Probabilistic Solution of Inverse Problems

In this thesis we study the general problem of reconstructing a function, defined on a finite lattice from a set of incomplete, noisy and/or ambiguous observations. The goal of this work is to demonstrate the generality and practical value of a probabilistic (in particular, Bayesian) approach to this problem, particularly in the context of Computer Vision. In this approach, the prior knowledge about the solution is expressed in the form of a Gibbsian probability distribution on the space of all possible functions, so that the reconstruction task is formulated as an estimation problem. Our main contributions are the following: (1) We introduce the use of specific error criteria for the design of the optimal Bayesian estimators for several classes of problems, and propose a general (Monte Carlo) procedure for approximating them. This new approach leads to a substantial improvement over the existing schemes, both regarding the quality of the results (particularly for low signal to noise ratios) and the computational efficiency. (2) We apply the Bayesian appraoch to the solution of several problems, some of which are formulated and solved in these terms for the first time. Specifically, these applications are: teh reconstruction of piecewise constant surfaces from sparse and noisy observationsl; the reconstruction of depth from stereoscopic pairs of images and the formation of perceptual clusters. (3) For each one of these applications, we develop fast, deterministic algorithms that approximate the optimal estimators, and illustrate their performance on both synthetic and real data. (4) We propose a new method, based on the analysis of the residual process, for estimating the parameters of the probabilistic models directly from the noisy observations. This scheme leads to an algorithm, which has no free parameters, for the restoration of piecewise uniform images. (5) We analyze the implementation of the algorithms that we develop in non-conventional hardware, such as massively parallel digital machines, and analog and hybrid networks.

Birkhoff normal forms in semi-classical inverse problems

Iantchenko, A.; Sj?strand, J.; Zworski, M., (2002) 'Birkhoff normal forms in semi-classical inverse problems', Mathematical Research Letters 9(3) pp.337-362 RAE2008

The sacred edict; with a translation of the colloquial rendering, notes and vocabulary, by F. W. Baller. Prepared for the use of junior members of the China Inland Mission.

http://www.archive.org/details/thesacrededict01kanguoft

The Specialized Mappings Architecture

A probabilistic, nonlinear supervised learning model is proposed: the Specialized Mappings Architecture (SMA). The SMA employs a set of several forward mapping functions that are estimated automatically from training data. Each specialized function maps certain domains of the input space (e.g., image features) onto the output space (e.g., articulated body parameters). The SMA can model ambiguous, one-to-many mappings that may yield multiple valid output hypotheses. Once learned, the mapping functions generate a set of output hypotheses for a given input via a statistical inference procedure. The SMA inference procedure incorporates an inverse mapping or feedback function in evaluating the likelihood of each of the hypothesis. Possible feedback functions include computer graphics rendering routines that can generate images for given hypotheses. The SMA employs a variant of the Expectation-Maximization algorithm for simultaneous learning of the specialized domains along with the mapping functions, and approximate strategies for inference. The framework is demonstrated in a computer vision system that can estimate the articulated pose parameters of a human’s body or hands, given silhouettes from a single image. The accuracy and stability of the SMA are also tested using synthetic images of human bodies and hands, where ground truth is known.

Synthesis, characterisation and applications of group IV nanocrystals

Group IV materials such as silicon nanocrystals (Si NCs) and carbon quantum dots (CQDs) have received great attention as new functional materials with unique physical/chemical properties that are not found in the bulk material. This thesis reports the synthesis and characterisation of both types of nanocrystal and their application as fluorescence probes for the detection of metal ions. In chapter 2, a simple method is described for the size controlled synthesis of Si NCs within inverse micelles having well defined core diameters ranging from 2 to 6 nm using inert atmospheric synthetic methods. In addition, ligands with different molecular structures were utilised to reduce inter-nanocrystal attraction forces and improve the stability of the NC dispersions in water and a variety of organic solvents. Regulation of the Si NCs size is achieved by variation of the surfactants and addition rates, resulting high quality NCs with standard deviations (σ = Δd/d) of less than 10 %. Large scale production of highly mondisperse Si NC was also successfully demonstrated. In chapter 3, a simple solution phase synthesis of size monodisperse carbon quantum dots (CQDs) using a room temperature microemulsion strategy is demonstrated. The CQDs are synthesized in reverse micelles via the reduction of carbon tetrachloride using a hydride reducing agent. CQDs may be functionalised with covalently attached alkyl or amine monolayers, rendering the CQDs dispersible in wide range of polar or non-polar solvents. Regulation of the CQDs size was achieved by utilizing hydride reducing agents of different strengths. The CQDs possess a high photoluminescence quantum yield in the visible region and exhibit excellent photostability. In chapter 4, a simple and rapid assay for detection of Fe3+ ions was developed, based on quenching of the strong blue-green Si NC photoluminescence. The detection method showed a high selectivity, with only Fe3+ resulting in strong quenching of the fluorescence signal. No quenching of the fluorescence signal was induced by Fe2+ ions, allowing for solution phase discrimination between the same ion in different charge states. The optimised sensor system showed a sensitive detection range from 25- 900 μM and a limit of detection of 20.8 μM

