Normalization and noise-reduction algorithm for fringe patterns

Optics Communications 270 (2007) 161–168 www.elsevier.com/locate/optcom

Normalization and noise-reduction algorithm for fringe patterns Noe´ Alcala´ Ochoa b

a,*

, A.A. Silva-Moreno

b

a Centro de Investigaciones en Optica Apartado Postal 1-948, Leo´n, Guanajuato, Mexico CIATEC, A. C. Omega 201, Fraccionamiento Industrial Delta, Leo´n 37545, Guanajuato Me´xico

Received 1 July 2006; received in revised form 5 September 2006; accepted 6 September 2006

Abstract This paper presents a fringe pattern normalization and noise-reduction algorithm. Locally the background noise is suppressed, the modulation normalized and the noise smoothed. An expression to calculate the cosine-only term is formulated. It is related to the directional derivatives of the intensity fringes. Two-dimensional Fourier series are used to calculate the parameters needed for the algorithm. Experimental work is presented using diffraction and ESPI images. The programming is relatively simple and involves mainly local convolutions. The processing time using a 2 GHz computer to normalize an image of 256 · 256 pixels is approximately one second.  2006 Elsevier B.V. All rights reserved. PACS: 42.25.Hz; 42.30.ms; 42.30.Rx; 43.40.+s Keywords: Smoothing; Phase extraction; Vibrations; Electronic speckle pattern interferometry; Shearography

1. Introduction The use of light to analyze problems like stresses, cracks, deformations, topography, etc. involves the use of optical methods such as photoelasticity, interferometry, diffraction, moire´, ESPI, synthetic-aperture-radar [1,2], etc. In general, by using the above methods we obtain fringe patterns that have a phase function which contains the desired information. There are many methods to achieve the phase from the fringes. The most popular methods probably are those which use phase shifting (PS) [3] or fast Fourier transforms (FFT) with carrier [4] or without carrier [5]. The PS methods need various images and the FFT generally have an ambiguity on the resulting phase when closed fringes are analyzed. The method explained in Ref. [5] uses half-plane Fourier filtering to deduce the phase from single fringe patterns that includes closed fringes. By applying the cosine function to the resulting phase map a normalized and *

Corresponding author. Tel.: +52 4774414200; fax: +52 4774414209. E-mail addresses: [email protected] (N.A. Ochoa), [email protected] (A.A. Silva-Moreno). 0030-4018/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.09.062

smoothed fringe pattern is obtained. However, if the fringes are speckled, to obtain acceptable smoothing it requires the design of pass-band filters which does not always produce the best results, in particular for low-period fringes. An improvement of the above method can be found in Ref. [6], but again, it may distort the low-period fringes and the areas where no fringes are present act like distortion sources. There are some phenomena which are difficult to obtain fringes satisfying the conditions needed to apply the PS and FFT methods, for example transient phenomena and vibration analysis [7]. In general, we can say that an ideal fringe analysis method is relatively easy to program, it needs only one image, and calculates the phase quickly and accurately. Unfortunately, no such method exists currently. What the current fringe analysis methods have in common is that the calculated phase is more accurate if the fringes under analysis are noiseless and well contrasted. Furthermore, there are methods that require normalized fringes to obtain good results [8]. There are FFT methods to normalize sinusoidal fringes, for example, using a filter in the frequencies domain [9] or a quadrature operator using 1D Reisz filters [10]. However, these methods cannot smooth adequately the speckle

