Frame decompositions of bounded linear operators in Hilbert spaces with applications in tomography

Abstract

We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and ill-posedness of the problem and can be used as the basis for the design and implementation of efficient numerical solution methods. In contrast to the singular-value decomposition, the presented frame decompositions can be derived explicitly for a wide class of operators, in particular for those satisfying a certain stability condition. In order to show the usefulness of this approach, we consider different examples from the field of tomography.

Minimax detection of localized signals in statistical inverse problems

“…frames or wavelets, and set F σ = {u k : k ∈ I σ } for subsets I σ of I. The particular choices that we analyze in this paper will be built from such dictionaries, see also [7] and [12] for recent references in the context of estimation.…”

On Regularization via Frame Decompositions with Applications in Tomography

“…In order to remedy this situation, several researchers have studied generalizations of the SVD such as the Wavelet-Vaguelette Decomposition (WVD) [1,6,7,15,16] or the Frame Decomposition (FD) [8,12,13]. The idea is that by weakening some of the requirements of the SVD such as the eigenvalue properties and the resulting orthogonality of the functions u k and v k , one may end up with decompositions similar to (1.2) and (1.5) which are easier to derive explicitly for a given operator.…”

“…The properties of the operator A and its use for obtaining (approximate) solutions of (1.1) was studied in detail in [13]. In particular, it was investigated in which cases Ay is either a minimum-coefficient or a minimum-norm (least-squares) solution of (1.1).…”

“…In particular, it was investigated in which cases Ay is either a minimum-coefficient or a minimum-norm (least-squares) solution of (1.1). While we refer to [13] for details, we here only want to mention the special case that A satisfies a stability condition of the form…”

“…and it then follows that {e k } k∈N forms a frame over X satisfying (1.12) with λ k = 1/α k [13]. A typical example for an operator satisfying condition (1.16) is given by the Radon transform [17,18], and (1.17) can e.g.…”