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One of the challenges in phase measuring deflectometry is to retrieve the wavefront from objects that present discontinuities or non-differentiable gradient fields. Here, we propose the integration of such gradients fields based on an Lp-norm minimization problem. The solution of this problem results in a nonlinear partial differential equation, which can be solved with a fast and well-known numerical methods and doesn't depend on external parameters. Numerical reconstructions on both synthetic and experimental data are presented that demonstrate the capability of the proposed method.

In this paper, we introduce a total variation based variational model for denoising wrapped phase images. Our model improves on former methods by preserving discontinuities of the phase map and enforcing the fundamental Pythagorean trigonometric identity between the real and imaginary parts of the phase map enhancing the quality of the restored phase. The existence and uniqueness of the solution of our model is proven using standard methods. Further, we provide a fast fixed point method for finding the numerical solution and prove its convergence. Experiments on both synthetic and real patterns verify our findings.

In this paper, we analyze and evaluate suitable preconditioning techniques to improve the performance of the $L^p$-norm phase unwrapping method. We consider five preconditioning techniques commonly found in the literature, and analyze their performance with different sizes of wrapped-phase maps. Keywords.- Phase unwrapping, $L^p$-norm based method, Preconditioning techniques.

We present a variational method for recovering the phase term from the information obtained from phase-shifting methods. First we introduce the new method based on a variational approach and then describe the numerical solution of the proposed cost function, which results in a simple algorithm. Numerical experiments with both synthetic and real fringe patterns shows the accuracy and simplicity of the resulting algorithm., Comment: Published in International Journal of Computer Mathematics Volume 96, 2019 - Issue 6

In this work we reformulate the method presented in App. Opt. 53:2297 (2014) as a constrained minimization problem using the augmented Lagrangian method. First we introduce the new method and then describe the numerical solution, which results in a simple algorithm. Numerical experiments with both synthetic and real fringe patterns show the accuracy and simplicity of the resulting algorithm., Comment: Article submitted to Journal of Algorithms and Computational Technology for publication

Let $\psi$ be a positive function defined near the origin such that $\lim_{t\to 0^{+}}\psi(t)=0$. We consider the operator \begin{equation*} T_\theta f(x) = \lim_{\varepsilon\to 0^+} \int_\varepsilon^1 e^{i\gamma(t)}f(x-t) \frac{dt}{t^{\theta}\psi(t)^{1-\theta}}, \end{equation*} where $\gamma$ is a real function with $\lim_{t\to 0^+}|\gamma(t)| = \infty$ and $0 \le \theta \le 1$. Assuming certain regularity and growth conditions on $\psi$ and $\gamma$, we show that $T_1$ is of weak type $(1,1)$., Comment: 10 pages

In this paper we study the harmonic functions and the Dirichlet eigenfunctions of the Hata set, and their restrictions to the interval $[0,1]$, its main edge. We prove that these restrictions of the harmonic functions are singular, \ie monotone and with zero derivatives almost everywhere, and provide numerical evidence that this is also the case for the eigenfunctions., Comment: 7 figures

In this paper we study the boundary limit properties of harmonic functions on $\mathbb R_+\times K$, the solutions $u(t,x)$ to the Poisson equation \[ \frac{\partial^2 u}{\partial t^2} + \Delta u = 0, \] where $K$ is a p.c.f. set and $\Delta$ its Laplacian given by a regular harmonic structure. In particular, we prove the existence of nontangential limits of the corresponding Poisson integrals, and the analogous results of the classical Fatou theorems for bounded and nontangentially bounded harmonic functions., Comment: 22 pages

We show that it is possible to obtain numerical solutions to quantum mechanical problems involving a fractional Laplacian, using a collocation approach based on Little Sinc Functions (LSF), which discretizes the Schr\"odinger equation on a uniform grid. The different boundary conditions are naturally implemented using sets of functions with the appropriate behavior. Good convergence properties are observed. A comparison with results based on a WKB analysis is performed., Comment: 13 pages, 5 figures, 3 tables

In this paper we generalize and improve a method for calculating the period of a classical oscillator and other integrals of physical interest, which was recently developed by some of the authors. We derive analytical expressions that prove to be more accurate than those commonly found in the literature, and test the convergence of the series produced by the approach., Comment: 11 pages, 5 figures

We develop a simple method to obtain approximate analytical expressions for the period of a particle moving in a given potential. The method is inspired to the Linear Delta Expansion (LDE) and it is applied to a large class of potentials. Precise formulas for the period are obtained., Comment: 5 pages, 4 figures