Fringe Order, Image Processing, Experimental Mechanics. 1. Introduction. Photoelasticity is an effective whole-field experimental method for stress analysis.
Journal of JSEM, Vol.9, Special Issue (2009) 100-105 Copyright Ⓒ 2009 JSEM
Unwrapping of Isochromatic Fringe Orders Using Fringe Gradient Method Yudai NOMURA1 and Eisaku UMEZAKI2 1
2
Graduate Student, Nippon Institute of Technology, Saitama 345-8501, Japan Department of Mechanical Engineering, Nippon Institute of Technology, Saitama, 345-8501, Japan (Received 1 March 2009; received in revised form 28 May 2009; accepted 4 July 2009)
Abstract A new phase unwrapping algorithm for the automatic determination of the isochromatic parameter (absolute fringe order) from a single-color isochromatic image using the fringe gradient and RGB wavelengths techniques is developed. To evaluate the effectiveness of the proposed method, a single-color isochromatic image for a circular disk subjected to a concentrated load was used. Numerical simulation results showed that the proposed method yields accurate absolute fringe orders of isochromatics. Key words Photoelasticity, RGB Photoelasticity, Isochromatics, Fringe Order, Image Processing, Experimental Mechanics 1. Introduction Photoelasticity is an effective whole-field experimental method for stress analysis. This method represents stress or strain visually. The method yields two types of fringe patterns, namely, isoclinic and isochromatic parameters. The former corresponds to the principal stress direction whereas the latter correlates to the contours of principal stress differences. A number of methods have been proposed for the determination of the isochromatic parameter, namely, fractional fringe orders [1-3]. A recent tendency of the determination is to use the phase-shifting technique, the principle of which is to change the incremental phase shift in the light intensity equation to acquire a sufficient number of equations (photoelastic fringe images). The phase-shifting technique can give fractional fringe orders from the combination of photoelastic fringe images. Experimental stress analysis needs absolute fringe orders, which are obtained by connecting fractional fringe orders using an unwrapping technique. The procedure of unwrapping depends on the kind of polariscope, namely, circular, semicircular or plane polariscope, used to obtain photoelastic fringe images. For the methods using the circular polariscope [4-5], the analysis system [6] consisting of a quarter-wave plate and a polarizer, the fractional fringe order is given through the atan() operator; hence, its profile is a sawtooth and this eases phase unwrapping (PU) for the absolute fringe order. For the methods using the semicircular [7] and plane polariscopes [8-13], the profiles of the fractional fringes obtained are triangular owing to the use of asin() and acos() operators, respectively. In this case, the absolute fringe order is determined with sign ambiguity, which makes PU much more difficult. For the ease of phase unwrapping, as described above, the circular polariscope is used by many researchers and
engineers. However, the constitution of the circular polariscope is more complex than that of the plane polariscope. Furthermore, the technique of using the circular polariscope needs an isoclinic parameter for the determination of the fractional fringe order. On the other hand, the technique of using the plane polariscope does not need an isoclinic parameter in the determination. Therefore, if the phase unwrapping of the fractional fringe order with the triangular profile is facilitated, the plane polariscope can be used for obtaining isochromatic fringe orders. In this study, a new PU algorithm for the automatic determination of the isochromatic parameter (absolute fringe order) from a single-color isochromatic image using the fringe gradient and RGB wavelengths techniques is developed. 2. Determination of Absolute Fringe Order 2.1 Light intensity of isochromatics The light intensity, I, with generic orientations, m, of the transmission axes of the polarizer and analyzer in the dark field coming out of the plane polariscope is given by [13] G
2 2(I Tm ) Ib,O , (1) I m,O aO sin2 O sin 2 where Ȝ denotes the plane-polarized R, G and B lights, aȜ is the intensity coming out of the polarizer, I is the isoclinic parameter or the angle of ı1 with respect to the reference axis and is counterclockwise, Tm is the induced phase shift angle at step m and is also counterclockwise, and Ib,Ȝ is the background intensity. Note that for the dark-field setup, the induced phase shift angle Tm is typically chosen to be constant. įȜ is the fractional phase retardation that is related to the fractional fringe order, NȜ, and the principal-stress difference, ı1-ı2, in the plane-stress state by G O N CO h (V V ) h (V V ) , 1 2 1 2 O 2S
O
f V ,O
(2)
where CȜ is the stress-optic coefficient, fı,Ȝ is the wellknown material stress fringe value obtained by calibration and h is the model thickness. When T1=0 and T2=ʌ/4, Eq. (1) becomes
-100-
I m, O
aO sin 2
I m, O
aO sin 2
GO 2
GO 2
sin 2 2I I b,O
(3)
cos2 2I I b,O .
