  Twenty years ago the anti-reflective numerical boundary conditions (BCs) were
introduced in a context of signal processing and imaging, for increasing the
quality of the reconstruction of a blurred signal/image contaminated by noise
and for reducing the overall complexity to that of few fast sine transforms
i.e. to $O(N\log N)$ real arithmetic operations, where $N$ is the number of
pixels. Here for quality of reconstruction we mean a better accuracy and the
elimination of boundary artifacts, called ringing effects. Now we propose
numerical anti-reflective BCs in the context of nonlocal problems of fractional
differential type: the goals are the same i.e. a smaller approximation error
and the reduction of boundary artifacts. In the latter setting, we compare
various types of numerical BCs, including the anti-symmetric ones considered in
the case of fractional differential problems for modeling reasons. More in
detail, given important similarities between anti-symmetric and anti-reflective
BCs, we compare them from the perspective of computational efficiency, by
considering nontruncated and truncated versions and also other standard
numerical BCs such as reflective/Neumann. Several numerical tests, tables, and
visualizations are provided and critically discussed. The conclusion is that
the truncated numerical anti-reflective BCs perform better, both in terms of
accuracy and low computational cost.
