Stabilized Reconstruction in Signal and Image Processing

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Abstract
Part II is essentially devoted to the iterative reconstruction procedures that can easily be implemented in signal and image processing when the stability conditions are fulfilled. The application of the regularization principle introduced in part I for deconvolution is examined in this context. As the convergence of the method of conjugate gradients is then superlinear, this technique proves to be very well suited to solving the least-squares problem without constraint. If need be, the non-negativity constraint can be taken into account by slightly modifying the algorithm of steepest descent with fixed step, i.e. the Bialy-Jacobi iteration. Implemented in an appropriate manner, these methods may also provide the interesting part of the spectrum of the operator A∗ A to be inverted. This last point is particularly useful for conducting the error analysis. The exploration of the eigenspaces is also possible. As far as the implementation of the regularization principle is concerned, the theoretical resolution limit of the reconstruction process is selected through an interactive decision procedure based on a progressive estimation of the size of the object-reconstruction error. The numerical implementation, which is illustrated with the aid of simulated one-dimensional examples, reinforces and completes, in a concrete manner, the overall analysis presented in part I. The transposition to the more general situation in which the object function is defined on a low-resolution background is outlined in this context. It is also indicated how this approach should lead to a better understanding of the other deconvolution methods.