On uniform approximation on subsets. (English. Russian original) Zbl 1337.94005
Math. Notes 98, No. 5, 860-863 (2015); translation from Mat. Zametki 98, No. 5, 797-800 (2015).
In the present paper, the author studies a problem in approximation theory that formalizes the problem of approximation of a document photograph’s background considered in the theory of image processing. The question under consideration is related to the problem of compressed measurement. It is well known that \(l_1\)-minimization and greedy algorithms (cf. [D. L. Donoho, IEEE Trans. Inf. Theory 52, No. 4, 1289–1306 (2006; Zbl 1288.94016); E. D. Livshits, Sb. Math. 203, No. 2, 183–195 (2012; Zbl 1259.94019); translation from Mat. Sb. 203, No. 2, 33–44 (2012)]) are among the most effective methods of solving the problem of compressed measurements. In order to solve the problem considered here, the author employs some analogues of the methods mentioned above.
Reviewer: H. P. Dikshit (Bhopal)
MSC:
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
| 41A50 | Best approximation, Chebyshev systems |
Keywords:
image processing; uniform approximation on subsets; background approximation problem; greedy algorithm; compressed measurement; Nikol’skiĭ’s inequalityReferences:
| [1] | Gabriel de Franca Pereira e Silva, Rafael Dueire Lins, and AndréRicardson Silva, in Lecture Notes in Computer Science, Vol. 7950: Image Analysis and Recognition (Springer-Verlag, Berlin, 2013), pp. 290-298. |
| [2] | Donoho, D. L., No article title, IEEE Trans. Inform. Theory, 52, 1289 (2006) · Zbl 1288.94016 · doi:10.1109/TIT.2006.871582 |
| [3] | Livshits, E. D., No article title, Mat. Sb., 203, 33 (2012) · doi:10.4213/sm7827 |
| [4] | S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems (Nauka, Moscow, 1977) [in Russian]. · Zbl 0496.46020 |
| [5] | Malykhin, Yu. V.; S. Ryutin, K., No article title, Mat. Sb., 205, 95 (2014) · doi:10.4213/sm8332 |
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