**
An
edge-preserving multilevel method for deblurring, denoising, and segmentation
**

CIRAM – Dept. of Mathematics,
University of Bologna, Italy

**Lothar Reichel**: Dept. of Mathematical
Sciences, Kent State University, OH, USA

We developed a cascadic multilevel image restoration method for reducing blur and noise in contaminated images that allows both spatially variant and spatially invariant point-spread functions (PSF).

The method requires the solution of a linear system of equations on each level. These systems are solved by an iterative method, the choice of which depends on properties of the PSF.

We introduce a thresholding updating strategy in order to suppress “ringing."

Constraints on the solution can be added in the cascadic environment. The restriction operators are defined by solving local weighted least-squares problems, and the prolongation operators are determined by piecewise linear prolongation followed by integrating a discretized nonlinear Perona-Malik diffusion equation for a few time-steps. The purpose of the integration is to reduce noise.

The cascadic multilevel method so obtained shares the computational efficiency and simplicity of truncated iteration for the solution of linear discrete ill-posed problems with the edge-preserving property of nonlinear models.

The multilevel method proceeds from coarser to finer levels, and regularizes by truncated iteration on each level.

For many image restoration problems, the multilevel method demands fewer matrix-vector product evaluations on the finest level than the corresponding 1-level truncated iterative method, and often determines restorations of higher quality. A benefit of our multilevel approach to image restoration is that it easily can be combined with image segmentation, as is illustrated in the images below.

**
[1]**
**S. Morigi, L. Reichel, F. Sgallari, and A. Shyshkov**,
*Cascadic multiresolution methods for image deblurring*, SIAM J. Imaging
Sci., Vol.1, no. 1, pp. 51-74, 2009.

**
[2]**
**S. Morigi,
L. Reichel and F. Sgallari**,
*An edge-preserving multilevel method for deblurring, denoising, and
segmentation*, X.-C.Tai et al. (Eds): SSVM 2009, LNCS 5567, pp. 427-439,
2009, Springer-Verlag Berlin Heidelberg 2009.

[3]
**S. Morigi, L. Reichel and F. Sgallari**, *
Cascadic Multilevel Methods for Fast nonsymmetric Blur- and Noise Removal*,
Applied
Numerical Mathematics,
2009 (in press).

*
Return to Fiorella Sgallari's Home Page*