×

zbMATH — the first resource for mathematics

Signal recovery via TV-type energies. (English) Zbl 1392.49003
St. Petersbg. Math. J. 29, No. 4, 657-681 (2018) and Algebra Anal. 29, No. 4, 159-195 (2017).
Summary: One-dimensional variants are considered of the classical first order total variation denoising model introduced by Rudin, Osher, and Fatemi. This study is based on previous work of the authors on various denoising and inpainting problems in image analysis, where variational methods in arbitrary dimensions were applied. More than being just a special case, the one-dimensional setting makes it possible to study regularity properties of minimizers by more subtle methods that do not have correspondences in higher dimensions. In particular, quite strong regularity results are obtained for a class of data functions that contains many of the standard examples from signal processing such as rectangle or triangle signals as a special case. The analysis of the related Euler-Lagrange equation, which turns out to be a second order two-point boundary value problem with Neumann conditions, by ODE methods completes the picture of this investigation.

MSC:
49J05 Existence theories for free problems in one independent variable
49N60 Regularity of solutions in optimal control
26A45 Functions of bounded variation, generalizations
49J45 Methods involving semicontinuity and convergence; relaxation
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad, Nonlinear total variation based noise removal algorithms, Phys. D, 60, 1-4, 259-268, (1992) · Zbl 0780.49028
[2] LJ M. A. Little and N. S. Jones, \emph Generalized methods and solvers for noise removal from piecewise constant signals. \rm I. Background theory, Proc. Math. Phys. Eng. Sci. \bf 467 (2011), no. 2135, 3088–3114. · Zbl 1239.94018
[3] BPS I. Selesnick, A. Parekh, and I. Bayram, \emph Convex \rm 1-D total variation denoising with non-convex regularization, IEEE Signal Process. Lett. \bf 22 (2015), no. 2, 141–144.
[4] FIMT A. Torres, A. Marquina, J. A. Font, and J. M. Ib\'a\~nez, \emph Total-variation-based methods for gravitational wave denoising, Phys. Rev. D. \bf 90 (2014), no. 8, 084029.
[5] Bildhauer, Michael; Fuchs, Martin, A variational approach to the denoising of images based on different variants of the TV-regularization, Appl. Math. Optim., 66, 3, 331-361, (2012) · Zbl 1260.49074
[6] BF2 M. Bildhauer and M. Fuchs, \emph On some perturbations of the total variation image inpainting method. Pt. \rm I: regularity theory, Probl. Mat. Anal. \bf  76 (2014), 39–52; English transl., J. Math. Sci. (N.Y.) \bf  202 (2014), no. 2, 154–169.
[7] BF3 M. Bildhauer and M. Fuchs, \emph On some perturbations of the total variation image inpainting method. Pt. \rm II: relaxation and dual variational formulation, Probl. Mat. Anal. \bf  77 (2015), 3–18; English transl., J. Math. Sci. (N.Y.) \bf  205 (2015), no. 2, 121–140.
[8] BF5 M. Bildhauer and M. Fuchs, \emph Image inpainting with energies of linear growth. A collection of proposals, Probl. Mat. Anal. \bf 74 (2014), 45–50; English transl., J. Math. Sci. (N.Y.) \bf  196 (2014), no. 4, 490–497.
[9] BF6 M. Bildhauer and M. Fuchs, \emph On some perturbations of the total variation image inpainting method. Pt. \rm III: Minimization among sets with finite perimeter, Probl. Mat. Anal. \bf  78 (2015), 27–30; English transl., J. Math. Sci. (N.Y.) \bf  207 (2015), no. 2, 142–146.
[10] Bildhauer, M.; Fuchs, M.; Tietz, C., \(C^{1,α}\)-interior regularity for minimizers of a class of variational problems with linear growth related to image inpainting, Algebra i Analiz. St. Petersburg Math. J., 27 27, 3, 381-392, (2016) · Zbl 1335.49058
[11] FT M. Fuchs and C. Tietz, \emph Existence of generalized minimizers and of dual solutions for a class of variational problems with linear growth related to image recovery, Probl. Mat. Anal. \bf 81 (2015), 107–120; English transl., J. Math. Sci. (N.Y.) \bf  210 (2015), no. 4, 458–475.
[12] Bredies, Kristian; Kunisch, Karl; Valkonen, Tuomo, Properties of \(L^1\)-\({\rm TGV}^2\): the one-dimensional case, J. Math. Anal. Appl., 398, 1, 438-454, (2013) · Zbl 1253.49024
[13] Strong, David; Chan, Tony, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 19, 6, S165-S187, (2003) · Zbl 1043.94512
[14] Papafitsoros, Konstantinos; Bredies, Kristian, A study of the one dimensional total generalised variation regularisation problem, Inverse Probl. Imaging, 9, 2, 511-550, (2015) · Zbl 1336.49044
[15] Buttazzo, Giuseppe; Giaquinta, Mariano; Hildebrandt, Stefan, One-dimensional variational problems. An introduction, Oxford Lecture Series in Mathematics and its Applications 15, viii+262 pp., (1998), The Clarendon Press, Oxford University Press, New York · Zbl 0915.49001
[16] Ad R. A. Adams, \emph Sobolev spaces, Pure Appl. Math., vol. 65, Acad. Press, New-York–London, 1975.
[17] Giusti, Enrico, Minimal surfaces and functions of bounded variation, Monographs in Mathematics 80, xii+240 pp., (1984), Birkh\"auser Verlag, Basel · Zbl 0545.49018
[18] Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, xviii+434 pp., (2000), The Clarendon Press, Oxford University Press, New York · Zbl 0957.49001
[19] Th1 H. B. Thompson, \emph Second order ordinary differential equations with fully nonlinear two-point boundary conditions. \rm I, Pacific J. Math. \bf 172 (1996), no. 1, 255–277.
[20] Th2 H. B. Thompson, \emph Second order ordinary differential equations with fully nonlinear two-point boundary conditions. \rm II, Pacific J. Math. \bf 172 (1996), no. 1, 279–297.
[21] De Coster, Colette; Habets, Patrick, Two-point boundary value problems: lower and upper solutions, Mathematics in Science and Engineering 205, xii+489 pp., (2006), Elsevier B. V., Amsterdam · Zbl 1330.34009
[22] Bildhauer, M.; Fuchs, M., A geometric maximum principle for variational problems in spaces of vector valued functions of bounded variation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI). J. Math. Sci. (N.Y.), 385 178, 3, 235-242, (2011) · Zbl 1319.49074
[23] Hewitt, Edwin; Stromberg, Karl, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, vii+476 pp., (1965), Springer-Verlag, New York · Zbl 0137.03202
[24] Anzellotti, G.; Giaquinta, M., Convex functionals and partial regularity, Arch. Rational Mech. Anal., 102, 3, 243-272, (1988) · Zbl 0658.49005
[25] Ekeland, Ivar; T\'emam, Roger, Convex analysis and variational problems, Classics in Applied Mathematics 28, xiv+402 pp., (1999), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA · Zbl 0939.49002
[26] Attouch, Hedy; Buttazzo, Giuseppe; Michaille, G\'erard, Variational analysis in Sobolev and BV spaces. Applications to PDEs and optimization, MPS/SIAM Series on Optimization 6, xii+634 pp., (2006), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA · Zbl 1095.49001
[27] Ro R. T. Rockafellar, \emph Convex analysis, Princeton Landmarks Math. Phys., Princeton Univ. Press, Princeton, 2015.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.