Abstract
Blake-Zisserman functional \(F_{\alpha ,\beta}^{g} \) achieves a finite minimum for any pair of real numbers α, β such that 0<β≤α≤2β and any g∈L 2(0,1). Uniqueness of minimizer does not hold in general. Nevertheless, in the 1D case uniqueness of minimizer is a generic property for \(F_{\alpha ,\beta }^{g}\) in the sense that it holds true for almost all gray levels data g and parameters α, β: we prove that, whenever α/β∉ℚ, the minimizer is unique for any g belonging to a dense G δ set of L 2(0,1) dependent on α and β.
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Notes
With the same arrangement we would have uniqueness by Remark 3, Theorem 3 and Theorem 7.
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Authors want to thank M.I.U.R. (Ministero Istruzione Università e Ricerca) which supported this research in 2008 as part of the P.R.I.N. (Progetti di Ricerca di Interesse Nazionale) project.
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Boccellari, T., Tomarelli, F. Generic uniqueness of minimizer for Blake & Zisserman functional. Rev Mat Complut 26, 361–408 (2013). https://doi.org/10.1007/s13163-012-0103-1
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DOI: https://doi.org/10.1007/s13163-012-0103-1