The work in this thesis is concerned with variational methods for two-phase segmentation problems. We are interested in both the obtaining of numerical solutions to the partial differential equations arising from the minimisation of a given functional, and forming variational models that tackle some practical problem in segmentation (e.g. incorporating prior knowledge, dealing with intensity inhomogeneity). With that in mind we will discuss each aspect of the work as follows. A seminal two-phase variational segmentation problem in the literature is that of Active Contours Without Edges, introduced by Chan and Vese in 2001, based on the piecewise-constant formulation of Mumford and Shah. The idea is to partition an image into two regions of homogeneous intensity. However, despite the extensive success of this work its reliance on the level set method means that it is nonconvex. Later work on the convex reformulation of ACWE by Chan, Esedoglu, and Nikolova has led to a burgeoning of related methods, known as the convex relaxation approach. In Chapter 4, we introduce a method to find global minimisers of a general two-phase segmentation problem, which forms the basis for work in the rest of the thesis. We introduce an improved additive operator splitting (AOS) method based on the work of Weickert et al. and Tai et al. AOS has been frequently used for segmentation problems, but not in the convex relaxation setting. The adjustment made accounts for how to impose the relaxed binary constraint, fundamental to this approach. Our method is analogous to work such as Bresson et al. and we quantitatively compare our method against this by using a number of appropriate metrics. Having dealt with globally convex segmentation (GCS) for the general case in Chapter 4, we then bear in mind two important considerations. Firstly, we discuss the matter of selective segmentation and how it relates to GCS. Many recent models have incorporated user input for two-phase formulations using piecewise-constant fitting terms. In Chapter 5 we discuss the conditions for models of this type to be reformulated in a similar way. We then propose a new model compatible with convex relaxation methods, and present results for challenging examples. Secondly, we consider the incorporation of priors for GCS in Chapter 8. Here, the intention is to select objects in an image of a similar shape to a given prior. We consider the most appropriate way to represent shape priors in a variational formulation, and the potential applications of our approach. We also investigate the problem of segmentation where the observed data is challenging. We consider two cases in this thesis; in one there is significant intensity inhomogeneity, and in the other the image has been corrupted by unknown blur. The first has been widely studied and is closely related to the piecewise-smooth formulation of Mumford and Shah. In Chapter 6 we discuss a Variant Mumford- Shah Model by D.Chen et al. that uses the bias field framework. Our work focuses on improving results for methods of this type. The second has been less widely studied, but is more commonly considered when there is knowledge of the blur type. We discuss the advantages of simultaneously reconstructing and segmenting the image, rather than treating each problem separately and compare our method against comparable models. The aim of this thesis is to develop new variational methods for two-phase image segmentation, with potential applications in mind. We also consider new schemes to compute numerical solutions for generalised segmentation problems. With both approaches we focus on convex relaxation methods, and consider the challenges of formulating segmentation problems in this manner. Where possible we compare our ideas against current approaches to determine quantifiable improvements, particularly with respect to accuracy and reliability.

This thesis has been devoted to investigate and improve ptychography, which is a newly developed coherent diffractive imaging technique that can achieve quantitative imaging (both modulus and phase) at diffraction-limited resolution without imaging lenses. In particular, this thesis has first looked into two solutions of partial coherence in ptychography: the Wigner distribution deconvolution method (WDDM) and mixed state decomposition. WDDM is a non-iterative solution and with it partial coherence was first mathematically demonstrated solvable. We have improved the performance of WDDM, especially in the presence of noise, by proposing three tools that can be used together: a projection strategy, design of a favourable probe, and an iterative method. Furthermore, the reconstruction of spatial partial coherence via WDDM has been successfully demonstrated using a model calculation for the first time. Mixed state decomposition is an iterative solution. It provides much more flexibility and is able to solve any experimental instability (not just partial coherence) that can be modelled as a set of mutually orthogonal states. According to the formation of the mixed states, it can be divided into spatially mixed state ptychography and temporally mixed state ptychography. For spatially mixed state reconstruction, we have mathematically and experimentally demonstrated an inherent linear ambiguity in the reconstructions and also that the ambiguity can be broken by using an orthogonality constraint or phase-only constraint. Besides, the effects of a diffused probe on the reconstructions have been investigated using a spatial partial coherent x-ray experiment. For temporally mixed state ptychography, we have mathematically and experimentally demonstrated the breakdown of the linear ambiguity. In addition, an iterative algorithm vi has been proposed to remove the static background noise from the measurements by treating the background as the diffraction pattern from an extra temporal state. Moreover, this thesis has also explored two ways to extend ptychography for three-dimensional (3D) imaging: multislice ptychography and ptychographic tomography. The multislice method has already been introduced into ptychography to provide 3D information before. In this thesis, we have further extended it into a Fourier variant of ptychography – Fourier ptychography – by applying a parallel update for the aperture reconstruction and reforming the iterative algorithm to involve the specimen plane. Also, the reconstruction resolution has been discussed via the Ewald sphere construction and demonstrated via model calculations. Ptychographic tomography utilises ptychography to acquire 2D projection images at different orientations and makes use of tomography to achieve isotropic 3D reconstruc-tions at high resolution. In this thesis, we have demonstrated this technique step by step via an x-ray experiment and shown how the inherent ptychographic reconstruction ambiguities are removed prior to the tomographic reconstruction. The possibility of electron ptychographic tomography is also discussed based on the scale calculation with the x-ray experiment.