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fringes noise, in which case additional smoothing techniques should be used after the normalization procedure. Other methods find the peaks and valleys of the fringes and use their coordinates to design filters, but they need good quality skeletons [11]. In Ref. [12] is described an algorithm to obtain smooth and normalized fringes generated by speckle-pattern interferometry according to the fringe intensity slope and the distance ratio to neighboring skeletons. The skeleton is calculated and manipulated using the fringes slopes map. So, the success of the method depends on the correct calculation of the skeletons. Unfortunately, the method described to find such skeletons, when closed fringes appear, requires the slopes map again to determine some spurious lines due to fringes slope changes. This is not an easy task and may require user intervention in some cases. Another method was proposed to improve the signal-noise ratio in the interferograms by using adapted algorithms for temporal and local filtering [13]. The purpose is to extract the skeletons of fringes to classify flaws. A division of the interferogram by and estimated background intensity was enough for shading correction. However, it is difficult to determine thresholds suitable for the entire image. Some other methods utilize iterative procedures requiring high programming amount and processing time [14–16]. In this paper we describe a method to filter the noise and normalize the fringe patterns simultaneously. This procedure is done locally allowing the normalization of nonsinusoidal fringe patterns. We formulate an expression to calculate the cosine profile of the fringes without the background noise and the modulation terms. This expression contains terms involving only directional derivatives of the intensity fringe pattern. During our research, we found that the best derivative direction is that which is normal to the fringes orientation. All the parameters needed by the algorithm are calculated from the coefficients of a twodimensional Fourier series. The proposed algorithm was tested using simulated and experimental fringe patterns. The simulated fringes have sinusoidal and Bessel profiles. The experimental fringes were obtained from diffraction and ESPI configurations of time-averaged vibration and out-of-plane deformations. In the following sections we will describe the proposed algorithm; the results obtained using experimental and simulated images, and the conclusions of the article. 2. Theoretical analysis of noise reduction and normalization 2.1. General algorithm Let I(x, y) be the intensity of a discrete fringe pattern having a cosine, Bessel or another profile function. We will also assume that inside a rectangular window W of dimensions KL can be expressed by a cosine term such as the following: Iðx1 ; y 1 Þ ¼ Aðx1 ; y 1 Þ þ Bðx1 ; y 1 Þ cos½uðx1 ; y 1 Þ;

ð1Þ

where (x1, y1) are the pixels of W in local Cartesians coordinates, u(x1, y1) is the phase that determines the fringes geometry, A(x1, y1) and B(x1, y1) are the background and modulation terms, respectively. We use the pixel intensities in the window to calculate some Fourier coefficients at its center (x1 = 0, y1 = 0), i.e. at (x, y) on the global coordinates. By shifting the window we obtain these coefficients for each pixel (x, y) on the full image. Assuming that inside the window W, A(x1, y1) and B(x1, y1) are constants and the phase u(x1, y1) is a plane, (which implies that ux(x1, y1) and uy(x1, y1) are both constants) we can do the following operations p1 ðx; yÞ ¼ I x ðx; yÞ cosðaÞ þ I y ðx; yÞ sinðaÞ; p2 ðx; yÞ ¼ I xx ðx; yÞ cos2 ðaÞ þ 2I xy ðx; yÞ cosðaÞ sinðaÞ þ I yy ðx; yÞ sin2 ðaÞ p3 ðx; yÞ ¼ I xxx ðx; yÞ cos3 ðaÞ þ 3I xxy ðx; yÞ cos2 ðaÞ sinðaÞ

ð2Þ

þ 3I xyy ðx; yÞ cosðaÞ sin2 ðaÞ þ I yyy ðx; yÞ sin3 ðaÞ; where the subscripted symbols mean derivation of I(x, y) along the x or y the times indicated, for example Ixy mean o2I/ox oy, and a(x, y) is a derivative direction function given by the user, to obtain p1 ðx; yÞ ¼ B½ux cosðaÞ þ uy sinðaÞ sinðuÞ;

ð3Þ

2

p2 ðx; yÞ ¼ B½ux cosðaÞ þ uy sinðaÞ cosðuÞ 3

p3 ðx; yÞ ¼ B½ux cosðaÞ þ uy sinðaÞ sinðuÞ; where the sub-index of p were set according to the powers of the directional derivatives. This last equation can be manipulated to obtain the cosine term without the background and the modulation function qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos½uðx; yÞ ¼ p2 ðx; yÞ= p22 ðx; yÞ  p1 ðx; yÞp3 ðx; yÞ: ð4Þ Notice that p1(x, y), p2(x, y) and p3(x, y) (Eq. (2)) depend only on the values that can be calculated from Eq. (1), i.e. a(x, y) and derivatives of I(x, y). At first glance, an expression to calculate the cosine term can be obtained without using the derivative direction a(x, y), however, if it is not introduced, the normalization algorithm becomes dependent on the fringes geometry. Notice that the root term in Eq. (4) vanishes when a(x, y) is perpendicular to the gradient direction, that is, when tg[a(x, y)] = ux(x, y)/uy(x, y)  Ix(x, y)/Iy(x, y); however, as we will see later, we choose a(x, y) along the gradient direction, therefore in theory the denominator does not vanish. It is known that the derivative operator is noise sensitive and that most of the fringe patterns have noise. Therefore it is essential to formulate a robust procedure to reduce its effects on the derivatives calculation. In the following section, we will explain a method to calculate the derivatives of the fringes and the proper values for a(x, y) based on the fringes orientation calculation.

