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3D Profile Filter Algorithm Based on Parallel Generalized B-Spline Approximating Gaussian

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3D Profile Filter Algorithm Based on Parallel Generalized B-Spline Approximating Gaussian CHINESE JOURNAL OF MECHANICAL ENGINEERING ·148· Vol. 28,aNo. 1,a2015 DOI: 10.3901/CJME.2014.1106.163, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn 3D Profile Filter Algorithm Based on Parallel Generalized B-spline Approximating Gaussian REN Zhiying1, 2, GAO Chenghui1, 2, *, and SHEN Ding3 1 School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350108, China 2 Tribology Research Institute, Fuzhou University, Fuzhou 350108, China 3 Fujian Institute of Metrology, Fuzhou 35000, China Received January 24, 2014; revised September 10, 2014; accepted November 6, 2014 Abstract: Currently, the approximation methods of the Gaussian filter by some other spline filters have been developed. However, these methods are only suitable for the study of one-dimensional filtering, when these methods are used for three-dimensional filtering, it is found that a rounding error and quantization error would be passed to the next in every part. In this paper, a new and high-precision implementation approach for Gaussian filter is described, which is suitable for three-dimensional reference filtering. Based on the theory of generalized B-spline function and the variational principle, the transmission characteristics of a digital filter can be changed through the sensitivity of the parameters (t1, t2), and which can also reduce the rounding error and quantization error by the filter in a parallel form instead of the cascade form. Finally, the approximation filter of Gaussian filter is obtained. In order to verify the feasibility of the new algorithm, the reference extraction of the conventional methods are also used and compared. The experiments are conducted on the measured optical surface, and the results show that the total calculation by the new algorithm only requires 0.07 s for 480´480 data points; the amplitude deviation between the reference of the parallel form filter and the Gaussian filter is smaller; the new method is closer to the characteristic of the Gaussian filter through the analysis of three-dimensional roughness parameters, comparing with the cascade generalized B-spline approximating Gaussian. So the new algorithm is also efficient and accurate for the implementation of Gaussian filter in the application of surface roughness measurement. Keywords: generalized B-spline, Gaussian filter, three-dimensional reference, cascade characteristic, parallel characteristic spline filter, and so on[2]. The Gaussian filtering algorithm 1 Introduction has been defined as the standard for two-dimensional and three-dimensional filters by ISO 11562[3] and ISO 16610[4] With the development of science and technology, the respectively, because of its superior zero phase and bilinear surface quality requirements have improved from the characteristics. However, there are several deficiencies with macro scale to the micro scale; roughness assessments have the Gaussian filtering, such as border and outlier progressed from the one or two-dimensional era into the era distortions[5]. To solve these problems, ISO 16610-22 of three-dimensions. In the description and evaluation of proposed the standard spline filter[6], but the computational surface features, filtration has always been the basis of the efficiency was low and the application was limited in assessment of 3D surface parameters[1]. The three- industrial engineering[7–8]. In addition, it has been found dimensional profile filter algorithm, compared to the that the results from Gaussian and spline filtering were not one-dimensional filter algorithm, has greater redundancy the same and gave the different roughness assessments for and is a more complicated calculation. Therefore, the same surface, and this resulted in discrepancies in researching a fast and highly accurate algorithm is very practical situations. A significant amount of research has meaningful. Currently, there are many established been done to try to unify the two[9]. traditional methods, such as the 2RC filter, Gaussian filter, The approximation of the Gaussian filter by some other spline filters has also been discussed[10–11]. Using the spline principle, the boundary distortion problem of Gaussian was * Corresponding author. E-mail: [email protected] [12] Supported by National Natural Science Foundation of China (Grant Nos. restrained ; by adding a corresponding smoothing 51175085, 51375094), Fujian Provincial Education Department algorithm, the redundancy of the computation was greatly Foundation of China (Grant No. JA13059), Open Fund of State Key reduced[13], and using a cascade characteristic Laboratory of Tribology of Tsinghua University, China (Grant No. [14] SKLTKF13B02), and Fuzhou Science and Technology plan Fund of China approximation of the standard Gaussian filter , a high (Grant No. 2014-G-74) precision of filtering was achieved. However, these © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2015 