Tangent least-squares fitting filtering method for electrical speckle pattern interferometry phase fringe patterns

Tangent least-squares fitting filtering method for electricalspeckle pattern interferometry phase fringe patterns

Chen Tang, Wenping Wang, Haiqing Yan, and Xiaohui Gu

An efficient method is proposed to reduce the noise from electrical speckle pattern interferometry (ESPI)phase fringe patterns obtained by any technique. We establish the filtering windows along the tangentdirection of phase fringe patterns. The x and y coordinates of each point in the established filteringwindows are defined as the sine and cosine of the half-wrapped phase multiplied by a random quantity,then phase value is calculated using these points’ coordinates based on a least-squares fitting algorithm.We tested the proposed methods on the computer-simulated speckle phase fringe patterns and theexperimentally obtained phase fringe pattern, respectively, and compared them with the improvedsine�cosine average filtering method [Opt. Commun. 162, 205 (1999)] and the least-squares phase-fittingmethod [Opt. Lett. 20, 931 (1995)], which may be the most efficient methods. In all cases, our results areeven better than the ones obtained with the two methods. Our method can overcome the main disad-vantages encountered by the two methods. © 2007 Optical Society of America

OCIS codes: 120.6160, 100.5010, 070.6110.

1. Introduction

Electronic speckle pattern interferometry (ESPI) is aconvenient method for various nondestructive testsand measurements of optically diffuse objects. Accu-rate extraction of phase values is very important forthe successful application of the ESPI technique. Inthe study of static phenomena, the phase-shiftingtechnique is one of the best methods for demodulationbecause of its high accuracy. However, in the case ofdynamic and fast transient processes that are unsuit-able for the phase-shifting technique, the need forprocessing a single fringe pattern arises. There aremany techniques to estimate the phase term from asingle pattern, such as the Fourier transform method,1the synchronous method,2 and the phase-locked loopmethod.3 However, these techniques work well only ifthe analyzed interferogram has a carrier frequencyand a narrow bandwidth and if the signal has lownoise. Moreover, these methods fail for phase calcu-lation of a closed-fringe pattern.4 Additionally, the

Fourier and synchronous methods estimate thephase wrapping because of the arctangent functionused in the phase calculation, so an additional un-wrapping process is required. Another technique forextraction of phase from a single fringe pattern usesa regularized phase tracker.5 However, this methodis time consuming and is affected by the fringesvisibility.6 In general, the problem of evaluatingphase fringe patterns can be attacked from two pointsof view. In the first, all efforts are put into improvingthe filter algorithm. If a sufficiently high degree ofquality can be achieved, then the unwrapping of theresulting perfectly filtered phase fringe pattern canbe accomplished with a common unwrapping algo-rithm. In the second point of view, all efforts are putinto improving the unwrapping algorithm.7

In this paper, we put emphasis on improving thefiltering algorithm for phase patterns. Owing to thenature of speckle, at many pixels of the speckle pat-terns, the intensity is close to or even below theelectronic noise level. The fringe-analysis or phase-analysis methods that are suitable for typical inter-ferometry with smooth wavefronts therefore cannot beapplied to ESPI, and special techniques are required.7–12

Among these methods, the least-squares phase-fittingmethod8 and sine�cosine average filter method7 maybe the better available representative methods.

The basic ideal of the least-squares phase-fittingmethod is that the points whose x and y coordinates arethe differences of the intensities are clustered as a line

C.Tang([email protected]),W.Wang,andH.Yan([email protected]) are with the Department of Applied Physics, University ofTianjin, Tianjin 300072, China. X. Gu is with the School of Avia-tion and Astronavigation Engineering, Shenyang Institute of Aero-nautical Engineering, Shenyang 110034, China.

Received 29 November 2006; revised 11 February 2007; accepted11 February 2007; posted 12 February 2007 (Doc. ID 77535); pub-lished 1 May 2007.

