Bias Error Reduction of Digital Image Correlation Based on Kernel

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1 Vol.81 (CST 15),.16- htt://dx.doi.org/1.1457/astl Bias Error Redution of Digital Image Correlation Based on Kernel Huan Shen 1,, eize Zhang 1, and Xiang Shen 1 Energy and ower College, anjing Uniersity of Aeronautis and Astronautis, o. 9, Yudao Street, Baixia Distrit, anjing 116, China Aeronautis Siene and Tehnology Key Laboratory of full sale airraft struture and fatigue, o. 85, Dianzi nd Road, Yanta Distrit, Xi an 7165, China Huan Shen: huan_shen@nuaa.edu.n Abstrat. Kernel based Digital Image Correlation (KDIC) method is roosed to imroe auray of the dislaement field alulation. With the effet of the image noise onsidered, a new kernel based similarity oeffiient is defined for robust grid data math. Different tyes of kernel funtion mean different weighted forms. Two kinds of kernel funtion are alied for dislaement alulation. One is Eanehniko kernel, and KDIC is equialent to the traditional digital image orrelation (TDIC). The other is a Gaussian kernel used in KDIC to hel get more aurate sub-ixel dislaement measurement. Simulation analyses alidate the effetieness of this new method in this aer. Keywords: Kernel; Bias error redution; image noise; Digital image orrelation 1 Introdution As an effetie otial measurement method, Digital Image Correlation (DIC) tehnique has been deeloing raidly in reent 3 years. Owing to the adantages of DIC, suh as non-ontat measurement, simle to build the system, full-field data aquisition and so on, DIC tehnique is widely used in exerimental mehanis [1-]. At the same time, lots of methods are ut forward to imroe DIC for higher measurement auray and faster roessing seed. Howeer, results obtained ia traditional DIC (TDIC) [3] are not that satisfatory in ratial use. Fators like sekle attern, image subset size, orrelation funtion, onergene ondition of the iteration and so on, affet the testing methods. Some researhers hae analyzed these roblems in detail and roosed some riniles for ratial aliation [4-5]. As we all know, image noise is ineitable in atual measurement. TDIC tehnique deends on the intensity of the image whih is affeted and hanged by the image noise. So, the results alulated by TDIC are not that aurate. Some smoothing methods based on retreatment [6-7] or ostroessing tehnologies [8-9] are resented to derease the bad effet of the image noise. In atual measurement, the smoothing arameters in these tehnologies need to be hosen artifiially but with true deformation results of the materials unknown. A new DIC method, with kernel ISS: ASTL Coyright 15 SERSC

2 Vol.81 (CST 15) funtion inluded (KDIC), is roosed to deal with the image noise for high-reision dislaement alulation. Simulation results alidate the effetieness and robustness of the KDIC tehnique that is insensitie to the image noise. Introdutions of KDIC Kernel funtion is added to the orrelation funtion to imroe anti-noise erformane of the traditional DIC. Considering the ontrol ability of the kernel funtion in radial diretion, intensity differenes of the images before and after deformation in gien neighborhood S are transformed into the feature sae entered at zero. ew orrelation funtion based on kernel an be defined as r ( s, ) k (1) s s h Where is the normalization onstant; h( h ) resents the bandwidth of the kernel; r ( s, ) f ( s ) g ( s, ). () First-order gradient of k stands for the kernel funtion. and the Hessian matrix are shown as follows: ( ) i i 1,..., 7 r ( s, ) g ( s, ) ' k r ( s, ) h s s h i i 1,..., 6 (3) ( ) i j i 1,..., 6 j 1,..., 6 r ( s, ) g ( s, ) g ( s, ) ' k h s s h i j i 1,..., 6 j 1,..., 6 (4) The iteration equation of ewton-rahson method is alied: = - (5) At last, Eq.3 and Eq.4 an be substituted into Eq.5. Then, the Eq. is minimized to get the otimum deformation arameters for full-field dislaement alulation. Comared with the gradient and Hessian matrix of the two orrelation funtion, KDIC is a kind of weighted TDIC [3] in format. Therefore, different kinds of kernel funtions orresond to different weighted forms [1]. If Eanehniko kernel is alied, KDIC is the same to TDIC in format. In Gaussian kernel ase, the rofile is Coyright 15 SERSC 17

3 Vol.81 (CST 15) k x x ex x x (6) Where x is the enter of the funtion; stands for the bandwidth and ontrols radial atuating range of the kernel funtion. Gaussian kernel is alied in this aer. 3 Simulation Analysis In order to alidate the effetieness of the new KDIC method, syntheti sekle images are rodued to analyze the deformation of the seimen more objetiely. Based on the referene image, twenty deformation images are generated with.5 ixel dislaement inrement along y diretion. oise with 1%, % and 4% leels is searately added to this grou of images whose dislaements range from -1ixel. Different noise leels may hae different imat on the dislaement alulation obtained ia traditional DIC. Four sets of images (noiseless, 1% leel, % leel, 4% leel) are dealt with TDIC. The alulated dislaements are ealuated with e (7) tru e 1 ( ) (8) i 1 i 1 Where 1 ; i tru e i 1 resents the number of all the grid oints. is the true dislaement of the simulated image; Mean bias error(ixel) oise leel 4% oise leel % oise leel 1% Standard deiation error(ixel) oise leel 4% oise leel % oise leel 1% Fig 1. Mean bias errors of the dislaements alulated by TDIC Fig. SD errors of the dislaements alulated by R of the four image sets Fig 1 and Fig show the mean bias errors and standard deiation errors of the dislaements alulated by TDIC searately. From Fig 1 we an see that, with the 18 Coyright 15 SERSC

