Biomedical Imaging GroupSTI
English only   BIG > Publications > Consistent Sampling

 Home Page
 News & Events
 Tutorials and Reviews
 Download Algorithms

 All BibTeX References

Consistent Sampling and Signal Recovery

平林 晃 (A. Hirabayashi), M. Unser

IEEE Transactions on Signal Processing, vol. 55, no. 8, pp. 4104-4115, August 2007.

An attractive formulation of the sampling problem is based on the principle of a consistent signal reconstruction. The requirement is that the reconstructed signal is indistinguishable from the input in the sense that it yields the exact same measurements. Such a system can be interpreted as an oblique projection onto a given reconstruction space. The standard formulation requires a one-to-one relationship between the input measurements and the reconstructed model. Unfortunately, this condition fails when the cross-correlation matrix between the analysis and reconstruction basis functions is not invertible; in particular, when there are less measurements than the number of reconstruction functions. In this paper, we propose an extension of consistent sampling that is applicable to those singular cases as well, and that yields a unique and well-defined solution. This solution also makes use of projection operators and has a geometric interpretation. The key idea is to exclude the null space of the sampling operator from the reconstruction space and to enforce consistency on its complement. We specify a class of consistent reconstruction algorithms corresponding to different choices of complementary reconstruction spaces. The formulation includes the Moore-Penrose generalized inverse, as well as other potentially more interesting reconstructions that preserve certain preferential signals. In particular, we display solutions that preserve polynomials or sinusoids, and therefore perform well in practical applications.

AUTHOR="Hirabayashi, A. and Unser, M.",
TITLE="Consistent Sampling and Signal Recovery",
JOURNAL="{IEEE} Transactions on Signal Processing",

© 2007 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IEEE.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.