By Peter W. Hawkes (Ed.)
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Extra resources for Advances in Electronics and Electron Physics, Vol. 87
It is important to note that the Moore-Penrose inverse itself would not be useful for image restoration because the condition number of the matrix may IMAGE RESTORATION ON THE HOPFIELD NEURAL NETWORK 23 still be very large and small fluctuations in g would be amplified. It is useful in the context of the pseudo-inverse content addressable memory, however, since it can more precisely discriminate between similar images in recall. An approach of greater generality to the image restoration problem allows one to incorporate more detailed information on the known or desired characteristics of the object o r image; for example, to predispose the reconstruction to a particular shape, or to control its behavior in the neighborhood of the prescribed boundaries.
It is because of this apodization of the singular values that the neural matrix inverse is both useful and robust. 35 IMAGE RESTORATION ON THE HOPFIELD NEURAL NETWORK . 0001 0-0 D FIGURE 13. Singular value spectra of the SVD inverses The neural network inverses do not rely heavily upon a specific value of a regularization parameter or on the choice of settling accuracy used. Clearly, even a large settling accuracy using the neural inversion method is providing a matrix inverse with suitable properties for this restoration problem.
26) The grey-scale mapping we are considering associates a specific neuron with one and only one image pixel. Hence the columns of S each contain only one element, and it is found that the diagonal elements of T take the form Tkk = - 2 s $ ( A T A f (27) where sjk is the nonzero element of the kth column of S. negative, since the diagonal elements of A T A are positive and quantity. Hence we can rewrite Eq. (26) in the form Tkk is always 0 is a positive AEk = -Ak[(Tv+b)k-iITkklAklr (28) and thus A E k will be negative provided where This condition will be met if lAvkI < (Akl and sgn(Avk) = sgn(Ak).
Advances in Electronics and Electron Physics, Vol. 87 by Peter W. Hawkes (Ed.)