N.A. Ochoa, A.A. Silva-Moreno / Optics Communications 270 (2007) 161–168

2.2. Noise reduction

I y ðx; yÞ ¼ fy ðx1 ¼ 0; y 1 ¼ 0Þ ¼

Now we represent the intensity I(x1, y1) in the W-window by the following two-dimensional Fourier series of M-degree. f ðx1 ; y 1 Þ ¼

M M X X

Amn cosðmx1 þ ny 1 Þ

m¼M n¼M

þ Bmn sinðmx1 þ ny 1 Þ:

ð5Þ

The equation that we modified in order to have independent basis functions, i.e. to avoid situations like cos(mx1 + ny1) = cos(mx1  ny1), as follows: M m1 X X

f ðx1 ; y 1 Þ ¼ A00 þ

m¼1 n¼m

þ Bmn sinðmx1 þ ny 1 Þ þ Anm cosðnx1 þ my 1 Þ þ Bnm sinðnx1 þ my 1 Þ; ð6Þ where x1 and y1 are scaled to the [p, +p) interval, m and n are integers, and the coefficients Amn, Bmn, Anm and Bnm are calculated with the usual formulas, Amn ðx; yÞ ¼

1 KL

X

f ðx1 ; y 1 Þ cosðmx1 þ ny 1 Þ;

ðx1 ;y 1 Þ2W

n ¼ m; . . . ; m  1; m ¼ 1; . . . ; M; 1 X Bmn ðx; yÞ ¼ f ðx1 ; y 1 Þ sinðmx1 þ ny 1 Þ; KL ðx ;y Þ2W 1

ð7aÞ

1

n ¼ m; . . . ; m  1;

m ¼ 1; . . . ; M;

ð7bÞ

where K and L are the number of divisions of the [p, +p) interval along the horizontal and vertical directions, respectively, i.e., KL is the number of pixels in W. The coefficients A00, Anm and Bnm are also calculated with Eqs. (7a) and (7b) doing the appropriate index changes. Note that Eq. (6) has fewer coefficients than Eq. (5), 8M + 1 instead of 2(2M + 1)2, for the same value of M. The +p frontier is excluded in the calculations to preserve the discrete orthogonality of the basis functions. The set of Fourier coefficients are assigned to the central point of the window (x1 = 0, y1 = 0), which is (x, y) on the full image. M is the degree of the approximation. The product KL must be greater than the number of calculated coefficients 8M + 1. In general, K 5 L but better noise rejection is obtained setting K = L. Once the coefficients are known it is straightforward to calculate the derivatives of I(x, y), by deriving Eq. (6) respect x1 and y1 the times needed and evaluating it at (0,0). The expressions obtained will depend only on the calculated coefficients. For example, for Ix(x, y) and Iy(x, y) we have: I x ðx; yÞ ¼ fx ðx1 ¼ 0; y 1 ¼ 0Þ ¼

M X m1 X

mBmn ðx; yÞ  nBnm ðx; yÞ;

m¼1 n¼m

ð8aÞ

nBmn ðx; yÞ þ mBnm ðx; yÞ:

m¼1 n¼m

ð8bÞ

Observe that the coefficients are determined just once for each window and that they are used to calculate all the derivatives needed. What remains is the calculation for the angle a(x, y). We found that the best noise rejection and normalization results for a given W are obtained when the a(x, y) direction is chosen as the gradient orientation, that is, normal to the fringe. The angle a(x, y) is evaluated using the arctg function: aðx; yÞ ¼ arctg

