evaluation of the areal material distribution of paper from its...

Eur. Phys. J. Appl. Phys. 55, 20701 (2011) DOI: 10.1051/epjap/2011100366

Evaluation of the areal material distribution of paper from itsoptical transmission image

J. Takalo, J. Timonen, J. Sampo, S. Siltanen, and M. Lassas

The title “The European Physical Journal” is a joint propertyof EDP Sciences, Società Italiana di Fisica (SIF) and Springer

Eur. Phys. J. Appl. Phys. 55, 20701 (2011)DOI: 10.1051/epjap/2011100366

THE EUROPEANPHYSICAL JOURNALAPPLIED PHYSICS

Regular Article

Evaluation of the areal material distribution of paperfrom its optical transmission image

J. Takalo1,a, J. Timonen1, J. Sampo2,3, S. Siltanen2,b, and M. Lassas2

1 Department of Physics, University of Jyväskylä, Finland2 Department of Mathematics and Statistics, University of Helsinki, Finland3 Department of Mathematics and Physics, Lappeenranta University of Technology, Finland

Received: 15 September 2010 / Received in final form: 20 December 2010 / Accepted: 4 April 2011Published online: 11 August 2011 – c© EDP Sciences 2011

Abstract. The goal of this study was to evaluate the areal mass distribution (defined as the X-ray trans-mission image) of paper from its optical transmission image. A Bayesian inversion framework was used inthe related deconvolution process so as to combine indirect optical information with a priori knowledgeabout the type of paper imaged. The a priori knowledge was expressed in the form of an empirical Besovspace prior distribution constructed in a computationally effective way using the wavelet transform. Theestimation process took the form of a large-scale optimization problem, which was in turn solved using thegradient descent method of Barzilai and Borwein. It was demonstrated that optical transmission imagescan indeed be transformed so as to fairly closely resemble the ones that reflect the true areal distributionof mass. Furthermore, the Besov space prior was found to give better results than the classical Gaussiansmoothness prior (here equivalent to Tikhonov regularization).

1 Introduction

The concept of formation has been introduced in paperresearch [1–4] so as to describe the small-scale uniformityof the basis weight (areal distribution of solid material)in paper. In the papermaking process wood fibers, min-eral fillers, and other additives, together form the basicstructure of paper. Fibers themselves form a more or lessrandom network with predominantly planar orientationof fibers. The resulting structure displays variation in itsareal mass density, which depends on the inhomogeneityat small length scales of the distributions of the individualsolid components, and on changes in process conditions atlarge length scales. The small-scale density variation, i.e.,formation, has traditionally been one of the quality pa-rameters of paper, originally measured by visual inspec-tion and later by analyzing in different ways its opticaltransmission image [1,3]. Paper formation has also beenmeasured using transmission images of X-rays and betaradiation (electrons) [1,4]. In these latter two cases radia-tion passes through the sample with very little scattering,while visible electromagnetic radiation undergoes scatter-ing in addition to absorption from interfaces inside thepaper structure. In the visible region absorption is weakin comparison with scattering, i.e., the scattering coeffi-cient is much larger than the absorption coefficient [5].

a e-mail: [email protected] e-mail: [email protected]

In optical transmission of light through paper, (multi-ple) scattering events result in its effectively diffusive mo-tion around the direction of propagation [5], the resultingimage is ‘blurred’, and structural features, which wouldappear sharp if only absorption would take place, are notwell detectable in the resulting transmission image. It hasthus become evident that ‘optical formation’ is differentfrom those determined by other means [1]. Apart from dif-fusive motion, fibers may also act as wave guides for lightrays, thus adding another component to blurring. We donot consider here paper structures with anisotropic ori-entation of fibers, and both these contributions are thusisotropic [5]. It has not been possible so far to estimatethe true areal mass distribution from optical transmissionimages. This would, however, be desirable as scanning oflarge areas on-line with X-rays or beta radiation is tech-nically much more difficult than obtaining similar opticaltransmission data (such a technology already exists).

