9.4 Applications to Image Processing
361
{(¯ v, ¯
q) ∈ R
2n
: ¯
v = 0, ¯
q ∈ R
n
}.
It is then a structural property of the SEs describing the dynamics of a M-SCNN
in the (v, i)-domain to possess a manifold (a continuum) of nonisolated EPs. Note
that at any EP the capacitor voltages vanish.
Theorem 9.2 Suppose that (9.16) is satisfied and G is symmetric. Then, any
solution (v C (t), q(t)) of (9.21) is bounded for t ≥ 0 and converges to an EP
of (9.21) as t → +∞, i.e., (9.21) is convergent. In particular, v C (t) → 0 as
t → +∞, i.e., the capacitor voltages tend to 0.
Proof Let (v(t), q(t)), t ≥ 0, be the solution of (9.21) with initial condition
(v(0), ϕ(0)) = (v 0 , q 0 ). On the basis of FCAM, if we choose Q 0 as in (9.15),
then q(t), t ≥ 0, is the unique solution of (9.13) with initial condition q(0) = q 0 .
By Theorem 9.1 we have that q(t) → ¯
q and dq(t)/dt → 0 as t → +∞, where ¯
q is
an EP of (9.13). The proof is concluded by noting that v C (t) = k q R(dq(t)/dt).
Remark 9.4 We point out that the results in Theorems 9.1, 9.2 are consistent with
each other. In fact, the latter theorem shows that capacitor voltages tend to 0 as
t → +∞, hence, due to (9.20), also memristor currents vanish as t → +∞. This
is in agreement with Theorem 9.1, according to which memristor charges, i.e., the
integral of memristor currents, tend to a constant value as t → +∞. It is not difficult
to verify from Fig. 9.8 that for a convergent M-SCNN all voltages and currents in
the network vanish when a steady state is reached. This means that voltages and
currents are not useful for processing purposes. However, their integrals, i.e., fluxes
and charges, tend to finite constants in steady state and in particular the memristor
charges can be used for processing purposes, as discussed via the examples in
Sect. 9.4.
9.4 Applications to Image Processing
We have seen in Sect. 9.2.2 that the SEs (9.13) describing the M-SCNN model
are a good approximation, in the (ϕ, q)-domain, of the SEs describing the SCNN
model (9.1) in the (v, i)-domain, i.e., the two models are formally analogous. In this
section we show by numerical means that the two models indeed display similar
processing capabilities when applied to the solution of some image processing tasks
in real time.
361
{(¯ v, ¯
q) ∈ R
2n
: ¯
v = 0, ¯
q ∈ R
n
}.
It is then a structural property of the SEs describing the dynamics of a M-SCNN
in the (v, i)-domain to possess a manifold (a continuum) of nonisolated EPs. Note
that at any EP the capacitor voltages vanish.
Theorem 9.2 Suppose that (9.16) is satisfied and G is symmetric. Then, any
solution (v C (t), q(t)) of (9.21) is bounded for t ≥ 0 and converges to an EP
of (9.21) as t → +∞, i.e., (9.21) is convergent. In particular, v C (t) → 0 as
t → +∞, i.e., the capacitor voltages tend to 0.
Proof Let (v(t), q(t)), t ≥ 0, be the solution of (9.21) with initial condition
(v(0), ϕ(0)) = (v 0 , q 0 ). On the basis of FCAM, if we choose Q 0 as in (9.15),
then q(t), t ≥ 0, is the unique solution of (9.13) with initial condition q(0) = q 0 .
By Theorem 9.1 we have that q(t) → ¯
q and dq(t)/dt → 0 as t → +∞, where ¯
q is
an EP of (9.13). The proof is concluded by noting that v C (t) = k q R(dq(t)/dt).
Remark 9.4 We point out that the results in Theorems 9.1, 9.2 are consistent with
each other. In fact, the latter theorem shows that capacitor voltages tend to 0 as
t → +∞, hence, due to (9.20), also memristor currents vanish as t → +∞. This
is in agreement with Theorem 9.1, according to which memristor charges, i.e., the
integral of memristor currents, tend to a constant value as t → +∞. It is not difficult
to verify from Fig. 9.8 that for a convergent M-SCNN all voltages and currents in
the network vanish when a steady state is reached. This means that voltages and
currents are not useful for processing purposes. However, their integrals, i.e., fluxes
and charges, tend to finite constants in steady state and in particular the memristor
charges can be used for processing purposes, as discussed via the examples in
Sect. 9.4.
9.4 Applications to Image Processing
We have seen in Sect. 9.2.2 that the SEs (9.13) describing the M-SCNN model
are a good approximation, in the (ϕ, q)-domain, of the SEs describing the SCNN
model (9.1) in the (v, i)-domain, i.e., the two models are formally analogous. In this
section we show by numerical means that the two models indeed display similar
processing capabilities when applied to the solution of some image processing tasks
in real time.