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Licensed Unlicensed Requires Authentication Published by De Gruyter August 15, 2014

The PPI value of open foams and its estimation using image analysis

Joachim Ohser , Claudia Redenbach and Ali Moghiseh


The mean number of pores per inch (PPI value) is one of the most important geometric characteristics of cellular materials such as open or closed foams. It is defined as the number of pores (cells) along a straight test line related to the line length. Counting cells along a test line sounds very simple, but on the surface of an open foam specimen it is often hard to decide whether a cell hits the line or not. Thus, there exists no quick and safe method to estimate the PPI value from an optical image taken from the specimen's surface. In this article, we present a very efficient method of estimating the PPI value of foams from optical dark-field images of the surface of a foam sample. The method is based on the computation of the spectral density of the (two-dimensional) dark field images. It turns out that the radius of the first interference ring in the spectral density is proportional to the PPI value. The constant of proportionality can be determined from geometric models for open foams or analysis of three-dimensional images of foam samples. These techniques allow calibration of the estimation of the PPI value from the spectral density of two-dimensional dark field images.

*Correspondence address, Prof. Joachim Ohser, University of Applied Sciences, Dept. Math. & Nat. Sci., Schöfferstraße 3, D-64295 Darmstadt, Germany, Tel.: +49(0)6151 16 8655, Fax: +49(0)6151 16 8754, E-mail:


[1] M.Schaffer, P.Colombo: Cellular Ceramics, Wiley-VCH, Weinheim (2005).Search in Google Scholar

[2] C.Redenbach: Comp. Mat. Sci.44 (2009) 1397. 10.1016/j.commatsci.2008.09.018Search in Google Scholar

[3] J.Ohser, K.Schladitz: 3D Images of Materials Structures - Processing and Analysis, Wiley VCH, Weinheim, Berlin (2009). 10.1002/9783527628308Search in Google Scholar

[4] A.Liebscher, C.Redenbach: Int. J. Mater. Res.103 (2012) 155. 10.3139/146.110667Search in Google Scholar

[5] J.Ohser, in: H.Schumann and H.Oettel (Eds.), Metallographie, 14th Ed., Wiley-VCH, Weinheim, Berlin (2005) 250.Search in Google Scholar

[6] R.Hosemann, N.S.Bagchi: Direct analysis of matter by diffraction, North Holland Publishing Company, Amsterdam (1962).Search in Google Scholar

[7] J.Ohser, F.Mücklich: Statistical Analysis of Microstructures in Materials Science, J Wiley & Sons, Chichester, New York (2000).Search in Google Scholar

[8] C.H.Arns, J.Mecke, K.R.Mecke, D.Stoyan: Eur. Phys. J. B47 (2005) 397. 10.1140/epjb/e2005-00338-5Search in Google Scholar

[9] C.Redenbach, C.Thäle: Statistics47 (2013) 237. 10.1080/02331888.2011.586458Search in Google Scholar

[10] S.N.Chiu, D.Stoyan, W.S.Kendall, J.Mecke: Stochastic Geometry and Its Applications, 3rd Ed., Wiley Series in Probability and Statistics, J. Wiley & Sons, Chichester (2013). 10.1002/9781118658222Search in Google Scholar

[11] K.Unverzagt: Spectral Analysis of Random Closed Sets: The Surface Measure Associated with a Random Closed Set, Diploma thesis, Universität Siegen (2005).Search in Google Scholar

[12] M.Abramowitz, I.A.Stegun: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Number 55 in Appl. Math, Nat. Bureau of Standards, Washington DC (1964).Search in Google Scholar

[13] W.Gille: Experimentelle Technik in der Physik36 (1988) 197.Search in Google Scholar

[14] F.Aurenhammer: SIAM J. Comput.16 (1987) 7896. 10.1137/0216006Search in Google Scholar

[15] J.Ohser, K.Schladitz, K.Koch, M.Nöthe: Z. Metallkunde96 (2005) 731737. 10.3139/146.101094Search in Google Scholar

[16] C.Redenbach, J.Ohser, A.Moghiseh: submitted (2013).Search in Google Scholar

[17] F.Natterer: The mathematics of computerized tomography, Teubner, Stuttgart (1986).10.1007/978-3-663-01409-6Search in Google Scholar

[18] D.Stoyan: Math. Operationsforsch. Statist., Ser. Statist.15 (1984) 421. 10.1080/02331888408801790Search in Google Scholar

[19] K.H.Hanisch, P.Klimanek, D.Stoyan: Cryst. Res. Technol.20 (1985) 921. 10.1002/crat.2170200712Search in Google Scholar

[20] S.Torquato: Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer, New York (2002). 10.1007/978-1-4757-6355-3Search in Google Scholar

[21] J.W.Goodman: Introduction to Fourier Optics, Roberts & Co Publishers, New York (2005).Search in Google Scholar

Received: 2013-09-10
Accepted: 2013-11-26
Published Online: 2014-08-15
Published in Print: 2014-07-14

© 2014, Carl Hanser Verlag, München

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