for AIT, APIT, AIT-GP, and APIT-GP: (a) Barbara test case, (b) Cell test case, (c) Satellite test case. In black with stars AIT, in blue with circles APIT, in red with
105
Chapter 7
Multigrid iterative regularization
method for image deblurring with
arbitrary boundary conditions
In this chapter we turn again our attention to the finite dimensional case of the linear system of equations (2.3). We are going to discuss a multigrid algorithm designed for image deblur- ring and denoising. In particular, we want to construct an iterative regularization method using the multigrid framework.
Multigrid methods are very powerful algorithms that are able to achieve very fast computa- tions and high accuracy, see e.g. [24,120]. Multigrid methods have been initially developed for solving linear systems of equations derived from partial differential equations (PDEs) [22] and later successfully applied to more general linear systems [118].
Multigrid methods have already been considered to solve ill-posed problems [36,37,55,81, 92, 94,114], but usually as solvers for Tikhonov like regularized models. The first attempt of using multigrid methods as iterative regularization methods has been probably done in [56], where the authors combined an iterative regularization method used as pre-smoother with a low-pass filter coarsening. Later the same authors furtherly discussed in [55] the regularizing properties of this method. A different multilevel strategy based on the cascadic approach was proposed in [110]. Nonlinear “corrections” to the previous multigrid methods were introduced in [100] using a total variation-type regularization and in [45,63] combining multigrid and wavelets. More recently, also the blind deconvolution has been successfully approached in [62]. Note that, these multigrid methods have been defined to preserve the BTTB structure of the blurring matrix at each coarser level. This is crucial for the definition of the algorithm and for preserving a fast and simple matrix vector product.
The main novelty in [45], with respect to [56], was the addition of a soft-thresholding de- noising as post-smoother. We are going to define our method by starting from the idea in [45] of combining framelet denoising and multigrid. Firstly, differently from the previous works, we define a general coarsening strategy independently of the boundary conditions. In practice, the Galerkin projection of the operator is applied to the PSF instead of to the coefficient matrix and at each coarser level we can apply the favorite boundary conditions. Furthermore, our proposal differs from the one in [45] also for the use of framelet denoising as a pre-smoother instead of as post-smoother and for the use of APIT, described in Chap- ter 6, instead of CGLS as inner iterative regularization method. This choice let us ensure the nonnegativity of the provided approximation, because APIT projects each iteration into the nonnegative cone and so using it as post-smoother it is equivalent to project each multigrid iteration into the nonnegative cone. Finally we also give a theoretical proof of convergence
of the algorithm in the two grid case (as usual for multigrid methods [120]) under some restrictive, but reasonable, hypothesis.
Using the knowledge of δ the proposed algorithm is able to achieve very high accuracy with- out tuning any parameter.
This chapter is structured as follows: in Section 7.1 we briefly describe the multigrid algo- rithm and the framelet denoising, which are both needed for the formulation of our algo- rithm, in Section 7.2 we describe our algorithmic proposal, in Section 7.3 we discuss the convergence of the method and, finally, in Section 7.4 we give some numerical examples.
7.1
Preliminaries
In this section we present some tools which are needed in the construction of our algorithm.
7.1.1 Multigrid Methods
The first tool which we have to describe is the Multigrid approach.
Multigrid methods have been developed for solving partial differential equation and, more in general, for solving linear systems of equations of large size. The basic idea of the multi- grid is to create a sequence of linear systems which get smaller and smaller by consecutive projection. In this way the computational effort can be reduced and the convergence speed can be improved up if the smaller linear systems are properly chosen.
Let us start with the linear system of equation
Ax = b, (7.1)
where A ∈ Rn×nis invertible and x, b ∈ Rn.
It is well known that iterative methods first converge in the well-conditioned space and that the convergence can be very slow in the ill-conditioned space. Differently, while direct meth- ods are not affected by this kind of problem, they are usually much more expensive and are more sensible to error propagation.
Remark 7.1. The definition of well- and ill-conditioned space is not formal.
Let V ⊂ Rnbe a linear subspace of Rn. We define the conditioning number of A restricted to V by
κV = sup x∈V
kAxk kxk .
The well-conditioned space is the space W such that κW is not too large, whereas the ill-conditioned space I is the one where κI is very large.
For matrices deriving from the discretization of compact integral operator we have that W corresponds to the low frequency space and I is the high frequency space.
The idea of the Multigrid method is to combine the positive aspects of both direct and itera- tive method.
7.1. Preliminaries 107
The TGM is an iterative algorithm. Let xkbe an approximation of the solution of (7.1) at the kth step, apply ν1steps of an iterative method to xkobtaining
˜
xk=Pre-Smooth(A, b, xk, ν1).
This step is called pre-smoothing, since it is done before everything else and, in the context of differential equations, damps the error in the high frequencies, i.e., it smooths the error. We then compute the residual
rk= b− A˜xk,
since we are moving to the error equation in order to compute a refinement term for ˜xk. Let 0 < n1 < n, we call R ∈ Rn1×nthe restriction operator. This operator projects a vector from a grid of size n to a grid of size n1.
Define P ∈ Rn×n1 the interpolation operator. This operator interpolates a vector from a grid
of size n1to a grid of size n. Usually R = Pt.
We can now define the restricted operator using the Galerkin approach as
A1 = RAP ∈ Rn1×n1.
Let us assume that both R and P are of full rank. This implies that A1is invertible. Comput- ing the refinement term for ˜xkas
hk= P A−11 Rrk, we obtain the refined version of ˜xkas
ˆ
xk= ˜xk+ hk= ˜xk+ P A−11 R(b− A˜xk).
The procedure that computes ˆxkfrom ˜xkis called Coarse Grid Correction (CGC). Let us call C the iteration matrix of the CGC, i.e.,
C = I − P (RAP )−1RA,
it is possible to show that C is a projector and hence λ(C) = {0, 1}, where λ(C) denotes the spectrum of C. Therefore, the TGM algorithm, without any smoothing step, can not converge to the solution of (7.1), cf. [24].
Finally, to obtain the (k + 1)th approximation, we apply ν2 steps of an iterative method
xk+1 =Post-Smooth(A, b, ˆxk, ν2),
which can be different from the pre-smoother. This is called post-smoothing.
It is possible to show that, under mild conditions, this method converges to the solution of (7.1).
The problem of the TGM algorithm is obviously the computation of hksince it requires the inversion of A1 which, if n1 is large, can be extremely expensive. The multigrid method stems from this observation. Since A1 can be very large the idea is to restrict consecutively the grid until it is so small that the inversion can be easily performed.