Neural Abstraction Pyramid Architecture - Hierarchical Neural Networks for Image Interpretation - page 95

Layer 0 (16×1) Layer 1 (8×2) Layer 2 (4×4) Layer 3 (2×8)
Fig. 4.21. Invariant feature extraction - hierarchical decomposition. Two moving signals
are transformed hierarchically by recursive application of the basic decomposition (see Fig-
ure 4.20). With height the number of features and their invariance to translation increases.
The different input patterns yield different invariant representations.

and at the top by 16 border cells, updated using wrap-around copying of activities.
All other feature arrays of the figure have a vertical resolution of eight cells, framed
by an eight cell wide border. The filter responses S sin and S cos are squared, added
together, and passed through the square root transfer function of the output unit
C H . Its response represents the local energy of the high-frequency components of
I . This feature is complemented by a feature C L , representing the low-frequency
components of I . It is produced by setting the projection weights of C L to the low-
pass filter

1

128 B 8 , as illustrated in the lower part of the figure.
Both the high-frequency energy C H and the low frequency part C L lack high
frequencies. They move with half the speed of I and intermediate responses can
be interpolated easily from the responses of the 8 cells as visible in the diagonal
line structure. The one-dimensional invariant feature extraction can be generalized
to two dimensions in the same way as two-dimensional wavelets or the 2D DFT are
constructed.
As shown in Figure 4.21, the basic decomposition into two invariant features can
be applied recursively. This yields a sequence of representations with decreasing
resolutions and increasing invariance to translations. The figure shows the response
of the invariance hierarchy to two different moving input patterns. The eight Layer 3
responses of length two are almost constant as the pattern moves, but change con-
siderably between patterns. This shows that the high-level features not only have
a high degree of invariance to translations, but also possess high representational
power.