Formation and characterisation of ordered porous vanadium oxide inverse opal materials for Li-ion batteries

This thesis presents several routes towards achieving artificial opal templates by colloidal self-assembly of polystyrene (PS) or poly(methyl methacrylate) (PMMA) spheres and the use of these template for the fabrication of V2O5 inverse opals as cathode materials for lithium ion battery applications. First, through the manipulation of different experimental factors, several methods of affecting or directing opal growth towards realizing different structures, improving order and/or achieving faster formation on a variety of substrates are presented. The addition of the surfactant sodium dodecyl sulphate (SDS) at a concentration above the critical micelle concentration for SDS to a 5 wt% solution of PMMA spheres before dip-coating is presented as a method of achieving ordered 2D PhC monolayers on hydrophobic Au-coated silicon substrates at fast and slow rates of withdrawal. The effect that the degree of hydrophilicity of glass substrates has on the ordering of PMMA spheres is next investigated for a slow rate of withdrawal under noise agitation. Heating of the colloidal solution is also presented as a means of affecting order and thickness of opal deposits formed using fast rate dip coating. E-beam patterned substrates are shown as a means of altering the thermodynamically favoured FCC ordering of polystyrene spheres (PS) when dip coated at slow rate. Facile routes toward the synthesis of ordered V2O5 inverse opals are presented with direct infiltration of polymer sphere templates using liquid precursor. The use of different opal templates, both 2D and 3D partially ordered templates, is compared and the composition and arrangement of the subsequent IO structures post infiltration and calcination for various procedures is characterised. V2O5 IOs are also synthesised by electrodeposition from an aqueous VOSO4 solution at constant voltage. Electrochemical characterisation of these structures as cathode material for Li-ion batteries is assessed in a half cell arrangement for samples deposited on stainless steel foil substrates. Improved rate capabilities are demonstrated for these materials over bulk V2O5, with the improvement attributed to the shorter Li ion diffusion distances and increased electrolyte infiltration provided by the IO structure.

Synthesis, functionalisation and characterisation of germanium nanocrystals & their applications

In the last two decades, semiconductor nanocrystals have been the focus of intense research due to their size dependant optical and electrical properties. Much is now known about how to control their size, shape, composition and surface chemistry, allowing fine control of their photophysical and electronic properties. However, genuine concerns have been raised regarding the heavy metal content of these materials, which is toxic even at relatively low concentrations and may limit their wide scale use. These concerns have driven the development of heavy metal free alternatives. In recent years, germanium nanocrystals (Ge NCs) have emerged as environmentally friendlier alternatives to II-VI and IV-VI semiconductor materials as they are nontoxic, biocompatible and electrochemically stable. This thesis reports the synthesis and characterisation of Ge NCs and their application as fluorescence probes for the detection of metal ions. A room-temperature method for the synthesis of size monodisperse Ge NCs within inverse micelles is reported, with well-defined core diameters that may be tuned from 3.5 to 4.5 nm. The Ge NCs are chemically passivated with amine ligands, minimising surface oxidation while rendering the NCs dispersible in a range of polar solvents. Regulation of the Ge NCs size is achieved by variation of the ammonium salts used to form the micelles. A maximum quantum yield of 20% is shown for the nanocrystals, and a transition from primarily blue to green emission is observed as the NC diameter increases from 3.5 to 4.5 nm. A polydisperse sample with a mixed emission profile is prepared and separated by centrifugation into individual sized NCs which each showed blue and green emission only, with total suppression of other emission colours. A new, efficient one step synthesis of Ge NCs with in situ passivation and straightforward purification steps is also reported. Ge NCs are formed by co-reduction of a mixture of GeCl4 and n-butyltrichlorogermane; the latter is used both as a capping ligand and a germanium source. The surface-bound layer of butyl chains both chemically passivates and stabilises the Ge NCs. Optical spectroscopy confirmed that these NCs are in the strong quantum confinement regime, with significant involvement of surface species in exciton recombination processes. The PL QY is determined to be 37 %, one of the highest values reported for organically terminated Ge NCs. A synthetic method is developed to produce size monodisperse Ge NCs with modified surface chemistries bearing carboxylic acid, acetate, amine and epoxy functional groups. The effect of these different surface terminations on the optical properties of the NCs is also studied. Comparison of the emission properties of these Ge NCs showed that the wavelength position of the PL maxima could be moved from the UV to the blue/green by choice of the appropriate surface group. We also report the application of water-soluble Ge NCs as a fluorescent sensing platform for the fast, highly selective and sensitive detection of Fe3+ ions. The luminescence quenching mechanism is confirmed by lifetime and absorbance spectroscopies, while the applicability of this assay for detection of Fe3+ in real water samples is investigated and found to satisfy the US Environmental Protection Agency requirements for Fe3+ levels in drinkable water supplies.

