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A method is presented for estimating unknown Fourier domain (k-space) datausing a small number of samples in that space. The method is derived fromBochners Theorem, and is termed: Bochner Inequality Completion of K-Space(BICKS). It is suitable for filling the k-space of a real and nonnegativeunknown quantity, and applicable even when the sampling rate is substantiallylower than the Nyquist sampling rate. The BICKS method is demonstrated in thecontext of medical imaging, but it is also applicable to many other scientificareas that utilize signal processing in Fourier domain. The results indicatethat filling a highly undersampled k-space using BICKS enables high qualityimage reconstruction.

Utilizing Bochners Theorem for Constrained Evaluation of Missing Fourier Data