Modelling and analysing oriented fibrous structures

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Modelling and analysing oriented fibrous structures Rantala, Maaria; Lassas, Matti; Sampo, Jouni; Takalo, Jouni; Timonen, Jussi; Siltanen, Samuli Rantala, M., Lassas, M., Sampo, J., Takalo, J., Timonen, J., & Siltanen, S. (2014). Modelling and analysing oriented fibrous structures. In 2nd International Conference on Mathematical Modeling in Physical Sciences 2013 (IC-MSQUARE 2013) (Article 012089). Institute of Physics Publishing Ltd (IOP). Journal of Physics: Conference Series, 490. https://doi.org/10.1088/1742-6596/490/1/012089 2014 This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 130.234.74.31 This content was downloaded on 16/09/2014 at 03:43 Please note that terms and conditions apply. Modelling and analysing oriented fibrous structures View the table of contents for this issue, or go to the journal homepage for more 2014 J. Phys.: Conf. Ser. 490 012089 (http://iopscience.iop.org/1742-6596/490/1/012089) Home Search Collections Journals About Contact us My IOPscience Modelling and analysing oriented fibrous structures M Rantala1, M Lassas1, J Sampo2, J Takalo3, J Timonen3and S Siltanen1 1Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FI-00014 University of Helsinki, Finland 2Department of Mathematics and Physics, Lappeenranta University of Technology, P.O. Box 20, FI-53851 Lappeenranta, Finland 3Department of Physics, University of Jyväskylä, P.O. Box 35, FI-40014 University of Jyväskylä, Finland E-mail: [email protected] Abstract. A mathematical model for fibrous structures using a direction dependent scaling law is presented. The orientation of fibrous nets (e.g. paper) is analysed with a method based on the curvelet transform. The curvelet-based orientation analysis has been tested successfully on real data from paper samples: the major directions of fibrefibre orientation can apparently be recovered. Similar results are achieved in tests on data simulated by the new model, allowing a comparison with ground truth. 1. Introduction The quality of paper and also other fibre-based products depends essentially on how wood fibres are distributed in a more or less random network of predominantly planar orientation. For this reason, it would be important to be able to measure and control orientation and other properties of the fibre network already during the manufacturing process. The formation of paper has traditionally been inspected visually and later by analysing in different ways its optical transmission image [1]. However, there are artefacts in optical transmission images due to strong scattering of visible electromagnetic radiation in paper-like fibrous structures. Also, it is not certain as yet from which part of the paper structure an optical transmission image contains information. Therefore, it is of utmost importance to properly calibrate the information gained from optical images. There are two especially suitable methods of calibration: x-ray tomography and simulated networks with precisely known properties. A method to transform an optical transmission image to one that fairly closely resembles that of x-ray transmission has already been formulated [2]. In this work we present a mathematical model for simulating simple fibre nets using a direction dependent scaling law. The orientation of simulated data is analysed with a method based on the curvelet transform [3, 4], which has been successfully tested in analysing real data from paper samples [5]. The aim of the work is to design an algorithm that simulates a fibrous system from a given orientation distribution. 2nd International Conference on Mathematical Modeling in Physical Sciences 2013 IOP Publishing Journal of Physics: Conference Series 490 (2014) 012089 doi:10.1088/1742-6596/490/1/012089 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 2. Theory 2.1. Analysing orientation with curvelets The orientation of complex patterns has traditionally been analysed by applying the Fourier transform [6] or gradient-based methods like the structure tensors [7, 8]. However, more sophisticated methods, especially the wavelet transform, [9] have become popular in the last two decades. Furthermore, in recent years transforms like the curvelet, contourlet and shearlet transform have been developed and proved to be well-suited for some applications [3, 4, 10, 11]. The curvelet transform is tightly localised in both space and frequency domain, and has in addition an angle parameter that makes it an optimal tool for orientation analysis. The mother curvelet is defined at each scale 0< a < a0in the frequency domain as bγa00(rcos(ω), r sin(ω)) = a3 4W(ar)V(ω/√a), where r≤0and 0≤ω≤2π, the radial window Wis a non-negative, infinitely smooth realvalued function supported inside the interval (1 2,2), and the angular window Vis a non-negative, infinitely smooth real-valued function supported on the interval [−1,1] (see Figure 1). The whole curvelet is achieved as γabθ(x) = γa00(R−θ(x−b)), where 0≤θ≤2πis a rotation parameter, b∈R2is a translation parameter, and Rθis the matrix of planar counter-clockwise rotation by the angle θ. The curvelet transform is then defined by Γf(a, b, θ) := hγabθ, fi=ZR2 f(x)γabθ(x)dx, for all 0< a < a0, b ∈R2and 0≤θ≤2π. Curvelets follow the parabolic scaling law in the aspect ratio of the area that contains most of their energy. If we take a piece of smooth curve (corresponding to an edge of a fibre) with a length of about a, then the whole piece will fit into a rectangle with the side lengths aand √a. We can think of γabθ as a sensor that tries to detect if there is a fibre with orientation θin the neighbourhood of b. If fdenotes a fibrous image, then the inner product hγabθ, fipresents the response of sensor γabθ.A small value of parameter ameans we focus into a part of a fibre, while its larger values can embed a whole fibre. If there is no fibre with the orientation angle θlocated at point b, the value of |hγabθ, fi| is very small. 