Variational approach to tomographic reconstruction

Jan Kybic; Thierry Blu; Michael A. Unser

doi:10.1117/12.431108

3 July 2001 Variational approach to tomographic reconstruction

Jan Kybic, Thierry Blu, Michael A. Unser

Proceedings Volume 4322, Medical Imaging 2001: Image Processing; (2001) https://doi.org/10.1117/12.431108
Event: Medical Imaging 2001, 2001, San Diego, CA, United States

Abstract

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.

Citation Download Citation

Jan Kybic, Thierry Blu, and Michael A. Unser "Variational approach to tomographic reconstruction", Proc. SPIE 4322, Medical Imaging 2001: Image Processing, (3 July 2001); https://doi.org/10.1117/12.431108

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