½Amn cosðmx1 þ ny 1 Þ

M X m1 X

163

I y ðx; yÞ : I x ðx; yÞ

ð9Þ

Due to the fringes noise or low modulation points, for example at the fringes extreme, the calculations may have some inaccuracies that result in errors. To diminish this possibility, although media filtering may be used, we found that it is more accurate to calculate a dominant orientation of the angle a(x, y) in a neighborhood W2 of (x, y) as follows, PyþL=2 PxþK=2 1 yL=2 xK=2 2I x ðx2 ; y 2 ÞI y ðx2 ; y 2 Þ aðx; yÞ ¼ atan2 PyþL=2 PxþK=2 2 ; ð10Þ 2 2 yL=2 xK=2 I x ðx2 ; y 2 Þ  I y ðx2 ; y 2 Þ where K and L are the horizontal and vertical widths of the window W2 and (x2, y2) 2 W2 . This expression was derived in Ref. [17] and re-expressed like Eq. (10) in Ref. [18] with a sign error in the denominator, which is corrected here. For comfort, we set the dimensions of W2 equal to the product KL but it may be different. Notice that the calculation of a(x, y) makes use of only derivatives calculated previously with Eqs. (8). In summary, the normalizing process is performed in the following four steps: i. Smooth the original fringe pattern to reduce noise, for example by a direct convolution or with Gaussian filters [18]. ii. Calculate the Fourier coefficients on each pixel using Eq. (7). iii. Obtain the fringes direction with Eqs. (8) and (10). iv. Use Eqs. (2) and (4) to get the only cosine term. For computational purposes, especially if software like Mat Lab is used, it is better to express Eqs. (7) and (10) as convolutions and take the central part. In the following section, we will show the behavior of the algorithm with simulated and experimental images. 3. Results and discussion The experimental and simulated images used in this section have the following characteristics: they were smoothed with a direct convolution of 3 · 3 pixels five times; the size of the square window W, (L = K), used for the calculations

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of the Fourier coefficients was the same as the one used for the dominant orientation angle calculation W2; M = 1 was used in the Fourier expression, which means the calculation of only nine coefficients for each point (x, y); the software was programmed in Mat Lab 7 using a 2 GHz PC and the processing time to deal with the steps iiv was about one second per image. To understand the behavior of the algorithm under modulation and background variations we will show some simulation on Section 3.1 and experiments in Section 3.2.

3.1. Simulations We computer-generated various fringe patterns to demonstrate the performance of the proposed algorithm. The first one is an ESPI image (Fig. 1a) simulated as follows: Iðx; yÞ ¼ rAðx; yÞ þ ð1  rÞBðx; yÞmðx; yÞ cos2 ½uðx; yÞ;

ð11Þ

with r = 0.1, A(x, y) and B(x, y) randomly distributed on the interval [0,127], and m(x, y) a smooth function to change the image contrast. The result is a circular fringe pattern with random background, constant fringe spacing,

Fig. 1. Normalization results of a computer generated sinusoidal image (size 256 · 256). (a) Noisy image with background and modulation terms randomly distributed. It has fringe spacing of 32 pixels and maximum contrast of 0.8 at the edges which decreases toward its center, (b) normalized with the proposed method using L = 31 pixels.

Fig. 2. These curves represent the correlation of the ideal fringe pattern with the fringes obtained with our algorithm. Correlation values near 1.0 mean the image is better normalized and smoothed. The processed fringes were like 1a, using m(x,y) = 1, and contrasts, C, of 1.0 (continuous), 0.8 (dots), 0.6 (circles), 0.4 (plus) and 0.2 (asterisks). Each one of the five images was normalized using different window sizes. The maximum correlation obtained was 0.992 for C = 1.0.

N.A. Ochoa, A.A. Silva-Moreno / Optics Communications 270 (2007) 161–168

P, of 32 pixels and maximum contrast, C = (1  r)/(1 + r), at the edges of about 0.8. After following steps iiv with L = 31, the cosine only term is presented in Fig. 1b. Note that the contrast increases from the centre outwards to have the lowest modulation on the small ring. We did it in this way to emphasize on this fringe pattern the relationship between the distortions of the fringes with the contrast variation. The noise rejection on the entire image and the distortion at its center caused by the low modulation in that area can be seen. This distortion is not observed in setting m(x, y) = 1. Generally speaking, we can consider a non-normalized fringe pattern as normalized plus local variations of its contrast. To understand the performance of the algorithm with such contrast variations, we carried out the following simulation test using m(x, y) = 1. Various fringe patterns, like the one of Fig. 1a were simulated with contrasts of 0.2, 0.4,

165

0.6, 0.8 and 1.0. Each fringe pattern was analyzed with the algorithm using windows sizes that vary from 15 to 51 in steps of 2 pixels, i.e., approximately P ± P/2. The correlation coefficient of the ideal image and the one normalized with the algorithm was calculated for each case to assess the filtering and normalization amount. Values nearer to one mean better results. The results are plotted in Fig. 2. The four curves, continuous, dots, circles and plus, signs corresponding to 1, 0.8, 0.6 and 0.4 contrast values respectively, show a correlation maximum near the window size value of 32 pixels, which is the fringe spacing introduced. The curve corresponding to 0.2, (asterisks), also have maximum values near the 32 but with oscillations. It can be noticed that the pixel interval centered on 32 have a corresponding correlation interval of tolerable values, i.e. [0.98 0.99], where we can consider the fringe pattern successfully normalized and filtered.

