Interferogram phase evaluation through a differential equation technique

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Abstract

A technique for the evaluation of the unknown phase of a spatial carrier interferogram is proposed in this paper. This technique is based on the formulation of a differential equation for the unknown phase. The coefficients of this equation are known functions. They are formulated with the aid of a modified Fourier transform method and Hilbert transform where they are functions of the measured interferogram. This equation is made discrete and solved through the application of a pseudospectral method. An illustrative example is given in which this technique is applied. Results generated from the application of this technique are compared with exact phase values, and exemplary agreement between the exact and approximate phase values is demonstrated.

Introduction

The phase of an interferogram is related to the physical quantity being measured in applications such as depth measurement, strain analysis, temperature gradients measurement, surface deformation assessment [1], evaluation of stress, vibration and displacement [2], [3], computation of real imagery [4], optical signal processing, magnetic resonance imaging [5], [6], and optical metrology operation [7]. In many interferogram phase-evaluation techniques, only the wrapped phase values, which are the principal phase values, are measured or directly computed [4], and accordingly, phase unwrapping has been under considerable study over the past two decades [8]. Computations that produce wrapped phase values experience the phase modulo 2π effect, where resulting phase values lie between −π and +π [9], [10]. Given the wrapped phase values, phase unwrapping is the task of finding the true phase values [4]. In general, phase unwrapping consists of two steps: the first is the estimation of phase gradients from the interferograms, and the second step is the integration of the gradient estimates to obtain the unwrapped phase values [7]. Furthermore, often it is desired to evaluate the phase of a digitally recorded interferogram with a spatial carrier. Examples of methods used for the evaluation of the phase of a given interferogram with a spatial carrier are: the Fourier transform [11], the carrier phase-shifting [12], the sinusoidal fitting [13], the spatial synchronous [14], the logical moiré [15], the multiplicative analogical moiré [16], the direct spatial reconstruction optical phase [17], and the iterative synchronous [18] methods. Moreover, methods are presented to evaluate the maps of phase derivatives in holographic interferometry and electronic speckle pattern interferometry by the direct manipulation of three or more phase-shifted interferograms [19], [20]. In this case, phase derivatives can be obtained without prior evaluation of displacements and complex unwrapping methods are not needed. The idea of direct evaluation of phase derivatives and the evasion of the usage of complex phase unwrapping are considered for the formulation of the technique presented in this paper. This technique utilizes a pseudospectral discrete approach that allows for the use of standard numerical integration procedure. In addition to its rapid rate of convergence, the ability of providing results with high accuracy and solution stability, the pseudospectral discretization method used in the technique presented in this paper utilizes discrete points that are non-uniformly distributed [21], [22]. The latter feature provides the ability to perform computations with higher density at regions where the solution relatively changes rapidly [21]. A carrier frequency interferogram with a linear spatial frequency, ω0, can be presented as [18],I(x,y)=a(x,y)+b(x,y)cos[ω0x+ϕ(x,y)]where x and y represent pixel directions, a(x, y) is the background intensity, b(x, y) is an amplitude modulating term, ϕ(x, y) is the phase that is needed to be found. The assumptions under which Eq. (1) is considered areω0>ϕx(x,y)ω0>1|a(x,y)|ax(x,y)ω0>1|b(x,y)|bx(x,y)where the subscript represents the partial derivative with respect to x. In this paper we propose a technique for the evaluation of the phase in Eq. (1) under the conditions presented in Eqs. (2), (3), (4). With the aid of the applications of Hilbert Transform (HT) a differential equation for the unknown phase is formulated and discretely solved in this technique. The basics of solving differentiated equations for the benefit of evaluating the phase of an interferogram have been utilized [23], [24].

The sections of the paper are organized as follows: The formulation of the differential equation for the unknown phase is outlined in Section 2. This equation is made discrete with the aid of a pseudospectral method in Section 3. In Section 4, the performance of the proposed technique is demonstrated through the solution of an illustrative example.

Section snippets

Formulation of the differential equation for the unknown phase

A differential equation is formulated in this section for the unknown phase. For this purpose the operational principal performed in [25] is utilized where the differentiation is taken in the direction of the carrier, i.e. the x-direction. Also the operational principle adapted in [26] where a slice-wise HT is applied to provide rapid signal processing is performed. In this case, slicing is made perpendicular to the direction of differentiation. Let the formula of the nth slice of the

Pseudospectral Legendre method

Let LM(τ), −1  τ  1 denote the Legendre polynomial of order M; then the Legendre–Gauss–Lobatto (LGL) nodes are defined in [21] byτ0=-1,τM=1,τlarethezerosofL˙M(τ),1lM-1where L˙M(τ) denotes the first derivative of LM(τ). No explicit formula of the nodes in Eq. (9) is known. However, they can be computed numerically [21], [32]. Define the polynomial approximation of a function F(τ) asFM(τ)=l=0Mflψl(τ)where fl is a coefficient andψk(τ)=(τ2-1)L˙M(τ)M(M+1)(τ-τk)LM(τk),k=0,1,,Mare Lagrange

An illustrative example

To test the performance of the proposed technique we used a phase given in [35] where the phase map produced from the corresponding interferogram could be under-sampled. This phase isϕ(x,y)=120×exp[-0.5r2(x,y)(μ1+μ2c(x,y))]1x,y128withr(x,y)=[(x-35.5)2+(y-65.5)2]1/2c(x,y)=x-35.5r(x,y)μ1=0.01μ2=0.0004where ϕ(x, y) is in radians. A value of 1.5 was assigned to ω0, while a(x, y) and b(x, y) were selected to be 128 and 127, respectively [18]. Accordingly, the interferogram of Eq. (1) was:I(x,y)=127+

Conclusion

A technique that generates a differential equation for the unknown phase of an interferogram is proposed in this paper. This technique utilizes the measured interferogram along with a modified Fourier transform method and Hilbert transform for this formulation. The differential equation is decomposed into two coupled differential equations and solved numerically with the aid of the pseudospectral Legendre method. Consequently, values of the unknown phase are generated at all pixel positions.

Acknowledgments

This work was supported in part by the Department of Defense, Engineering Research and Development Center, ERDC, Vicksburg, Mississippi under contract W912HZ-05-P-0026 and the Maxheim Faculty Fellowship from the University of North Carolina, Charlotte. The authors would like to extend their sincere thanks to Mrs. Pamela Kinnebrew, Chief, Survivability Engineering Branch, ERDC. The Chief of Engineers from ERDC granted permission to publish this information.

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