Alejandro Téllez-Quiñones

Inhomogeneous phase shifting: an algorithm for nonconstant phase displacements

In this work, we have developed a different algorithm than the classical one on phase-shifting interferometry. These algorithms typically use constant or homogeneous phase displacements and they can be quite accurate and insensitive to detuning, taking appropriate weight factors in the formula to recover the wrapped phase. However, these algorithms have not been considered with variable or inhomogeneous displacements. We have generalized these formulas and obtained some expressions for an implementation with variable displacements and ways to get partially insensitive algorithms with respect to these arbitrary error shifts.

Phase-shifting algorithms for a finite number of harmonics: first-order analysis by solving linear systems

From generalized phase-shifting equations, we propose a simple linear system analysis for algorithms with equally and nonequally spaced phase shifts. The presence of a finite number of harmonic components in the fringes of the intensity patterns is taken into account to obtain algorithms insensitive to these harmonics. The insensitivity to detuning for the fundamental frequency is also considered as part of the description of this study. Linear systems are employed to recover the desired insensitivity properties that can compensate linear phase shift errors. The analysis of the wrapped phase equation is carried out in the Fourier frequency domain.

Nonlinear differential equations for the wavefront surface at arbitrary Hartmann-plane distances

In the Hartmann test, a wave aberration function W is estimated from the information of the spot diagram drawn in an observation plane. The distance from a reference plane to the observation plane, the Hartmann-plane distance, is typically chosen as z=f, where f is the radius of a reference sphere. The function W and the transversal aberrations {X,Y} calculated at the plane z=f are related by two well-known linear differential equations. Here, we propose two nonlinear differential equations to denote a more general relation between W and the transversal aberrations {U,V} calculated at any arbitrary Hartmann-plane distance z=r. We also show how to directly estimate the wavefront surface w from the information of {U,V}. The use of arbitrary r values could improve the reliability of the measurements of W, or w, when finding difficulties in adequate ray identification at z=f.

Phase recovering without phase unwrapping in phase-shifting interferometry by cubic and average interpolation

A simple phase estimation employing cubic and average interpolations to solve the oversampling problem in smooth modulated phase images is described. In the context of a general phase-shifting process, without phase-unwrapping, the modulated phase images are employed to recover wavefront shapes with high fringe density. The problem of the phase reconstruction by line integration of its gradient requires a form appropriate to the calculation of partial derivatives, especially when the phase to recover has higher-order aberration values. This is achieved by oversampling the modulated phase images, and many interpolations can be implemented. Here an oversampling procedure based on the analysis of a quadratic cost functional for phase recovery, in a particular case, is proposed.

Basic Fourier properties for generalized phase shifting and some interesting detuning insensitive algorithms

In this manuscript, some interesting properties for generalized or nonuniform phase-shifting algorithms are shown in the Fourier frequency space. A procedure to find algorithms with equal amplitudes for their sampling function transforms is described. We also consider in this procedure the finding of algorithms that are orthogonal for all possible values in the frequency space. This last kind of algorithms should closely satisfy the first order detuning insensitive condition. The procedure consists of the minimization of functionals associated with the desired insensitivity conditions.

Equations to estimate the wavefront surface in the Hartmann test for lenses: comparison between two wavefront estimations when the Hartmann screen is close to the test lens

Abstract. When testing lenses with Hartmann methods, a wave aberration function W is typically estimated. This W represents the deviations of the wavefront surface w with respect to an ideal wavefront E. In this test, the distance r from the observation screen to the second lens surface is considered, and, as in the case of W, by considering paraxial approximations, two estimations of w can be directly constructed from Hartmann test data without calculating W. We have compared these two estimations by taking into account small r values; a possible and suitable condition to measure some relatively high-power lenses. The importance of estimating w can be useful for improving some optical measurements as power map reconstructions.

Polynomial fitting model for phase reconstruction: interferograms with high fringe density

A data fitting model is proposed to estimate phases from its cosine and sine. The a priori assumption is that the phases to be reconstructed should be expressed by polynomials. The cosine and sine of the phases are obtained from interferograms with high fringe density by generalized phase-shifting techniques The proposed method is employed for phase reconstrution by line integration of the phase gradient or any other phase-unwrapping technique and the fit is achieved by a least-squares minimization.

Built-up index methods and their applications for urban extraction from Sentinel 2A satellite data: discussion

Several built-up indices have been proposed in the literature in order to extract the urban sprawl from satellite data. Given their relative simplicity and easy implementation, such methods have been widely adopted for urban growth monitoring. Previous research has shown that built-up indices are sensitive to different factors related to image resolution, seasonality, and study area location. Also, most of them confuse urban surfaces with bare soil and barren land covers. By gathering the existing built-up indices, the aim of this paper is to discuss some of their advantages, difficulties, and limitations. In order to illustrate our study, we provide some application examples using Sentinel 2A data.

Differentiability of a projection functional in ray-tracing processes: applied study to estimate the coefficients of a single lens with conic surfaces

In optical design, many error functions can be used to generate an optical system with desired characteristics. These error functions are optimized by iterative algorithms. However, these error functions should be theoretically and mathematically differentiable to be optimized. In this paper, the differentiability of an error function is partially justified. The error function herein is called the projection functional. This proposed projection functional can be used to estimate the coefficients of an arbitrary lens with conic surfaces by means of the spot distributions on two planes produced by a fixed Hartmann plate. The differentiability of the projection functional is required to guarantee the existence of its Jacobian matrix, which is a suitable condition to minimize this functional by iterative methods. Numerical examples of the functional minimization are given.

Compensation of the two-stage phase-shifting algorithms in the presence of detuning and harmonics.

The Fourier analysis of two-stage phase-shifting (TSPS) algorithms is growing in interest as a research topic, specifically, the algorithms insensitivity properties to various error sources. The main motivation of this paper is to propose TSPS algorithms that perform well in the face of detuning and harmonics for each of the two sets of interferograms with different or equal reference frequencies. TSPS algorithms based on the development of generalized equations consider both the frequency sampling functions that represent them and nonconstant phase shifts.