Numerical stability and near-field reconstruction

Cabayan, H.S.; Murphy, R. Kim; Pavlasek, T.

doi:10.1109/tap.1973.1140501

Cited by 16 publications

(9 citation statements)

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“…In particular, whenever logarithmic continuity holds true, it should be possible to estimate a resolution, attainable in the inversion procedure, which is rather weakly dependent on the data accuracy . This is the case for restoration of coherent and incoherent objects as well as, for instance, for the problem of near-field reconstruction [8] .…”

Section: Discussionmentioning

confidence: 99%

“…We apply here the analysis made in § § 3 .2 and 3 .3 . We focus on the estimate of relative errors as defined in equation (3 .11) and we consider only the case B = 1, which is the most popular in many analysis of regularization methods [8,11,13,17] . For our problem, condition (2.2) with B=1 has the following physical meaning : we shall restore objects whose energy does not exceed one .…”

Section: 2 Errors On Smeared Objectsmentioning

confidence: 99%

“…Let us mention a few examples : object reconstruction from radiographs (transaxial tomography) [1,2] ; epicardial potential calculation from body surface maps [3] ; radar target shape estimation [4][5][6] ; near-field reconstruction from the scattered far-field [7,8] and stepwise analytic continuation in order to identify an unknown scatterer [7,9]. Further examples can be found in [10] .…”

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confidence: 99%

“…The remedy is to make a guess on some of the properties of the function f which has to be identified . This is the basis of regularization methods whose relevance for linear inverse problems has been already emphasized by many authors [11][12][13][14]8] . In particular [11] is also a good tutorial paper, and we will try to derive our results without going beyond the mathematical tools used in that paper.…”

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confidence: 99%

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Resolution Beyond the Diffraction Limit for Regularized Object Restoration

Bertero

Viano

Mol

1980

Optica Acta: International Journal of Optics

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Section: Discussionmentioning

confidence: 99%

Section: 2 Errors On Smeared Objectsmentioning

confidence: 99%

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Resolution Beyond the Diffraction Limit for Regularized Object Restoration

Bertero

Viano

Mol

1980

Optica Acta: International Journal of Optics

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“…2) the well-known truncation method for eliminating the noise amplification due to eigenvalues Xk very close to zero. 1 ",1 5 Both estimates fi and f2 can be shown to satisfy the inequality 9 11fi -f || • N M(e,E), (i = 1,2), and therefore they converge to the true object f when M(, ,E) tends to zero.…”

Section: F2(x) = E -Uk(x)mentioning

confidence: 99%

Restoration of optical objects using regularization

1978

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Using the regularization theory for improperly posed problems, we discuss object restoration beyond the diffraction limit in the presence of noise. Only the case of one-dimensional coherent objects is considered. We focus attention on the estimation of the error on the restored objects, and we show that, in most realistic cases, it is at best proportional to an inverse power of I In t l, where E is the error on the data (logarithmic continuity). Finally we suggest the extension of this result to other inverse problems.Let us consider an ideal, diffraction-limited, spaceinvariant imaging system and a one-dimensional, coherent object f, identically zero outside the interval (-1,1). Its noiseless image is given by Af, where A is the linear integral operator,The quantity d = 7r/c is the Rayleigh-resolution distance. To restore the object, one has to invert the operator A. As is known, this problem is improperly posed. When the image is known only approximately, we have numerical instability. Hence, in order to control error propagation, one has to introduce a priori constraints (as far as possible of physical origin) that restrict the class of admitted solutions. This is the concept of regularization of Tikhonov. Its importance for the practical solution of linear inverse problems has been emphasized by many authors.'-5 Various regularization methods have been proposed.6-1 0 They are essentially equivalent, but most of them overlook the problem of obtaining precise estimates on error propagation: usually the stability of the regularized solution is tested only numerically.'-3 ' 7 Our aim is to focus on error valuation, and therefore we shall use regularization in a formulation of Miller, 9 which is particularly suited for deriving stability estimates. In order to formulate our problem more precisely, we have to introduce suitable functional spaces for the solutions and for the data. Now, if the object fis taken to have finite energy, then it belongs to L 2 (-1,1) (hereafter abbreviated as L 2 ). We denote by (fig) the scalar product of two functions of L 2 , i.e.,where g* is the complex conjugate of g, and by lIf 0 = (ff)112 the L 2 norm of f. Moreover, we suppose that the optical image is observed in the interval (-1,1) so that we can again take L 2 as data space. If g denotes a noisy image corresponding to the object f, then we assume that g = Af + n, where n is an L2 function describing any kind of additive noise and errors. This particular model is unrealistic; nonetheless it is the only one found usable to date. 1 ' Obviously n is not known. However, we assume an upper bound for its L 2 norm, i.e., 11 n 11 < e. Besides, according to Miller,9 we assume that the additional information on the object f (which is required in order to get numerical stability) may be expressed by means of a constraint operator B as follows: IIBf 11 < E, where E is a known constant.Thus we have:The simplest choice for B is B = 1 (the identity operator).t 2 A4$8 Then, in object restoration, condition (3b) means that we assume an upper b...

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