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Multi-Channel Deconvolution: The Generalized Sampling Approach

M.J. Vrhel, M. Unser

Proceedings of the SPIE Conference on Image Reconstruction and Restoration, San Diego CA, USA, July 24-29, 1994, vol. 2302, pp. 188-199.



We investigate the problem of signal restoration and reconstruction in a multi-channel system with the constraint that the entire system acts as a projection operator. This projection requirement is optimal in the noise free case since an input signal which is contained in the reconstruction space is recovered exactly. We find a general optimization problem which gives rise to a large class of projection operators. This formalization allows optimization of various criteria while enforcing the projection constraint. In this paper, we consider the projection operator which minimizes the noise power at the system output. The significance of this work is that it incorporates knowledge of the final reconstruction method which can include splines, wavelets, or display devices. In addition, unlike most classical formulations, the input signal is not required to be band-limited; it can be an arbitrary finite energy function. The approach requires no a priori information about the input signal, but does require knowledge of the impulse responses of the input channels. The projection method is compared to a generalized multi-channel Wiener filter which uses a priori signal information. At best the projection approach achieves the least squares solution which is the orthogonal projection of the input signal onto the space defined by the reconstruction method.


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