Simple connections between structure and optical response empower us with essential intuition to engineer complex optical functionalities. In this thesis, I study photonic crystals, quasicrystals and amorphous materials to quantify the structural properties that give rise to photonic band gaps (PBGs). Along the way, I develop two novel perspectives on the optical analysis of arbitrarily-structured media: generalised photonic band structure and local self-uniformity (LSU). Generalised photonic band structures reproduce the Bloch-wave band structure of photonic crystals but, crucially, also yield naturally unfolded complete dispersion relations for aperiodic materials. Using generalised band structures, I demonstrate that the overall form of a material's dispersion relation is determined by the non-zero momentum transfers of its structure factor. I observe in great detail the fractal-like PBG spectra of a pair of Penrose photonic quasicrystals. Here, I demonstrate that, in most cases, the Penrose PBGs form through a mechanism of Bragg scattering-induced standing wave formation. The fundamental gap of each structure, however, is attributed to spatially localised scattering resonances. I also fabricate and characterise hyperuniform gold metasurfaces. Fluorescence emission characterisation reveals a statistically isotropic distribution of momentum states within the light cone; this property is shown to result from the metasurface structure factor. In the second part of this thesis I introduce LSU as a continuous measure of the extent to which a network possesses an optimal PBG-forming structure. Specifically, LSU measures the geometrical and topological similarities of the local vertex environments in a network of uniform valency. I demonstrate that both known optimal photonic crystal structures and disordered PBG-forming networks possess significant LSU. Further, I produce the first known designs of amorphous gyroid networks. Amorphous gyroids possess significant LSU and can exhibit a sizeable complete PBG; these PBGs are validated experimentally by performing microwave transmission experiments on centimetre-scale alumina prototypes (εr = 9:5 ± 0.3 at 22 GHz). Using ensembles of both amorphous gyroids and planar hyperuniform networks, I demonstrate the striking correlation between LSU and PBG-forming ability. I rationalise the success of LSU by advancing a picture of photonic tight binding in high index connected networks. This picture explains the origin of PBGs in both ordered and disordered connected networks, and suggests why the diamond architecture possesses the largest known PBG. To conclude, I explore the possibility that amorphous gyroids exist in the wing scales of butteries. I reveal that the microstructure in the scales of Pseudolycaena marsyas possesses substantial amorphous gyroid character and demonstrate that the buttery's reflectance spectrum can be effectively reproduced by amorphous gyroid microstructures.

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This thesis presents detailed investigation on the fabrication, spectral characterisation and applications of UV-inscribed optical fibre gratings devices. Of prominent significance is the characterisation of the optical fibre gratings devices with nanoparticles and biological recognition elements for novel developments in the field of optical biosensing. A major contribution detailed in this thesis is the systematic study on fabrication, spectral characterisation and applications of different UV-inscribed in-fibre gratings. Specifically, uniform and apodized Fibre Bragg gratings (FBGs), normal and dual-peak long period fibre gratings (LPFGs), small-angle tilted fibre gratings (S-TFGs) and excessively tilted fibre gratings (Ex-TFGs) are presented. The holographic, phase-mask scanning and point-by-point methods are employed to fabricate these advanced optical fibre gratings using 244nm frequency-doubled Ar+ laser. Particular emphasis is laid on fabrication of dual-peak LPFGs in SMF-28 and thin-cladding single mode fibres of grating periods 140μm and 300μm respectively. Also, Ex-TFGs of different tilt angles are inscribed in single mode fibres using amplitude masks of different periods: 5.0μm, 6.6μm and 25μm. Another important contribution from this study is the nano-structural characterisation of the in-fibre gratings with nanoparticles such as carbon nanotubes (CNT), zinc oxide (ZnO) and gold nanoparticles for power demodulation, sensitivity enhancement and polarisation dependent SPR excitation respectively. Refractive index (RI) sensors based on 81° Ex-TFGs with carbon nanotube (CNT) overlay deposition have been investigated. The CNT, a dark material, with high absorption of light and high RI is responsible for the power demodulation of the attenuation band while the 81°-TFG induces the wavelength shift as the surrounding medium RI changes. Results show high sensitivities of 557.29 nm/RIU and 95.54 dB/RIU for the wavelength shift and power demodulation respectively. Also, nano-deposition of zinc oxide (ZnO) on Ex-TFGs inscribed in two different fibre types has been investigated using dissimilar morphologies (direct ZnO overlay and PSZnO overlay) for enhanced RI sensing. Significant improvement in sensitivity of ~ 21% (~ 522 nm/RIU) is obtained. The polarisation dependence of Au-coated S-TFGs on excitation of surface plasmon resonance (SPR) has also been investigated. Finally, the in-fibre gratings are surface-functionalized with bioreceptor elements such as enzymes (glucose oxidase) and antibody/antigen (Trx, IL-6). Enzyme functionalized biosensor based on dual-peak LPFG has been investigated for sugar concentration level and specific glucose detection and high sensitivities of ~4.67 nm/% and 12.21 ± 0.19 nm/ (mg/ml) are obtained respectively. Also, fibre optic biosensors based on antibodyfunctionalized 81º-TFGs have been presented for label-free specific recognition of interleukin-6 (IL-6) and thioredoxin (Trx) proteins. High saturation values (∧λ max ) of 35.05nm and 33.19nm are obtained respectively. The specificity validation of the biosensors in the presence of other interfering proteins is investigated using human plasma and results show high specificity.

