Geometric Characterizations for Variational Minimization Solutions of the 3-Body Problem

In this paper, we prove that for any given positive masses the variational minimization solutions of the 3-body problem in R^3 or R^2 are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772, and that the variational minimization solutions of the circular rostricted 3-body problem in R^3 or R^2 are also planar equilateral triangle circular solutions.

Adaptive variational models for image decomposition combining staircase reduction and texture extraction

New models for image decomposition are proposed which separate an image into a cartoon, consisting only of geometric objects, and an oscillatory component, consisting of textures or noise. The proposed models are given in a variational formulation with adaptive regularization norms for both the cartoon and texture parts. The adaptive behavior preserves key features such as object boundaries and textures while avoiding staircasing in what should be smooth regions. This decomposition is computed by minimizing a convex functional which depends on the two variables u and v, alternatively in each variable. Experimental results and comparisons to validate the proposed models are presented.

Seislet transform denoising based on total variation minimization

Attenuation of noise is a persistent problem in seismic exploration. The authors use conventional denoising method to remove noise which may cause vibration near the discontinuity called pseudo-Gibbs artifact.In order to remove the artifact,the study proposed a method combining the seislet transform and total variation minimization. Firstly,the data are converted into the seislet transform domain. Secondly,the hard threshold was used for eliminating the noise and keep useful signal,which is the initial input for the next step. Finally,total variation minimization dealed with denoised data to recover boundary information and further eliminated the noise. Synthetic data examples show that the method has feasibility in eliminating random noise and protecting detailed signal,and also shows better results than the classic f-x deconvolution. The field data example also shows effective in practice. It can remove the noise and preserve the discontinuity signal at the same time.

Wavelet inverse scale space for image restoration

This paper proposes a model for image restoration by combining the wavelet shrinkage and inverse scale space （ISS） method. The ISS is applied to the wavelet representation to modify the retained wavelet coefficients, and the coefficients smaller than the threshold are set to zero. The curvature term of the ISS can remove the edge artifacts and preserve sharp edges. For the multiscale interpretation of the ISS and the multiscale property of the wavelet representation, small details are preserved. This paper illustrates that the wavelet ISS model can be deduced from the wavelet based on a total variation minimization problem. A stopping criterion is obtained from this minimization in the sense of the Bregman distance in the wavelet domain. Numerical examples show the improvement for the image denoising with the proposed method in the sense of the signal to noise ratio and with fewer details remained in the residue.

Sparse Representation by Frames with Signal Analysis

The use of frames is analyzed in Compressed Sensing (CS) through proofs and experiments. First, a new generalized Dictionary-Restricted Isometry Property (D-RIP) sparsity bound constant for CS is established. Second, experiments with a tight frame to analyze sparsity and reconstruction quality using several signal and image types are shown. The constant is used in fulfilling the definition of D-RIP. It is proved that k-sparse signals can be reconstructed if by using a concise and transparent argument1. The approach could be extended to obtain other D-RIP bounds (i.e. ). Experiments contrast results of a Gabor tight frame with Total Variation minimization. In cases of practical interest, the use of a Gabor dictionary performs well when achieving a highly sparse representation and poorly when this sparsity is not achieved.

Comparison of MRI Under-Sampling Techniques for Compressed Sensing with Translation Invariant Wavelets Using FastTestCS: A Flexible Simulation Tool

A sparsifying transform for use in Compressed Sensing (CS) is a vital piece of image reconstruction for Magnetic Resonance Imaging (MRI). Previously, Translation Invariant Wavelet Transforms (TIWT) have been shown to perform exceedingly well in CS by reducing repetitive line pattern image artifacts that may be observed when using orthogonal wavelets. To further establish its validity as a good sparsifying transform, the TIWT is comprehensively investigated and compared with Total Variation (TV), using six under-sampling patterns through simulation. Both trajectory and random mask based under-sampling of MRI data are reconstructed to demonstrate a comprehensive coverage of tests. Notably, the TIWT in CS reconstruction performs well for all varieties of under-sampling patterns tested, even for cases where TV does not improve the mean squared error. This improved Image Quality (IQ) gives confidence in applying this transform to more CS applications which will contribute to an even greater speed-up of a CS MRI scan. High vs low resolution time of flight MRI CS re-constructions are also analyzed showing how partial Fourier acquisitions must be carefully addressed in CS to prevent loss of IQ. In the spirit of reproducible research, novel software is introduced here as FastTestCS. It is a helpful tool to quickly develop and perform tests with many CS customizations. Easy integration and testing for the TIWT and TV minimization are exemplified. Simulations of 3D MRI datasets are shown to be efficiently distributed as a scalable solution for large studies. Comparisons in reconstruction computation time are made between the Wavelab toolbox and Gnu Scientific Library in FastTestCS that show a significant time savings factor of 60×. The addition of FastTestCS is proven to be a fast, flexible, portable and reproducible simulation aid for CS research.

Variational minimizing parabolic and hyperbolic orbits for the restricted 3-body problems Dedicated to my Teacher Professor Yang Wannian on the Occasion of his 75th Birthday

Using variational minimizing methods,we prove the existence of the odd symmetric parabolic or hyperbolic orbit for the restricted 3-body problems with weak forces.

Geometric characterizations for variational minimizing solutions of charged 3-body problems

We study the charged 3-body problem with the potential function being （-a）-homogeneous on the mutual distances of any two particles via the variational method and try to find the geometric characterizations of the minimizers. We prove that if the charged 3-body problem admits a triangular central configuration, then the variational minimizing solutions of the problem in the τ/2-antiperiodic function space are exactly defined by the circular motions of this triangular central configuration.

CT local reconstruction of solid rocket motor based on PI-POCS-TVM

In order to test the defects of solid rocket motor(SRM)accurately and efficiently,the computed tomography(CT)inspection scheme for SRM's defects was investigated and a CT local reconstruction algorithm called prior information-projection onto convex sets-total variation minimization(PI-POCS-TVM)was developed.The SRM was first inspected by industrial CT(ICT)to generate a low-resolution SRM image used as the prior information,then high-resolution local inspection was carried out for SRM's defects.To validate the effectiveness of the CT inspection scheme for SRM's defects,one SRM was inspected by a narrow fan beam ICT system.Filtered back projection(FBP)algorithm,POCS,POCS-TVM and PI-POCS-TVM algorithms were applied to reconstruct CT images.The performance of these algorithms was also compared.Results demonstrated the effectiveness of the CT inspection scheme for SRM's defects.Moreover,the PI-POCS-TVM algorithm has better local reconstruction image quality than the other three algorithms,showing great significance for accurate measurement of SRM's defects.