Let $\alpha, \beta > 0$ and $\Delta := \nabla^T \nabla$ be the discrete laplacian operator, $$\nabla: \mathbb{R}^{n_x \times n_y \times n_z} \rightarrow \mathbb{R}^{3 \times n_x \times n_y \times n_z}, \hspace{.5em}w \mapsto (\nabla_xw, \nabla_yw, \nabla_zw) $$ being the discrete spatial gradient operator in $3$D. You can see $\Delta$ as a linear operator of shape $p \times p$, where $p := n_xn_yn_z$.
My intuition is that the operator $(\alpha I + \beta \Delta)^{-1}$ corresponds to some kind of Gaussian smoothing. Is this true ? If yes, what's the width of the corresponding kernel in terms of $\alpha$ and $\beta$ ?
