Donatelli, M.; Serra-Capizzano, S. Anti-reflective boundary conditions and re-blurring. (English) Zbl 1088.94510 Inverse Probl. 21, No. 1, 169-182 (2005). Summary: Anti-reflective boundary conditions have been introduced recently in connection with fast de-blurring algorithms: in the noise-free case, it has been shown that they reduce substantially artefacts called ringing effects with respect to other classical choices (zero Dirichlet, periodic, Neumann) and lead to algorithms costing \(O(n^d\log(n))\) arithmetic operations where \(n^d\) is the size of the signal if \(d=1\) or of the image if \(d=2\). Here we limit our analysis to the case of signals i.e. \(d=1\). More precisely, our study considers the role of the noise and how to connect the choice of appropriate boundary conditions with classical regularization schemes. It turns out that a successful approach is close to the Tikhonov technique: we call it re-blurring where the normal equations product \(A^TA\) is replaced by \(A^2\) with \(A\) being the blurring operator. A wide numerical experimentation confirms the effectiveness of the proposed idea. Cited in 2 ReviewsCited in 20 Documents MSC: 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 94A12 Signal theory (characterization, reconstruction, filtering, etc.) PDFBibTeX XMLCite \textit{M. Donatelli} and \textit{S. Serra-Capizzano}, Inverse Probl. 21, No. 1, 169--182 (2005; Zbl 1088.94510) Full Text: DOI