Fibers used in paper are chemically and structurallyheterogeneous, and more than one type of fiber are alsocommonly used, so that their dielectric properties vary.Because the effective diffusive motion of light around itsdirection of propagation results from multiple scattering,we however assume that the system of fibers can be de-scribed with its average dielectric properties (scatteringand absorption coefficient) [1,5]. The dielectric propertiesof fibers, fillers and coating pigments are as well differ-ent, which adds another complication to the problem. Atthis stage we do not address this complication, and only

20701-p1

The European Physical Journal Applied Physics

consider uncoated paper in which the concentration offillers is relatively low.

Previously the effect of scattering on X-ray transmis-sion has been described by a convolution process [6], andwe adopt the same approach here. For the case of a sin-gle scattering during transmission (thin sample), it canbe shown that the convolution kernel is Gaussian [7]. Forthicker samples with multiple scattering the analyticalform of the kernel is not known, but the Gaussian kernelhas successfully been used in this case as well [6,8]. In op-tical tomography based on recording optical transmissionimages a similar distribution has also been applied [9,10].Therefore, we assume in the following that a Gaussiankernel can be used when reducing scattering effects in thetransmission of visible light by deconvolution.

For deconvolution we use the Bayesian inversion frame-work [11], and, as discussed above, the aim is to esti-mate the areal mass distribution of paper from itsoptical transmission image. As the scattering coefficientin paper of X-rays is very small, the true areal mass dis-tribution is assumed to be obtained by an X-ray trans-mission image. The basic idea in Bayesian inversion is toaugment the ‘indirect’ optical data with a priori knowl-edge about the mass distribution. Such knowledge isgained here by microtomographic reconstructions of papersamples. The prior distributions are constructed from thetomographic reconstructions using the recently introducedwavelet-based Besov space approach [12]. These distribu-tions are generic enough so that they need to be deter-mined only for the type of paper of interest.

2 Materials and methods

2.1 Mathematical model for measured data

Let A′ be a mapping from X-ray transmission data,f : R2 → R, to the corresponding optical transmissiondata, m : R2 → R. Although the form of A′ in princi-ple unknown and may even be nonlinear, we will assumehere, as discussed above, that it is a convolution operatorA such that

(Af)(x) :=∫

R2f(u)g(x− u)du. (1)

Following the above discussion we assume furthermorethat the convolution kernel is a Gaussian bell-shaped func-tion g : R2 → R with a width determined empirically fromthe measured edge-spread function (esf), see Figures 1and 2.

2.2 Optical transmission data

The optical measurements of paper were done by illumi-nating the sample with a parallel light beam on one sideand recording the light that passes through to the otherside. The light source was a Durst CLS 450 darkroomenlarger. Images were recorded with a Canon 5D Mark IIdigital single lens reflex camera equipped with a Canon EF

Fig. 1. A schematic setup of the experiment for measuringthe edge-spread function (esf). The recorded esf image has ablurred boundary between the dark and light areas due mainlyto scattering of light inside the paper.

Fig. 2. (Color online) An optical transmission image (upperleft), X-ray transmission image (upper right), both of a size of500 × 400 pixels, and their normalized esfs (lower panel) of asample of handsheet paper.

100-mm f2.8 USM macro-objective with aperture stoppeddown to f16 for optimal resolution.

For measuring the esf as illustrated in Figure 1 thesample was partly eclipsed with a strip of opaque metaltape. The resulting optical intensity profile was then usedto determine the esf of the sample as shown in Figure 2.

20701-p2

J. Takalo et al.: Evaluation of the areal material distribution of paper from its optical transmission image

2.3 X-ray transmission and tomographic data

We used a SkyScan 1172 X-ray scanner to take a radi-ograph (X-ray transmission image) of the same partlyeclipsed paper sample as in Section 2.2. The resolutionwas chosen to be 8.5 μm such that individual fibers wouldbe visible in as large a scanned area as possible. The im-age together with the respective intensity profile acrossthe edge of the eclipsing tape is shown in Figure 2.

Furthermore, we collected three-dimensional (3D)information about the paper structure by making X-raytomographic reconstructions of samples. X-ray micro-computed tomography (μct) is based on analyzing theabsorption of X-rays in the sample while illuminating itwith them from (up to about a thousand) different di-rections, and using the resulting radiographs to form a3D reconstruction (image) of the sample. To this end weused an Xradia Micro XCT-400 device with a resolutionof 1.2 μm and a standard filtered back-projection recon-struction algorithm.

