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Variational Approach to Tomographic Reconstruction

J. Kybic, T. Blu, M. Unser

Proceedings of the SPIE International Symposium on Medical Imaging: Image Processing (MI'01), San Diego CA, USA, February 17-22, 2001, vol. 4322, part I, pp. 30-39.


We formulate the tomographic reconstruction problem in a variational setting. The object to be reconstructed is considered as a continuous density function, unlike in the pixel-based approaches. The measurements are modeled as linear operators (Radon transform), integrating the density function along the ray path. The criterion that we minimize consists of a data term and a regularization term. The data term represents the inconsistency between applying the measurement model to the density function and the real measurements. The regularization term corresponds to the smoothness of the density function.

We show that this leads to a solution lying in a finite dimensional vector space which can be expressed as a linear combination of generating functions. The coefficients of this linear combination are determined from a linear equation set, solvable either directly, or by using an iterative approach.

Our experiments show that our new variational method gives results comparable to the classical filtered back-projection for high number of measurements (projection angles and sensor resolution). The new method performs better for medium number of measurements. Furthermore, the variational approach gives usable results even with very few measurements when the filtered back-projection fails. Our method reproduces amplitudes more faithfully and can cope with high noise levels; it can be adapted to various characteristics of the acquisition device.

@INPROCEEDINGS(http://bigwww.epfl.ch/publications/kybic0102.html,
AUTHOR="Kybic, J. and Blu, T. and Unser, M.",
TITLE="Variational Approach to Tomographic Reconstruction",
BOOKTITLE="Progress in Biomedical Optics and Imaging, vol. 2, no.
	27",
YEAR="2001",
editor="Sonka, M. and Hanson, K.M.",
volume="4322",
series="Proceedings of the {SPIE} International Symposium on Medical
	Imaging: {I}mage Processing ({MI'01})",
pages="30--39",
address="San Diego CA, USA",
month="February 19-22,",
organization="",
publisher="",
note="Part {I}")
© 2001 SPIE. 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 SPIE. 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.
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