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We consider image denoising as the problem of removing spurious oscillations due to noise while preserving edges in the images. We will suggest here how to directly make infinitesimal adjustment to standard variational methods of image denoising, to enhance desirable target assumption of the noiseless image. The standard regularization method is used to define a suitable energy functional to penalize the data fidelity and the smoothness of the solution. This energy functional is tailored so that the region with small gradient is isotropically smoothed whereas in a neighborhood of an edge presented by a large gradient smoothing is allowed only along the edge contour. The regularized solution that arises in this fashion is then the solution of a variational principle. To this end the associated Euler - Lagrange equation needs to be solved numerically and the half-quadratic minimization is generally used to linearize the equation and to derive an iterative scheme. We describe here a method to modify Euler - Largrange equation from commonly used energy functionals, in a way to enhance certain desirable preconceived assumptions of the image, such as edge preservation. From an algorithmic point of view, we may deem this algorithm as a smoothing by a local average with an adaptive gradient-based weight. However, this algorithm may result in noisy edges although the edge is preserved and noise is suppressed in the low-gradient regions of the image. The main focus here is to present an edge-preserving regularization in the aforementioned view point, and to provide an alternative and simple way to modify the existing algorithm to mitigate the phenomena of noisy edges without explicitly defining step where we specify an energy functional to be minimized. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the transport and mixing process of the batch mixers with two different configurations, the centered-blade and offset-blade mixers, by using a dynamical system approach. The 2-D velocity fields of the mixers measured using Particle Image Velocimetry (PIV) are used to identify the Lagrangian coherent structures (LCSs). The results show that the LCSs separate the physical space into two portions. In the case of the center-blade mixer the portion bounded inside the LCS experiences a relatively slow mixing relative to the portion outside of the LCS boundary. However, when the blade position is located near the wall, the LCS becomes more complicated but it still separates regions of fast mixing from a slower one. We develop a heuristic dynamical system model of our mixers to understand how the vorticity strength at the blade tips influences the variation of the LCSs. Finally, we define an appropriate notion of mixing to study the mixing rate of our mixing devices. [ABSTRACT FROM AUTHOR]

In this paper, we present an approach to approximate the Frobenius-Perron transfer operator from a sequence of time-ordered images, that is, a movie dataset. Unlike time-series data, successive images do not provide a direct access to a trajectory of a point in a phase space; more precisely, a pixel in an image plane. Therefore, we reconstruct the velocity field from image sequences based on the infinitesimal generator of the Frobenius-Perron operator. Moreover, we relate this problem to the well-known optical flow problem from the computer vision community and we validate the continuity equation derived from the infinitesimal operator as a constraint equation for the optical flow problem. Once the vector field and then a discrete transfer operator are found, then, in addition, we present a graph modularity method as a tool to discover basin structure in the phase space. Together with a tool to reconstruct a velocity field, this graph-based partition method provides us with a way to study transport behavior and other ergodic properties of measurable dynamical systems captured only through image sequences. [ABSTRACT FROM AUTHOR]