# JOINT PRIOR MODELS OF MUMFORD-SHAH REGULARIZATION FOR BLUR IDENTIFICATION AND SEGMENTATION IN VIDEO SEQUENCES

### Hongwei Zheng, Olaf Hellwich

#### Abstract

We study a regularized Mumford-Shah functional in the context of joint prior models for blur identification, blind image deconvolution and segmentation. For the ill-posed regularization problem, it is hard to find a good initial value for ensuring the soundness of the convergent value. A newly introduced prior solution space of point spread functions in a double regularized Bayesian estimation can satisfy such demands. The Mumford-Shah functional is formulated using Γ-convergence approximation and is minimized by projecting iterations onto an alternating minimization within Neumann conditions. The pre-estimated priors support the Mumford-Shah functional to decrease of the complexity of computation and improve the restoration results simultaneously. Moreover, segmentation of blurred objects is more difficult. A graph-theoretic approach is used to group edges which driven from the Mumford-Shah functional. Blurred objects with lower gradients and objects with stronger gradients are grouped separately. Numerical experiments show that the proposed algorithm is robust and efficiency in that it can handle images that are formed in different environments with different types and amounts of blur and noise.

#### References

- Ambrosio, L. and Tortorelli, V. M. (1990). Approximation of functionals depending on jumps by elliptic functionals via ?-convergence. Communications on Pure and Appled Mathematics, 43:999-1036.
- Aubert, l. and Kornprobst, P. (2002). Mathematical problems in image processing: partical differential equations and the Calculus of Variations. Springer.
- Bertero, M., Poggio, T. A., and Torre, V. (1988). Ill-posed problems in early vision. Proceedings of the IEEE, 76(8):869-889.
- Blake, A. and Zisserman, A. (1987). Visual Reconstruction. MIT Press, Cambridge.
- Bouman, C. and Sauer, K. (1993). A generalized gaussian image model for edge-preserving map estimation. IEEE Transactions of Image Processing, 2:296- 310.
- Chan, T. F. and Vese, L. A. (2001). Active contours without edges. IEEE Trans. on Image Processing,, 10(2):266- 277.
- Chan, T. F. and Wong, C. K. (2000). Convergence of the alternating minimization algorithm for blind deconvolution. Linear Algebra Appl., 316:259-285.
- Green, P. (1990). Bayesian reconstruction from emission tomography data using a modified em algorithm. IEEE Tr. Med. Imaging, 9:84-92.
- Katsaggelos, A., Biemond, J., Schafer, R., and Mersereau, R. (1991). A regularized iterative image restoration algorithm. IEEE Tr. on Signal Processing, 39:914- 929.
- Kundur, D. and Hatzinakos, D. (1996). Blind image deconvolution. IEEE Signal Process. Mag., May(5):43-64.
- Lagendijk, R., Biemond, J., and Boekee, D. (1988). Regularized iterative image restoration with ringing reduction. IEEE Tr. on Ac., Sp., and Sig. Proc., 36(12):1874-1888.
- Molina, R., Katsaggelos, A., and Mateos, J. (1999). Bayesian and regularization methods for hyperparameters estimate in image restoration. IEEE Tr. on S.P., 8.
- Molina, R. and Ripley, B. (1989). Using spatial models as priors in astronomical image analysis. J. App. Stat., 16:193-206.
- Mumford, D. and Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42:577-684.
- Poggio, T., Torre, V., and Koch, C. (1985). Computational vision and regularization theory. Nature, 317:314- 319.
- Rudin, L., Osher, S., and Fatemi, E. (1992). Nonlinear total varition based noise removal algorithm. Physica D, 60:259-268.
- Shi, J. and Malik, J. (2000). Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, Aug.:888-905.
- Tikhonov, A. and Arsenin, V. (1977). Solution of Ill-Posed Problems. Wiley, Winston.
- Vogel, R. V. and Oman, M. E. (1998). Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Trans. on Image Processing, 7:813-824.
- Weiser, A. and Wheeler, M. F. (1988). On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal., 25:351-275.
- Zheng, H. and Hellwich, O. (2006). Double regularized bayesian estimation for blur identification in video sequences. In LNCS of Asian Conference on Computer Vision, ACCV06, pages 943-952.

#### Paper Citation

#### in Harvard Style

Zheng H. and Hellwich O. (2006). **JOINT PRIOR MODELS OF MUMFORD-SHAH REGULARIZATION FOR BLUR IDENTIFICATION AND SEGMENTATION IN VIDEO SEQUENCES** . In *Proceedings of the First International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP,* ISBN 972-8865-40-6, pages 56-63. DOI: 10.5220/0001371400560063

#### in Bibtex Style

@conference{visapp06,

author={Hongwei Zheng and Olaf Hellwich},

title={JOINT PRIOR MODELS OF MUMFORD-SHAH REGULARIZATION FOR BLUR IDENTIFICATION AND SEGMENTATION IN VIDEO SEQUENCES},

booktitle={Proceedings of the First International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP,},

year={2006},

pages={56-63},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001371400560063},

isbn={972-8865-40-6},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the First International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP,

TI - JOINT PRIOR MODELS OF MUMFORD-SHAH REGULARIZATION FOR BLUR IDENTIFICATION AND SEGMENTATION IN VIDEO SEQUENCES

SN - 972-8865-40-6

AU - Zheng H.

AU - Hellwich O.

PY - 2006

SP - 56

EP - 63

DO - 10.5220/0001371400560063