Ismael de la Rosa

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.

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.

Demodulation of single interferograms using a sliding 2-D continuous wavelet transform method

In this paper, we present an alternative technique for the demodulation of single interferograms using a sliding 2-D continuous wavelet transform (2-D CWT) method. The sliding strategy proposed in this work is used with two purposes: to reduce the processing time when the 2-D CWT is applied, and to solve the problem of the phase ambiguity when closed fringes are present. Experimental results with real and simulated interferograms show that the proposed multiresolution method is a proper alternative for many applications of interferogram demodulation with closed fringes.

Towards the automatization of the Foucault knife-edge quantitative test

Given the increasing necessity of simple, economical and reliable methods and instruments for performing quality tests of optical surfaces such as mirrors and lenses, in the recent years we resumed the study of the long forgotten Foucault knife-edge test from the point of view of the physical optics, ultimately achieving a closed mathematical expression that directly relates the knife-edge position along the displacement paraxial axis with the observable irradiance pattern, which later allowed us to propose a quantitative methodology for estimating the wavefront error of an aspherical mirror with precision akin to interferometry. In this work, we present a further improved digital image processing algorithm in which the sigmoidal cost-function for calculating the transient slope-point of each associated intensity-illumination profile is replaced for a simplified version of it, thus making the whole process of estimating the wavefront gradient remarkably more stable and efficient, at the same time, the Fourier based algorithm employed for gradient integration has been replaced as well for a regularized quadratic cost-function that allows a considerably easier introduction of the region of interest (ROI) of the function, which solved by means of a linear gradient conjugate method largely increases the overall accuracy and efficiency of the algorithm. This revised approach of our methodology can be easily implemented and handled by most single-board microcontrollers in the market, hence enabling the implementation of a full-integrated automatized test apparatus, opening a realistic path for even the proposal of a stand-alone optical mirror analyzer prototype.

Matlab Graphic User Interface for image segmentation using Markov random fields and entropy estimation with parallel processing

In this work it is presented, described and tested a new Matlab Graphic User Interface (GUI) for image segmentation of degraded images using two probabilistic techniques, Markov random fields (MRF) and nonparametric entropy estimation. This GUI was created in order to integrate a series of steps needed for the segmentation process into a single visual environment to allow an easier handling of input images and saving of results. It is also used a powerful utility of the Matlab software concerning to parallel processing, with the aim of reduce the computational time because of the high time consumption of this kind of algorithms. Results show a very satisfactory performance of this tool, allowing us to make this task easier and faster.

A 2D continuous wavelet transform method for InSAR phase-maps denoising

Analysis of Interferometric Synthetic Aperture Radar (InSAR) phase–maps with large wide band is still a challenging problem that requires the development of robust methods. This paper presents a 2–D Continuous Wavelet Transform method for denoising InSAR phase-maps. Owing to its high directionality, sensitivity and anisotropy, multiresolution analysis with 2–D Continuous Wavelet Transform (2–D CWT) is potentially a useful tool to construct appropriate filtering algorithms for detecting and identifying images with specific features, transient information content, or other properties. The 2-D Gabor wavelets naturally model the phase fringes, which means that can properly reconstruct the image. We describe the theoretical basis of the proposed technique and some experimental results with real InSAR phase–maps. As can be verified the proposal is robust and effective.

Directional filters for fringe pattern denoising

For a successful phase demodulation it is important to have a good quality fringe pattern image. For this reason preprocessing fringe patterns is, many times, an unavoidable task. Often, noise removal is the main problem to be solved, however, the use of ordinary linear filters is not always a proper procedure specially in the presence of high density fringes because the signal and noise are mixed in the Fourier space. Also, as fringe pattern images are two-dimensional functions, frequencies are two-component vectors which requires consider the filtering direction. We present a new denoising technique for preprocessing fringe pattern images which requires to previously estimate the fringe orientation. For cases of high noise levels we modify the proposed technique by means of a regularized local cost function in order to get a better noise response. We present a noise response analysis of the proposed technique, some experimental results and its application to wrapped phase maps denoising.