Jesús Villa

Phase recovery from a single fringe pattern using an orientational vector-field-regularized estimator

Recent studies have demonstrated that the phase recovery from a single fringe pattern with closed fringes can be properly performed if the modulo 2pi fringe orientation is estimated. For example, the fringe pattern in quadrature can be efficiently obtained in terms of the orientational phase spatial operator using fast Fourier transformations and a spiral phase spectral operator in the Fourier space. The computation of the modulo 2pi fringe orientation, however, is by far the most difficult task in the global process of phase recovery. For this reason we propose the demodulation of fringe patterns with closed fringes through the computation of the modulo 2pi fringe orientation using an orientational vector-field-regularized estimator. As we will show, the phase recovery from a single pattern can be performed in an efficient manner using this estimator, provided that it requires one to solve locally in the fringe pattern a simple linear system to optimize a regularized cost function. We present simulated and real experiments applying the proposed methodology.

Regularized quadratic cost function for oriented fringe-pattern filtering

We use the regularization theory in a Bayesian framework to derive a quadratic cost function for denoising fringe patterns. As prior constraints for the regularization problem, we propose a Markov random field model that includes information about the fringe orientation. In our cost function the regularization term imposes constraints to the solution (i.e., the filtered image) to be smooth only along the fringes tangent direction. In this way as the fringe information and noise are conveniently separated in the frequency space, our technique avoids blurring the fringes. The attractiveness of the proposed filtering method is that the minimization of the cost function can be easily implemented using iterative methods. To show the performance of the proposed technique we present some results obtained by processing simulated and real fringe patterns.

Foucault test: shadowgram modeling from the physical theory for quantitative evaluations

The physical theory of the Foucault test has been investigated to represent the complex amplitude and irradiance of the shadowgram in terms of the wavefront error; however, most of the studies have limited the treatment for the particular case of nearly diffraction-limited optical devices (i.e., aberrations smaller than the wavelength). In this paper we discard this restriction, and in order to show a more precise interpretation from the physical theory we derive expressions for the complex amplitude and the irradiance over an optical device with larger aberrations. To the best of our knowledge, it is the first time an expression is obtained in closed form. As will be seen, the result of this derivation is obtained using some properties of the Hilbert transform that permit representing the irradiance in a simple form in terms of the partial derivatives of the wavefront error. Additionally, we briefly describe from this point of view a methodology for the quantitative analysis of the test.

Semi-Huber potential function for image segmentation

In this work, a novel model of Markov Random Field (MRF) is introduced. Such a model is based on a proposed Semi-Huber potential function and it is applied successfully to image segmentation in presence of noise. The main difference with respect to other half-quadratic models that have been taken as a reference is, that the number of parameters to be tuned in the proposed model is smaller and simpler. The idea is then, to choose adequate parameter values heuristically for a good segmentation of the image. In that sense, some experimental results show that the proposed model allows an easier parameter adjustment with reasonable computation times.

A tour of nonlocal means techniques for image filtering

The present work proposes a review and comparison of different Nonlocal Means (NLM) methods in the task of digital image filtering. Some different alternatives to change the classical exponential kernel function used in NLM methods are explored. Moreover, some approaches that change the geometry of the neighborhood and use dimensionality reduction of the neighborhood or patches onto principal component analysis (PCA) are also analyzed, and their performance is compared with respect to the classic NLM method. Mainly, six approaches were compared using quantitative and qualitative evaluations, to do this an homogenous framework has been established using the same simulation platform, the same computer, and same conditions for the initializing parameters. One will notice that particularly, the BM3D SAPCA approach gives the best denoising results, but in contrast, the computation times of this method were the longest.

Electropyroelectric technique: A methodology free of fitting procedures for thermal effusivity determination in liquids

This paper describes an alternative methodology to determine the thermal effusivity of a liquid sample using the recently proposed electropyroelectric technique, without fitting the experimental data with a theoretical model and without having to know the pyroelectric sensor related parameters, as in most previous reported approaches. The method is not absolute, because a reference liquid with known thermal properties is needed. Experiments have been performed that demonstrate the high reliability and accuracy of the method with measurement uncertainties smaller than 3%.

N-dimensional regularized fringe direction-estimator

It has been demonstrated that the vectorial fringe-direction field is very important to demodulate fringe patterns without a dominant (or carrier) frequency. Unfortunately, the computation of this direction-filed is by far the most difficult task in the full interferogram phase-demodulation process. In this paper we present an algorithm to estimate this fringe-direction vector-field of a single n-dimensional fringe pattern. Despite that our theoretical results are valid at any dimension in the Euclidean space, we present some computer-simulated results in three dimensions because it is the most useful case in practical applications. As herein demonstrated, our method is based on linear matrix and vector analysis, this translates into a low computational cost.

Regularized quadratic cost-function for integrating wave-front gradient fields

From the Bayesian regularization theory we derive a quadratic cost-function for integrating wave-front gradient fields. In the proposed cost-function, the term of conditional distribution uses a central-differences model to make the estimated function well consistent with the observed gradient field. As will be shown, the results obtained with the central-differences model are superior to the results obtained with the backward-differences model, commonly used in other integration techniques. As a regularization term we use an isotropic first-order differences Markov Random-Field model, which acts as a low-pass filter reducing the errors caused by the noise. We present simulated and real experiments of the proposal applied in the Foucault test, obtaining good results.

Foucault test: a quantitative evaluation method

Reliable and accurate testing methods are essential to guiding the polishing process during the figuring of optical telescope mirrors. With the natural advancement of technology, the procedures and instruments used to carry out this delicate task have consistently increased in sensitivity, but also in complexity and cost. Fortunately, throughout history, the Foucault knife-edge test has shown the potential to measure transverse aberrations in the order of the wavelength, mainly when described in terms of physical theory, which allows a quantitative interpretation of its characteristic shadowmaps. Our previous publication on this topic derived a closed mathematical formulation that directly relates the knife-edge position with the observed irradiance pattern. The present work addresses the quite unexplored problem of the wavefronts gradient estimation from experimental captures of the test, which is achieved by means of an optimization algorithm featuring a proposed ad hoc cost function. The partial derivatives thereby calculated are then integrated by means of a Fourier-based algorithm to retrieve the mirrors actual surface profile. To date and to the best of our knowledge, this is the very first time that a complete mathematical-grounded treatment of this optical phenomenon is presented, complemented by an image-processing algorithm which allows a quantitative calculation of the corresponding slope at any given point of the mirrors surface, so that it becomes possible to accurately estimate the aberrations present in the analyzed concave device just through its associated foucaultgrams.

Hilbert vs. exponential Kernel functionals for Nonlocal Means image filtering

The present work introduces a new alternative to change the classical exponential kernel function used in Nonlocal Means (NLM) methods to deal with digital image filtering. The idea is based on the premise that making a good selection or estimation of the bandwidth parameter h is difficult and there are some other kernels which have another equivalent parameters to be selected into a more easiest way. A First method is obtained, when using an optimal manner proposed in nonparametric estimation literature to estimate h to tune the exponential kernel function. And a second proposed method, is to change the exponential function by the so called Hilbert function where one must to choose a parameter d. This Hilbert function is used for the first time in the NLM framework. Finally, the obtained filtering results reveals, that the NLM Hilbert kernel approach gives similar performance to other approaches according to recent reported results in literature, and the first proposed methodology is restricted by the estimation of h.