# AN EFFICIENT NUMERICAL RESOLUTION FOR MRI RICIAN DENOISING

### A. Martín, J. F. Garamendi, E. Schiavi

#### Abstract

We consider a variational Rician denoising model for Magnetic Resonance Images (MRI) that we solve by a semi-implicit numerical scheme, which leads to the resolution of a sequence of Rudin, Osher and Fatemi (ROF) models. This allows to implement efficient numerical gradient descent schemes based on the dual formulation of the ROF model which are compared with a direct semi-implicit approach for the primal problem recently proposed for model validation. In this new framework the total variation operator is exactly solved as opposed to the approximating problems which must be considered when the primal problem is dealt with. The comparison among the above methods is performed using synthetic and real MR brain images and the results show the effectiveness of the new method in both, the accuracy and the speeding up of the algorithm.

#### References

- Ambrosio, L., Fusco, N., and Pallara, D. (2000). Functions of Bounded Variation and free discontinity problems. The Clarendon Press, Oxford University.
- Aubert-Broche, B., Griffin, M., Pike, G., Evans, A., and Collins, D. (2006). Twenty new digital brain phantoms for creation of validation image data bases. In IEEE transactions on Medical Imaging, volume 25, pages 1410-1416.
- Basu, S., Fletcher, T., and Whitaker, R. (2006). Rician noise removal in diffusion tensor mri. Medical Image Computing and Computer-Assisted Intervention, 9(Pt 1):117-125.
- Casas, E., Kunisch, K., and Pola, C. (1998). Some applications of bv functions in optimal control and calculus of variations. In ESAIM: Proceedings. Control and partial differential equations, volume 4, pages 83-96.
- Chambolle, A. (2004). An algorithm for total variation minimization and applications. Journal Mathematical Imaging and Vision, 20:89-97.
- DK Jones, MA Horsfield, A. S. (1999). Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging. Journal of Magnetic Resonance in Medicine, 42 (3):515-525.
- Gudbjartsson, H. and Patz, S. (1995). The rician distribution of noisy mri data. Magnetic Resonance in Medicine, 34(6):910-914.
- Henkelman, R. M. (1985). Measurement of signal intensities in the presence of noise in mr images. Medical Physics, 12(2):232-233.
- Lassey, K. R. (1982). On the computation of certain integrals containing the modified bessel function i0(x). Mathematics of Computation, 39.
- Lysaker, M., Lundervold, A., and cheng Tai, X. (2003). Noise removal using fourth-order partial differential equations with applications to medical magnetic resonance images in space and time. IEEE Trans. Imag. Proc, 12:1579-1590.
- Martín, A., Garamendi, J., and Schiavi, E. (2011). Iterated rician denoising. In IPCV'11 Proceedings, Las Vegas, Nevada, USA. CSREA Press.
- Nikolova, M., Esedoglu, S., and Chan, T. F. (2006). Algorithms for finding global minimizers of image segmentation and denoising models. SIAM Journal of Applied Mathematics, 66(5):1632-1648.
- Perona, P. and Malik, J. (1990). Scale-space and edge detection using anisotropic diffusion. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 12(7):629-639.
- Rudin, L. I., Osher, S., and Fatemi, E. (1992). Nonlinear total variation based noise removal algorithms. Physica D Nonlinear Phenomena, 60:259-268.
- Sijbers, J., Dekker, A. D., Audekerke, J. V., Verhoye, M., and Dyck, D. V. (1998). Estimation of the noise in magnitude mr images. Magnetic Resonance Imaging, 1(16):87-90.

#### Paper Citation

#### in Harvard Style

Martín A., F. Garamendi J. and Schiavi E. (2012). **AN EFFICIENT NUMERICAL RESOLUTION FOR MRI RICIAN DENOISING** . In *Proceedings of the International Conference on Bio-inspired Systems and Signal Processing - Volume 1: BIOSIGNALS, (BIOSTEC 2012)* ISBN 978-989-8425-89-8, pages 15-24. DOI: 10.5220/0003734000150024

#### in Bibtex Style

@conference{biosignals12,

author={A. Martín and J. F. Garamendi and E. Schiavi},

title={AN EFFICIENT NUMERICAL RESOLUTION FOR MRI RICIAN DENOISING},

booktitle={Proceedings of the International Conference on Bio-inspired Systems and Signal Processing - Volume 1: BIOSIGNALS, (BIOSTEC 2012)},

year={2012},

pages={15-24},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0003734000150024},

isbn={978-989-8425-89-8},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Bio-inspired Systems and Signal Processing - Volume 1: BIOSIGNALS, (BIOSTEC 2012)

TI - AN EFFICIENT NUMERICAL RESOLUTION FOR MRI RICIAN DENOISING

SN - 978-989-8425-89-8

AU - Martín A.

AU - F. Garamendi J.

AU - Schiavi E.

PY - 2012

SP - 15

EP - 24

DO - 10.5220/0003734000150024