Evaluation of the areal material distribution of paper from its optical transmission image

¹ Department of Physics, University of Jyväskylä, Finland
² Department of Mathematics and Statistics, University of Helsinki, Finland
³ Department of Mathematics and Physics, Lappeenranta University of Technology, Finland

^a e-mail: jouni.j.takalo@jyu.fi
^b e-mail: samuli.siltanen@helsinki.fi

Received: 15 September 2010
Revised: 20 December 2010
Accepted: 4 April 2011
Published online: 11 August 2011

Abstract

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