12 3 4 5 1 2345 Figure 1. Schematic illustration of curvelet functions, left: curvelet functions of one angle θin frequency domain, right: the same curvelet functions in spatial domain. 2.2. The H model for fibrous systems The statistics of physical parameters (e.g. density) of fibrous structures measured on lines depend on the line direction. If we assume that the physical parameters on a line follow the fractional 2nd International Conference on Mathematical Modeling in Physical Sciences 2013 IOP Publishing Journal of Physics: Conference Series 490 (2014) 012089 doi:10.1088/1742-6596/490/1/012089 2 Brownian motion statistics, then the Hurst index models the order of the fractional Brownian motion [12]. We create a random model for objects of controllable lengths, where a greater value of the Hurst index means larger features in the object. Assume now that Y(x)is a stationary two-dimensional Gaussian random field with zero mean, which satisfies a non-standard scaling law for the covariance operator. Here, a Gaussian random field Y(x)is stationary if the covariance function KY(x, y) = E(Y(x)Y(y)) satisfies KY(x+a, y +a) = KY(x, y).We consider two one-dimensional stationary Gaussian processes p1 and p2having covariances E(p1(t)p1(s)) ∼1−|t−s|2H1and E(p2(t)p2(s)) ∼1−|t−s|2H2as t→s, and zero mean, Epj= 0. This means that p1and p2are independent fractional Brownian motions with Hurst indices H1and H2. The random field Y(x)is obtained by taking oriented fibres having one end point at (y1, y2),i.e. the functions (x1, x2)7→ Ky1,y2(x1, x2) = k1(x1−y1)k2(x2−y2), and summing such functions with all possible (y1, y2)together with random weights Y(x1, x2) = ZR2 k1(x1−y1)k2(x2−y2)W(y1, y2)dy1dy2, where W(y1, y2)is a 2D Gaussian white noise. The Fourier transform of the function K(x1, x2) = k1(x1)k2(x2)satisfies b K(ξ1, ξ2) = b k1(ξ1)b k2(ξ2)∼c1c2(1 + |ξ1|)−1/2−H1(1 + |ξ2|)−1/2−H2as ξ→ ∞. 3. Results and discussion Figure 2. Left: fibre net with Hurst indices H1= 0.3and H2= 0.5,right: fibre net with Hurst indices H1= 0.5and H2= 0.3. The H model gives us fibre nets with two perpendicular orientations. Two samples of fibre nets simulated with the model are shown in Figure 2. A common example of this type of fibre net is newsprint. The simulated fibre nets can then be transformed to change the orientation of the fibres. Examples of the transformed fibre nets are shown in Figure 3. Note that the vertical fibres are turned to the same orientation in both transformations, but the horizontal fibres are transformed differently in rotation and shearing. The orientation analysis of these fibre nets is 2nd International Conference on Mathematical Modeling in Physical Sciences 2013 IOP Publishing Journal of Physics: Conference Series 490 (2014) 012089 doi:10.1088/1742-6596/490/1/012089 3 Figure 3. Left: the left fibre net of Figure 2 rotated 30◦clockwise, right: the same fibre net sheared horizontally 30◦clockwise and vertically 10◦counter-clockwise. 0 30 60 90 120 150 180 0 5 0 30 60 90 120 150 180 0 30 60 90 120 150 180 0 30 60 90 120 150 180 (a) (b) (c) (d) Figure 4. (a): orientation analysis of the left fibre net of Figure 2, (b): orientation analysis of the left rotated fibre net of Figure 3, (c): orientation analysis of the right sheared fibre net of Figure 3, (d): orientation curves (a)-(c) shifted to match the strongest orientation. shown in Figure 4. The strongest orientations are found in the expected positions, although the transformations vary the orientation strengths. The numerical evidence shows that we get the desired orientations in the fibre nets simulated with the H model. A further goal is to analyse the orientations in real fibre data and then simulate data with the exact same orientations. 4. References [1] Niskanen K 1998 Papermaking Science and Technology (Jyväskylä, Fapet) [2] Takalo J, Timonen J, Sampo J, Siltanen S and Lassas M 2011 Evaluation of the areal material distribution of paper from its optical transmission image Europ. Phys. J. - Appl. Phys. 55 20701 [3] Candès E J and Donoho D L 2005 Continuous curvelet transform. I: Resolution of the wavefront set Appl. Comput. Harmon. Anal. 19 (2) 162–97 [4] Candès E J and Donoho D L 2005 Continuous curvelet transform. II: Discretization and frames Appl. Comput. Harmon. Anal. 19 (2) 198–222 [5] Sampo J, Takalo J, Siltanen S, Lassas M, Miettinen A and Timonen J 2013 Curvelet-based method for orientation estimation of particles Submitted [6] Enomae T, Han Y-H and Isogai A 2006 Nord. Pulp Pap. Res. J. 21 (2) 253–9 [7] Brox T, van den Boomgaard R, Lauze F, van de Weijer J, Weickert J, Mrzek P and Kornprobst P 2006 Visualization and Processing of Tensor Fields ed J Weickert, H Hagen (Berlin, Springer-Verlag) pp 17–47 [8] Krause M, Hausherr J M, Burgeth B, Herrmann C and Krenkel W 2010 J. Mater. Sci. (45) 888–96 [9] Daubechies I 1992 Ten lectures on wavelets (Philadelphia, SIAM) [10] Do M N and Vetterli M 2005 IEEE T. Image Process. 14 (12) 2091–106 [11] Guo K and Labate D 2007 SIAM J. Math. Anal. 39 (1) 298–318 [12] Kallenberg O 1997 Foundations of Modern Probability (New York, Springer-Verlag) 2nd International Conference on Mathematical Modeling in Physical Sciences 2013 IOP Publishing Journal of Physics: Conference Series 490 (2014) 012089 doi:10.1088/1742-6596/490/1/012089 4