Making use of a partial order in solving inverse problems

Abstract

In many applications, the concepts of inequality and comparison play an essential role, and the nature of the objects under consideration is better described by means of partial order relations. To reflect this nature, the conventional problem statements in normed spaces have to be modified. There is a need to enrich the structure of the functional spaces employed. In this paper, we consider inverse problems in Banach lattices-functional spaces endowed with both norm and partial order. In this new problem statement, we are able to construct a linearly constrained set that contains all admissible approximate solutions given the approximate data, the approximation errors and prior restrictions that are available. We do not suppose the problem to be either well posed or ill posed since we believe that the concepts and techniques we describe here can be useful in both cases. The range of applications includes medical imaging (CT, PET, SPECT, photo-and thermoacoustics), geophysics (e.g., inverse gravimetry problems), engineering design and other inverse problems.

Variational regularisation for inverse problems with imperfect forward operators and general noise models

“…An alternative approach to modelling operator errors using order intervals in Banach lattices was proposed in [ 21 – 23 ]. It assumes that the spaces and have a lattice structure [ 24 ] and, instead of ( 1.4 ), lower and upper bounds for the operator are available where the inequalities are understood in the sense of a partial order for linear operators, i.e.…”

Image Reconstruction with Imperfect Forward Models and Applications in Deblurring

“…the forward operator is exact), the set U h,δ is non-convex and the residual method results in a non-convex optimisation problem even for convex regularisation functionals. An alternative approach to modelling uncertainty in A and f using partially ordered spaces was proposed in [15,16]. Assume that U and F are Banach lattices, i.e.…”

“…An alternative approach to modelling uncertainty in A and f using partially ordered spaces was proposed in [15,16]. Assume that U and F are Banach lattices, i.e.…”

Diffusion Tensor Imaging with Deterministic Error Bounds

“…The data may also contain process artefacts from black box devices [41]. Partial order in Banach lattices has therefore recently been investigated in [30,32,31] as a less-assuming error modelling approach for inverse problems. This framework allows the representation of errors in the data as well as in the forward operator of an inverse problem by means of order intervals (i.e., lower and upper bounds by means of appropriate partial orders).…”