0003-6935/07/152907-07$15.00/0© 2007 Optical Society of America

20 May 2007 � Vol. 46, No. 15 � APPLIED OPTICS 2907

within a small window. Fitting this line, then twicethe inclination angle of the line should be the meanphase of the pixels. With the least-squares phase-fitting method, the noise reduction is automaticallyaccomplished when each phase value is estimatedfrom a distribution of intensity differences by the least-squares fit. However, this method is only suitable forthese phase maps obtained by the phase-shifting tech-nique with four phase-shifted original speckle pat-terns. Furthermore, the process of this techniquecannot be repeated, so sometimes phase fringe pat-terns obtained by this method still have noise. Al-though the author of this method has made someimprovements on this method,9,12 the improved al-gorithms are complex.

For phase fringe patterns, owing to the speckle noiseand the 2� discontinuities, applying a conventionalfiltering method often smear out the discontinuities.For example, we have presented the partial differen-tial equations (PDEs) filtering methods to reduce noiseof the ESPI fringe patterns,13 and we also have utilizedthe differential equations enhancement method to en-hance contrast of the ESPI fringe patterns.14 Althoughthe PDEs filtering method can improve the quality ofESPI fringes significantly, like other usual filteringmethods, this technique cannot produce the so-calledsawtooth jumps of phase fringe patterns. This problemis solved by the sine�cosine average filter method.7 Inthis method, the sine and cosine of the wrapped phasefringe pattern is first calculated, which leads to con-tinuous fringe patterns. Then these sine and cosinefringe patterns are filtered individually by applying anaverage filter. The phase fringe pattern is finally ob-tained by the inverse tangent of the filtered sine andcosine fringe pattern. This method is suitable for allphase maps that may be obtained by any technique.Aebischer et al. have improved this method.7 One im-provement is to apply a sine�cosine average filter 20 to30 times rather than only two or three times as previ-ously practiced. In addition, depending on the patternto be filtered, they allow for an anisotropic averagefilter for best results. This anisotropic filtering is sim-ply realized by using the rectangular filtering windowsinstead of square filtering windows. They have shownthat their method clearly outperformed the best cur-rently available filter algorithms, like the scale-spacefilter and Pfister’s partially recursive filter.7 However,according to our own experience in the method, it isdifficult to choose an appropriate filtering window forsome patterns.

Here we present a filtering method for ESPI phasefringe patterns that is inspired by spin filtering15 andthe least-squares phase-fitting method. We first calcu-late the tangent direction of the phase map and estab-lish filtering windows along the calculated tangentdirection of intensity. Here the x and y coordinates ofthe points within the established filtering windows aredefined as the sine and cosine of the half-wrappedphase multiplied by a random quantity. Then thephase value is calculated using these points’ coor-dinates. Our method can be repeated and is suitablefor all phase patterns obtained by any technique. We

also present a method for establishing an appropriatefilter window, so our method can overcome the above-mentioned drawbacks that both the least-squaresphase-fitting method and the improved sine�cosineaverage filtering method encounter. We name theproposed method the tangent least-squares fittingfiltering method (TLSFF), which differs from theformer PDEs filtering method completely.13 This fil-tering method is suitable for ESPI phase fringe pat-terns, while the PDEs filtering method is suitable forESPI fringe patterns. Furthermore, the basic princi-ple of this filtering method is completely differentfrom that of the PDEs filtering method. The mainidea of the PDEs filtering method is transforming theimage processing to solving partial differential equa-tions. The image is regarded as a continuous signal,and the filtering function is treated as a partial dif-ferential operator, and then the PDE can be seen asthe iteration of local filters with an infinitesimalneighborhood.

We tested the proposed method on the computer-simulated speckle phase fringe patterns and experi-mentally obtained phase fringe pattern, respectively,and compared it with the improved sine�cosineaverage filtering method and the least-squaresphase-fitting method. We quantitatively evaluatethe performance of our method with two compara-tive parameters, the image fidelity and the speckleindex. Experimental results have confirmed thatthe proposed method is capable of removing noise inESPI phase images effectively.

