Abstract: The work deals with the development of a two-dimensional isotropic spline filter with application to the separation of components of the surface topography, i.e. roughness, waviness and form. An appropriate variational problem was formulated to define the mapping of primary surface data into filtered surface data with a transfer function of a circularly symmetric low-pass Butterworth filter of a given order. The variational problem was solved by approximating the filtered surface data by means of two-dimensional B-spline functions. The Fourier transform of the filter impulse response was determined to estimate the quality of approximation of the Butterworth filter and to select the appropriate filter cutoff. The paper presents one application of the designed filter to determine 3D roughness of an inner ring race of a ball bearing.
B I B L I O G R A F I A[1] Raja J, Muralikrishnan B, Fu S. Recent advances in separation of roughness, waviness and form. Precision Engineering 2002 26: 222–35.
[2] Mathia T.G., Pawlus P., Wieczorowski M., Recent trends in sutface metrology. Wear 271 (2011) 494-508.
[3] ISO/TS 16610-28:2010. Geometrical product specifications (GPS) -- Filtration -- Part 28: Profile filters: End effects.
[4] Schoenberg I.J., Spline functions and then problem of graduation, Proc. Nat. Acad. Sci., vol 52 (1964), pp. 947-950.
[5] Reinsh C.H., Smoothing by spline functions, Numer. Math., vol 10 (1967), pp. 177-193.
[6] D'Haeyer J.P.F., Gaussian filtering of images: a regularization approach. Signal Processing, vol. 18 (1989), pp. 169-181.
[7] Krystek M., Form filtering by splines. Measurement, vol. 18 (1996), pp. 9-15.
[8] Seewig J., Linear and robust Gaussian regression filters. Journal of Physics: Conference Series, vol. 13 (2005), pp. 254–257.
[9] Brinkmam S., Bodschwinna H., Lemke H.W., Accessing roughness in threedimensions using Gaussian regression filtering. International Journal of Tools Manufacture, vol. 41 (2001), pp. 2153-2161.
[10] ISO/TS 16610–22:2006. Geometrical product specifications (GPS) - Filtration. Part 22. Linear profile filters: Spline filters.
[11] ISO/TS 16610–31:2010. Geometrical product specifications (GPS) - Filtration. Part 31. Robust profile filters: Gaussian regression filters.
[12] Dobrzanski P, Pawlus P. Digital filtering of surface topography. Part I. Separation of one-process surface roughness and waviness by Gaussian convolution, Gaussian regression and spline filters. Precision Engineering 2010 34:647–50.
[13] Hanada H., Saito T., Hasegawa M., Yanagi K., Sophisticated filtration technique for 3D surface topography data of rectangular area, Wear, vol. 264 (2008) 422–427.
[14] Zhang H, Yuan Y, Piao W. A universal spline filter for surface metrology. Measurement, 2010 43:1575–82.
[15] Janecki D., Gaussian filters with profile extrapolation, Precision Engineering 35 (2011) 602– 606.
[16] Janecki D., Edge effect elimination in the recursive implementation of Gaussian filters, Precision Engineering 36 (2012) 128– 136.
[17] Goto T, Miyakurab J, Umedaa K, Kadowakib S, Yanagic K. A Robust Spline Filter on the basis of L2-norm. Precision Engineering 2005 29:157–61.
[18] Numada M, Nomura T, Yanagi K, Kamiya K, Tashiro H. High-order spline filter and ideal low-pass filter at the limit of its order. Precision Engineering 2007 31:234–42.
[19] Janecki D.. A generalized L2-spline filter. Measurement,.vol. 42 (2009), pp. 937–943.
[20] Zeng W., Jiang X., Scott P., A generalised linear and nonlinear spline filter, Wear 271 (2011) 544–547.
[21] Goto T., Yanagi K., An optimal discrete operator for the two-dimensional spline filter, Meas. Sci. Technol. 20 (2009) 125105 (4pp).
[22] Zhang H, Yuan Y, Piao W. The spline filter: A regularization approach for the Gaussian filter, Precision Engineering 36 (2012) 586– 592.
[23] Schoenberg I.J., Cardinal interpolation and spline functions, J. Approximation Theory, vol. 2 (1969), pp. 167-206.
[24] Unser M., Aldroubi A., Eden M., B-spline signal processing, IEEE Trans. Signal Processing, vol. 41 (1993), pp. 821-848. 
Pełny tekst 
DOI 
Web of Science