# Spatially Varying Blur Recovery - Diagonal Approximations in the Wavelet Domain

### Paul Escande, Pierre Weiss, Francois Malgouyres

#### Abstract

Restoration of images degraded by spatially varying blurs is an issue of increasing importance. Many new optical systems allow to know the system point spread function at some random locations, by using microscopic luminescent structures. Given a set of impulse responses, we propose a fast and efficient algorithm to reconstruct the blurring operator in the whole image domain. Our method consists in finding an approximation of the integral operator by operators diagonal in the wavelet domain. Interestingly, this method complexity scales linearly with the image size. It is thus applicable to large 3D problems. We show that this approach might outperform previously proposed strategies such as linear interpolations (Nagy and O’Leary, 1998) or separable approximations (Zhang et al., 2007). We provide various theoretical and numerical results in order to justify the proposed methods.

#### References

- Beylkin, G., Coifman, R., and Rokhlin, V. (1991). Fast wavelet transform and numerical algorithm. Commun. Pure and Applied Math., 44:141-183.
- Chambolle, A. and Pock, T. (2011). A first-order primaldual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision, 40(1):120-145.
- Chang, E.-C., Mallat, S., and Yap, C. (1999). Wavelet foveation. Applied and Computational Harmonic Analysis, 9:312-335.
- Coifman, R. and Meyer, Y. (1997). Wavelets, calderónzygmund and multilinear operators. Cambridge Studies in Advanced Math, 48.
- Hansen, P., Nagy, J., and O'leary, D. (2006). Deblurring images: matrices, spectra, and filtering. Siam.
- Kirshner, H., Sage, D., and Unser, M. (2011). 3D PSF models for fluorescence microscopy in ImageJ. In Proceedings of the Twelfth International Conference on Methods and Applications of Fluorescence Spectroscopy, Imaging and Probes (MAF'11), page 154, Strasbourg, France.
- Nagy, J. and O'Leary, D. (1998). Restoring images degraded by spatially variant blur. SIAM Journal on Scientific Computing, 19:1063.
- Preibisch, S., Saalfeld, S., Schindelin, J., and Tomancak, P. (2010). Software for bead-based registration of selective plane illumination microscopy data. Nature methods, 7(6):418-419.
- Temerinac-Ott, M., Ronneberger, O., Ochs, P., Driever, W., Brox, T., and Burkhardt, H. (2011). Multiview deblurring for 3-d images from light sheet based fluorescence microscopy. Image Processing, IEEE Transactions on, (99):1-1.
- Wahba, G. (1990). Spline models for observational data, volume 59. Society for Industrial Mathematics.
- Zhang, B., Zerubia, J., and Olivo-Marin, J. (2007). Gaussian approximations of fluorescence microscope point-spread function models. Applied Optics, 46(10):1819-1829.

#### Paper Citation

#### in Harvard Style

Escande P., Weiss P. and Malgouyres F. (2013). **Spatially Varying Blur Recovery - Diagonal Approximations in the Wavelet Domain** . In *Proceedings of the 2nd International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,* ISBN 978-989-8565-41-9, pages 222-228. DOI: 10.5220/0004308202220228

#### in Bibtex Style

@conference{icpram13,

author={Paul Escande and Pierre Weiss and Francois Malgouyres},

title={Spatially Varying Blur Recovery - Diagonal Approximations in the Wavelet Domain},

booktitle={Proceedings of the 2nd International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,},

year={2013},

pages={222-228},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004308202220228},

isbn={978-989-8565-41-9},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Conference on Pattern Recognition Applications and Methods - Volume 1: ICPRAM,

TI - Spatially Varying Blur Recovery - Diagonal Approximations in the Wavelet Domain

SN - 978-989-8565-41-9

AU - Escande P.

AU - Weiss P.

AU - Malgouyres F.

PY - 2013

SP - 222

EP - 228

DO - 10.5220/0004308202220228