2. Least-Squares Phase-Fitting Method

Hong has presented a least-squares phase-fittingmethod.8 The intensities of the (i, j) pixel of the fourphase-shifted original speckle patterns are thengiven by

I1,i, j � I0,i, j � Ir,i, j � 2�I0,i, j Ir,i, j cos �i, j � n0,i, j, (1a)

I2,i, j � I0,i, j � Ir,i, j � 2�I0,i, j Ir,i, j cos��i, j ��

2�� n0,i, j,

(1b)

I3,i, j � I0,i, j � Ir,i, j � 2�I0,i, j Ir,i, j cos��i, j � �i, j ��

2�� n0,i, j,

(1c)

I4,i, j � I0,i, j � Ir,i, j � 2�I0,i, j Ir,i, j cos��i, j � �i, j � �� n0,i, j,(1d)

where I0,i, j and Ir,i, j are the intensities of the objectand the reference beams, respectively; �i, j is the ran-dom interferometric phase of the speckle field; �i, j isthe phase change that is due to deformation of thesurface of the tested object; n0,i, j is the random noise.Usually, the deformation phase �i, j is calculated from

�i, j � 2 tan�1�I2,i, j � I3,i, j

I1,i, j � I4,i, j�. (2)

2908 APPLIED OPTICS � Vol. 46, No. 15 � 20 May 2007

Let

xi, j � I1,i, j � I4,i, j � 4�Ir,i, j I0,i, j cos��i, j � �i, j�2�� cos��i, j�2�, (3)

yi, j � I2,i, j � I3,i, j � 4�Ir,i, j I0,i, j cos��i, j � �i, j�2�� sin��i, j�2�. (4)

The noise of the phase map is usually reduced byaveraging the phase values of the neighboring pixelsor by processing the map with a median filter. Inthese processes the information of the uncertaintiesis lost, and all the phase values are treated equally.The least-squares phase-fitting method can overcomesuch drawbacks.8 Within a m � m window (for ex-ample, m � 5), the points whose x and y coordinatesare the differences, given by Eqs. (3) and (4), of the256 gray-level intensities of the pixels, are clusteredas a line. If we fit a line passing through the origin tothem, then twice the inclination angle of the lineshould be the mean phase of the pixels. By a naturalcriterion of the best-fit line, the inclination angle ofthe line should be obtained, by the sum

k,l

�� k,l

�xk,l2 sin2 � � 2xk,lyk,l sin � cos �

� yk,l2 cos2 ��, (5)

which is minimized, where k and l are the subscriptsof the pixel point in the fitting window, i � (m � 1)�2 k i � (m � 1)�2 and j � (m � 1)�2 l j �(m � 1)�2.

When the derivative of �� to � is zero, the sumis minimized, i.e.,

��

� 0.

From the above equation, the inclination angle of thefitted line is obtained by

�i, j �12 tan�1 2

k,lxk,lyk,l

k,l

�xk,l2 � yk,l

2��. (6)

So the deformation phase �i, j is obtained by

�i, j � 2�i, j. (7)

3. Main Principle of the Proposed Method

Our method differs from the least-squares phase-fitting method as outlined in Section 2 on twopoints: (1) The x and y coordinates of the pointswithin fitting windows are the sine and cosine of thehalf-wrapped phase multiplied by a random quan-tity instead of intensity differences of four specklepatterns. (2) Fitting windows are along the tangentdirection of phase fringe patterns. Using such asimple process, our method can be repeated andproduces an excellent effect for filtering speckle-

interferometric phase fringe patterns obtained byany technique.

Equations (3) and (4) have the sine and cosine ofthe half-wrapped phase and the random quantities.For corresponding to Eqs. (3) and (4), here we let

xi, j � Ri, j � cos��i, j�2�, (8)

yi, j � Ri, j � sin��i, j�2�, (9)

where Ri, j is a random quantity.In the least-squares phase-fitting method,8 x and

y coordinates are the intensity differences, so theleast-squares phase-fitting method cannot be iter-ated. While using our definition, our method can berepeated.