4 Vol.81 (CST 15) inrement of the noise leel, effets on the dislaements beome more aarently. The higher the noise leel is, the bigger the two errors are. For the maximum mean bias error, it inreases from.77ixel to.994ixel. Similarly, Fig shows that the standard deiation error inreases from.9ixel to.133ixel with the raise of the noise leels. Then, kernel based DIC resented in this aer is alied to deal with the four simulated image sets. And the two kinds of errors are alulated and listed in Fig 3 and Fig 4. Mean bias error(ixel) oise leel 4% - TDIC oise leel 4% - KDIC oise leel % - KDIC oise leel 1% - KDIC Standard deiation error(ixel) oise leel 4% oise leel % oise leel 1% Fig 3. Mean bias errors of the dislaements alulated by KDIC Fig 4. Standard deiation errors of the dislaements alulated by KDIC As shown in Fig 3, omared with the dislaement mean bias error of 4% noise leel alulated by TDIC, the dislaement errors of the four image sets alulated by KDIC are all dramatially redued. For examle, as for images with 4% noise leel, the maximum error dereases from.94ixel to.71ixel, whih alidates the remarkable effetieness of the new KDIC method. Maximum mean bias errors and standard deiation errors of the four image sets are listed in Table1. The results demonstrate the effetieness of the KDIC method to derease the mean bias error. Howeer, there is no imroement for reduing the standard deiation error of the dislaement alulation. For 1% noise leel, whih means the SR (Signal to noise ratio) of the image is 4dB, the maximum mean bias error of the dislaement alulation is 6. For the industrial ameras, the SR alues an reah 4dB and some may be higher. So, reision with 1 of the KDIC method an satisfy the requirement of the high-reision dislaement alulation. Table 1. Maximum bias and standard deiation errors of TDIC and KDIC alulation oise leel Maximum bias error alulated by TDIC (L) and KDIC (R) Mean SD error Calulated by TDIC (L) and KDIC (R).7/-.65/- 1%.77/.6.9/.48 %.3/.97.64/.7 4%.994/ /.119 Coyright 15 SERSC 19

5 Vol.81 (CST 15) 4 Conlusion In onlusion, onsidering bad effets of the random image noise on the dislaement alulated by TDIC, a kernel based DIC method whih is a generalization exression of TDIC, is roosed in this aer to imroe the robustness of DIC without sarifiing the intensity gradient. As for large intensity differenes aused by noise, smaller weights, near to zero, should be gien to them to redue the influene of the noise. And large weights are enouraged for these small intensity differenes. Obiously, KDIC is a noel method for bias error redution without hanging the image information. Simulation analyses erify the effetieness of KDIC to redue the bias error for a high-reision dislaement alulation. Aknowledgments. This work was suorted by the Aeronautial Siene Foundation of China (Grant o. 195) and the Fundamental Researh Funds for the Central Uniersities (Grant o. S14) Referenes 1. ierron, F., Sutton, M.A., Tiwari, V.: Ultra high seed DIC and irtual fields method analysis of a three oint bending imat test on an aluminum bar. J. Ex. Meh. 51, 4(11).. Hild, F. and Roux. S.: Digital Image Correlation from Dislaement Measurement to Identifiation of Elasti roerties-a Reiew.J. Strain 4, (6). 3. Vendroux, G. and Knauss, W. G.: Submiron deformation field measurements - art. Imroed digital image orrelation. J. Ex. Meh. 38, (1998). 4. Trionnet, K., Derrien, K., Hild, F. O., et. al: arameter hoie for otimized digital image orrelation. J. Ot. Lasers Eng. 47 (9) 5. Laa,., Cooreman, S. and Debruyne, D.: Study of systemati errors in strain fields obtained ia DIC using heterogeneous deformation generated by lasti FEA. J. Ot. Lasers Eng. 48, 4(1). 6. an, B.: Bias error redution of digital image orrelation using Gaussian re-filtering. J. Ot. Lasers Eng. 51, 1(13). 7. Mazzoleni,., Matta, F., Zaa, E. et.: Gaussian re-filtering for unertainty minimization in digital image orrelation using numerially-designed sekle atterns. J. Ot. Lasers Eng. 66(15). 8. an, B., Yuan, J. Y. and Xia, Y.: Strain field denoising for digital image orrelation using a regularized ost-funtion. J. Ot. Lasers Eng. (14). 9. an, B., Xie, H. M., Guo, Z. Q., and Hua, T.: Full-field strain measurement using a twodimensional Saitzky-Golay digital differentiator in digital image orrelation. J. Ot. Eng. 46,3(7). 1. Comaniiu, D., Meer,.: Mean shift: A robust aroah toward feature sae analysis. J. IEEE Transations on attern Analysis and Mahine Intelligene, 4(5): (). Coyright 15 SERSC