Incorporating Prior Knowledge in Image Reconstruction through the Choice of the Hilbert Space and Inner Product

The problem is to reconstruct a function f : R → C from finitely many linear functional values. To model the operator that transforms f into the data vector, we need to select an ambient space containing f . Typically, we choose a Hilbert space. The selection of the inner product provides an opportunity to incorporate prior knowledge about f into the reconstructon. The inner product induces a norm and our reconstruction is the function, consistent with the data, for which this norm is minimized. The method is illustrated using Fourier-transform data and prior knowledge about the support of f and about its overall shape. 1 The Basic Problem We want to reconstruct a function f : R → C from finitely many linear functional values, g1, ..., gN . Although we may reasonably view f as part of objective reality, once we embed f in a Hilbert space we are imposing theory that, while reasonable, is not given a priori, and is not part of objective reality. As we shall see, the selection of the ambient Hilbert space provides one of the few opportunities we have to incorporate prior knowledge about f , and therefore is an important step in the reconstruction. Because the problem is highly underdetermined, there will be infinitely many reconstructions consistent with the finite data. A common approach to solving such problems is to select the reconstruction with the smallest norm; how we select the norm in the first place is the important step.