Fast Hyperparameter Calibration of Sparsity Enforcing Penalties in Total Generalised Variation Penalised Reconstruction Methods for XCT Using a Planted Virtual Reference Image

The reconstruction problem in X-ray computed tomography (XCT) is notoriously difficult in the case where only a small number of measurements are made. Based on the recently discovered Compressed Sensing paradigm, many methods have been proposed in order to address the reconstruction problem by leveraging inherent sparsity of the object’s decompositions in various appropriate bases or dictionaries. In practice, reconstruction is usually achieved by incorporating weighted sparsity enforcing penalisation functionals into the least-squares objective of the associated optimisation problem. One such penalisation functional is the Total Variation (TV) norm, which has been successfully employed since the early days of Compressed Sensing. Total Generalised Variation (TGV) is a recent improvement of this approach. One of the main advantages of such penalisation based approaches is that the resulting optimisation problem is convex and as such, cannot be affected by the possible existence of spurious solutions. Using the TGV penalisation nevertheless comes with the drawback of having to tune the two hyperparameters governing the TGV semi-norms. In this short note, we provide a simple and efficient recipe for fast hyperparameters tuning, based on the simple idea of virtually planting a mock image into the model. The proposed trick potentially applies to all linear inverse problems under the assumption that relevant prior information is available about the sought for solution, whilst being very different from the Bayesian method.

On the Existence of Non-Spurious Solutions to Second Order Dirichlet Problem

Using the direct variational method together with the monotonicity approach we consider the existence of non-spurious solutions to the following Dirichlet problem −x¨t =ft,xt, x0 =x1 =0, where f: 0,1 × R→R is a jointly continuous and not necessarily convex function. A new approach towards deriving the discrete family of approximating problems is proposed.

The International Criminal Police Organization And Its Role In The Defense Of The Organization Crime

The international organization is the criminal clause, an international organization specialized in the fight against crime in general and the organized crime , the term in particular, and this is due to the seriousness of this crime in various fields, and states can no longer can front it individually, given technological development, and with limited police competencies the interior the international community sought to find a mechanism capable of police cooperation among countries to confront organized crime . As this organization specializes in extraditing criminals and un covering unknown searches through coordination of security services between member states and plays a prominent role in the area of fighting transnational organized crime, the organization also works to prevent international crime through field and technical support to the police in the world working to confront growing international criminal challenges and I suggest spurious solutions to tackle these crimes and simplify the procedures for extraditing them to justice so that they can be punished without escaping them.

Constructing equivalence-preserving Dirac variational integrators with forces

The dynamical motion of mechanical systems possesses underlying geometric structures and preserving these structures in numerical integration improves the qualitative accuracy and reduces the long-time error of the simulation. For a single mechanical system, structure preservation can be achieved by adopting the variational integrator construction (Marsden, J. & West, M. (2001) Discrete mechanics and variational integrators. Acta Numer., 10, 357–514). This construction has been generalized to more complex systems involving forces or constraints as well as to the setting of Dirac mechanics (Leok, M. & Ohsawa, T. (2011) Variational and geometric structures of discrete Dirac measures. Found. Comput. Math., 11, 529–562). Forced Lagrange–Dirac systems are described by a Lagrangian and an external force pair, and two pairs of Lagrangians and external forces are said to be equivalent if they yield the same equations of motion. However, the variational discretization of a forced Lagrange–Dirac system discretizes the Lagrangian and forces separately, and will generally depend on the choice of representation. In this paper we derive a class of Dirac variational integrators with forces that yield well-defined numerical methods that are independent of the choice of representation. We present a numerical simulation to demonstrate how such equivalence-preserving discretizations avoid spurious solutions that otherwise arise from poorly chosen representations.

Singular perturbations in noisy dynamical systems

Consider a deterministic dynamical system in a domain containing a stable equilibrium, e.g., a particle in a potential well. The particle, independent of initial conditions, eventually reaches the bottom of the well. If however, the particle is subjected to white noise, due, e.g., to collisions with a population of smaller, lighter particles comprising the medium through which the particle travels, a dramatic difference in the behaviour of the Brownian particle occurs. The particle will exit the well. The natural questions then are how long will it take for it to exit and from where on the boundary of the domain of attraction of the deterministic equilibrium (the rim of the well) will it exit. We compute the mean first passage time to the boundary and the mean probabilities of the exit positions. When the noise is small each quantity satisfies a singularly perturbed deterministic boundary value problem. We treat the problem by the method of matched asymptotic expansions (MAE) and generalizations thereof. MAE has been used successfully to solve problems in many applications. However, there exist problems for which MAE does not suffice. Among these are problems exhibiting boundary layer resonance, i.e., the problem of ‘spurious solutions’, which led some to conclude that this was ‘the failure of MAE’. We present a physical argument and four mathematical arguments to modify or augment MAE to make it successful. Finally, we discuss applications of the theory.