Fig. 3. Normalization results from computer generated Bessel images. (a) Noisy image, (b) normalized with the proposed method (L = 15), (c) profiles of (b) (continuous) and ideal without noise (dots). The normalization is correct but there is still noise on the broad fringes.

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where J0 is the zeroth-order Bessel function of first kind, j j means absolute value and we set r = 1 to achieve a maximum contrast of 1.0 (Fig. 3a). We also chose L = 15 (the lower spaced fringes became blurred with L = 21). After following the four steps mentioned in Section 2.2, the normalized image obtained is shown in Fig. 3b. As expected, the noise was not rejected completely in the zone of low fringes density, but it was normalized anyway. The modulation variation is clearly seen in Fig. 3c, where the dotted line corresponds to the ideal profile of J 20 without noise and the continuous is the normalized with our algorithm. 3.2. Experimental results

Fig. 4. (a) Experimental Out-of-plane ESPI image of a plate under compression, (b) normalized using L = 25. Some dark areas are seen where jcosj > 1 + 0.025 due to low modulation.

The pixel interval is broad and almost the same for the four curves, being narrower for fringes contrast of about 0.2. With this result, we conclude that the algorithm is sensitive to the fringes spacing and that the optimal noise reduction is obtained using windows sizes near the local fringe period. We also find that the fringes may be distorted a bit if their contrast near 0.2 even using the optimum window size. The second simulated image (Fig. 3a) has local variations of modulation. It is a Bessel fringe pattern with random background and modulation functions given by: Iðx; yÞ ¼ rAðx; yÞ þ ð1  rÞBðx; yÞJ 20 ½juðx; yÞj;

ð12Þ

Fig. 4a is an image (size 256 · 128) of an aluminum rectangular disk while being compressed by a compression machine. It was obtained using an out-of-plane ESPI configuration. The fringes are low-contrasted with areas of small modulation. The image obtained from the normalization process is shown in Fig. 4b. L = 25 pixels was used. Fig. 5a is a time-averaged ESPI image of a vibrating square plate (256 · 256). We can represent this image by Eq. (12) and locally by Eq. (1). After processing it with the proposed method (L = 25), the normalized image is shown in Fig. 5b. Note that even while not using the optimal window size, corresponding to the fringes separation, the image is normalized and smoothed successfully. Fringe patterns like this do not have well defined line jumps between +p/2 and p/2, so it may be may difficult to use methods like the one proposed in Ref. [12] to obtain high quality skeletons without user intervention. Also, the low frequency fringes complicate the use of FFT techniques to normalize it [5]. The high contrast variations create problems in the selection of a background value for the full image [13]. Fig. 6a is a Fraunhofer diffraction image (256 · 256) of a rectangular aperture illuminated with a laser line. The

Fig. 5. (a) Bessel profiles of a rectangular metal plate excited with a loudspeaker using time-averaged ESPI, (b) normalized with the proposed method. The errors at the saddle and low modulations points are clearly seen.

N.A. Ochoa, A.A. Silva-Moreno / Optics Communications 270 (2007) 161–168

167

Fig. 6. (a) Fraunhofer diffraction image of a narrow rectangular aperture, (b) normalized with the proposed method, (c) using a constant derivative direction a = 0, (d) profiles of (b) (continuous) and smoothed (a) (dots) to show the background and modulation terms.

aperture is much larger on one side than in the other at a proportion of about 1/200. Fig. 6b shows the normalized image obtained with the proposed algorithm (L = 25). Some ripples can be seen due to a non-isotropic noise of the fringes. It is interesting to note that fringes whose orientations do not vary greatly do not need the calculation process of the fringes orientation (a(x, y)). This value may be introduced by the user instead. Obviously, the reduction of one step in the normalization process also removes the errors due to the a calculation. For example Fig. 6c is the result of using Eqs. (2) and (4) with a = 0; the image is better filtered and normalized. Fig. 6d contains the profiles of the smoothed patterns before (dotted line) and after normalization, 6(b) (continuous line). The variations of the background and modulation terms can be seen. The smoothing filter applied to the images (step i) do not completely remove the speckle noise of the fringes; however its use helps to calculate the intensity derivatives more precisely. The algorithm produces reliable results without using the filtering process, but the fringes with higher periods are noisier. The application of better smoothing filters will improve the noise reduction of the low frequency fringes, but in any case, the fringes shape will depend on the window size, as shown in Fig. 2.

