A numerical example

As numerical example we study a two component mixture model for image completion. Assume we expect an only partially known image (corresponding to pixel-wise training data drawn with Gaussian noise from the original image) to be similar to one of the two template images shown in Fig.4. Next, we include hyperparameters parameterising deformations of templates. In particular, we have chosen translations ($\theta_1$, $\theta_2$) a scaling factor $\theta_3$, and a rotation angle (around template center) $\theta_4$.

Interestingly, it turned out that due to the large number of data ( $\tilde n\approx$ 1000) it was easier to solve Eq.(21) for the full discretized image than to invert (32) in the space of training data. A prior operator ${\bf K}_0$ has been implemented as a $3\times3$ negative Laplacian filter. (Notice that using a Laplacian kernel, or another smoothness measure, instead of a straight template matching using simply the squared error between image and template, leads to a smooth interpolation between data and templates.) Completed images $h$ for different $\beta$ have been found by iterating according to

$$h^{k+1} = h^k + \eta {\bf A}^{-1} \Big[ {\bf K}_T (t_T-h^k) + {\bf K}_0 \Big(\sum_j a_j^k t_j -h^k\Big) \Big] ,$$ (47)

performed alternating with $\theta$-minimisation. A Gaussian learning matrix ${\bf A}^{-1}$ (implemented by a $5\times 5$ binomial filter) proved to be successful. Typically, the relaxation factor $\eta$ has been set to $0.05$.

Being a mixture model with $m=2$ the situation is that of Fig.3. Typical solutions for large and small $\beta$ are shown in Fig.4.

Figure 4: Top row, from left to right: Data points sampled with Gaussian noise, two template functions $t_1$, $t_2$. Bottom row, from left to right: Original, reconstructed solutions (regression function $h$, 180$\times$240 pixels) at low and at high temperature.