Variational methods on the space of functions of bounded Hessian for convexification and denoising. (English) Zbl 1098.49022

The authors study variational principles on the space of functions of bounded Hessian. First some basic properties of convex functions, functions of bounded variation and relations between convex functions and the space of bounded Hessians are recalled. Then the existence of minimizers of variational problems for denoising, approximation by convex functions, and for calculation of the convex envelope are proved. Some numerical results are given.


49K20 Optimality conditions for problems involving partial differential equations
49J25 Optimal control problems with equations with ret. arguments (exist.) (MSC2000)
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
49J40 Variational inequalities
65K10 Numerical optimization and variational techniques
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[10] Chan, T. F., Wong, C. K.: Convergence of the alternating minimization algorithm for blind deconvolution. Linear Algebra Appl. 316, 259–285 (2000). Conf. Celebrating the 60th Birthday of Robert J. Plemmons (Winston-Salem, NC, 1999). · Zbl 0993.65149
[11] Chan, T. F., Wong, C. K.: Total variation blind deconvolution. IEEE Trans. Image Proc. 7, (1998).
[21] Golub, G. H., Van Loan, Ch. F.: Matrix computations, 3rd ed. Baltimore: The Johns Hopkins University Press 1996. · Zbl 0865.65009
[22] Isakov, V.: Inverse source problems. Providence, RI: American Mathematical Society, 1990. · Zbl 0721.31002
[23] Isakov, V.: Inverse problems for partial differential equations. Appl. Math. Sci. 127, (1998). · Zbl 0908.35134
[32] Nashed, M. Z., Scherzer, O. (eds.): Interactions on inverse problems and imaging. Contemp. Math. 313, (2002). · Zbl 1003.00013
[40] Scherzer, O.: Explicit versus implicit relative error regularization on the space of functions of bounded variation. In [32], 171–198 (2002). · Zbl 1028.65147
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