Lixin Shen

Fluid flow and optical flow

The connection between fluid flow and optical flow is explored in typical flow visualizations to provide a rational foundation for application of the optical flow method to image-based fluid velocity measurements. The projected-motion equations are derived, and the physics-based optical flow equation is given. In general, the optical flow is proportional to the path-averaged velocity of fluid or particles weighted with a relevant field quantity. The variational formulation and the corresponding Euler-Lagrange equation are given for optical flow computation. An error analysis for optical flow computation is provided, which is quantitatively examined by simulations on synthetic grid images. Direct comparisons between the optical flow method and the correlation-based method are made in simulations on synthetic particle images and experiments in a strongly excited turbulent jet.

Global Luminescent Oil-Film Skin-Friction Meter

This paper describes a global luminescent oil-film skin-friction meter that is particularly useful in global skinfriction diagnostics for complex flows. This method is developed based on the relationship between the oil-film thickness and luminescent intensity of an oil film doped with luminescent molecules. The projected thin oil-film equation is given in the image plane, which relates skin friction with the normalized luminescent intensity. A variational formulation with a smoothness constraint on skin friction is given, and the corresponding Euler– Langrage equations are solved to obtain a snapshot solution for a relative skin-friction field. Successive snapshot solutions are superposed to reconstruct a complete relative skin-friction field, and the corresponding absolute field can be further determined by in situ calibration. This method is examined through numerical simulation and experiments.

An analysis of physics-based optical flow

A variational solution of the physics-based optical flow equation is studied for extraction of high-resolution velocity fields from flow visualization images. The solution to preserve discontinuities in a fluid velocity field is mathematically justified. The uniqueness of the solution and the convergence of a successive approximation sequence are proven. A numerical algorithm is developed and examined through numerical experiments.

Separation of Undersampled Composite Signals Using the Dantzig Selector with Overcomplete Dictionaries

In many applications one may acquire a composition of several signals that may be corrupted by noise, and it is a challenging problem to reliably separate the components from one another without sacrificing significant details. Adding to the challenge, in a compressive sensing framework, one is given only an undersampled set of linear projections of the composite signal. In this paper, we propose using the Dantzig selector model incorporating an overcomplete dictionary to separate a noisy undersampled collection of composite signals, and present an algorithm to efficiently solve the model. nThe Dantzig selector is a statistical approach to finding a solution to a noisy linear regression problem by minimizing the

A fast and accurate algorithm for ℓ 1 minimization problems in compressive sampling

ell_1

Finding Dantzig selectors with a proximity operator based fixed-point algorithm

norm of candidate coefficient vectors while constraining the scope of the residuals. If the underlying coefficient vector is sparse, then the Dantzig selector performs well in the recovery and separation of the unknown composite signal. In the following, we propose a proximity operator based algorithm to recover and separate unknown noisy undersampled composite signals through the Dantzig selector. We present numerical simulations comparing the proposed algorithm with the competing Alternating Direction Method, and the proposed algorithm is found to be faster, while producing similar quality results. Additionally, we demonstrate the utility of the proposed algorithm in several experiments by applying it in various domain applications including the recovery of complex-valued coefficient vectors, the removal of impulse noise from smooth signals, and the separation and classification of a composition of handwritten digits.

Nesterov's algorithm solving dual formulation for compressed sensing

An accurate and efficient algorithm for solving the constrained ℓ1-norm minimization problem is highly needed and is crucial for the success of sparse signal recovery in compressive sampling. We tackle the constrained ℓ1-norm minimization problem by reformulating it via an indicator function which describes the constraints. The resulting model is solved efficiently and accurately by using an elegant proximity operator-based algorithm. Numerical experiments show that the proposed algorithm performs well for sparse signals with magnitudes over a high dynamic range. Furthermore, it performs significantly better than the well-known algorithm NESTA (a shorthand for Nesterov’s algorithm) and DADM (dual alternating direction method) in terms of the quality of restored signals and the computational complexity measured in the CPU-time consumed.

Computing the proximity operator of the ℓ p norm with 0 < p < 1

A simple iterative method for finding the Dantzig selector, designed for linear regression problems, is introduced. The method consists of two stages. The first stage approximates the Dantzig selector through a fixed-point formulation of solutions to the Dantzig selector problem; the second stage constructs a new estimator by regressing data onto the support of the approximated Dantzig selector. The proposed method is compared to an alternating direction method. The results of numerical simulations using both the proposed method and the alternating direction method on synthetic and real-world data sets are presented. The numerical simulations demonstrate that the two methods produce results of similar quality; however the proposed method tends to be significantly faster.

One-bit compressive sampling via ℓ 0 minimization

We develop efficient algorithms for solving the compressed sensing problem. We modify the standard @?1 regularization model for compressed sensing by adding a quadratic term to its objective function so that the objective function of the dual formulation of the modified model is Lipschitz continuous. In this way, we can apply the well-known Nesterov algorithm to solve the dual formulation and the resulting algorithms have a quadratic convergence. Numerical results presented in this paper show that the proposed algorithms outperform significantly the state-of-the-art algorithm NESTA in accuracy.

Minimizing Compositions of Functions Using Proximity Algorithms with Application in Image Deblurring

Sparse modelling with the l p norm of 0 ≤ p ≤ 1 requires the availability of the proximity operator of the l p norm. The proximity operators of the l0 and l1 norms are the well-known hard- and soft-thresholding estimators, respectively. In this study, the authors give a complete study on the properties of the proximity operator of the l p norm. Based on these properties, explicit formulas of the proximity operators of the l1/2 norm and l2/3 norm are derived with simple proofs; for other values of p, an iterative Newtons method is developed to compute the proximity operator of the l p norm by fully exploring the available proximity operators of the l0, l1/2, l2/3, and l1 norms. As applications, the proximity operator of the l p norm with 0 ≤ p ≤ 1 is applied to the l p -regularisation for compressive sensing and image restoration.