January 15, 2008 / Vol. 33, No. 2 / OPTICS LETTERS 183

Nearly preprocessing-free method forskeletonization of gray-scale electronic speckle

pattern interferometry fringe patterns via partialdifferential equations

Chen Tang,1,* Wenjing Lu,1 Yuanxue Cai,1 Lin Han,1 and Gao Wang2

1Department of Applied Physics, University of Tianjin, Tianjin, 300072, China2National Key Lab For Electronic Measurement Technology, North University of China, Xueyuan Road 3, Taiyuan,

Shanxi 030051, China*Corresponding author: tangchen@tju.edu.cn

Received October 17, 2007; revised November 21, 2007; accepted November 21, 2007;posted November 29, 2007 (Doc. ID 88718); published January 14, 2008

We describe a novel method for skeletonization of gray-scale electronic speckle pattern interferometry(ESPI) fringe patterns. Our method is based on the gradient vector field (GVF). We propose a new partialdifferential equation model for calculating the GVF of ESPI fringe patterns. Further, we propose rules usedto measure the possibility of each pixel on the skeleton based on the topological analysis of the GVF. Thefinal skeletons are traced, which mimics the behavior of edge detection based on these rules. The proposedmethod works directly on the gray-scale images. © 2008 Optical Society of America

OCIS codes: 120.6160, 110.6150, 100.2650.

Electronic speckle pattern interferometry (ESPI) is awell-known, nondestructive, whole-field techniquefor measuring displacements. The fringe skeletonmethod may be the most straightforward approach tofringe analysis [1]. It is well known that skeletoniza-tion of fringes has played an important role in thefringe skeleton methods. There are many techniquesto extract the fringe skeletons, which can be classi-fied into two main groups: fringe extreme trackingmethods and thresholding binary-fringe and thin-ning methods. However, these methods are likely tobe affected by the fringes’ noise and visibility. Nomatter which type of technique is used for extractionof skeletons of ESPI fringe images, a prerequisite ismuch preprocessing of original ESPI fringe imagesby some special techniques [2,3].

Recently, some efforts have been made toward thecomputation of skeletons directly from a gray-scaleimage by partial differential equation (PDE) imageprocessing methods. In [4], Tari et al. calculated theedge strength function by using a linear diffusionequation. The skeletons were extracted from a set oflevel-set curves of this edge strength function. In [5],Jang and Hong calculated a pseudo-distance-map byusing a nonlinear governing equation and extractedthe skeletons from this pseudo-distance-map. In [6],the skeleton strength map was generated from an an-isotropic diffusion vector field. The skeleton wastraced through the skeleton strength map. Chen andFarag [7] investigated topological features of the vec-tor field. Most of these methods are based on the gra-dient vector field (GVF) of a given image, obtained bya set of linear or nonlinear governing PDEs. Thenthese gradient vector fields are used for the extrac-tion of skeletons by various technologies.

Unfortunately, owing to speckle noise, previousgoverning equations cannot be used to calculateGVFs of ESPI fringe patterns. In this Letter, we de-

rive a new coupled nonlinear governing equation

0146-9592/08/020183-3/$15.00 ©

based on the variational method for calculating theGVFs of ESPI fringe patterns. Then we normalizethe GVF, calculate the strength of the partial deriva-tive of the normalized GVF (NGVF), and detect thesign of the real parts of eigenvalues of the Jacobianmatrix of the NGVF, thus determining the possibilityof each pixel on the skeleton. The final skeletons aretraced, which mimics the behavior of edge detection.

Originally, the GVF was proposed by Xu andPrince [8], where the GVF is defined as the vectorfield V�x ,y�= �u�x ,y� ,v�x ,y�� which minimizes the en-ergy function

E = ��

���ux2 + uy

2 + vx2 + vy

2� + ��F�2�V − �F�2�dxdy,

�1�

where ux and uy are the first-order partial derivativesof u�x ,y�, and vx and vy are the first-order partial de-rivatives of v�x ,y�. F is an edge strength map of theoriginal image I; � is a constant.

In the past decade, many different PDE denoisingmodels have been proposed [2,9–11]. Among these de-noising models, the anisotropic diffusion model [9]and the coupled nonlinear denoising model [10] aretwo representative and effective PDE denoising mod-els. Inspired by this work, here we redefine the GVFto be the vector field V�x ,y�= �u�x ,y� ,v�x ,y�� thatminimizes the energy functional

E1 = ��

���g��V����V� + ��V − �I�2� + �f���V���dxdy.

�2�

In this equation, � ,�, and � are weighting param-eters, and g�s�= �1+ks2�−1 is a nonincreasing functionwith, g�0�=1, g�s��0, and g�s�→0 at infinity. It is re-

quired that the function f�·��0 and be an increasing

2008 Optical Society of America

184 OPTICS LETTERS / Vol. 33, No. 2 / January 15, 2008

function. ��V � =ux2+uy

2+vx2+vy

2. V� is chosen to mini-mize ��V� �dxdy, and at the same time V� is not toofar removed from V, so V� is chosen to minimize theenergy functional

E2 = ��

�a��V�� + �b/2��V� − V�2�dxdy, �3�

where a�t� is a parameter that changes with time,and b remains constant.

