Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

Diffusion, convection and erosion on R3 x S2 and their application to the enhancement of crossing fibers

In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations (erosions) on the space R3 x S2 of 3D-positions and orientations naturally embedded in the group SE(3) of 3D-rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on R3 x S2 and can be solved by R3 x S2-convolution with the corresponding Green’s functions or by a finite difference scheme. The left-invariant Hamilton-Jacobi equations are Bellman equations of cost processes on R3 x S2 and they are solved by a morphological R3 x S2-convolution with the corresponding Green’s functions. We will reveal the remarkable analogy between these erosions/dilations and diffusions. Furthermore, we consider pseudo-linear scale spaces on the space of positions and orientations that combines dilation and diffusion in a single evolution. In our design and analysis for appropriate linear, non-linear, morphological and pseudo-linear scale spaces on R3 x S2 we employ the underlying differential geometry on SE(3), where the frame of left-invariant vector fields serves as a moving frame of reference. Furthermore, we will present new and simpler finite difference schemes for our diffusions, which are clear improvements of our previous finite difference schemes. We apply our theory to the enhancement of fibres in magnetic resonance imaging (MRI) techniques (HARDI and DTI) for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. We provide experiments of our crossing-preserving (non-linear) left-invariant evolutions on neural images of a human brain containing crossing fibers

Adopting Scenario-Based approach to solve optimal reactive power Dispatch problem with integration of wind and solar energy using improved Marine predator algorithm

The penetration of renewable energy resources into electric power networks has been increased considerably to reduce the dependence of conventional energy resources, reducing the generation cost and greenhouse emissions. The wind and photovoltaic (PV) based systems are the most applied technologies in electrical systems compared to other technologies of renewable energy resources. However, there are some complications and challenges to incorporating these resources due to their stochastic nature, intermittency, and variability of output powers. Therefore, solving the optimal reactive power dispatch (ORPD) problem with considering the uncertainties of renewable energy resources is a challenging task. Application of the Marine Predators Algorithm (MPA) for solving complex multimodal and non-linear problems such as ORPD under system uncertainties may cause entrapment into local optima and suffer from stagnation. The aim of this paper is to solve the ORPD problem under deterministic and probabilistic states of the system using an improved marine predator algorithm (IMPA). The IMPA is based on enhancing the exploitation phase of the conventional MPA. The proposed enhancement is based on updating the locations of the populations in spiral orientation around the sorted populations in the first iteration process, while in the final stage, the locations of the populations are updated their locations in adaptive steps closed to the best population only. The scenario-based approach is utilized for uncertainties representation where a set of scenarios are generated with the combination of uncertainties the load demands and power of the renewable resources. The proposed algorithm is validated and tested on the IEEE 30-bus system as well as the captured results are compared with those outcomes from the state-of-the-art algorithms. A computational study shows the superiority of the proposed algorithm over the other reported algorithms

Oxygen transport enhancement by functionalized magnetic nanoparticles (FMP) in bioprocesses

The enhancement of fluid properties, namely thermal conductivity and mass diffusivity for a wide range of applications, through the use of nanosized particles’ suspensions has been gathering increasing interest in the scientific community. In previous studies, Olle et al. (2006) showed an enhancement in oxygen absorption to aqueous solutions of up to 6-fold through the use of functionalized nanosized magnetic particles with oleic acid coating. Krishnamurthy et al. (2006) showed a remarkable 26-fold enhancement in dye diffusion in water. These two publications are landmarks in mass transfer enhancement in chemical systems through the use of nanoparticles. The central goal of this Ph.D. thesis was to develop functionalized magnetic nanoparticles to enhance oxygen transport in bioprocesses. The experimental protocol for magnetic nanoparticles synthesis and purification adopted in this thesis is a modification of that reported by Olle et al. (2006). This is facilitated by employing twice the quantity of ammonia, added at a slower rate, and by filtering the final nanoparticle solution in a cross-flow filtration modulus against 55 volumes of distilled water. This modification in the protocol resulted in improved magnetic nanoparticles with measurably higher mass transfer enhancement. Magnetic nanoparticles with oleic acid and Hitenol-BC coating were screened for oxygen transfer enhancement, since these particles are relatively inexpensive and easy to synthesize. A glass 0.5-liter reactor was custom manufactured specifically for oxygen transport studies in magnetic nanoparticles suspensions. The reactor geometry, baffles and Rushton impeller are of standard dimensions. Mass transfer tests were conducted through the use of the sulphite oxidation method, applying iodometric back-titration. A 3-factor central composite circumscribed design (CCD) was adopted for design of experiments in order to generate sufficiently informative data to model the effect of magnetic nanoparticles on interfacial area and mass transfer coefficient. The parameters ranges used were: 250-750 rpm for stirring speed, 0-2 vvm for aeration and 0-0.00120 g g−1 magnetic nanoparticles mass fraction. It was found that 36 nm-sized nanoparticles produced during the course of this dissertation enhanced the volumetric mass transfer coefficient up to 3.3-fold and the interfacial area up to 3.3-fold in relation to gas-liquid dispersions without nanoparticles. These results are concordant with previously published enhancement data (kLa enhancement by 7.1-fold and a enhancement by 4.1-fold) (Olle et al. 2006). The magnetic nanoparticles synthesized in this thesis were stable (constant diameter) over a 1wide pH range (2-9). Statistical regression models showed that both kLa and a have high sensitivity to the nanoparticles loading. Empirical correlation models were derived for kLa and for interfacial area, a, as function of physical properties and nanoparticles loading. These correlations lay out a methodology that can help the scientific community to design and scale-up oxygen transfer systems that are based on nanoparticle suspensions