Intervals in Linear and Nonlinear Problems of Image Reconstruction

In the “series expansion” approach (see, e.g., [15]) to image reconstruction, the mathematical formulation takes the form of a system of equations, linear or nonlinear, (1)$${\rm{\{ }}{{\rm{f}}_{\rm{i}}}{\rm{\} (x) = \{ }}{{\rm{p}}_{\rm{i}}}{\rm{\} ,i = 1,2,}}...{\rm{,m,}}$$where the functions fi:ℝn → ℝ describe the physics and geometry of the model, x ∈ ℝn (the Euclidean n-space) represents the unknown image in digitized form, and pi are the values of the measurements taken. Typically, the Jacobian matrix of the system is huge, sparse, and lacks structure in its sparsity pattern. Moreover, due to various limitations, practical restrictions, and features which are inherent in the real problem and/or in the “pixel by pixel” model, the system at hand is inevitably inconsistent. It is highly over- or under-determined, mostly ill-conditioned and describes the real problem only approximately because discretization takes place at the very beginning of the series expansion approach.