zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Denoising with higher order derivatives of bounded variation and an application to parameter estimation. (English) Zbl 0891.65103
The paper is devoted to the generalization of bounded variation regularization to functions with higher-order derivatives of bounded variation. This concept is applied to locate discontinuities in derivatives, which has important applications in parameter estimation problems. The main objective is the analysis of an algorithm for locating discontinuities in derivatives of a function, which is an important question arising in inverse conductivity problems. Based on an overview on denoising algorithms the idea of the work is presented. In contrast to other papers on bounded variation regularization the main emphasis is to reconstruct discontinuities of the derivatives in smooth images (images which do not have edges). The numerical solution of a one-dimensional denoising problem is considered for both higher-order bounded variation regularization and standard Tikhonov regularization. A comparison of the numerical results is performed. Two-dimensional denoising problems are also studied. The way to extend all results theoretically for problems which involve determination of jumps in higher-order derivatives is proposed for future exercise.

MSC:
65M30Improperly posed problems (IVP of PDE, numerical methods)
35R30Inverse problems for PDE
35K05Heat equation
WorldCat.org
Full Text: DOI
References:
[1] Barrett, R., Berry, M., Chan, T., Demmel, J., Donato, J., Dongarra, J., Eijkhoui, V., Pozo, R., Romine, Ch., van der Vorst, H.: TEMPLATES for the solution of linear systems: building blocks for iterative methods. Software Package for the solution of linear systems. · Zbl 0814.65030
[2] Deimling, K.: Nonlinear functional analysis. Berlin, Heidelberg, New York: Springer 1980. · Zbl 1257.47059
[3] Dobson, D., Santosa, F.: An image enhancement technique for electrical impedance tomography. Inverse Problems10, 317--334 (1994). · Zbl 0805.35149 · doi:10.1088/0266-5611/10/2/008
[4] Dobson, D., Scherzer, O.: Analysis of regularized total variation penalty methods for denoising. Inverse Problems12, 601--617 (1996). · Zbl 0866.65041 · doi:10.1088/0266-5611/12/5/005
[5] Dobson, D., Vogel, C. R.: Convergence of an iterative method for total variation denoising. SIAM J. Numer. Anal.34, 1779--1791 (1997). · Zbl 0898.65034 · doi:10.1137/S003614299528701X
[6] Engl, H. W., Kunisch, K., Neubauer, A.: Convergence rates for Tikhonov regularization of non-linear ill-posed problems. Inverse Problems5, 523--540 (1989). · Zbl 0695.65037 · doi:10.1088/0266-5611/5/4/007
[7] Evans, L. C., Gariepy, R. F.: Measure theory and fine properties of functions. Ann Arbor: CRC Press 1995. · Zbl 0804.28001
[8] Groetsch, C. W.: Spectral methods for linear inverse problems with unbounded operators. J. Approx. Theory70, 16--28 (1992). · Zbl 0759.41021 · doi:10.1016/0021-9045(92)90053-Q
[9] Isakov, V.: Inverse source problems. Rhode Island: AMS 1990. · Zbl 0721.31002
[10] Kohn, R. V., Lowe, B. D.: A variational method for parameter identification. RAIRO. Math. Modell. Numer. Anal.22, 119--158 (1988). · Zbl 0636.65127
[11] Kunisch, K.: Inherent identifiability of parameters in elliptic differential equations. J. Math. Anal. Appl.132, 453--472 (1988). · Zbl 0673.35098 · doi:10.1016/0022-247X(88)90074-1
[12] Morel, J.-M., Solimini, S.: Variational Methods in Image Segmentation. Basel: Birkhäuser 1995.
[13] Neubauer, A.: Tikhonov regularisation for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation. Inverse Problems5, 541--557 (1989). · Zbl 0695.65038 · doi:10.1088/0266-5611/5/4/008
[14] Osher, S., Rudin, L.: Feature oriented image enhancement using shock filters. SIAM J. Numer. Anal.27, 919--940 (1990). · Zbl 0714.65096 · doi:10.1137/0727053
[15] Scherzer, O., Engl, H. W., Kunisch, K.: Optimal a posteriori parameter choice for Tikhonov regularization for solving nonlinear ill-posed problems. SIAM J. Numer. Anal.30, 1796--1838 (1993). · Zbl 0799.65060 · doi:10.1137/0730091
[16] Vainikko, G.: On the discretization and regularization of ill-posed problems with noncompact operators. Num. Funct. Anal. Opt.13, 381--396 (1992). · Zbl 0759.65030 · doi:10.1080/01630569208816485
[17] Vogel, C., Oman, M.: Iterative methods for total variation denoising. SIAM J. Sci. Comput.17, 227--238 (1996). · Zbl 0847.65083 · doi:10.1137/0917016
[18] Weidmann, J.: Linear operators in Hilbert spaces. Berlin, Heidelberg, New York, Tokyo: Springer 1990. · Zbl 1025.47001
[19] Ziemer, W. P.: Weakly Differentiable functions. Berlin, Heidelberg, New York, Tokyo: Springer 1980.