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Representation of High Resolution Images from Low Sampled Fourier Data: Applications to Dynamic MRI

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Abstract

In this work we propose the use of B-spline functions for the parametric representation of high resolution images from low sampled data in the Fourier domain. Traditionally, exponential basis functions are employed in this situation, but they produce artifacts and amplify the noise on the data. We present the method in an algorithmic form and carefully consider the problem of solving the ill-conditioned linear system arising from the method by an efficient regularization method.

Two applications of the proposed method to dynamic Magnetic Resonance images are considered. Dynamic Magnetic Resonance acquires a time series of images of the same slice of the body; in order to fasten the acquisition, the data are low sampled in the Fourier space. Numerical experiments have been performed both on simulated and real Magnetic Resonance data. They show that the B-splines reduce the artifacts and the noise in the representation of high resolution Magnetic Resonance images from low sampled data.

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Authors and Affiliations

  1. Department of Mathematics, University of Bologna, Bologna

    G. Landi & E. Loli Piccolomini

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  1. G. Landi
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  2. E. Loli Piccolomini
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Correspondence to G. Landi.

Additional information

This work was supported by the Italian MIUR project Inverse Problems in Medical Imaging 2004–2006 (grant no 2004015818).

Germana Landi received the BS degree in Mathematics from the University of Bologna in 1997 and the Ph.D. degree in Computational Mathematics from the University of Padova in 2000. She is currently a postdoctoral researcher in Numerical Analysis at the Department of Mathematics of the University of Bologna. Her research interests include medical imaging and inverse ill-posed problems.

Elena Loli Piccolomini received the BS degree in Mathematics from the University of Bologna in 1988. She is an associate professor in Numerical Analysis at the Department of Mathematics of the University of Bologna. Her research interests include numerical methods for the regularization of discrete ill-posed problems with application to medical imaging (MR, TAC, SPECT, PET).

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Landi, G., Piccolomini, E.L. Representation of High Resolution Images from Low Sampled Fourier Data: Applications to Dynamic MRI. J Math Imaging Vis 26, 27–40 (2006). https://doi.org/10.1007/s10851-006-7617-4

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