Tomographic reconstruction using prior with adaptively binary neighborhood

Bayesian approaches can be used to improve ill-posed image reconstruction by regularizing the inverse solution using spacial or temporal neighborhood information. This paper proposes a prior with adaptively binary neighborhood (ABN prior) for statistical tomographic reconstruction. With binary weight map adapted to local image features, the proposed prior can lead to improved reconstruction by including relevant pixels belonging to similar structures and excluding those not. A two-step algorithm is also put forward for tomographic reconstruction using the proposed prior. Experiments using both simulated and clinical computerized tomography (CT) data are performed to validate the reconstructions with the proposed ABN prior.     

A statistical method for characterizing the noise in nonlinearly reconstructed images from undersampled MR data: The POCS example

The projection-onto-convex-sets (POCS) algorithm is a powerful tool for reconstructing high-resolution images from undersampled k-space data. It is a nonlinear iterative method that attempts to estimate values for missing data. The convergence of the algorithm and its other deterministic properties are well established, but relatively little is known about how noise in the source data influences noise in the final reconstructed image. In this paper, we present an experimental treatment of the statistical properties in POCS and investigate 12 stochastic models for its noise distribution beside its nonlinear point spread functions. Statistical results show that as the ratio of the missing k-space data increases, the noise distribution in POCS images is no longer Rayleigh as with conventional linear Fourier reconstruction. Instead, the probability density function for the noise is well approximated by a lognormal distribution. For small missing data ratios, however, the noise remains Rayleigh distributed. Preliminary results show that in the presence of noise, POCS images are often dominated by POCS-enhanced noise rather than POCS-induced artifacts. Implicit in this work is the presentation of a general statistical method that can be used to assess the noise properties in other nonlinear reconstruction algorithms.     

The stress concentration factor for slightly roughened random surfaces: Analytical solution

We derive an analytical solution to the stress concentration factor (_kt)$(k_t)$ for slightly roughened random surfaces. Topology is assumed to possess Gaussian distribution of heights and auto correlation length, ACL  . For our development, we combine Gao’s first-order perturbation method, the Hilbert transform, and an energy conservation principal related to the Parseval theorem.The root-mean-square (RMS) value of _kt$k_t$ results in a function of the ratio RMS-roughness to ACL. The derived formula agrees with experimental results previously reported. The results provide insight for more efficient design.     

Change-of-variable interval stochastic perturbation method for hybrid uncertain structural-acoustic systems with random and interval variables

The response analysis from hybrid uncertain structural-acoustic systems with random and interval variables (HUSAS) plays an important role in the optimal design of structural-acoustic systems. In this work, a hybrid uncertain numerical method known as the change-of-variable interval stochastic perturbation method (CVISPM) is proposed to predict the interval of the response probability density function and the response confidence interval of a HUSAS. This method is based on perturbation analysis and the change-of-variable technique. In the proposed method, the response of a HUSAS is approximated as a linear function of random variables using the stochastic perturbation analysis. According to the approximated linear relationships between the response and the random variables, the change-of-variable technique is introduced to calculate the response probability density function. Based on the response probability density function, the interval perturbation approach is used to predict the interval of the response probability density function and the response confidence interval. A numerical example of a shell structural-acoustic system with random and interval variables was employed to verify the effectiveness and precision of the proposed method.     

On Exit Times of a Multivariate Random Walk and Its Embedding in a Quasi Poisson Process

Abstract

We introduce and analyze a delayed renewal process  = {τ₀,τ₁,…} marked by a multivariate random walk (,) and its behavior about fixed levels to be crossed by one of the components of (,). We derive the joint distribution of first passage time τ_ρ, pre-exit time τ_ρ?1 (i.e., the instant one phase prior to the first passage time), and the respective values of (,) at τ_ρ and τ_ρ?1 in a closed form. The results obtained are then applied to a multivariate quasi Poisson process Π, forming a random walk ((Π),) embedded in Π over . Processes like these can model various phenomena including stock market and option trading.

One of the central issues in the investigation of ((Π),) is to obtain the information about Π at any moment of time in random vicinities of τ_ρ and τ_ρ?1 previously available only upon . The results offer, again, closed form functionals. Numerous examples throughout the paper illustrate introduced constructions and connect the results with real-world applications, most prominently the stock market.     

Limit at Zero of the First-Passage Time Density and the Inverse Problem for One-Dimensional Diffusions

Abstract

We study the limit at zero of the first-passage time density of a one-dimensional diffusion process over a moving boundary and we also deal with the inverse first-passage time problem, which consists of determining the boundary shape when the first-passage density is known. Our results generalize the analogous ones already known for Brownian motion. We illustrate some examples for which the results are obtained analytically and by a numerical procedure.     

Attractors for a Damped Stochastic Wave Equation of Sine–Gordon Type with Sublinear Multiplicative Noise

Abstract

The existence of compact random attractors is proved for a damped stochastic wave equation of Sine–Gordon type with sublinear multiplicative noise under homogeneous Dirichlet boundary condition. To be important, in this note a precise estimate of upper bound of Hausdorff dimension of the random attractors is obtained in lower dimension.     

An Impact of Stochastic Dynamic Boundary Conditions on the Evolution of the Cahn-Hilliard System

Abstract

Nonlinear systems are often subject to random influences. Sometimes the noise enters the system through physical boundaries and this leads to stochastic dynamic boundary conditions. A dynamic, as opposed to static, boundary condition involves the time derivative as well as spatial derivatives for the system state variables on the boundary. Although stochastic static (Neumann or Dirichet type) boundary conditions have been applied for stochastic partial differential equations, not much is known about the dynamical impact of stochastic dynamic boundary conditions. The purpose of this article is to study possible impacts of stochastic dynamic boundary conditions on the long term dynamics of the Cahn-Hilliard equation arising in the materials science. We show that the dimension estimation of the random attractor increases as the coefficient for the dynamic term in the stochastic dynamic boundary condition decreases. However, the dimension of the random attractor is not affected by the corresponding stochastic static boundary condition.