Images acquired using different imaging devices are often corrupted due to the physics of the image formation model. MRI imaging [1], [2], computed tomography [3], [4], confocal microscopy [5] and widefield microscopy [6] are examples of imaging modalities that are vulnerable to such image degradation. An estimation scheme [7] that employs knowledge of the image formation forward model to generate a better quality estimate is known as image restoration [8]. One of the classical approaches to image restoration is by regularized image restoration [9]. It models the restored image f^ˆ as the solution of an optimization problem :

where the image restoration cost J(f) is often modelled as sum of a data fitting functional G(f,g) and a regularization functional R(f). The data fitting term ensures that the estimate f^ˆ is not too far away from the measured image g. The structure of data fitting term is often dependent on the forward model of the imaging device. The regularization functional [10] captures the prior information we have about the class of images we are trying to restore. The constant term λ is an algorithm parameter that is used to tradeoff between the imposition of regularization term against the data fitting term.

The Hessian-Schatten variation functional helps regularize the energy in the singular values of the Hessian of pixels in the estimated image. The TGV in its standard form do not have this control over image derivatives. The second order Total Generalized variation, in its definition employs a mixed matrix vector norm .It may be noted that the matrix norm applied here is the Frobenius norm. It is a special case of the class of Schatten norms acting on the Hessian matrix at each pixel position which we saw applied successful in The HessianSchatten norm based regularization. In this work , we consider the extension of this mixed matrix vector norm to include the family of Schatten norms. We propose a generalized Hessian-Schatten regularization (TGSV) regularization which may be defined as follows :

Here D1 and D2 represents derivative filter operators and q represents the order of Schatten norm employed in the regularization scheme. Experiments performed on under sampled MRI data demonstrate the superior reconstruction quality of algorithm in comparison to existing regularization based methods for image restoration.

References

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[2] Sathish Ramani, Zhihao Liu, Jeffrey Rosen, JonFredrik Nielsen, and Jeffrey A Fessler, “Regularization parameter selection for nonlinear iterative image restoration and mri reconstruction using gcv and surebased methods,” IEEE Transactions on Image Processing, vol. 21, no. 8, pp. 3659–3672, 2012.

[3] SD Rathee, Zoly J Koles, and Thomas R Overton, “Image restoration in computed tomography: Estimation of the spatially variant point spread function,” IEEE transactions on medical imaging, vol. 11, no. 4, pp. 539–545, 1992.

[4] Jianhua Ma, Jing Huang, Qianjin Feng, Hua Zhang,

Hongbing Lu, Zhengrong Liang, and Wufan Chen, “Low-dose computed tomography image restoration using previous normal-dose scan,” Medical physics, vol. 38, no. 10, pp. 5713–5731, 2011.

[5] Nicolas Dey, Laure Blanc-Feraud, C Zimmer, Z Kam,´ J-C Olivo-Marin, and J Zerubia, “A deconvolution method for confocal microscopy with total variation regularization,” in 2004 2nd IEEE International Symposium on Biomedical Imaging: Nano to Macro (IEEE Cat No. 04EX821). IEEE, 2004, pp. 1223–1226.

[6] Muthuvel Arigovindan, Jennifer C Fung, Daniel Elnatan, Vito Mennella, Yee-Hung Mark Chan, Michael Pollard, Eric Branlund, John W Sedat, and David A Agard, “High-resolution restoration of 3d structures from widefield images with extreme low signal-tonoise-ratio,” Proceedings of the National Academy of Sciences, vol. 110, no. 43, pp. 17344–17349, 2013.

[7] Athanasios Papoulis and S Unnikrishna Pillai, Probability, random variables, and stochastic processes, Tata McGraw-Hill Education, 2002.

[8] Mark R Banham and Aggelos K Katsaggelos, “Digital image restoration,” IEEE signal processing magazine, vol. 14, no. 2, pp. 24–41, 1997.

[9] Per Christian Hansen, Discrete inverse problems: insight and algorithms, SIAM, 2010.