CHINESE JOURNAL OF MECHANICAL ENGINEERING ·149· n methods are only suitable for the study of one-dimensional where s ()xi is the interpolation function. Cx()i is the filtering, not three-dimensional filtering. When these interpolation coefficient. It is found that the measured methods were used for three-dimensional filtering, it was signal can be represented by a linear series of generalized 3r found that in every part a rounding error and quantization B-spline basis functions 1 ()()ikkZ-Îin Eq. (3). error would be passed to the next. In Ref. [15], the B-spline When f ()i is the measured signal, the process which filter was used to achieve the approximation of a Gaussian determines the actual decomposition coefficients Ci()is filter, in which each part of the smoothing filter was broken called a direct B-spline transformation, (()fi ( B31r ()) z - l down into two second-order IIR filters in series form. Ci()); and the process that represents the measured signal However, when used for the three-dimensional filtering, f ()i by the generalized B-spline basis function is known because of its redundancy, computational complexity and as an indirect B-spline transformation, ( fi() 3r calculated dimension, it would result in errors of Bl ()zCi ()). accumulation, and ultimately would affect the accuracy of the approximation. On the other hand, compared to the 2.2 Two-dimensional Gaussian filter discussed spline filters, the generalized B-spline smoothing According to the ISO 11562[3], two-dimensional filter could simplify the calculation and improve its Gaussian filtering weights function as follows: efficiency as well as its strong versatility when extracting [16] the reference and waviness . æöéù2 2 ç æöæöt ÷ 1 ç êúç tx ÷ ç y ÷ ÷ To solve this problem, this paper proposes a 3D profile gt(, t )=-+= expç π êúç ÷ ç ÷ ÷ xy 2 ç ç ÷ ç ÷ ÷ a ç êúèøçaacxèøç cy ÷ ÷ fast filter algorithm based on the parallel generalized cx cy èøç ëûêú÷ B-spline of a three order approximating Gaussian filter. The æö2 æö2 ç æö÷ ç æöt ÷ results show that the new approximation filter is efficient 1 ç ç tx ÷ ÷ ç ç y ÷ ÷ expç--πç ÷ ÷ expç πç ÷ ÷, (4) a2 ç çaa÷ ÷ç ççç ç ÷ ÷ and accurate for the implementation of Gaussian filter in cx cy èøç èøcx÷ èøç èø cy ÷ the application of surface roughness measurement. tt, are respectively the independent variables in 2 Materials Used in Analysis where xy the directions of x, y ; , are respectively the cut-off cx cy wavelengths in the directions of x, y; 2.1 Generalized B-spline filter a ==ln 2 π 0.469 7; and when tt== , the Using a linear differential equation, the generalized xcxycy transmission efficiency of the filter is 50%. According to B-spline function nr ()x , is given as [16]: m the separability of the Gaussian function, the Gaussian n+1 weight function can be split into the right hand part of Eq. nr nr mjmjj()x =--HLgxxUxxå ()(), (1) (4). The two-dimensional Gaussian filtering process can be j=0 written as follows: nr where H is the constant for normalized B-spline, gm ()x ¥¥ is the (piecewise interpolant) B-spline generating function yn(,xy n )=--=åågn(x m xy , n m y )(, xm x m y ) of order n and type r expanded by an integer factor of mmxy=-¥ =-¥ m , Ux()is the unit step function and L is the constant of æöéù2 2 j ¥¥ ç æöæönm- ÷ 1 ç êúçnmxx- ÷ ç yy÷ ÷ multiplication in the jth segment of the B-spline. The expç-+π êúç ÷ ç ÷ ÷ åå2 ç ç ÷ ç ÷ ÷ =-¥ =-¥ a ç êúèøaacxèøç cy ÷ ÷ Laplace transform of the basis function g()x must have mmxy cx cy èøç ëûêú÷ four-quadrant symmetry when the sampling interval is xm(, m ), (5) identical. Nine different types of third degree generalized xy B-splines functions can be obtained by self-convolution of g()x . In general, the transfer function of the three orders of where x(,mmxy )is the measured data and yn(,xy n ) is generalized B-spline can be defined as follows [17–18]: the filtered reference. The transfer function of the filter is obtained by the Z-translation of the Gaussian weight ztz-1 ++ function: Bz3r ()=Î+¥2 , tt , (0, ). (2) 112t 1 = Gz(,xy z ) g xxyy () z g (), z (6) Actually, the generalized B-spline filtering process is an ¥ approximation of interpolation. The generalized B-spline 1 2 where g ()zn=- expéùπ()a z-n , function can construct a series of functions to approximate xcx11å ëûêú n=-¥ acx the objective function within a given range [,x x ]. ¥ 0 n 2 [17] 1 éù-n B-spline interpolation is redefined as follows : gzycy()22=- expêúπ()na z . å a ëû n=-¥ cy n nr3 It was supposed that the cutoff angle- frequencies s ()xCxxxjnjiji=-Îå ()1 ( ),[] 0,, (3) i=0 wxwycx ==2π cx,2 cy π cy and zzxy, was replaced ·150· Y REN Zhiying, et al: 3D Profile Filter Algorithm Based on Parallel Generalized B-spline Approximating Gaussian by exp(jwwxy ), exp(j ). The corresponding frequency where yx()is the measured data; s()x is the filtered data; domain function of the standard Gaussian function can be is the Lagrange constant; ||¶¶22s (xx ) || is the norm of obtained, as follows: curve bending energy; ¶¶s()xxiiis the velocity item of the approximation function; and is the adjustment parameter.
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