When recovering smooth phases by phase unwrapping algorithms, many noniterative algorithms are available. However, normally those algorithms offer approximations of the current phase that cannot be accurate enough. This is because the majority of them are based on global approaches instead of local ones. Although smooth estimations are not often expected in phase reconstructions for real applications, a smooth initial guess could be useful for robust iterative techniques. Therefore, based on the most recent local polynomial approaches, we propose a simple least-squares fitting of the partial derivatives of the phase, normally estimated from the wrapped operator, by considering local polynomial models of the phase up to the third order. Synthetic and real data of wrapped phases are considered in our work.

Many works have been implemented to describe how seismic vertical displacement maps can be generated from synthetic aperture radar (SAR) images. The theory that supports these works is a very interesting field in remote sensing which is called differential interferometric SAR (DInSAR). However, behind this theory, there are many limitations and ambiguities that should be clarified before concluding specific results from the interpretation of subsidence maps. In this work, some of these drawbacks will be discussed, by following a typical subsidence map reconstruction with Sentinel-1 satellite images of a seismic region in Mexico. Particularly, the correct estimation of the unwrapped phase, is a crucial step for future analysis, like in time-series studies on subsidence ratios. In order to do this, a sophisticated routine called statistical-cost network-flow phase-unwrapping algorithm (SNAPHU), provided by the European Space Agency (ESA) in free software tools, is commonly applied to general SAR interferograms. However, in this particular study case, focused on simple dual-pair interferometry, atypical outputs obtained from SNAPHU were observed. Due to this inconsistency, the alternate solution proposed here is based on the so-called phase unwrapping by an accumulation of residual maps (ARM) algorithm. Our comparative analysis suggests that ARM outperforms SNAPHU and other tested methods in terms of unwrapped estimates.

Recently, a theory on local polynomial approximations for phase-unwrapping algorithms, considering a state space analysis, has been proposed in Appl. Opt. 56,29(2017)APOPAI 0003-6935 10.1364/AO.56.000029 . Although this work is a suitable methodology to deal with relatively low signal to noise ratios observed in the wrapped phase, the methodology has been developed only for local-polynomial phase models of order 1. The resultant proposal is an interesting Kalman filter approach for estimating the coefficient or state vectors of these local plane models. Thus, motivated by this approach and simple Bayesian theory, and considering our previous research on local polynomial models up to the third order [ Appl. Opt. 58,436(2019)APOPAI 0003-6935 10.1364/AO.58.000436 ], we propose an equivalent methodology based on a simple maximuma posterioriestimation, but considering a different state space: difference vectors of coefficients for the current high-order polynomial models. Specific estimations of the covariance matrices for difference vectors, as well as noise covariance matrices involved with the correct estimation of coefficient vectors, are proposed and reconstructions with synthetic and real data are provided.

In this work, we present a panoramic digital holographic system for the first time capable of obtaining 3D information of a quasi-cylindrical object by using a conical mirror. The proposed panoramic digital holographic system is able to scan the entire surface of the object to determine the amplitude and phase simultaneously. This paper demonstrates the feasibility of analyzing quasi-cylindrical objects in a short time (0.5 s) with a single camera and a minimum number of optical components. In addition, it can be applied to determine not only topographic measurement of the cylindrical surface but also measurements of radial deformations. Experimental results are presented at different magnifications, thus illustrating its capabilities and versatility.

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 $z=f$.

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.