We used the 3D reconstructions for the construction ofBayesian prior distributions: virtual X-ray transmissionimages of the samples were then created by simulatingprojections of the tomographic reconstructions in the ap-propriate direction, yielding information about the Besovspace properties of the samples as explained below.

One 3D reconstruction is shown on the left panel ofFigure 3. Different grayscales represent different localX-ray absorption coefficients. As absorption depends onmaterial density, different material components, such assolid components and the void space between them, can beidentified in the image by their varying ranges of grayscalevalues using computational image analysis methods.

2.4 Bayesian inversion

The aim is to extract information about the X-ray trans-mission map f of a paper sample from the indirect (opti-cal) measurement modeled by

m = Af + �, (2)

in which m is the recorded optical transmission imageand � is random measurement noise. This is an ill-posed

Fig. 3. 3D-tomographic reconstruction of a paper sample (left)and the coefficients of its wavelet transform (right; the lighterthe grayscale color the bigger the magnitude of the transformcoefficient).

deconvolution problem, so a priori information is neededfor its successful inversion in addition to the measurementdata.

We first discretize the problem by letting m and f bed× d pixel images. We then follow the Bayesian inversionapproach, wherem and f are modeled as random variablesthat take values in Rd×d. Also, we model the measurementnoise � as a Gaussian random variable with independent,identically distributed components with zero mean andvariance σ2 > 0. The complete solution to the inverseproblem is given by the posterior distribution:

π(f |m) = π(f)π(m | f)π(m)

, (3)

in which π(m) can be understood as a normalization con-stant, π(f) is called the prior distribution and π(m | f) iscalled the likelihood distribution.

The likelihood distribution is essentially a measure-ment model that, for the assumptions made, takes aGaussian form,

π(m | f) = C exp(

− 12σ2

‖m−Af‖22). (4)

Here ‖ · ‖2 denotes the standard Euclidean norm in Rd×d.The prior distribution π(f) is used to express our a

priori knowledge in mathematical form. It should assignhigh probabilities to images f ∈ Rd×d that are expectedin light of the a priori information, and low probability tounusual images. In this work we use Besov space priors asexplained in Section 2.5.

The posterior distribution equation (3) contains fullinformation about the ill-posed inverse problem at hand.However, for practical purposes we need to find a repre-sentative estimate for the unknown f from the posterior.There are many useful choices of estimates (and their con-fidence limits) to choose from; in this work we concentrateon the maximum a posteriori (map) estimate defined by

fmap = arg maxf∈Rd×d

π(f |m), (5)

i.e., fmap is the image in Rd×d, which gives the largestvalue for the posterior distribution π(f |m) evaluated witha fixed (measured) realization of the random variable m.

2.5 Wavelet-based Besov space priors

Classical choices for prior distributions are Gaussian andare (formally) of the form π(f) = C exp(−δ‖f‖L2(R2)) orπ(f) = C exp(−δ‖∇f‖L2(R2)). These choices lead, how-ever, to rather smooth estimates. In the present prob-lem, prior distributions that promote variable degrees ofsmoothness are, however, desirable. We proceed to dis-cuss the so-called Besov space priors (formally) of theform π(f) = C exp(−δ‖f‖Bspq ) with parameters s ∈ Rand 1 ≤ p, q ≤ ∞. These priors have been shown to bediscretization invariant [12], ensuring numerical reliabilitywhen working at different resolutions.

Out of the three parameters of a Besov space, s andp are the most important. Roughly speaking, if f ∈ Bspqthen we could say that ‘derivatives up to order s are in

20701-p3


Lp’, see, e.g., [13,14]. The parameter q is for fine-tuning,and we choose here p = q which leads to a simpler formfor the norm.

Evaluation of the Besov space norm is most easily donein the wavelet domain. We give here only a brief descrip-tion of wavelets; more details can be found, e.g., [13,14].The wavelet expansion for a function f : Rn → R isgiven by

f =∞∑

j=0

∑G∈Gj

∑m∈Zn

c(G)j,mψ

(G)j,m, (6)

where the coefficients are defined by

c(G)j,m =

〈f, ψ

(G)j,m

〉, (7)

and the index set Gj has 2n elements when j = 0, and2n − 1 elements when j > 0. The functions ψ(G)j,m forman orthogonal basis for L2(Rn) and can be chosen to becompactly supported in cubes of sizes 2−j .

