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 (
, 
)
a scaling factor 
,
and a rotation angle (around template center)
.
Interestingly,
it turned out that due to the large number of data
(
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 
has been implemented
as a 
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 
for different 
have been found by iterating according to
![\begin{displaymath}
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]
,
\end{displaymath}](img221.gif) |
(47) |
-minimisation.
A Gaussian learning matrix 
(implemented by
a 
binomial filter) proved to be successful.
Typically, the relaxation factor 
has been set to 
.
Being a mixture model with 
the situation is that of Fig.3.
Typical solutions for large and small
are shown in Fig.4.
 |
, 
.
Bottom row, from left to right:
Original, reconstructed solutions
(regression function 
240 pixels) at low and at
high temperature.