EPFL
 Biomedical Imaging GroupSTI
EPFL
  Publications
English only   BIG > Publications > B-Spline Transforms


 CONTENTS
 Home Page
 News & Events
 People
 Publications
 Tutorials and Reviews
 Research
 Demos
 Download Algorithms

 DOWNLOAD
 PDF
 Postscript
 All BibTeX References

Fast B-Spline Transforms for Continuous Image Representation and Interpolation

M. Unser, A. Aldroubi, M. Eden

IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 3, pp. 277-285, March 1991.



Efficient algorithms for the continuous representation of a discrete signal in terms of B-splines (direct B-spline transform) and for interpolative signal reconstruction (indirect B-spline transform) with an expansion factor m are described. Expressions for the z-transforms of the sampled B-spline functions are determined and a convolution property of these kernels is established. It is shown that both the direct and indirect spline transforms involve linear operators that are space invariant and are implemented efficiently by linear filtering. Fast computational algorithms based on the recursive implementations of these filters are proposed. A B-spline interpolator can also be characterized in terms of its transfer function and its global impulse response (cardinal spline of order n). The case of the cubic spline is treated in greater detail. The present approach is compared with previous methods that are reexamined from a critical point of view. It is concluded that B-spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner.

Errata

  • p. 282, Equation (3.18) has two errors. One sign is wrong, and an index is off by one. The corrected system of equations should read {d+(k) = ƒ'(k) + α d+(k - 1), (k = 2, …, K); c(K) = (-α ⁄ (1 - α2)) (2 d+(K) - ƒ'(K)); c(k) = α (c(k + 1) - d+(k)), (k = K - 1, …, 1)} instead of {d+(k) = ƒ'(k) - α d+(k - 1), (k = 2, …, K); c(K) = (-α ⁄ (1 - α2)) (2 d+(K) - ƒ'(K)); c(k) = α (c(k + 1) - d+(k + 1)), (k = K - 1, …, 1)}.

  • p. 282, the second definition on the line just below Equation (3.18) is not general enough. The corrected definition should read d+(k) = 6 c+(k) instead of d+(1) = 6 c+(1).


@ARTICLE(http://bigwww.epfl.ch/publications/unser9102.html,
AUTHOR="Unser, M. and Aldroubi, A. and Eden, M.",
TITLE="Fast \mbox{{B}-Spline} Transforms for Continuous Image
        Representation and Interpolation",
JOURNAL="{IEEE} Transactions on Pattern Analysis and Machine
        Intelligence",
YEAR="1991",
volume="13",
number="3",
pages="277--285",
month="March",
note="")

© 1991 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.