Aristotelian prior boundary conditions

The selection of boundary conditions when restoring a finite 1D or 2D signal has received a lot of interest in recent times. Dirichlet, period, reflective and antireflective boundary conditions may be appropriate selection for some problems, while wholly unsuitable for others. In this paper we show that, in an Aristotelian approach to knowledge, when it is not known a priori which boundary conditions should be chosen, by admitting our lack of information it is possible to let the data itself determine them. We consider 1D and 2D smoothness prior conditions. The application of this Aristotelian approach to boundary conditions to the solution of linear discrete ill-posed problems with Tikhonov regularization or truncated iterative methods is discussed. Computed examples of deblurring and limited angle tomography showing the effectiveness of Aristotelian boundary conditions are presented.