Advances in lithographic technology have made it possible to fabricate systems in which electrons are confined magnetically. With an inhomogeneous circularly symmetric magnetic field, Bz, that was modulated so as the magnetic field was zero in the centre, electrons could, theoretically, be confined to a disk region. These systems are referred to as magnetic quantum dots, and the purpose of this thesis is to investigate their properties. The eigenstates of the single electron system are calculated using new methods based on wave function matching. These enable the eigenstates to be determined for all values of Bz. Exact numerical diagonalisation is used to calculate the N-electron eigenstates, and new procedures are derived to evaluate the Coulomb matrix elements. It is shown that a dot is able to confide interacting electrons, and is therefore stable. Numerical results for GaAs and InSb dots indicate the existence of a stability boundary as a function of the dot radius and Bz. The form of this boundary is investigated and an analytic expression for it is obtained. The stability of the system is enhanced in a homogeneous external magnetic field, Bext. Results are also presented for the electron density, the pair distribution, and the pair correlation function. The response of GaAs and InSb dots to far infrared radiation (FIR) is investigated as a function of Bz and Bext. The FIR response of the one and two electron systems are dissimilar, and this is shown to be a consequence of the interaction. Results for an InSb system with two electrons show a large splitting of the spectrum. This is investigated and an explanation is given. As a function of Bext, the single electron FIR response is similar to that of an electrostatic quantum dot in a magnetic field. The FIR spectrum of the equivalent two electron system is shown to have a rich structure, which should be experimentally verifiable.

In the thesis light intensity attenuation-based fibre-optic chemical sensors are investigated. The investigation proceeds along two lines: theoretical analyses and design studies. Both intrinsic and extrinsic types of sensors are discussed from both aspects. The theoretical analyses are concentrated on the construction of theoretical models and numerical calculations, which were previously not well developed but have been attracting a great deal of attention of researchers for some time. Under this topic, fibre-optic evanescent wave absorption sensors and fibre-optic surface plasmon resonance (SPR) sensors are studied as examples of the intrinsic type of sensors, and two-wavelength Beer-Lambert law based fibre-optic sensors as the extrinsic example. The design studies are concentrated on the design of novel types of sensors which have potentially valuable applications. As the intrinsic type of sensors, novel types of fibre-optic surface plasmon resonance (SPR) chemical sensors are proposed and studied in both point and distributed systems. A fibre-optic sensor for concentration determination of the infusible anaesthetic Propofol is studied in detail as an example of an extrinsic type of sensor.

