Ill-Posed Problems of Mathematical Physics
                    and Analysis

Лаврентьев, М. М.; Романов, В. Г.; Shishatskiĭ, S. P.

doi:10.1090/mmono/064

Cited by 580 publications

(554 citation statements)

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“…This problem has been discussed in the literature for S being a plane, a sphere, etc, [1][2][3][4], [6], [7]. Inversion formulas, which allow one to calculate f given m(y, r) for all y ∈ S and all r > 0, were obtained for the above cases.…”

Section: This Assumption Is Relaxed Below: It Is Sufficient To Assumementioning

confidence: 99%

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Injectivity of the spherical means operator

Ramm

2002

Comptes Rendus. Mathématique

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Abstract. Let S be a surface in R n which divides the space into two connected components D1 and D2. Let f ∈ C0(R n ) be some real-valued compactly supported function with supp f ⊂ D1. Consider M f := m(y, r) := R n f (z)δ(|y − z| − r)dz, where δ is the delta-function, y ∈ S and r > 0 are arbitrary. A general, local at infinity, condition on S is given, under which M is injective, that is, M f = 0 implies f = 0. The injectivity result is extended to the case when the Fourier transform of f is quasianalytic, so that compactness of support of f is not assumed. A sufficient condition on S is given, under which M −1 can be analytically constructed. Two examples of inversion formulas are given: when S is a plane, and when S is a sphere. These formulas can be used in applications. c 2002 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS Injectivité de l'operateur de moyen spheriqueRésumé. Soit S une surface de R n qui divise l'espace en deux composantes connectées D1 and D2.

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Section: This Assumption Is Relaxed Below: It Is Sufficient To Assumementioning

confidence: 99%

“…These formulas are obtained by a method different from the one used in [1][2][3][4]. The injectivity result we obtain, is based on the results in [6], see also [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

Injectivity of the spherical means operator

Ramm

2002

Comptes Rendus. Mathématique

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“…In such contexts algorithms developed to deal explicitly with such issues, such as those in [15,18], may be more appropriate. On the other hand, for the not insignificant class of problems for which the Horn and Schunck formulation is well-suited, such as [32] and the many ill-posed and variational problems arising in fields ranging from image processing and tomography to meteorology, seismology and oceanography [5,31,47,22,46,26], our method will also work well and also provides the advantages described previously: computational efficiency, multiresolution estimates and multiscale error covariance information. Moreover, even in cases in which Horn and Schunck-type global smoothness constraints are inappropriate, there are reasons to believe that algorithms based on our formulation may provide the basis for promising new solutions.…”

Section: Introductionmentioning

confidence: 88%

Efficient Multiscale Regularization with Applications to the Computation of Optical Flow

Luettgen¹,

Karl²,

Willsky³

1993

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A new aplproach to regularization methods for image processing is iltro(duced and developed using as a. vehicle the problem of coml)uting dleilse optical flow fields in an image sequence. Standard formulations of this problemi re(luire the computationally intensive solution of an ellil)tic pa.rtial differential equation which arises froin the often used "smootlhness constraint' type regularization. We utilize the illterl)reta.tion of' .he smoot.hness constraint as a. "fractal prior" to motivate regularization based on a. recently introduced class of multiscale stochastic models. The solution of the new l)robleln formulat.ion is computed with an efficient multiscale algorithm. Experiiments on several ima.ge sequences (demonstrate the sutl)sta.nftial compult.a.tional savings tlia.t. canll I)e achieved due to the fact tha.t the algorit.hm is non-iterative and in fact has a. per pixel computational complexity which is independent. of imiage size. Tlie new a.pp)l)roach also has a. numbler of other important. advantages. Specifically, nitltiresolution flow field estimates are availabl)le, allowing grea.t. flexibility in dealing wvithl the t.radeoff b.)et.ween resolution a.nd(l accuracy. NIultiscale error covariance information is also availabLle. which is of considerable use inll assessing the accuracy of the estimates. Ini particular, these error statistics calln be used as the b)asis for a rational procedure for dletermlining the spatially-varying optimal reconstruction resolution. Furthermore, if there are conipelling reasons to insist upon a standard smoothness constraint, our algorithm provides an excellent initialization for the iterative algoritlhms associated with the smoot. ness constraint l)roblemi formulation. Finally, the usefulness of otir appI)roach should extend t-o a. wide variety of ill-posed inverse problems inll which variational techniques seeking a. "smooth" solution are generally usedl. EDICS category 1.11. Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

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“…The problem considered in this article belongs to the class of inverse problems of control system dynamics and in a more general context it fits into the framework of ill-posed problems [1,2]. The theory of such problems has been developed with sufficient detail, but only in a posterior setting -without requiring a dynamic reconstruction algorithm [3].…”

Section: Introductionmentioning

confidence: 99%

“…The latter in turn combines the methods of the theory of ill-posed problems [1,2] and positional control theory [8]. The essence of this theory is that the reconstruction algorithm is represented as a control algorithm for some auxiliary dynamical system -a model; the output of this algorithm in particular is a realization of the model control and it is by definition a dynamic algorithm.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction of unknown characteristics in a third-order system

Keesman

Максимов

2013

Comput Math Model

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The article considers dynamic identification of unknown characteristics in a third-order system. A realtime algorithm is proposed for the solution of the problem. The algorithm relies on constructs from stable dynamical inversion theory, which combines methods of the theory of ill-posed problems and positional control theory. In the proposed procedure, the reconstruction algorithm is represented as a control algorithm for some artificial dynamical system -a model. The model control is adapted to current observations so that its realization eventually "approximates" the unknown input.

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Ill-Posed Problems of Mathematical Physics and Analysis

Cited by 580 publications

References 0 publications

Injectivity of the spherical means operator

Injectivity of the spherical means operator

Efficient Multiscale Regularization with Applications to the Computation of Optical Flow

Reconstruction of unknown characteristics in a third-order system

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