Using the calculus of variations [8,11], it can beshown that the GVF can be found by solving the fol-lowing PDEs:

�u

�t= �g���u�����u�div� �u

��u�� + � � �g���u���� � u

− ��u − Ix���u� + � div�g���u��� � u�,

�u�

�t= a�t�div� �u�

��u��� − b�u� − u�,

�v

�t= �g���v�����v�div� �v

��v�� + � � �g���v���� � v

− ��v − Iy���v� + � div�g���v��� � v�,

�v�

�t= a�t�div� �v�

��v��� − b�v� − v�, �4�

with the initial conditions

V0 = V0� = � �I

�x,�I

�y� . �5�

After calculating the GVF by using Eq. (4), we nor-malize it by means of

uN =u

u2 + v2, vN =

v

u2 + v2. �6�

The NGVF can describe the geometric structures ofthe boundary and the skeleton of an object very well[7]. Therefore, to locate the skeleton points, we com-pute what we call the strength of the partial deriva-tive of the NGVF and detect the sign of the real partsof eigenvalues of the Jacobian matrix of the NGVF,and thus measure the possibility of each pixel on theskeleton.

Let Sx,y denote the strength of the partial deriva-tive of NGVF for pixel �x ,y� defined by

Sx,y =

� �ux,yN

�x�2

+ � �vx,yN

�y�2

max�� �ux,yN

�x�2

+ � �vx,yN

�y�2� . �7�

The Jacobian matrix of the NGVF for pixel �x ,y� is

�uN

�x

�uN

�y

�vN

�x

�vN

�y�

�x,y�

. �8�

For detecting the sign of the real of eigenvalues ofthe Jacobian matrix for pixel �x ,y�, we calculate the

following equation:

JMx,y = � �ux,yN

�x+

�vx,yN

�y� −

real� �ux,yN

�x�2

− 2� �ux,yN

�x�� �vx,y

N

�y� + � �vx,y

N

�y�2

+ 4� �ux,yN

�y�� �vx,y

N

�x�

�, �9�

where � is a constant.From the topology point of view, the pixels on the

skeleton of white fringes should exist where thestrength of the partial derivative of the NGVF has alarge value and the sign of the real parts of eigenval-ues of the Jacobian matrix is negative. Conversely,the pixels on the skeleton of black fringes exist wherethe strength of the partial derivative of the NGVFhas a large value and the sign of the real parts of ei-genvalues of the Jacobian matrix is positive. The fi-nal skeletons can be traced, which mimics the behav-ior of edge detection. Given the threshold, say T̄, theskeleton image of white fringes fW�x ,y� and the skel-eton image of black fringes fB�x ,y� are, respectively,

obtained through the following rules:

fW�x,y�

=�1, skeleton pixel if Sx,y � T̄, JMx,y 0

0, nonskeleton pixel otherwise,

�10�

fB�x,y�

=�1, skeleton pixel if Sx,y � T̄, JMx,y � 0

0, nonskeleton pixel otherwise,

�11�

January 15, 2008 / Vol. 33, No. 2 / OPTICS LETTERS 185

Figure 1(a) is a computer-simulated original ESPIfringe pattern. Its black and white fringe skeletonsobtained by our method are shown in Figs. 1(b) and1(c), respectively. Figure 1(d) is its filtered image, us-ing 33 window mean filtering only once, and itsblack and white fringe skeletons obtained by ourmethod are shown in Figs. 1(e) and 1(f), respectively.

Fig. 1. Computer-simulated fringe pattern and its skel-etons obtained by our method: (a) initial image, (b) blackfringe skeletons of (a), (c) white fringe skeletons of (a), (d)filtered image of (a) using a 33 window mean filteringonly once, (e) black fringe skeletons of (d), (f) superimposi-tion of the white fringe skeletons of (d) onto (d).

Fig. 2. Experimentally obtained ESPI fringe pattern andits skeletons obtained by our method: (a) initial image, (b)black fringe skeletons of (a), (c) white fringe skeletons of(a), (d) filtered image of (a) using a 33 window mean fil-tering only once, (e) black fringe skeletons of (d), (f) super-imposition of the white fringe skeletons of (d) onto (d).

For better illustration of our method, we enhance thecontrast of Fig. 1(d) and superimpose its skeletons onFig. 1(d). Figure 2(a) is an experimentally obtainedoriginal ESPI fringe image. Results for Fig. 2(a) simi-lar to those for Fig. 1(a) are given in Figs. 2(b)–2(f).Here the parameters �, �, �, a�t�, b are chosen as �=0.27, �=0.0005, �=0.2, k=0.0001, b=0.02, time step�t=1.0; a�t� is chosen according to an=35 for 1�n�3, an=an−1−0.7 for 3�n�10, and an= �1/2�an−1 forn�10; n is iteration time T̄ is chosen as the meanvalue of Sx,y, and �=3.

As can be seen, the noise is high in the original andin its filtered images when a mean filter is used once.Although the preprocessing is only a mean filter ap-plied one time, our results are as desired.

In conclusion, we described a novel skeletonizationapproach for gray-scale ESPI fringe images. Themain advantage of the proposed method is its sim-plicity. It works directly on gray-scale images withoutmuch preprocessing, which is significant progress infringe analysis for ESPI.

We thank the reviewers for their helpful commentson this Letter.

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