The work presented in this thesis is concerned with the preparation of small supersonically-cooled alkaline-earth metal-containing polyatomic molecules by a laser ablation method, with subsequent detection using laser electronic spectroscopy. The results of spectroscopic studies involving three alkaline-earth metal-containing free radicals are presented. A new electronic transition, the D2T,+ -X2T,+ system, of the BaOH free radical is reported. The laser-induced fluorescence (LIF) spectrum is relatively straightforward, showing a simple vibrational progression in the Ba-0 stretching mode (v3). A re investigation of the C - X system of BaOH, carried out under supersonic jet conditions for the first time, is also presented. The LIF excitation spectrum is rather more complex than expected and possible explanations for these observations are considered. Five new electronic transitions of the SrOH free radical are reported for the first time. Three of these, the 52E+-r, £2S+-*2I+ and F2T1-X21,+ transitions, show simple structure in their vibrationally-resolved LIF excitation spectra. Another new transition, the C2TI- X2I,+ transition, is remarkably complex. It is proposed that a combination of Renner-Teller coupling and a 'reverse-polarised' n orbital in the C2TI electronic state is responsible for the added complexity in the LIF excitation spectrum. The 2'2S+ - X2I,+ electronic transition of SrOH shows evidence of vibronic interaction with the nearby C2Tl state. A dispersed fluorescence study of the ground state (X2I > +) of the MgCCH radical is also presented. The results of this study, in addition to a previous FTIR-matrix isolation study, have now accounted for four out of the five fundamental vibrational frequencies of ground state MgCCH. Finally, experiments involving a newly constructed REMPI-TOF mass spectrometer are discussed. Initial test experiments involving NO, CaOH, SrOH and SrCCH show that while the mass spectrometer detection aspect of the instrument is working well, supersonic cooling has proved elusive. Possible design aspects of the spectrometer which may be affecting the supersonic cooling are discussed, along with suggestions for future improvements.

Ultrasound elastography is investigated in this thesis based on processing backscattered raw ultrasonic data (radio frequency signals) using standard cross-correlation tool (SCC). The data are obtained using separate simulation, and experiment. In the simulation scenario, ultrasound imaging is simulated using the Field-II program, while the deformation object is simulated using finite element modelling (FEM) of ANSYS-11 program. A composition algorithm is proposed that combine Field-II and ANSYS-11 programs to produce backscattered raw data before and after compression. The experimental scenario was performed in collaboration with the Centre for Ultrasound Engineering, University of Strathclyde using DYNARY ultrasonic Phased Array Controller (Zetec, Québec, Québec, Canada) and Tissue Mimicking Material (TMM), which was manufactured in the Medical Physics Department, Edinburgh Royal Infirmary. In the experiments, uniform compression was applied over the TMM object using an adaptor that employs the 10MHz transducer and the head of an XYZ scanner, while freehand compression is applied using 5MHz transducer. A deformation estimation algorithm of displacement and a strain mapping algorithm have developed according to three levels of compression of 0.2%, 1%, and 2%-4%. The algorithm was assessed for the three stages based on the level of compression. In the first stage of 0.2% compression, backscattered raw data from a simulation model are utilized. Displacement and strain fields are estimated using the proposed deformation algorithm and then compared favourably with FEM numerical solution. In second stage of 1% compression, backscattered raw data of simulation and experiment models are utilized. Displacement and strain fields are estimated using the proposed novel deformation algorithm of refinement that includes the regularization for non-consistent measurements based on surrounding neighbours information. Results from simulation and experiment are compared, which shows a good agreement in terms of displacement field consistency and strain field contrast. In the third stage of 2%-4% compression, backscattered raw data of experiment are utilized. Displacement and strain fields are estimated using advance refinement process that involves an exclusion of estimation outliers. Displacement and strain fields are compared with two dimensional (2D) median filter operation. Results show a superiority of the proposed algorithm over 2D median operation in terms of signal-to-noise ratio (SNR) and contrast-to-noise ratio (CNR). In the proposed algorithm, a small correlation window length is used to ensure sufficient spatial resolution of strain estimation. Finally, the deformation estimation algorithm is validated using backscattered raw data of freehand compression. Result of freehand compression shows superiority against 2D median operation.

Modern astrophysical analysis requires increasing higher resolution laboratory spectra over a broader spectral range. The advances in ground- and space-based telescopes are highlighting the deficiencies in the current laboratory spectral database. This thesis reports on the research carried out on the high resolution laboratory measurements of the spectrum of neutral vanadium and the associated lifetime measurements to calculate transition probabilities. New measurements of radiative atomic lifetimes were made for 25 levels between 24648 cm-1 and 37518 cm-1 at the Lund Laser Center, Sweden, using the time resolved Laser-induced Fluorescence method. High Resolution Fourier Transform (FT) Spectroscopy is used to observe the hollow cathode lamp spectrum of neutral vanadium in the range 2000 cm-1 - 35000 cm-1. This range was covered using two high resolution FT Spectrometers in Imperial College London and Lund University (Sweden) with a resolving power of approximately 2 million and 1 million respectively. This resolving power enables line profiles to be fully resolved to the Doppler width. The high resolution spectra were used to measure the branching fractions of levels of interest which were then used in combination with the lifetime measurements to calculate the oscillator strengths. Across the UV, visible and IR spectral regions, radiative lifetimes and oscillator strengths for 208 lines between 3040 Å and 20000 Å from 39 upper energy levels are reported. 13 of these oscillator strengths have not been reported previously. Additionally new measurements were made of hyperfine structure splitting factors for 3 odd levels of V I lying between 24700 cm-1 and 28400 cm-1.