**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