A Function Space Model for Digital Image Sampling and Its Application in Image Reconstruction

Abstract A general mathematical model for digital signal processing, which is based onLp function spaces, is introduced. This model is used to derive a new class of reconstruction filters for the reconstruction ofN-dimensional images which are not necessarily band-limited. A basic proposition in this work is that the best reconstruction algorithm will depend on the pre-sample filter (or point spread) function. The reconstruction filters described here are optimal in the sense that they result in a reconstructed image which is as close as possible, with respect to a given measure of fidelity, to the unsampled image. The filter is similar in form to the optimal filter derived by Peterson and Middleton (Inform. and Control5, 1962 , 279–323) in their comprehensive paper on multidimensional sampling. However the reconstruction filter of Peterson and Middleton is optimal for random fields, whereas the filter described here is optimal for individual images. The optimal reconstruction filter has certain practical advantages over empirically derived reconstruction methods such as cubic interpolation. One of these advantages is that the method produces positive-valued images without loss of image resolution. The performance of the optimal reconstruction filter can be understood without reference to aliasing and truncation errors and is quantified in terms of a simple error metric.

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