As an example we show in Figure 3 a tomographic re-construction of about a 0.5 × 0.5 mm2 piece of paper (leftpanel) together with its wavelet coefficients as a 3D Mallatpyramid (right panel) [15]. The smallest cubes contain co-efficients of ψ(G)1,m and the size of the cubes grows togetherwith j. Notice that most of the large coefficients are re-lated to small values of j, which is natural for a wavelettransform.

The Besov space norm can now be defined using thewavelet coefficients such that

‖f‖pBspp :=∞∑

j=0

2j(s−np +

n2 )p

∑G∈Gj

∑m∈Zn

∣∣∣c(G)j,m∣∣∣p . (8)

Since log2(∑

G∈Gj∑

m∈Zn∣∣∣c(G)j,m

∣∣∣p)

tends to behave asαj + C for some constants α and C, a necessary boundfor s is given by

s ≤ np

− n2

− αp. (9)

Equation (8) concerns functions defined on a continuum.A discrete (and approximate) version of the norm can becomputed for pixel images or voxel volumes by truncatingthe wavelet transform. There are fast algorithms for eval-uating the coefficients equation (7) for discrete data sets.We will use the prior distribution

π(f) = C exp(

− δ ‖f‖pBspp)

(10)

for f ∈ Rd×d with a truncated wavelet transform and asuitable weighting parameter 0 < δ < ∞ together withthe corresponding normalization constant C.

Let us calculate the relevant region in the sp plane.Figure 4 shows the scaling of the magnitude of the waveletcoefficients as a function of wavelet scale for the X-raytransmission data of the inset in Figure 5 with p = 2.Now when α, which is the slope of the regression line,

Fig. 4. Magnitude of wavelet coefficients as a function ofwavelet scale as calculated with p = 2 for the X-ray trans-mission data inserted in Figure 5. The slope of the regressionline (i.e., α) is about 1.26.

Fig. 5. The upper bound for s as a function of p for theX-ray transmission image shown in the inset, and the regionof possible smoothness parameters (shaded area). The point(s, p) chosen for the Besov norm is marked by a cross.

is estimated from data, the inequality equation (9) givesthe boundary for the acceptable region of (s, p) pairs, asillustrated in Figure 5.

Notice that the smoothness parameter can be deter-mined for both a tomographic reconstruction of the sam-ple (n = 3) and its 2D projection or a radiograph (n = 2).However, these parameters might not be the same.

2.6 Computing the maximum a posteriori estimate

Determination of the map estimate equation (5) that cor-responds to a given Besov space prior is equivalent to solv-ing the minimization problem:

fmap = arg minf∈Rd×d

{‖Af −m‖2 + δ ‖f‖pBspp

}. (11)

The solution to equation (11) was also found in the waveletdomain since the evaluation of the Besov norm and its

20701-p4


gradient is done there. From the computational point ofview the values p > 1 should only be considered since inthe case p = 1 the objective functional is not differentiable.

The two-point gradient method of Barzilai and Bor-wein [16] was used for the optimization. This method al-ways chooses the step in the direction of negative gradientof the objective function. However, owing to an effectivestep length choice, it converges faster and is less affectedby ill-conditioning than the classical steepest-descentmethod. Also, the method only uses vectors in the searchspace and gradients of the objective function, both of thesize d2. No matrices of size d2 × d2 are explicitly needed,which allows large-scale implementation of the method.

The Barzilai-Borwein method is defined iteratively bythe updating rule:

xk+1 = xk − αk∇g(xk) , (12)where g : Rn → R and the increment αk is given by

αk =sTk sksTk hk

(13)

with sk = xk − xk−1, hk = ∇g(xk) − ∇g(xk−1), and sTkthe transpose of sk.

3 Results

In Figure 6 we show an optical transmission image (m,left) and its deconvoluted map (f , right) as determinedfrom equation (2) such that the inversion was performedby solving equation (11) in which the Barzilai-Borweinoptimization method with Besov regularization was used.The parameter p was chosen as 2, and the correspondings value 0.2 (see the point crossed in Fig. 5) was found togive the minimum SEME value of equation (14) for sev-eral test samples. The X-ray transmission image is shownin the middle panel of the figure. The effect of the methodapplied is, as desired, to remove some of the ‘diffusion’ thetransmitted light has undergone inside the sample as a re-sult of reflections from fiber surfaces. The degree of simi-larity of the three transmission images, optical, X-ray, anddeconvoluted optical, must be analyzed quantitatively inorder to see how much closer the deconvoluted optical im-age is to the X-ray image than the original optical image.

