 # Fiorella Sgallari

Full Professor

Department of Mathematics

## Research

Linear Algebra and Parallel Computing:

Iterative Methods for Large Linear Systems of equations

Inverse Problems and Image Processing :

Regularization Methods for Large Linear Discrete Ill-posed Problems

PDEs Models and Methods for image and Image Sequences Processing

Image Segmentation

Image Interpolation and Zooming

Image Deblurring

Medical Image Processing

Models and numerical methods for Engineering Problems

The problem of processing images or sequences of images in two (2D) or three dimensions (3D) can be efficiently pursued with techniques based on the solution of partial differential equations (PDE).
In many image processing problems, the reconstruction or restoration of the image from data is an ill posed problem: the knowledge of the (blurring) model is not sufficient to determine an accurate solution.   It is necessary to regularize the problem, i.e. to introduce an a priori condition on the solution.

An efficient regularization technique is the edge preserving regularization, that reduce the noise in smooth regions of the image while preserving edges.
There is a strong connection between regularization and diffusion schemes modelled by PDEs.

The first step to use PDEs for image processing was done in the beginning of eighties. By the simple observation that the Gauss function is a fundamental solution of the linear heat (diffusion) equation, it has been possible to replace the classical image processing operation-convolution of an image with a gaussian function by solving the linear heat equation with the initial condition given by the processed image. It is well known that Gaussian smoothing (linear diffusion) is generally not the best choice and consequently the nonlinear diffusion models are widely used. Due to the evolutionary character of the process which controls the processing using diffusion equations, application of any PDE to an initially given image is understood as its embedding in the so-called scale space. In the case of nonlinear PDEs one speaks about nonlinear scale space. The image multiscale analysis associates a given image u0(x) to a family u(t,x) of smoothed-simplified images depending on an abstract parameter t in [0,T], the scale. If such a family fulfils certain basic assumptions then u(t,x) can be represented as the unique viscosity solution of a general second order(degenerate) parabolic partial differential equation. This theoretical result has an important practical counterpart.

The equations of (degenerate) parabolic type have a smoothing property, so they are a natural tool for filtering (image simplification) by removing spurious structures, e.g. noise.
Moreover, the simplification should be image oriented, e.g. it should respect edges and not blur them. Or, it should recognize motion of a structure in an image sequence, and consequently the smoothing (diffusion) should respect the motion coherence in consecutive frames.

A lot of interest is now devoted to 3D PDE denoising and segmentation, for particular structures like heart or vessels, or more generally tubular-like structure images.

Such problems need new models and introduce strong nonlinearity into the parabolic PDEs, this make the field interesting not only because of the applications but also from a mathematical and numerical point of view.

From the numerical point of view these models involve two different aspects: the space discretization and the time-scale discretization. The most popular space approximation method is finite differences, but other more powerful and flexible discretization techniques such as finite elements, finite volumes-covolumes have been recently investigated by the numerical analysis community.  In the literature the most popular time-scale discretization approach employs explicit schemes, limiting the simulation to very small time-scale steps. Otherwise, semi-implicit schemes require the efficient solution of very large linear systems of equations at each step.

From a general perspective, the approximation of PDEs by means of powerful discretization techniques, usually leads to large linear systems that need to be solved with sufficient accuracy,  while maintaining high computational efficiency.
Preconditioning techniques as well as ad-hoc implementation strategies make iterative projection-type approaches such as Krylov subspace methods, very appealing and will studied in the next years.

Some of the previous PDEs models can be set in the framework of inverse problems and regularization theory, a typical example is the deblurring problem. A better understanding of the relations between these two approaches is an emerging research topic.

The numerical solution of  large-scale ill-posed problems is an interesting and challenging topic which  has receive a lot of attention in the past decade. The use of Tikhonov regularization requires the determination of a suitable value of the regularization parameter. When the dimensions of the problem are so large as to make factorization of the matrix impossible or infeasible, the selection of  a suitable value of the regularization parameter may require that the original problem is solved over and over again for different test values. We recently proposed that instead of exact quantities, difficult to compute, upper and lower bounds are used to determine suitable values of the regularization parameter.

Most recently, we have extended the methodology to the case where the solution is required to have nonnegative entries. The proposed methods is more efficient that others recently proposed for the solution of this problem.

Another way of damping the disastrous effects of amplified errors in the data on the computed solution is to use iterative methods equipped with suitable stopping rules. When successful, this approach is typically faster than using Tikhonov regularization and may yield solutions of higher quality.

Multilevel methods are popular for the solution of well-posed problems, such as certain boundary value problems for partial differential equations and Fredholm integral equations of the second kind. However, little is known about the behavior of multilevel methods when applied to the solution of linear ill-posed problems, such as Fredholm integral equations of the first kind, with a right-hand side that is contaminated by error.  Presently we are considering cascadic multilevel methods that are particularly efficient schemes based on truncated iteration that blend linear algebra and Partial Differential Equations techniques. We are studying their properties and the applications to image deblurring and denoising that are large-scale problems, because of the typically large number of pixels that make up an image.

Moreover, we are investigating the design of preconditioners which can be used to convey information about the desired solution into the iterative methods. This is a novel approach to preconditioning ill-posed  problems which has been shown to be remarkably effective in a few test problems.

These problems are very important in several applications, from medical to astronomical image processing. Several research projects have been planned with academic and industrial partners

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