Learning Iterative Image Reconstruction - Hierarchical Neural Networks for Image Interpretation - page 177

originate in the 3 × 3 hyper-neighborhood of a feature cell in the same layer. Back-
ward projections receive input from a single cell in the low-resolution input array of
Layer 2. For each of the four features, there are 2 × 2 different backward projections.

Output

Input

Layer 2 (16x16)
Fig. 9.3. Network for super-resolution. It is a small Neural Abstraction Pyramid with four
hidden feature arrays in the middle layer. Each pixel corresponds to a feature cell that is
connected to its neighbors. The gray values show the activities after the recurrent network
has iteratively reconstructed the high-resolution image.

Layer 0 (64x64) Layer 1 (32x32)

While all projection units have linear transfer functions, a sigmoidal transfer
function f sig ( β = 1 , see Fig. 4.5(a) in Section 4.2.4) is used for the output units
of the processing elements. The feature arrays are surrounded by a one pixel wide
border that is set to zero since the background of the input and the desired output
has this value.
The network's 235 free parameters are initialized randomly, and they are trained
for ten time steps on a fixed set of 200 randomly chosen example digits. The test
set consisted of 200 different randomly chosen examples. Training was done using
the back-propagation through time (BPTT) method in combination with the RPROP
learning algorithm, as described in Section 6. The weighting of the squared output
error increased linearly with time.

9.2.3 Experimental Results

Figure 9.4 shows how the output of the trained network develops over time when the
first five digits of the test set are presented at its input. After two iterations, the input
can influence the output, but no further interactions are possible yet. Thus, the output
looks like a copy of the low-resolution input. In the following iterations, the initial
reconstruction is refined. Black lines with sharp smooth borders are generated. After
iteration five, the network's output does not change significantly any more.
The outputs produced by the network are close to the targets. The differences
are shown in Figure 9.5. Some small details at the edges of the lines, probably
caused by noise that was amplified by the binarization procedure, have not been