From Eqs. (8) and (9), we have

�i, j � 2 tan�1�yi, j

xi, j�. (10)

A wrapped phase value �i, j obtained by any techniquedistributes over the interval �0, 2��. Although ourdefinition of the x and y coordinates is different fromthat of the least-squares method of Ref. 8, the ratio ofy to x is the same and equal to half-phase value.Figure 1 gives a real distribution of the points in asmall filtering window 3 � 7. Similar to the least-squares phase-fitting method of Ref. 8, we can alsoobtain a line passing through the origin based on theprinciple that the sum of the squares of the distancesfrom the points to the line is the minimum. The in-clination angle of the line should be given by Eq. (6).The inclination angle of the line AB in Fig. 1 is ob-tained based in Eq. (6) using the points’ coordinates.

Fig. 1. Distribution of the points whose x and y coordinates aregiven by Eqs. (8) and (9) within a small filtering window 3 � 7,where the inclination angle of the line AB is obtained based on Eq.(6) using these points’ coordinates.


Then twice the angle should be the mean phase of thepixel. In our method, the noise of the phase map isreduced by using the x and y coordinates of the neigh-boring pixels [see Eq. (6)]. The value of �i, j obtainedby Eq. (7) distributes over the interval ��2, ��2�.The correct quadrant of �i, j should be determinedwith the signs of the two sums in Eq. (6). The processof our method can be repeated. The small filteringwindow m � n can be either rectangular �m � n� orsquare �m � m� along the vertical direction. For ob-taining a better filtering effect, we establish filteringwindows along the tangential direction of the phasefringe pattern.

4. Construction of the Filtering Windows Along theTangential Direction

As we know, on a local area of a phase fringe pattern,the gray-level (or intensity) distribution has the max-imum gradient in the direction normal to the fringeand has the minimum gradient in the direction of thetangent of the fringe. A window of m � m �5 � 5or 7 � 7) pixels with its center on the current point(i, j) is considered, where its tangent direction valuesmust be inside the interval �0, ��. The continuousdirection values are quantified into eight isogonic di-rection values (see Fig. 2), where every direction isrespectively marked by d (0–7). In order to make sureat which direction the tangent direction is, we com-pute the sum of the absolute differences between theintensity of each pixel and the average intensity ineach direction. It can be expressed as

Ai, jd �

1m

l�1

m

fld, (11)

Ci, jd �

l�1

m

�fld � Aij

d�, (12)

where Aijd denotes the average intensity in the dth

direction; fld denotes the intensity of the lth�l �

0, 1, 2, . . . , 7� pixel in the dth direction; (i, j) denotesthe coordinate of the current pixel; d denotes thedirection number; and m denotes the total number ofpoints in the dth direction; see Fig. 2, m � 7.

Let d* be at the right angle to the direction d, that is,

d* � �d � 4�mod8. (13)

Seen from formula (12), Ci, jd is in direct ratio to the

intensity change in the dth direction; that is, thestronger the intensity changes in the dth direction,the bigger Ci, j

d is. The tangent direction for currentpixel (i, j) is the direction at which the image inten-sity changes the most weakly and approximates tothe direction at which Ci, j

d�Ci, jd* (or Ci, j

d) takes theminimum. So the tangent direction of current pixel(i, j) can be defined as the following:

If Ci, jd* � 0, when Ci, j

d�Ci, jd* �d � 0, 1, 2, . . . , 7� is

minimized, the corresponding direction is the tangentdirection of current pixel (i, j).

If Ci, jd* � 0, when Ci, j

d �d � 0, 1, 2, . . . , 7� is min-imized, the corresponding direction is the tangentdirection of current pixel (i, j).

We establish filtering windows along the calculatedtangent direction shown in Fig. 3.

5. Implementation and Verification of the ProposedMethods

We summarize the steps to implement the proposedmethod. We assume that the ESPI phase fringe pat-terns have been obtained.

