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Anti-reflective boundary conditions and re-blurring. (English) Zbl 1088.94510

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.

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.)
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