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Non-local total bounded variation scheme for multiple-coil magnetic resonance image restoration

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Abstract

In this paper, we design a variational model for restoring multiple-coil magnetic resonance images (MRI) corrupted by non-central Chi distributed noise. The energy functional corresponding to the restoration problem is derived using the maximum a posteriori (MAP) estimator. Optimizing this functional yields the solution, which corresponds to the restored version of the image. The non-local total bounded variation prior is being used as the regularization term in the functional derived using the MAP estimation process. Further, the split-Bregman iteration scheme is being followed for fast numerical computation of the model. The results are compared with the state of the art MRI restoration models using visual representations and statistical measures.

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Notes

  1. Hereafter we use the term Chi distribution to denote non-central Chi distribution, if not mentioned otherwise.

  2. Here we note that, the samples of noise on each pixel are assumed to be mutually independent and identically distributed, even though the noise is correlated with the pixel.

  3. http://brainweb.bic.mni.mcgill.ca/brainweb/selection_normal.html.

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Acknowledgements

Dr. Jidesh would like to thank the Department of Science and Technology (Science and Engineering Research Board), Government of India for providing the financial support under the Project Grant No. ECR/2017/000230. Mr. Shivaram Holla would like to thank the Ministry of Human Resource Development, Government of India, for providing the financial assistance to pursue Ph.D. research work at National Institute of Technology, Karnataka, India.

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Authors and Affiliations

  1. Department of Mathematical and Computational Sciences, National Institute of Technology, Mangalore, Karnataka, 575025, India

    P. Jidesh & Shivaram Holla

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  1. P. Jidesh
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Correspondence to P. Jidesh.

Appendix I

Appendix I

Here we analyze the uniqueness property of the solution of (19). Since the convexity of the TV norm is known very well from the literature, the regularization term need not require any further explanation. Consider the reactive or fidelity term in (19) (first term) and define it as \(\phi (u)\) i.e.

$$\begin{aligned} \phi (u)=(L-1) \log (u)+ (u^2)/(2\sigma ^2)-\log I_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) , \end{aligned}$$
(33)

let us take the first variation of \(\phi (u)\) with respect to u, we get

$$\begin{aligned} \phi ^{\prime }(u)=(L-1)/u+\frac{1}{\sigma ^2}\left[ u-\frac{I^{\prime }_{l-1}\left( \frac{u_0 u}{\sigma ^2}\right) }{I_{l-1}\left( \frac{u_0 u}{\sigma ^2}\right) } u_0\right] , \end{aligned}$$
(34)

where \(I^{\prime }(\cdot )\) is the first derivative of the Bessel function \(I(\cdot )\). Now let us take the second derivative of the function \(\phi (u)\)

$$\begin{aligned} \phi ^{\prime \prime }(u)=\frac{1-L}{u^2}+\frac{1}{\sigma ^2}\left[ 1-\frac{u_0^2}{\sigma ^2}\left[ \frac{I^{\prime \prime }_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }{I_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }-\left[ \frac{I^{\prime }_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }{I_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }\right] ^2\right] \right] , \end{aligned}$$
(35)

where

$$\begin{aligned} I_L(u)= & {} \sum _{k=0}^{\infty } \frac{(-1)^k \left( \frac{u}{2}\right) ^{2L+1}}{(k+L)!k!},\\ I^{\prime }_L(u)= & {} \frac{1}{2}\left[ I_{L-1}(u)+I_{L+1}(u)\right] , \end{aligned}$$

and

$$\begin{aligned} I^{\prime \prime }_L(u)=\frac{1}{4}\left[ I_{L-2}(u)+2I_L(u)+I_{L+2}(u)\right] . \end{aligned}$$

Now the condition for convexity of the functional \(\phi (u)\) is that \(\phi ^{\prime \prime }(u)>0\), therefore (for \(L=1\)),

$$\begin{aligned} \frac{u_0^2}{\sigma ^2}\left[ \frac{I^{\prime \prime }_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }{I_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }-\left[ \frac{I^{\prime }_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }{I_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }\right] ^2\right]< & {} 1\nonumber \\ \left[ \frac{I^{\prime \prime }_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }{I_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }-\left[ \frac{I^{\prime }_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }{I_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }\right] ^2\right]< & {} \frac{\sigma ^2}{u^2} \end{aligned}$$
(36)

it implies

$$\begin{aligned} \frac{I^{\prime \prime }_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }{I_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }<\left( \frac{I^{\prime }_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }{I_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }\right) ^2+\frac{\sigma ^2}{u^2}, \end{aligned}$$

Substituting the expressions of \(I^{\prime }(\cdot )\) and \(I^{\prime \prime }(u)\) in the above expression we get

$$\begin{aligned} \frac{\frac{1}{4}\left[ I_{L-3}\left( \frac{u_0u}{\sigma ^2}\right) +2I_{L-1}\left( \frac{u_0u}{\sigma ^2}\right) +I_{L+1}\left( \frac{u_0u}{\sigma ^2}\right) \right] }{I_{L-1}\left( \frac{u_0u}{\sigma ^2}\right) }<\left( \frac{\frac{1}{2}\left[ I_{L-2}\left( \frac{u_0u}{\sigma ^2}\right) +I_{L}\left( \frac{u_0u}{\sigma ^2}\right) \right] }{I_{L-1}\left( \frac{u_0 u}{\sigma ^2}\right) }\right) ^2+\frac{\sigma ^2}{u^2} \end{aligned}$$
(37)

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Jidesh, P., Holla, S. Non-local total bounded variation scheme for multiple-coil magnetic resonance image restoration. Multidim Syst Sign Process 29, 1427–1448 (2018). https://doi.org/10.1007/s11045-017-0510-z

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