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The design and performance of spatial localization filters in an MR imaging system with time-varying gradients is presented. The system is based on recording measurements of the imaged volume-done in the presence of cosinusoidal gradients. A postprocessing scheme is shown which enables every point to be imaged from a single FID signal. Methods are shown which achieve a three-demensional localization with arbitrarily small sidelobes. Those localization filters are implemented without significant computational penalty. The signal-to-noise ratio (SNR) in the suggested imaging system is shown to be space-invariant (same SNR for each imaged point). We find a closed-form relation between the localization filter and the resulting noise. It is shown that the gain achieved by the localization filters highly exceeds the relatively small loss in SNR.