Non-uniform sampling and spiral MRI reconstruction - Math ...

and consequently the ordinary distance, from any point on the spiral A k to Λ k is less than δ. Finally, we setΛ R = ∪ M−1k=0 Λ k. Thus, by the triangle inequality,∀ξ ∈ ̂R 2 , dist(ξ,Λ R ) ≤ dist(ξ,B) + dist(B,Λ R ) ≤ c2M + δ = ρ.Hence, Rρ < 1/4 by our choice of M and δ; and so we can invoke Beurling’s Covering Theorem, Theorem 7.2, or theoriginal formulation state in the Introduction, to conclude that Λ R is a Fourier frame for PW B(0,R).Note that since we are reconstructing signals on a space domain having area about R 2 , we require essentially Rinterleaving spirals. On the other hand, if we are allowed to choose the spiral(s) after we are given PW B(0,R), thenwe can choose Λ R contained in a single spiral A c for c>0 small enough.REFERENCES1. A. Beurling, “Local harmonic analysis with some applications to differential operators,” in Proc. Annual ScienceConference, pp. 109–125, Belfer Graduate School of Science, 1966.2. J. J. Benedetto, “Irregular sampling and frames,” in Wavelets–A Tutorial in Theory and Applications, C.K.Chui, ed., pp. 445–507, Academic Press, Boston, 1992.3. K. Grochenig, “Non-uniform sampling in higher dimensions: from trigonometric polynomials to bandlimitedfunctions,” in Modern Sampling Theory: Mathematics and Applications, J. Benedetto and P. Ferreira, eds.,p. Chapter 7, Birkhauser, Boston, 2000.4. A. Beurling, “On balayage of measures in Fourier transforms (Seminar, Inst. for Advanced Studies, 1959–60,unpublished),” in Collected Works of Arne Beurling, L. Carleson, P. Malliavin, J. Neuberger, and J. Wermer,eds., Birkhauser, Boston, 1989.5. M. Bourgeois, F. T. A. W. Wajer, D. van Ormondt, and D. Graveron-Demilly, “Reconstruction of MRI imagesfrom non-uniform sampling, application to intrascan motion correction in functional MRI,” in Modern SamplingTheory: Mathematics and Applications, J. Benedetto and P. Ferreira, eds., p. Chapter 16, Birkhauser, Boston,2000.6. J. Benedetto and H.-C. Wu, “The Beurling-Landau theory of Fourier frames and applications,” J. FourierAnalysis and Applications 6, 2000.7. J. J. Benedetto, Harmonic Analysis and Applications, CRC Press, Inc., Boca Raton, FL, 1997.8. E. Stein and G. Weiss, An Introduction to Fourier Analysis on Euclidean Space, Princeton University Press,Princeton, NJ, 1971.9. R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier series,” Trans. Amer. Math. Soc. 72, pp. 341–366, 1952.10. R. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloquium Publ. 19,New York, 1934.11. H. J. Landau, “Necessary density conditions for sampling and interpolation of certain entire function,” ActaMath. 117, pp. 37–52, 1967.12. S. Jaffard, “A density criterion for frames of complex exponentials,” Michigan Math. J. 38, pp. 339–348, 1991.