resulting set of shapes is formulated as a multiscale shape stack which is analysed and described for each scale level with a large set of shape descriptors to obtain and analyse shape changes across scales. This shape stack leads naturally to several questions in regard to variable sampling and appropriate levels of detail to investigate an image. The relationship between active contour sampling precision and scale-space is addressed. After a thorough review of modem shape description, multi-scale image processing and active contour model techniques, the novel framework for multi-scale active shape description is presented and tested on synthetic images and medical images. An interesting result is the recovery of the fractal dimension of a known fractal boundary using this framework. Medical applications addressed are grey-matter deformations occurring for patients with epilepsy, spinal cord atrophy for patients with Multiple Sclerosis, and cortical impairment for neonates. Extensions to non-linear scale-spaces, comparisons to binary curve and curvature evolution schemes as well as other hierarchical shape descriptors are discussed

Mapping and controlling nucleation

This thesis reports on an investigation of two non-trivial nucleation phenomena, with the goal that understanding these will enable greater understanding of nucleation generally. Non-photochemical laser-induced nucleation (NPLIN) phenomena have been investigated for more than twenty years but lack a complete explanation. The laser-induced phase separation and nucleation (LIPSaN) phenomenon has been discovered and it is proposed as the mechanism behind NPLIN. A laser generates a potential which, when incident on a binary mixture in proximity to its critical point, causes the high-refractive index component to migrate to the focus. The effect bears a similarity to optical trapping of particles, as the trapped particle has a higher refractive index than the medium it is in. It has been shown that nucleation can be induced in metastable binary mixtures, which is analogous to the metastable supersaturated solutions which are typical in NPLIN experiments. It is proposed that NPLIN can only work if there is a hidden liquid-liquid critical point in the supersaturated regime. The liquid-liquid transition (LLT) is a ubiquitous example of polyamorphism – the transition between one liquid state with no long-range ordering to another. There are several examples of LLTs, but none that are quite as fiercely debated as triphenyl phosphite (TPP). The debate can be summarised as two competing hypotheses: Hédoux – the second liquid does not exist; it is actually the untransformed liquid mixed with nano or micro scale crystals, or Tanaka – the second liquid state exists, but nano or micro crystals are also produced at higher LLT temperatures. It will be shown using a wide range of techniques that Tanaka is at least partially correct, but the two sides are two sides of the same coin. The second liquid state exists and there is a first order LLT, but the so-called nanocrystals are better described as locally favoured structures that are similar to the structure of the crystal. There are three crystal polymorphs of TPP and their distinct unit cells and conformers have been characterised with single crystal X-ray diffraction (XRD). There is a conformational change during the LLT which has been characterised using infrared, density functional theory (DFT) and XRD data. The LLT is associated by a flip of a phenoxy arm and change from parallel ‘sandwich’ to T-shaped π- π stacking. Both avenues of investigation emphasise the importance of critical points and their influence in how nucleation proceeds. The work presented here sheds some light on two poorly understood nucleation phenomena and will hopefully aid in a more robust understanding of nucleation generally

The Fifteenth Marcel Grossmann Meeting

The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity

Carbon Nanotubes:Adsorption, Point Defects and Structural Deformation from Quantum and Classical Simulations

This thesis studies carbon nanotubes using state-of-the-art computational methods. Using large-scale quantum-mechanical calculations, based on density-functional-theory, we investigate several important aspects of the physics and chemistry of single-walled-nanotubes. The focus is on the effect of defects, namely adparticles and vacancies, on the structural, electronic and dynmical properties of the nanotubes. We also present preliminary results of simulations exploring a possible route for the formation of SWNTs from graphene nanoflakes. Adparticles include atomic hydrogen, oxygen, sulfur at different concentrations as well as nitrogen--oxides. Chemisorption of hydrogen as well as oxygen and isoelectronic species results in the formation of clusters on the sidewall, with characteristic structures corresponding to characteristic signatures in the electronic spectra. Especially in the case of oxygen, we find that relatively high energy barriers separate different structures: this shows that not only thermodynamically favored configurations are relevant for the understanding of oxygen chemisorption but the presence of traps cannot be neglected. The fingerpint of these traps is confirmed by scanning-tunneling spectroscopy. Trends with size and chirality of the nanotubes and oxygen coverage are studied in detail and also explained in terms of simple chemical descriptors. The importance of large-scale atomistic models is also emphasized to obtain convergent results and thus reliable predictions, and comparison is made of results we obtain using different gradient-corrected exchange-correlation functionals and in part with hybrid functionals. The study of nitrogen-oxides faces the difficulty to correctly represent physisorption, also with empirically corrected gradient-corrected exchange-correlation and hybrid functionals. Still our calculations of vibrational frequencies of different molecules on the sidewall, once compared with experiment, are able to distinguish the specific species observed in infrared spectra. Part of our work is centered on the comparison of widely used classical potentials (reactive force fields) with DFT results. Specifically we use them to study hydrogen and oxygen chemisorption, and especially examine the validity of several different force-fields for the description of the structure and energetics of single and double vacancies. In all cases, we find rather limited agreement with DFT results, showing the intrinsic difficulty to represent the subtle and intrinsic quantum effects governing the physico-chemical behavior of carbon nanotubes. Still, the wealth of results we have obtained might be useful for an improvement of these classical schemes for a specific application or other semi-classical models

The Fifteenth Marcel Grossmann Meeting

The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity