without any further information. Nevertheless, by imposing the prior information of $u$ (in the continuous domain) into \[OurProblem\], we can achieve the continuous domain restoration, and reduce the basis mismatch. For this purpose, we follow [@G.Ongie2016] and consider the piecewise constant function model of the proton density $u$: $$\begin{aligned}
\label{uModel}
u(\x)=\sum_{j=1}^J\alpha_j1_{\Om_j}(\x),~~~~~~~\x\in\R^2,\end{aligned}$$ where $\alpha_j\in\C$ and $1_{\Om}$ denotes the characteristic function on a set $\Om$: $1_{\Om}(\x)=1$ if $\x\in\Om$, and $0$ otherwise. Here, we assume that each $\Om_j$ lies in $[-L/2,L/2)^2$. We further assume that \[uModel\] is expressed with the smallest number of characteristic functions such that $\Om_j$’s are pairwise disjoint. Under this setting, the discontinuities of $u$ agrees with $\Gamma=\bigcup_{j=1}^J\p\Om_j$, which is called the *edge set* of $u$ [@G.Ongie2016].

In the sense of distribution, the gradient of $u$ satisfies $$\begin{aligned}
\label{Gradu}
\na u=\left(\p_1 u,\p_2 u\right)=\sum_{j=1}^J\alpha_j\rd\nnu_j,~~~~~~~\text{where}~~~\rd\nnu_j=-\bn_j\rd\ssi\big|_{\p\Om_j},\end{aligned}$$ with $\bn_j$ being the outward normal vector on $\p\Om_j$ and $\ssi$ being the surface measure. Since $|\bn_j|=1$ a.e. on $\p\Om_j$ and $\ssi(\Om_j)<\infty$, $\na u$ in \[Gradu\] defines a finite Radon vector measure on $\R^2$ supported on $\Gamma$. In addition, since \[Gradu\] holds in the sense of tempered distribution, we can compute $\msF(\na u)(\xxi)$ as a Fourier transform of a measure (e.g. [@Folland1999]). Namely, we have $$\begin{aligned}
\label{FouGradu}
