1990
DOI: 10.1109/29.57544
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Regularization theory in image restoration-the stabilizing functional approach

Abstract: This paper presents several aspects of the application of regularization theory in image restoration. This is accomplished by extending the applicability of the stabilizing functional approach to 2-D ill-posed inverse problems. Image restoration is formulated as the constrained minimization of a stabilizing functional. The choice of a particular quadratic functional to be minimized is related to the a priori knowledge regarding the original object through a formulation of image restoration as a maximum a poste… Show more

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Cited by 87 publications
(46 citation statements)
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“…Thus, a necessary and sufficient condition for the Wiener filter to be well defined (i.e., bounded) is that this quantity be nonvanishing (a sufficient condition is ). Also, if we take the Fourier transform of (49), we see that the solution is of the same form as (30) where the Fourier transform of the equivalent basis function is given by (51) Once again, the optimal reconstruction space is generally not bandlimited, unless either or are bandlimited to start with. Here too, the reconstruction filters for the deterministic and stochastic cases can be made equivalent.…”
Section: B Unification With the Wiener Formulationmentioning
confidence: 99%
See 1 more Smart Citation
“…Thus, a necessary and sufficient condition for the Wiener filter to be well defined (i.e., bounded) is that this quantity be nonvanishing (a sufficient condition is ). Also, if we take the Fourier transform of (49), we see that the solution is of the same form as (30) where the Fourier transform of the equivalent basis function is given by (51) Once again, the optimal reconstruction space is generally not bandlimited, unless either or are bandlimited to start with. Here too, the reconstruction filters for the deterministic and stochastic cases can be made equivalent.…”
Section: B Unification With the Wiener Formulationmentioning
confidence: 99%
“…Since is strictly positive and because is bounded, the filter is strictly positive and bounded as well. Therefore, by writing the solution (28) in the Fourier domain, we obtain (30) where is the equivalent basis function that needs to be applied to the measurements to produce the continuous-space signal reconstruction, as illustrated in Fig. 2.…”
Section: ) Solution Of the Quadratic/tikhonov Problemmentioning
confidence: 99%
“…Restoration methods based on regularization theory [3] are widely used instead as they can successfully replace the ill-posed problem by a well-posed problem.…”
Section: Solving This Equation Directly To Get the Solutionmentioning
confidence: 99%
“…These splines can be represented as linear combinations of shifted radial basis functions (RBFs) called thin plate splines (for integer orders) (10) where these RBFs are, in fact, the Green functions (inverse filter) of the th-order symmetric differentiation operator ; they are given by (11) where and are appropriate constants [37] even otherwise.…”
Section: Polyharmonic B-splinesmentioning
confidence: 99%
“…Digital Object Identifier 10.1109/TIP.2006.877390 1) the Wiener formulation, where one minimizes the mean square estimation error (MMSE) for a given class of stochastic processes [5], [6]; 2) the Bayesian framework, where one searches for the maximum a posteriori solution given some prior knowledge of the statistical distribution of its parameters [7], [8]; 3) the variational approach, where one minimizes an energy functional that favors solutions with some desirable features (eg., nonoscillating, and/or with sharp edges) [9], [10]; formulations have been proposed for both discrete [11] and continuously defined signals [12], [13]. The Wiener formulation typically produces a linear solution (Wiener filter) which is simple to implement; however, it requires complete knowledge of the second-order statistics of the signal which is often not available.…”
mentioning
confidence: 99%