PPKE ITK PhD and MPhil Thesis Classes
PPKE ITK PhD and MPhil Thesis Classes
PPKE ITK PhD and MPhil Thesis Classes
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6 1. INTRODUCTION<br />
equation with FSR model is as follows:<br />
⎧<br />
⎨ 1 if v ij (n) > 1<br />
x ij (n + 1) = v ij (k) if |v ij (n)| ≤ 1<br />
⎩<br />
−1 if v ij (n) < −1<br />
(<br />
v ij (n) = (1 − h)x ij (n)+<br />
)<br />
∑<br />
∑<br />
+h<br />
A ij,kl x kl (n)+ B ij,kl u kl (n) + z ij<br />
C(kl)∈N r(i,j)<br />
C(kl)∈N r(i,j)<br />
(1.4)<br />
Now the x <strong>and</strong> y variables are combined by introducing a truncation, which is<br />
simple in the digital world from computational aspect. In addition, the h <strong>and</strong><br />
(1-h) terms are included into the A <strong>and</strong> B template matrices resulting templates<br />
Â, ˆB.<br />
By using these modified template matrices, the iteration scheme is simplified<br />
to a 3×3 convolution plus an extra addition:<br />
∑<br />
v ij (n + 1) = Â ij,kl x kl (n) + g ij<br />
(1.5a)<br />
g ij =<br />
C (kl)∈N r(i,j)<br />
∑<br />
C (kl)∈N r(i,j)<br />
ˆB ij,kl u kl + hz ij<br />
(1.5b)<br />
If the input is constant or changing slowly, g ij can be treated as a constant <strong>and</strong><br />
should be computed only once at the beginning of the computation.<br />
1.1.2 Nonlinear templates<br />
The implementation of nonlinear templates are very difficult on analog VLSI <strong>and</strong><br />
quite simple on emulated digital CNN. In some interesting spatio-temporal problems<br />
(Navier-Stokes equations) the nonlinear templates (nonlinear interactions)<br />
play key role. In general the nonlinear CNN template values are defined by an<br />
arbitrary nonlinear function of input variables (nonlinear B template), output<br />
variables (nonlinear A template) or state variables <strong>and</strong> may involve some time<br />
delays. The survey of the nonlinear templates shows that in many cases the<br />
nonlinear template values depend on the difference of the value of the currently<br />
processed cell (C ij ) <strong>and</strong> the value of the neighboring cell (C kl ). The Cellular<br />
Wave Computing Library [21] contains zero- <strong>and</strong> first-order nonlinear templates.