(1) Establish filtering windows along the calcu-lated tangent direction.

(2) Calculate x and y coordinates of the points inthe filter windows based on Eqs. (8) and (9).

(3) Calculate the angle �i, j based on Eq. (6) usingthe points’ coordinates; twice the angle is the phase�i, j, then the correct quadrant of �i, j should be deter-mined with the signs of the two sums in Eq. (6).

We have tested our method on the computer-simulated speckle phase fringe patterns and experi-mentally obtained phase fringe pattern, respectively,which include sparse and dense fringes. For compari-son with the least-squares phase-fitting method, herethe tested original speckle phase patterns are still ob-tained by the phase-shifting technique with Eq. (2).

Figures 4(a)–6(a) are the computer-simulated orig-inal speckle phase patterns based on Eq. (2). Here welet �, I0, Ir, and n0 in Eqs. (1a)–(1d) uniformly distrib-ute over the intervals ��, ��, �0, Im�, �0, Im�, and��In, In� and choose Im � 180 and � 0.2. Let the

Fig. 2. Eight quantified directions: (a)–(h) directions 0–7 in agiven mask (size 7 � 7), where the center point of the mask justcorresponds to the current pixel (i, j).

Fig. 3. Filtering window along the tangent direction.


random quantity Ri, j in Eqs. (8) and (9) distributeover the small interval [0.8, 1.2]. The phase �i, j cor-responding to Figs. 4(a)–6(a) is given as the following:

�i, j � 30 ��exp��i � 0.5M�2 � j2

10000 �� exp�

�i � 0.5M�2 � �j � N�2

10000 � ,

M � 244, N � 244, In � 80; (14)

�i, j � 25 ��exp��1.4i � 78�2 � �j � 78�2

45000 �� exp�

�1.4i � 128�2 � �j � 192�2

45000 � ,

M � 244, N � 244, In � 70; (15)

�i, j � 2��6 �i � 0.5M

M �2

� �6 �j � 0.5N

N �2�,M � 244, N � 244, In � 20, (16)

where M, N represent the sizes of the computer-simulated image.

Figure 7(a) shows an original speckle phase pat-tern based on Eq. (2) with the four phase-shiftedspeckle patterns, which depicts the derivative of theout-of-plane displacement of a square plate. The plateis rigidly clamped at its boundary and is subjected toa central load. These four phase-shifted speckle pat-terns are obtained using a shearing device in theinterferometer.

The various methods, including our tangent least-squares fitting filtering method (TLSFF), the im-

Fig. 4. Computer-simulated phase pattern corresponding to Eq.(14) and its filtered images: (a) Original image. (b) The filteredimage by TLSFF with a 3 � 7 window for three iterations. (c1), (c2)The filtered images by ISCAF with 3 � 7 and 7 � 3 windows forthree iterations. (d1), (d2) The filtered images by LSF with 3 � 7 and7 � 3 windows.

Fig. 5. Computer-simulated phase pattern corresponding to Eq.(15) and its filtered images: (a) Original image. (b) The filteredimage by TLSFF with a 3 � 7 window for three iterations. (c1), (c2)The filtered images by ISCAF with 3 � 7 and 7 � 3 windows forthree iterations. (d1), (d2) The filtered images by LSF with 3 � 7 and7 � 3 windows.


proved sine�cosine average filtering method (ISCAF),and the least-squares phase-fitting method (LSF),are applied to Figs. 4(a)–7(a). The filtered phase im-ages obtained by our proposed method with a 3 � 7window are shown in Figs. 4(b)–7(b). The results ofthe improved sine�cosine average filtering methodwith a 3 � 7 window and a 7 � 3 window are shownin Figs. 4(c1) and 4(c2) through Figs. 7(c1) and 7(c2).The filtered phase images obtained by the least-squares phase-fitting method (LSF) with a 3 � 7window and a 7 � 3 window are shown in Figs. 4(d1)and 4(d2) through Figs. 7(d1) and 7(d2).

