Proximal operator of image Laplacian

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 $\beta$ ?

• Wait; $\beta \Delta \in \mathbb K^{1\times N}$, but $I$ is inherently of $N\times N$ shape. How can you find an inverse of that operation? – Marcus Müller Feb 8 '16 at 15:36
• I think you should more explicitely define what $\nabla$ and $\Delta$ are, in your case. I don't think I know the definitions you're using. – Marcus Müller Feb 8 '16 at 15:41
• That inversion is well-defined. $\nabla$ is the discrete spatial gradient operator. – dohmatob Feb 8 '16 at 19:27
• ha, I was assuming your $\nabla$ was the classical divergence operator in vector form, hence $\nabla_\text{Müller} = \left(\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_N}\right)$! – Marcus Müller Feb 8 '16 at 19:29