Second-order Shape Optimization for Geometric Inverse Problems in Vision

We develop a method for optimization in shape spaces, i.e., sets of surfaces modulo re-parametrization. Unlike previously proposed gradient flows, we achieve superlinear convergence rates through a subtle approximation of the shape Hessian, which is generally hard to compute and suffers from a series of degeneracies. Our analysis highlights the role of mean curvature motion in comparison with first-order schemes: instead of surface area, our approach penalizes deformation, either by its Dirichlet energy or total variation. Latter regularizer sparks the development of an alternating direction method of multipliers on triangular meshes. Therein, a conjugate-gradients solver enables us to bypass formation of the Gaussian normal equations appearing in the course of the overall optimization. We combine all of the aforementioned ideas in a versatile geometric variation-regularized Levenberg-Marquardt-type method applicable to a variety of shape functionals, depending on intrinsic properties of the surface such as normal field and curvature as well as its embedding into space. Promising experimental results are reported

A Posteriori Error Control for the Binary Mumford-Shah Model

The binary Mumford-Shah model is a widespread tool for image segmentation and can be considered as a basic model in shape optimization with a broad range of applications in computer vision, ranging from basic segmentation and labeling to object reconstruction. This paper presents robust a posteriori error estimates for a natural error quantity, namely the area of the non properly segmented region. To this end, a suitable strictly convex and non-constrained relaxation of the originally non-convex functional is investigated and Repin's functional approach for a posteriori error estimation is used to control the numerical error for the relaxed problem in the $L^2$-norm. In combination with a suitable cut out argument, a fully practical estimate for the area mismatch is derived. This estimate is incorporated in an adaptive meshing strategy. Two different adaptive primal-dual finite element schemes, and the most frequently used finite difference discretization are investigated and compared. Numerical experiments show qualitative and quantitative properties of the estimates and demonstrate their usefulness in practical applications.Comment: 18 pages, 7 figures, 1 tabl

The Application of Preconditioned Alternating Direction Method of Multipliers in Depth from Focal Stack

Post capture refocusing effect in smartphone cameras is achievable by using focal stacks. However, the accuracy of this effect is totally dependent on the combination of the depth layers in the stack. The accuracy of the extended depth of field effect in this application can be improved significantly by computing an accurate depth map which has been an open issue for decades. To tackle this issue, in this paper, a framework is proposed based on Preconditioned Alternating Direction Method of Multipliers (PADMM) for depth from the focal stack and synthetic defocus application. In addition to its ability to provide high structural accuracy and occlusion handling, the optimization function of the proposed method can, in fact, converge faster and better than state of the art methods. The evaluation has been done on 21 sets of focal stacks and the optimization function has been compared against 5 other methods. Preliminary results indicate that the proposed method has a better performance in terms of structural accuracy and optimization in comparison to the current state of the art methods.Comment: 15 pages, 8 figure

Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications

We propose a new variational model in weighted Sobolev spaces with non-standard weights and applications to image processing. We show that these weights are, in general, not of Muckenhoupt type and therefore the classical analysis tools may not apply. For special cases of the weights, the resulting variational problem is known to be equivalent to the fractional Poisson problem. The trace space for the weighted Sobolev space is identified to be embedded in a weighted $L^2$ space. We propose a finite element scheme to solve the Euler-Lagrange equations, and for the image denoising application we propose an algorithm to identify the unknown weights. The approach is illustrated on several test problems and it yields better results when compared to the existing total variation techniques

A novel variational model for image registration using Gaussian curvature

Image registration is one important task in many image processing applications. It aims to align two or more images so that useful information can be extracted through comparison, combination or superposition. This is achieved by constructing an optimal trans- formation which ensures that the template image becomes similar to a given reference image. Although many models exist, designing a model capable of modelling large and smooth deformation field continues to pose a challenge. This paper proposes a novel variational model for image registration using the Gaussian curvature as a regulariser. The model is motivated by the surface restoration work in geometric processing [Elsey and Esedoglu, Multiscale Model. Simul., (2009), pp. 1549-1573]. An effective numerical solver is provided for the model using an augmented Lagrangian method. Numerical experiments can show that the new model outperforms three competing models based on, respectively, a linear curvature [Fischer and Modersitzki, J. Math. Imaging Vis., (2003), pp. 81- 85], the mean curvature [Chumchob, Chen and Brito, Multiscale Model. Simul., (2011), pp. 89-128] and the diffeomorphic demon model [Vercauteren at al., NeuroImage, (2009), pp. 61-72] in terms of robustness and accuracy.Comment: 23 pages, 5 figures. Key words: Image registration, Non-parametric image registration, Regularisation, Gaussian curvature, surface mappin

Application priority framework for fixed mobile converged communication networks

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The current prospects in wired and wireless access networks, it is becoming increasingly important to address potential convergence in order to offer integrated broadband services. These systems will need to offer higher data transmission capacities and long battery life, which is the catalyst for an everincreasing variety of air interface technologies targeting local area to wide area connectivity. Current integrated industrial networks do not offer application aware context delivery and enhanced services for optimised networks. Application aware services provide value-added functionality to business applications by capturing, integrating, and consolidating intelligence about users and their endpoint devices from various points in the network. This thesis mainly intends to resolve the issues related to ubiquitous application aware service, fair allocation of radio access, reduced energy consumption and improved capacity. A technique that measures and evaluates the data rate demand to reduce application response time and queuing delay for multi radio interfaces is proposed. The technique overcomes the challenges of network integration, requiring no user intervention, saving battery life and selecting the radio access connection for the application requested by the end user. This study is split in two parts. The first contribution identifies some constraints of the services towards the application layer in terms of e.g. data rate and signal strength. The objectives are achieved by application controlled handover (ACH) mechanism in order to maintain acceptable data rate for real-time application services. It also looks into the impact of the radio link on the application and identifies elements and parameters like wireless link quality and handover that will influence the application type. It also identifies some enhanced traditional mechanisms such as distance controlled multihop and mesh topology required in order to support energy efficient multimedia applications. The second contribution unfolds an intelligent application priority assignment mechanism (IAPAM) for medical applications using wireless sensor networks. IAPAM proposes and evaluates a technique based on prioritising multiple virtual queues for the critical nature of medical data to improve instant transmission. Various mobility patterns (directed, controlled and random waypoint) has been investigated and compared by simulating IAPAM enabled mobile BWSN. The following topics have been studied, modelled, simulated and discussed in this thesis: 1. Application Controlled Handover (ACH) for multi radios over fibre 2. Power Controlled Scheme for mesh multi radios over fibre using ACH 3. IAPAM for Biomedical Wireless Sensor Networks (BWSN) and impact of mobility over IAPAM enabled BWSN. Extensive simulation studies are performed to analyze and to evaluate the proposed techniques. Simulation results demonstrate significant improvements in multi radios over fibre performance in terms of application response delay and power consumption by upto 75% and 15 % respectively, reduction in traffic loss by upto 53% and reduction in delay for real time application by more than 25% in some cases