An invariant region for the collisional dynamics of two bodies on Keplerian orbits

We study the dynamics of two bodies moving on elliptic Keplerian orbits around a fixed center of attraction and interacting only by means of elastic or inelastic collisions. We show that there exists a bounded invariant region: for suitable values of the total energy and the total angular momentum (explicitly computable) the orbits of the bodies remain elliptic, whatever are the number and the details of the collisions. The invariant region exists also in the case of two bodies interacting by short range potential

A conjugate gradient like method for p-norm minimization in functional spaces.

We develop an iterative algorithm to recover the minimum p-norm solution of the functional linear equation Ax=b, where A:X⟶Y is a continuous linear operator between the two Banach spaces X=Lp, 11, with x∈X and b∈Y. The algorithm is conceived within the same framework of the Landweber method for functional linear equations in Banach spaces proposed by Schöpfer et al. (Inverse Probl 22:311–329, 2006). Indeed, the algorithm is based on using, at the n-th iteration, a linear combination of the steepest current “descent functional” A∗J(b−Axn) and the previous descent functional, where J denotes a duality map of the Banach space Y. In this regard, the algorithm can be viewed as a generalization of the classical conjugate gradient method on the normal equations in Hilbert spaces. We demonstrate that the proposed iterative algorithm converges strongly to the minimum p-norm solution of the functional linear equation Ax=b and that it is also a regularization method, by applying the discrepancy principle as stopping rule. According to the geometrical properties of Lp spaces, numerical experiments show that the method is fast, robust in terms of both restoration accuracy and stability, promotes sparsity and reduces the over-smoothness in reconstructing edges and abrupt intensity changes

Analysis of Reconstructions Obtained Solving lp-Penalized Minimization Problems

Most of the inverse problems arising in applied electromagnetics come from an underdetermined direct problem, this is the case, for instance, of spatial resolution enhancement. This implies that no unique inverse operator exists; therefore, additional constraints must be imposed on the sought solution. When dealing with microwave remote sensing, among the possible choices, the minimum p-norm constraint, with 1 < p 64 2, allows obtaining reconstructions in Hilbert (p = 2) and Banach (1 < p < 2) subspaces. Recently, it has been experimentally proven that reconstructions in Banach subspaces mitigate the oversmoothing and the Gibbs oscillations that typically characterize reconstructions in Hilbert subspaces. However, no fair intercomparison among the different reconstructions has been done. In this paper, a mathematical framework to analyze reconstructions in Hilbert and Banach subspaces is provided. The reconstruction problem is formulated as the solution of a p-norm constrained minimization problem. Two signals are considered that model abrupt and spot-like discontinuities. The study, undertaken in both the noise-free and the noisy cases, demonstrates that lp reconstructions for 1 < p < 2 significantly outperform the l2 ones when spot-like discontinuities are considered; when dealing with abrupt discontinuities, l2 and lp reconstructions are characterized by similar performance; however, lp reconstructions exhibit oscillations when the background is not properly accounted for

Conjugate gradient method in Hilbert and Banach spaces to enhance the spatial resolution of radiometer data

International audienceIn this paper, a new computer-time effective iterative method is proposed to enhance the spatial resolution of microwave radiometer data in both Hilbert and Banach spaces. The reconstruction method is based on the conjugate gradient (CG) method. CG is a well-known method in electromagnetics, where it is commonly used to address inverse problems in Hilbert spaces. However, the method has never been applied to inverse problems related to spatial resolution enhancement of microwave remotely sensed measurements. In this paper, CG is adapted to the microwave radiometer case and exploited to reconstruct the brightness field at enhanced spatial resolution. Then, CG is, for the first time, extended to Banach spaces by means of duality maps and then applied to enhance the radiometer spatial resolution. Experiments undertaken on both simulated and actual radiometer data confirm the soundness of the proposed approach in Banach spaces and demonstrate that CG is able to provide a reconstruction accuracy similar to the conventional Landweber method, but with a significantly reduced processing time

On the spatial resolution enhancement of microwave radiometer data in banach spaces

A reconstruction technique, mathematically based on a generalization of the gradient method in Banach spaces, is first proposed to enhance the spatial resolution of radiometer earth observation measurements. This approach allows reducing the over-smoothing effects and the oscillations that are often present in standard Hilbert-spaces procedures without any drawback on the numerical complexity. Experiments undertaken on a data set consisting of both simulated and actual 2-D special sensor microwave imager radiometer measurements show the accuracy and the effectiveness of the proposed technique. A typical radiometer scene is processed in few minutes by a standard PC processor. Furthermore, since the proposed approach is iterative, the processing time increases slowly with the problem's size

Lp‐norm regularization approaches in variational data assimilation

International audienceThis article presents a formulation of the 4DVar objective function using as a penalty term a Lp‐norm with 1 < p < 2. This approach is motivated by the nature of the problems encountered in data assimilation, for which such a norm may be more suited to tackle the generalized Gaussian distribution of the variables. It also aims at making a compromise between the L2‐norm that tends to oversmooth the solution, and the L1‐norm that tends to ”oversparsify” it, in addition to making the problem non‐smooth. We show the benefits of using this strategy on different setups through numerical experiments where the background and measurements noise covariance are known and a sharp solution is expected

MRI multicomponent relaxometry based on compressive sensing

A novel application of 1-norm minimization in the MRI field is presented. The novel methodology, called Intra Voxel Analysis (IVA), by combining different acquisitions with standard resolution, is able to investigate the presence of different contributions, i.e., of different tissues, inside each imaged voxel. The approach is somehow similar to spectroscopy, but instead of searching different resonance frequencies, it discriminates within each voxel the tissues characterized by different spin-spin relaxation times. The proposed methodology is able to work on MR images acquired at full resolution by using any acquisition scheme. A phantom has been built and images for testing the approach

Intra voxel analysis in magnetic resonance imaging

A technique for analyzing the composition of each voxel, in the magnetic resonance imaging (MRI) framework, is presented. By combining different acquisitions, a novel methodology, called intra voxel analysis (IVA), for the detection of multiple tissues and the estimation of their spin-spin relaxation times is proposed. The methodology exploits the sparse Bayesian learning (SBL) approach in order to solve a highly underdetermined problem imposing the solution sparsity. IVA, developed for spin echo imaging sequence, can be easily extended to any acquisition scheme. For validating the approach, simulated and real data sets are considered. Monte Carlo simulations have been implemented for evaluating the performances of IVA compared to methods existing in literature. Two clinical datasets acquired with a 3T scanner have been considered for validating the approach. With respect to other approaches presented in literature, IVA has proved to be more effective in the voxel composition analysis, in particular in the case of few acquired images. Results are interesting and very promising: IVA is expected to have a remarkable impact on the research community and on the diagnostic field