When the denominator is zero or when the division goes beyond ±1, Eq. (4), there may be some inaccuracies. This situation may happen at low modulation points such as fringe extreme or saddle points. A solution to the extreme points is the use of a tolerance parameter t in Eq. (4), i.e. when jcosj > 1 + t the points are excluded. The tolerance we used for all the normalization results shown in this article was t = 0.025, which introduced a maximum error of only 5% on peaks and valleys pixels of the fringes. For the saddle points we do not have a general answer and we just marked them, as in Fig. 5b. The Eq. (4) cannot be obtained unless the phase is assumed to be linear locally, that is why it can be expressed by only 9 Fourier coefficients (M = 1) in Eq. (6). The use of additional coefficients (M > 1) do not improve the normalization results significantly. We explained a general procedure to calculate the only cosine term and calculate the derivatives; however a similar procedure must be used if another approximation function is proposed. The concept is to relate the coefficients with the derivatives. If non-orthogonal basis functions are used, the coefficients may be calculated with a least squares procedure. It is clear from Eq. (4) that the cosine only term depends on the derivatives of I(x, y), more specifically, on

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the correct calculation of them. We propose one method to calculate the derivatives using Fourier series but other methods may be developed as well. It must be noted that in order to normalize the fringes it is only necessary to know the fringes orientation, not their direction.

his useful comments about the writing. We want to thank the anonymous reviewers who made comments, which we found extremely valuable.

4. Conclusions

[1] G.L. Cloud, Optical Methods of Engineering Analysis, Cambridge University Press, 1995. [2] P.H. Eichel, D.C. Ghiglia, C.V. Jakowatz Jr., Opt. Lett. 14 (1989) 1. [3] K. Creath, Appl. Opt. 24 (1985) 3503. [4] M. Takeda, H. Ina, S. Kobayasi, J. Opt. Soc. Am. 72 (1982) 156. [5] Th. Kreis, J. Opt. Soc. Am. A 3 (1986) 847. [6] J.A. Quiroga, J.A. Gomez Pedrero, A. Garcia Botella, Opt. Commun. 197 (2001) 43. [7] F. Chen, T.E. Allen, C.T. Griffen, in: P.K. Rastogi, D. Inaudi (Eds.), Trends in Optical Nondestructive Testing and Inspection, Elsevier, 2000, p. 241. [8] J. Villa, I. De la Rosa, G. Miramontes, J.A. Quiroga, J. Opt. Soc. Am. A 22 (2005) 2766. [9] A. Federico, G.H. Kauffmann, Appl. Opt. 44 (2005) 2728. [10] J.A. Quiroga, M. Servin, Opt. Commun. 224 (2003) 221. [11] Ch. Quan, Ch.J. Tay, F. Yang, X. He, Appl. Opt. 44 (2005) 4814. [12] Q. Yu, X. Yang, S. Fu, X. Sun, Appl. Opt. 44 (2005) 7050. [13] U. Mieth, W. Osten, W. Juptner, in: W. Juptner, W. Osten, (Eds.) Fringe ’93 Automatic Processing of Fringe Patterns, 1993, p. 365. [14] R. Legarda-Sae´nz, W. Osten, W. Juptner, Appl. Opt. 41 (2002) 5519. [15] J.A. Guerrero, J.L. Marroquı´n, M. Rivera, J.A. Quiroga, Opt. Lett. 22 (2005) 3018. [16] R. Robin, V. Valle, .F. Bre´mand, Appl. Opt. 44 (2005) 7261. [17] A. Rao, B.G. Schunck, CVGIP Graph. Model. Image Process. 53 (1991) 157. [18] X. Zhou, J.P. Baird, J.F. Arnold, Appl. Opt. 38 (1999) 795.

This study has proven the possibility to simultaneously normalize and reduce the noise of fringe patterns that can be represented locally by an expression involving a cosine term. The theory was developed and demonstrated with experimental and computer generated images. Its advantages and limitations were also analyzed. The experimental images were obtained from diffraction and from ESPI using deformation and time-averaged vibration. The derivatives needed for our algorithm were calculated with Fourier series; however, other orthogonal or not orthogonal basis functions may be used as well. The algorithm is easily-programmable. The processing time to normalize each image is about one second with a 2 GHz computer. Acknowledgements This research is supported by the CONACyT of Mexico under project SEP-2004-C01-46970 and CONCyTEG. The authors thank Juan Antonio Rayas for providing the images 4 and 5 used in this study and Clift Williams for

References