(4)
Adding Eqs. (3) and (4) yields IO
aO sin 2
GO 2
2 I b,O .
(5)
Journal of JSEM, Vol.9, Special Issue (2009)
Equation (5) gives the light intensity of isochromatics for wavelength Ȝ. In this study, the isochromatics with aȜ =1 and Ib,Ȝ=0 in Eq. (5) were used for determining the fractional phase retardation. Hereafter, subscript Ȝ is omitted unless otherwise noted. 2.2 Fractional phase retardation The proposed method for determining fractional phase retardation is based on the technique developed for the demodulation of interferograms [14]. In this technique, one interferogram is used to determine the fractional phase retardation. The first step towards obtaining fractional phase retardation is to determine the gradient of the isochromatic fringe pattern, I, as follows: I
ª wI wI º « , » ¬ wx wy ¼
>Gx , Gy @ ,
(6)
where Gx and Gy are the gradients of the pattern in the x- and y-directions, respectively. The finite difference, which utilizes the difference in light intensities at pixels before and after a noticed pixel, and the first-order differential filter, such as a Prewitt filter and Sobel filter [15], may be used for obtaining the gradient. Using the gradients, Gx and Gy, the fringe orientation angle, Tʌ, is obtained as
TS ( x, y)
§ Gy · ¸ . tan 1¨¨ ¸ © Gx ¹
G k 1 ( x, y) G k ( x, y)
ª n ( x, y) x I ( x, y) º tan 1 « G ,S » . I ( x, y) ¬ ¼
(8)
(9)
cos>G ( x, y )@ I ( x, y )
0
(10) 0
that is closest to the phase value, į , obtained above. The iteration in the Halley method is given by
A
ʌ į
0 -ʌ
The phase retardation obtained by Eq. (9) may have a small phase error where G z 1. Therefore, to obtain the correct phase value, Halley’s method [14], which is an improved version of the Newton-Raphson technique, is used. The correct phase value is found as the solution to the nonlinear equation f ( x, y )
, (11)
A
Using I shown in Eq. (6), nį,ʌ in Eq. (8) and I, the fractional phase retardation, į0, for wavelength Ȝ is obtained by
G 0 ( x, y )
2 fG ( x, y ) fGG ( x, y) f ( x. y )
(12) The values of the fractional phase retardation obtained from Eq. (12) are continuously distributed in the range of (-ʌ, ʌ] in a model. As a result, the increasing distribution of the values in the range of (-ʌ, ʌ] and the decreasing distribution in the range of (ʌ, -ʌ] are obtained, as shown in Fig. 1(a). In general, the increasing or decreasing distributions are repeated, and several regions having the repeated distributions exist in a model. However, whether the distribution is correct is ambiguous because the fringe orientation angle, Tʌ, obtained using Eq. (7) is limited in the range of (-ʌ/2, ʌ/2]. In some cases, the increasing distribution may be modified to the decreasing one and vice versa, as shown in Fig. 1(b). Therefore, the ambiguity must be resolved before the determination of the absolute fringe order. Servin et al. [14] have resolved the ambiguity using the fringe orientation angle, T2ʌ, in the range of (-ʌ, .
(7)
cos>TS ( x, y)@i sin>TS ( x, y )@ j .
2 f ( x, y) fG ( x, y )
where fį and fįį are the first and second derivatives of f with respect to į, which gives 2 f ( x, y ) sin[G k ( x, y )] G k 1 ( x, y ) G k ( x, y ) . 2 k 2 sin [G ( x, y)] cos[G k ( x, y)] f ( x. y )
In Eq. (7), the range of Tʌ is (-ʌ/2, ʌ/2] (this represents the range of -ʌ/2
