Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 1576735, 11 pages

http://dx.doi.org/10.1155/2016/1576735

## FSD-HSO Optimization Algorithm for Closed Fringes Interferogram Demodulation

^{1}Departamento de Ciencias Exactas y Tecnología, Centro Universitario de los Lagos, Universidad de Guadalajara, 47460 Lagos de Moreno, JAL, Mexico^{2}Departamento de Ingeniería Eléctrica y Electrónica, Instituto Tecnológico de Aguascalientes, 20256 Aguascalientes, AGS, Mexico^{3}Departamento de Posgrado e Investigación, Universidad Politécnica de Aguascalientes, 20342 Aguascalientes, AGS, Mexico

Received 1 December 2015; Revised 1 February 2016; Accepted 8 March 2016

Academic Editor: Hassan Askari

Copyright © 2016 Ulises H. Rodriguez-Marmolejo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Due to the physical nature of the interference phenomenon, extracting the phase of an interferogram is a known sinusoidal modulation problem. In order to solve this problem, a new hybrid mathematical optimization model for phase extraction is established. The combination of frequency guide sequential demodulation and harmony search optimization algorithms is used for demodulating closed fringes patterns in order to find the phase of interferogram applications. The proposed algorithm is tested in four sets of different synthetic interferograms, finding a range of average relative error in phase reconstructions of 0.14–0.39 rad. For reference, experimental results are compared with the genetic algorithm optimization technique, obtaining a reduction in the error up to 0.1448 rad. Finally, the proposed algorithm is compared with a very known demodulation algorithm, using a real interferogram, obtaining a relative error of 1.561 rad. Results are shown in patterns with complex fringes distribution.

#### 1. Introduction

Interferometry includes a series of techniques used in the measurement of aberrations, deformations, flatness, and perturbations. It may also be applied to measure variables as temperature gradients, strain analysis, depth measurement, and so forth. Widely known techniques to demodulate interferograms are phase shifting algorithms for several images and the Fourier method for a single linear carrier fringe pattern [1]. However, these techniques are difficult to implement when the object under study changes fast and continuously, and the dynamic range of the phase does not allow the use of a large linear carrier without infringing the sampling theorem. In those situations a single interferogram with closed and possibly complex fringe distribution must be analyzed to recover the phase information related to the physical phenomena being measured. This is known as a difficult task since there are many solutions that are compatible with the measured data but lacks physical meaning. The accuracy of measurements carried out from a single fringe pattern that includes closed fringes is thus intensely dependent on the phase distribution of the recorded interferogram being estimated. Recently, many phase recovering methods have been developed as combination of genetic algorithms and parametric methods [2–4], soft computing techniques applied to Zernike polynomials [5], combination of genetic algorithms and frequency guided sequential demodulation [6], particle swarm optimization [7], unwrapping of phase maps with sign changes [8], two-dimensional regularized phase-tracking technique [9], and so forth. In general, there are not particular processes that succeed in obtaining the phase for any given interferogram, but all of them are limited to specific features of the fringe pattern.

The phase demodulation trouble has been formulated as an optimization challenge, where soft computing procedures may be used to find the phase solution that best matches the nonlinear equation represented by fringe patterns. Few years ago, genetic algorithms have been tested [2]; the authors developed a parametric method for fringe pattern demodulation using a genetic algorithm (GA). A parametric estimation of the coefficients of a 15th degree Zernike polynomial is used in order to approximate the phase; a population of chromosomes is programmed within the coefficients to calculate the phase. A cost function is then employed considering the number of the observed fringes and the fringes that result from the recovered phase match, the phase softness, and the prior knowledge of the object. Normally, the final solution of the GA is based on a cost function, which is stated as the comparison between the better individual in the population and the target (real fringes); a population evolution process is allowed until a cost function average threshold is achieved. The authors reported a root mean square (rms) error of 0.12 radians. This method was applied to noisy fringe patterns and to a single closed fringe image. Additional improvements and variations of this work were subsequently presented by the same research team [3, 4].

Another soft computing technique used for phase reconstruction is particle swarm optimization (PSO). This algorithm was introduced by Kennedy and Eberhart in 1995 [10], as an evolving optimization technique. In 2012 Jiménez et al. [7] used PSO for phase recovery; they compared a GA and a PSO for phase recovery on several fringe patterns, obtaining errors of 0.4281 and 0.313 rad., respectively, showing an improvement in accuracy of PSO over GA; processing time improvements were announced, but no results were shown.

As mentioned before, the demodulation of a single interferogram often involves a combination of methods (GA + Zernike, PSO + Zernike, Neuronal networks + others, etc.). In 2009 Wang and Kemao reported a new hybrid method; they used frequency guided sequential demodulation (FSD) as interferogram demodulator, combined with Levenberg Marquardt (LM) optimization [11], method implemented by their quickness and efficiency in fringes demodulation.

In this work a FSD with harmony search optimization (HSO) is investigated in order to test the performance in a single interferogram with closed fringes. The main motivation is the advantages of the HSO technique over other soft computing techniques already reported. The HSO technique was inspired in the observation of musical composition to search a perfect harmony and was introduced by Geem et al. in 2001 [12] and has found its way in several applications as diverse as engineering, math, industrial process, biology, and so forth [13–19]. An excellent recent review and categorization of the applications of HSO was conducted by Manjarres et al. in 2013 [20]. Some advantages of this method are that it uses simple algebraic equations and real values, while the derivative information is unnecessary unlike GA and other optimization techniques.

In the following section, the physical theory of the interferograms is presented as well as the concepts of the HSO and FSD algorithms. In the next section, the image-processing techniques and the experimental setup used to implement the soft computing proposed method are described. Finally, in the last two sections the results and the conclusions are presented, respectively.

#### 2. Theory

Metrology has techniques such as fringe projection profilometry and optical interferometry to measure physical quantities in many areas of engineering and science, but the importance of these methods lies in the fact that they are noninvasive procedures [21]. Recently, advances in computational techniques have the potential to extend the measuring capabilities of optical metrology applications. In the present section the optical metrology basis, the harmony search optimization model, and the frequency guided sequential demodulation process are shown.

##### 2.1. Optical Interferometry

Interferometry studies the engagement of two or more light waves, where one of them has suffered a modification by one characteristic of an object being tested [22]. Demodulation of the phase is the most important task in interferometry measurements; the phase is related to a physical quantity to be measured.

The optical arrangement, shown in Figure 1, is a Twyman-Green interferometer setup (a Michaelson interferometer modification). The interference is produced by the difference of optical path between the two arms of the interferometer. The interferogram is reordered by a photo detector array (e.g., a charged coupled display camera) and then digitized for show on a monitor or stored for further processing with computational algorithms like unwrapped phase, digital filtering, demodulation phase, and so forth [23]. The optical components Lc, Lf, and Bs are a positive collimating lens, a positive focusing lens, and a beam splinter, respectively. The fringe pattern intensity is modeled bywhere , and are the interference fringe pattern intensity, the background illumination, the modulation amplitude, the phase term, and the spatial coordinates of the surface under test, respectively.