To this end, the entropies of the error signals (i.e.,pointwise grayscale differences between two compared im-ages) were determined by seme as defined in equation (14)[17]. The SEME function was chosen as the measure ofsimilarity as it is specifically designed for determining thedegree of image distortion unlike the popular measuressuch as, e.g., mean squared error (MSE) and peak signal-to-noise ratio (PSNR) [17]. The absolute values of themaximum and minimum intensities were used in this ex-pression so as to prevent certain types of differences fromcanceling out each other:

seme=1

k1k2

k2∑j=1

k1∑i=1

[∣∣Iwmax;i,j∣∣−∣∣Iwmin;i,j∣∣∣∣Iwmax;i,j∣∣+∣∣Iwmin;i,j

∣∣mse(Iwi,j

)], (14)

in which I is the error signal, max and min values werecalculated for 4 × 4 pixel blocks, i.e.,

Iwmax;i,j = max{I1,1i,j , I

1,2i,j , ..., I

4,3i,j , I

4,4i,j

},

Iwmin;i,j = min{I1,1i,j , I

1,2i,j , ..., I

4,3i,j , I

4,4i,j

},

and

mse(Iwi,j

)=

116

[4∑

m,n=1

(Im,ni,j

)2].

The smaller the seme value, the more similar are the twoimages.

For seme we found a value of 38.98 (2.00) when com-paring the optical (deconvoluted optical) image with theX-ray transmission image. It is evident that when thepresent deconvolution method is applied to optical trans-mission images, the deconvoluted images resemble the cor-responding X-ray transmission images significantly better(for the measure of similarity used) than the original op-tical images.

To compare the images further, we also applied agradient-based orientation analysis method, the so-calledstructure tensor (ST) analysis [18,19]. ST attempts to findthe directions θm for which the L2 norms of the local direc-tional derivatives are maximized. The orientations of thefibers are then perpendicular to the directions of the gradi-ents. The orientation distribution curves such as those de-termined for the original optical, X-ray and deconvolutedoptical images are depicted in Figure 7. It is evident that

Fig. 6. Optical transmission image (left) of a paper sample, its X-ray transmission image (middle), and a deconvoluted(corrected) optical transmission image (right).

20701-p5


Fig. 7. (Color online) Orientation distribution for original op-tical, deconvoluted optical, and X-ray transmission images asdetermined by a structure tensor analysis.

there is no particular orientation visible in the distributiondetermined for the original optical image, although thereis a clear preferred orientation of fibers in a direction closeto zero degrees (horizontal direction in the image) in thedistribution determined for the X-ray transmission image.In the distribution determined for the deconvoluted opti-cal image this preferred orientation is however recoveredto a fairly high accuracy. Only the orientational anisotropyis somewhat smaller than in the distribution for the X-rayimage.

We also made a comparison of the Besov space basedregularization with the more traditional Tikhonovregularization method [20], or equivalently, used π(f) =C exp(−δ‖∇f‖L2(R2)) as the prior. The results of this com-parison are shown in Table 1. It is evident from this tablethat the Besov space based method gives better resultsfor the deconvoluted images. We found in particular thatthe Besov space based method can reproduce the finestdetails of the figure better than the Tikhonov method.

Table 1. seme values for the similarity of X-ray transmissionand corrected optical transmission images as determined bythe Besov space based and Tikhonov methods. The values forthe two methods with the same δ value are not directly compa-rable; the main feature is that the Besov space approach givesthe smallest seme value.

