Abstract
We address the problem of estimating a random vector X from two sets of measurements Y and Z, such that the estimator is linear in Y. We show that the partially linear minimum mean squared error (PLMMSE) estimator requires knowing only the second-order moments of X and Y, making it of potential interest in various applications. We demonstrate the utility of PLMMSE estimation in recovering a signal, which is sparse in a unitary dictionary, from noisy observations of it and of a filtered version of it. We apply the method to the problem of image enhancement from blurred/noisy image pairs. In this setting the PLMMSE estimator performs better than denoising or deblurring alone, compared to state-of-the-art algorithms. Its performance is slightly worse than joint denoising/deblurring methods, but it runs an order of magnitude faster.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Costa, O.L.V.: Linear minimum mean square error estimation for discrete-time Markovian jump linear systems. IEEE Trans. Autom. Control 39(8), 1685–1689 (1994)
Schniter, P., Potter, L.C., Ziniel, J.: Fast Bayesian matching pursuit. In: Information Theory and Applications Workshop (ITA 2008), pp. 326–333 (2008)
Soussen, C., Idier, J., Brie, D., Duan, J.: From bernoulli-gaussian deconvolution to sparse signal restoration. IEEE Trans. Signal Process. (99), 4572–4584 (2010)
Girolami, M.: A Variational method for learning sparse and overcomplete representations. Neural Computation 13(11), 2517–2532 (2001)
Härdle, W., Liang, H.: Partially linear models. In: Statistical Methods for Biostatistics and Related Fields, pp. 87–103 (2007)
Michaeli, T., Sigalov, D., Eldar, Y.: Partially linear estimation with application to sparse signal recovery from measurement pairs. IEEE Trans. Signal Process. (2011) (accepted)
Yuan, L., Sun, J., Quan, L., Shum, H.Y.: Image deblurring with blurred/noisy image pairs. In: ACM SIGGRAPH 2007 Papers, pp. 1–10. ACM (2007)
Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-d transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007)
Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image restoration by sparse 3d transform-domain collaborative filtering. In: SPIE Electronic Imaging (2008)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Michaeli, T., Sigalov, D., Eldar, Y.C. (2012). Partially Linear Estimation with Application to Image Deblurring Using Blurred/Noisy Image Pairs. In: Theis, F., Cichocki, A., Yeredor, A., Zibulevsky, M. (eds) Latent Variable Analysis and Signal Separation. LVA/ICA 2012. Lecture Notes in Computer Science, vol 7191. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28551-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-28551-6_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28550-9
Online ISBN: 978-3-642-28551-6
eBook Packages: Computer ScienceComputer Science (R0)