Further, we quantitatively evaluate the perfor-mance of our method with two comparative param-eters, the image fidelity f and the speckle index s.16,17

The image fidelity f, which is a parameter thatquantifies how good image details are preservedafter noise removal, is defined as

f � 1 � �I0 � I�2

I02 , (17)

where I0 and I are the noiseless image and the esti-mated image, respectively. A high fidelity value willindicate that the processed image is very similar to thenoiseless one, i.e., has good fidelity. The speckle indexs is used to quantify the local smoothness of the filteredfringe patterns. This parameter is evaluated as thesum of the ratios of the local standard deviation to itsmean. The speckle index can be regarded as an aver-age reciprocal signal-to-noise ratio, where the signal isthe mean value and the noise is the standard devia-tion. Therefore a low speckle index should be regardedas an indication of local smoothness of the fringe pat-

Fig. 6. Computer-simulated phase pattern corresponding to Eq.(16) and its filtered images: (a) Original image. (b) The filteredimage by TLSFF with a 3 � 7 window for two iterations. (c1), (c2)The filtered images by ISCAF with 3 � 7 and 7 � 3 windows for twoiterations. (d1), (d2) The filtered images by LSF with 3 � 7 and7 � 3 windows.

Fig. 7. Experimentally obtained phase fringe pattern and its fil-tered images: (a) Original image. (b) The filtered image byTLSFF with a 3 � 7 window for three iterations. (c1), (c2) Thefiltered images by ISCAF with 3 � 7 and 7 � 3 windows for threeiterations. (d1), (d2) The filtered images by LSF with 3 � 7 and7 � 3 windows.


tern. Since the noiseless image I0 for the experimen-tally obtained speckle phase pattern shown in Fig. 7is unknown, here the image fidelity f is calculated forthe filtered images of computer-simulated phase pat-terns (shown in Figs. 4–6) and the speckle index s iscalculated for all filtered images (shown in Figs. 4–7).The results are given in Table 1.

As we can see from the original phase images, thenoise in the original phase patterns is very high. Inall cases, our method gives desired results. All fringesare perfectly preserved, and the noise is very effec-tively suppressed. From Table 1, one can find that theimage fidelity f for the filtered image obtained by ourmethod is the highest and the speckle index is thelowest in each test. Therefore our results are evenbetter than the ones obtained with improved sine�cosine average filtering method and the least-squaresphase-fitting method.

6. Conclusion

We propose a very efficient method for filtering elec-trical speckle pattern interferometry phase fringepatterns. Our method can be repeated and is suit-able for all phase maps obtained by any technique.We also present a method for establishing filterwindows, which is suitable for other filter meth-ods. Our method overcomes the main disadvantagesthat the best known improved sine�cosine averagefiltering method and the least-squares phase-fittingmethod encounter. We test, the introduced method onthe computer-simulated speckle phase fringe patternsand experimentally obtain phase fringe pattern, re-spectively, which include sparse and dense fringes. In

all cases, our method gives the desired results. Exper-imental results confirm that the proposed method iscapable of removing noise in ESPI phase images effec-tively.

This work was supported by Liu Hui AppliedMathematics Center of Nankai-Tianjin University.We thank Jinglong Chen at Tianjin University forhis kind help of providing some images. We alsothank the reviewers for their valuable suggestionsfor improvement of the paper.

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Table 1. Performance Evaluation Results Based on Phase PatternsShown in Figs. 4–7

Images f s

Fig. 4(b) 0.9070 0.1744Fig. 4(c1) 0.8934 0.2070Fig. 4(c2) 0.8947 0.2016Fig. 4(d1) 0.8769 0.2531Fig. 4(d2) 0.8811 0.2448



Fig. 7(b) 0.0914Fig. 7(c1) 0.1575Fig. 7(c2) 0.1594Fig. 7(d1) 0.2235Fig. 7(d2) 0.2225


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Tangent least-squares fitting filtering method for electrical speckle pattern interferometry phase fringe patterns