The inverse limit of GIT quotients of Grassmannians by the maximal torus

There are finitely many GIT quotients of

3D vanadium oxide inverse opal growth by electrodeposition

Three-dimensional vanadium pentoxide (V2O5) material architectures in the form of inverse opals (IOs) were fabricated using a simple electrodeposition process into artificial opal templates on stainless steel foil using an aqueous solution of VOSO4.χH2O with added ethanol. The direct deposition of V2O5 IOs was compared with V2O5 planar electrodeposition and confirms a similar progressive nucleation and growth mechanism. An in-depth examination of the chemical and morphological nature of the IO material was performed using X-ray crystallography, X-ray photoelectron spectroscopy, Raman scattering and scanning/transmission electron microscopy. Electrodeposition is demonstrated to be a function of the interstitial void fraction of the artificial opal and ionic diffusivity that leads to high quality, phase pure V2O5 inverse opals is not adversely affected by diffusion pathway tortuosity. Methods to alleviate electrodeposited overlayer formation on the artificial opal templates for the fabrication of the porous 3D structures are also demonstrated. Such a 3D material is ideally suited as a cathode for lithium ion batteries, electrochromic devices, sensors and for applications requiring high surface area electrochemically active metal oxides.

Artificial opal photonic crystals and inverse opal structures - fundamentals and applications from optics to energy storage

Photonic crystals (PhCs) influence the propagation of light by their periodic variation in dielectric contrast or refractive index. This review outlines the attractive optical qualities inherent to most PhCs namely the presence of full or partial photonic band gaps and the possibilities they present towards the inhibition of spontaneous emission and the localization of light. Colloidal self-assembly of polymer or silica spheres is one of the most favoured and low cost methods for the formation of PhCs as artificial opals. The state of the art in growth methods currently used for colloidal self-assembly are discussed and the use of these structures for the formation of inverse opal architectures is then presented. Inverse opal structures with their porous and interconnected architecture span several technological arenas - optics and optoelectronics, energy storage, communications, sensor and biological applications. This review presents several of these applications and an accessible overview of the physics of photonic crystal optics that may be useful for opal and inverse opal researchers in general, with a particular emphasis on the recent use of these three-dimensional porous structures in electrochemical energy storage technology. Progress towards all-optical integrated circuits may lie with the concepts of the photonic crystal, but the unique optical and structural properties of these materials and the convergence of PhC and energy storage disciplines may facilitate further developments and non-destructive optical analysis capabilities for (electro)chemical processes that occur within a wide variety of materials in energy storage research.

A regularized iterative algorithm for limited-angle inverse Radon transform

The tomography problem is investigated when the available projections are restricted to a limited angular domain. It is shown that a previous algorithm proposed for extrapolating the data to the missing cone in Fourier space is unstable in the presence of noise because of the ill-posedness of the problem. A regularized algorithm is proposed, which converges to stable solutions. The efficiency of both algorithms is tested by means of numerical simulations. © 1983 Taylor and Francis Group, LLC.

Stability and resolution in electromagnetic inverse problems

info:eu-repo/semantics/published

On the regularization of linear inverse problems in Fourier optics

info:eu-repo/semantics/published

Ill-posed Inverse Problems and Logarithmic Continuity in Electromagnetics

info:eu-repo/semantics/published