This paper is concerned with the uniqueness of the solution of a nonlinear
equation, named discrepancy equation. For the restoration problem of data corrupted
by Poisson noise, we have to minimize an objective function that combines a
data-fidelity function, given by the generalized Kullback–Leibler divergence, and a
regularization penalty function. Bertero et al. recently proposed to use the solution
of the discrepancy equation as a convenient value for the regularization parameter.
Furthermore they devised suitable conditions to assure the uniqueness of this solution
for several regularization functions in 1D denoising and deblurring problems.
The aim of this paper is to generalize this uniqueness result to 2D and 3D problems
for several penalty functions, such as an edge preserving functional, a simple case of
the class of Markov Random Field (MRF) regularization functionals and the classical
Tikhonov regularization
