The shell-based inpainting Gaussian-blur conjecture

Let the inpainting domain $D$ and undamaged area $\mathcal U$, together with their discrete counterparts $D_h$ and $\mathcal U_h$, be as described in the paper. Let $u_0:\mathcal U\to\mathbb R^d$ be non-negative and bounded, and let $u_h:D_h\to\mathbb R^d$ be produced by Algorithm 1 or its semi-implicit extension. Write $a_r^*$ for the stencil, let $Z_i=(V_i,W_i)$ have support in $\operatorname{Supp}(a_r^*)$ and mean $(\mu_x,\mu_y)$, and let $\Pi_{\theta_r^*}:D\to\partial D$ be the transport operator with direction $\mathbf g_r^*=(\mu_x,\mu_y)$. Define the discrete mollification by

$$\left(u_0\big|_{y=0}*g_{\sigma(h)}\right)(x):=\sum_{i=1}^N\left[\int_{(i-1)h}^{ih}g_{\sigma(h)}(x-t)\\,dt\right]u_0\big|_{y=0}(ih),$$

where

$$\sigma(h)^2=\frac{\gamma^2yh}{|\mu_y|^3},\qquad \gamma^2=\operatorname{Var}(\mu_xW_1-\mu_yV_1).$$

The shell-based inpainting Gaussian-blur conjecture. If $u_0$ is independent of its $y$-coordinate, then

$$u_h(x,y)\to\left(u_0\big|_{y=0}*g_{\sigma(h)}\right)(\Pi_{\theta_r^*}(x,y))$$

asymptotically as $h\to0$, at a rate dependent on $y$ but independent of $x$. For general $u_0$, the same conclusion holds with $u_0\big|_{y=0}$ replaced by $\tilde u_0$, where $\tilde u_0(ih)=\sum_{j=-r-1}^{0}\alpha_j(ih)u_0(ih,jh)$, $0\leq\alpha_j(ih)\leq1$, and $\sum_{j=-r-1}^{0}\alpha_j(ih)=1$. The context is numerical evidence that shell-based geometric inpainting produces a blur described by a one-dimensional Gaussian whose variance depends on the distance travelled in the inpainting direction; the claim concerns the asymptotic behaviour of the algorithm as the grid spacing tends to zero.