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Optimal Sampling of Non-Bandlimited Signals with Applications to Image Processing

M. Unser, A. Aldroubi

Final Program and Advance Printing of Papers, Fourty-Eighth Annual Conference of the Society for Imaging Science and Technology: Imaging on the Information Superhighway (IS&T'95), Washington DC, USA, May 7-11, 1995, pp. 370-375.



Sampling is the process of representing continuous-time (or space) functions by sequences of numbers (discrete signal representation). Traditionally, both the signal and its representation are assumed to be bandlimited. Here, we abandon this hypothesis and present a general procedure for the approximation of arbitrary finite energy signals from their measurements sampled at the output of a given analog prefilter (e.g., non-ideal acquisition device). The approximation spaces that we consider are generated by translation of a generating kernel phi , a special case being the conventional sinc interpolator. This function may correspond to the impulse response of a display device, or may be selected to specify a certain spline or wavelet representation space. We show that a consistent signal approximation can be obtained by appropriate digital filtering of the discrete measurements. This approximation is essentially equivalent to the initial signal in the sense that it would result in exactly the same measurements if it was re-injected into the system. We present the conditions under which this scheme yields the optimal least squares solution and provide general error bounds. The theory is illustrated with the design of a digital filtering algorithm for the improvement of image display, and a spline-based procedure for the minimum error scale-conversion of images (with an arbitrary scaling factor).


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