δ Besov Tikhonov0.1 9.03 13.800.01 5.72 9.620.001 2.00 16.640.0001 16.66 17.930.00001 17.13 27.200.000001 17.05 24.32

4 Conclusions

Disparity of optical transmission images of sheets of inho-mogeneous materials such as paper with the correspondingtransmission images obtained by X-rays or beta radiationis an old unsolved problem. In order to at least partly solvethis problem a new inversion method was introduced bywhich the optical transmission images can be corrected soas to better resemble, e.g., the X-ray transmission images.In this method a prior based on a Besov space classifi-cation of the actual structure was used for defining theregularization scheme. This regularization is known to beless affected by sharp edges (singularities) in the structure.In the optimization the Barzilai-Borwein method was ap-plied, which has proved to converge faster and to need lessresources than more traditional optimization methods.

It was demonstrated for paper samples that, with themethods introduced, optical transmission images can in-deed be transformed so that they fairly closely resemblethe ones that reflect the true areal distribution of mass,determined here by X-ray transmission. Similar resultswere obtained for other samples, but those results are notshown here. It was also demonstrated that the Besov spacebased regularization introduced gives better results thanthe traditional Tikhonov regularization. As a further mea-sure of similarity an orientational analysis of the threedifferent transmission images was performed. The clearpreferred orientation of fibers evident in the X-ray trans-mission image was lost in the optical transmission image,but it was fairly accurately recovered in the deconvolutedoptical image. We can conclude that the method intro-duced here can provide rather reliable results for prop-erties related to the areal mass distribution even whenthey are determined from optical transmission images. It isevident, however, that this method is only the first step to-ward an actual retrieval of the true (2D) material distribu-tion from optical transmission data, especially for systemswith multiple material components with varying dielectricproperties.

References

1. K. Niskanen, Papermaking Science and Technology (Fapet,Jyväskylä, 1998)

2. B. Norman, in Fundamentals in Papermaking, vol. 3,edited by C.F. Baker, V.W. Punton (Mech. Eng. Publ.,London, 1989), p. 674

3. J.-P. Bernie, W.J.M. Douglas, Tappi J. 79, 193 (1996)4. A. Hasan, D.S. Keller, J. Pulp Paper Sci. 35, 74 (2009)5. D.A. Zimnyakov, L.V. Kuznetsova, O.V. Ushakova,

R.Myllylä, Quantum Electron. 37, 9 (2007)6. J.A. Seibert, J.M. Boone, Med. Phys. 14, 567 (1988)7. H.H. Barret, W. Swindell, Radiological Imaging, vol. 2

(Academic Press, New York, 1981)8. J.M. Boone, B.A. Arnold, J.A. Seibert, Med. Phys. 13,

254 (1986)

20701-p6


9. J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross,J. Hecksher-Sorensen, R. Baldock, D. Davidson, Science296, 541 (2002)

10. J. Sharpe, Annu. Rev. Biomed. Eng. 6, 209 (2004)11. J.P. Kaipio, E. Somersalo, Applied Mathematical Sciences,

vol. 160 (Springer-Verlag, New York, 2005)12. M. Lassas, E. Saksman, S. Siltanen, IPI 3, 87

(2009)13. H. Triebel, Function Spaces and Wavelets on Domains

(EMS, Zürich, 2008)14. S. Mallat, IEEE Trans. Pattern Anal. Mach. Intell. 11, 674

(1989)15. S. Mallat, A Theory for Multiresolution Signal Decomposi-

tion: The Wavelet Representation (Elsevier, Amsterdam,2009)

16. J. Barzilai, J.M. Borwein, IMA J. Numer. Anal. 8, 141(1988)

17. E.A. Silva, K.A. Panetta, S.S. Agaian, Quantify simi-larity with measurement of enhancement by entropy, inProceedings: Mobile Multimedia/Image Processing for Mil-itary and Security Applications, SPIE Defence and Secu-rity Symposium 2007, vol. 6579, Paper #6579-0U, 2007

18. F. Lauze, J. van de Weijer, J. Weickert, P. Mrzek,P. Kornprobst, T. Brox, R. van den Boomgaard, in Vi-sualization and Processing of Tensor Fields, edited by H.Hagen, J. Weickert (Springer-Verlag, 2006), pp. 17–47

19. K.N. Sumit, K. Palaniappan, Lect. Notes Comput. Sci.3804, 445 (2005)

20. G.H. Golub, P.C. Hansen, D.P. O’Leary, SIAM J. MatrixAnal. Appl. 21, 185 (1999)

20701-p7

evaluation of the areal material distribution of paper from its...

Documents