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

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.

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)