where x
(k)
and x =(x
1
,x
2
,...,x
J
)
T
are state vec-
tors in R
J
, each of which corresponds to the succes-
sive estimate of the reconstructed value of an iterative
algorithm in image reconstruction. The PMART can
be written in the mapping form with the following ele-
ments f
j
s for j =1, 2,...,J: f
j
= f
I
j
◦f
I−1
j
◦···◦f
1
j
with ith submap
f
i
j
= x
j
q
i
p
i
x
γp
i
j
(8)
where p
i
=
p
i
1
,p
i
2
,...,p
i
J
is the normalized projec-
tion operator applied on the x image, q
i
and p
i
x are
respectively the projection and the reprojection val-
ues, corresponding to the ith ray, for i =1, 2,...,I,
and γ is a positive real parameter.
The derivative of f with respect to x is given by
∂f
∂x
=
∂f
I
∂x
∂f
I−1
∂x
···
∂f
1
∂x
and the derivative of each
submap f
i
can be obtained by
∂f
i
∂x
= diag
j
q
i
p
i
x
γp
i
j
×
E −
γ
p
i
x
diag
j
{x
j
}·p
i
· p
i
T
for i =1, 2,...,I, where E denotes the J×J identity
matrix. By direct calculation, we see that the eigen-
values of the Jacobian ∂f
i
/∂x at a fixed point of f
are (1 − γ) and (J − 1) ones. Therefore when the
value of γ is 2 or the critical power (Badea and Gor-
don, 2004), the absolute values of the determinants of
the derivatives for the map f as well as each submap
f
i
at the fixed point are all 1, and every characteristic
multiplier of the fixed point is located on the unit cir-
cle in the complex plane. Moreover the fixed point is
stable and unstable when the values of γ are less and
greater than 2, respectively.
We now consider a dynamical system defined by
g : R
J
→ R
J
; x → (1 − λ)x + λf(x) (9)
where f denotes the map of the PMART in Eq.(7)
with Eq.(8), and λ ∈ R is a parameter. Note that
the expression Eq.(9) includes the algorithm of the
MART (Badea and Gordon, 2004) when λ =1and
γ =1.
As discussed above, in the case of J ≥ 2, a higher
codimension bifurcation satisfying multiple generic
bifurcation conditions including the Neimark-Sacker
bifurcation, of the nonhyperbolic fixed point occurs
in the dynamical system f at γ =2. Then an in-
variant closed curve (ICC) generates around the fixed
point in the state space. When γ<2, the ICC disap-
pears and iterative points converge to the stable fixed
point along the center manifold that is qualitatively
equivalent to the spiral. Due to the attracting spiral
behavior, it is expected that g(x), which is considered
Figure 1: Phantom image of 5 × 5 pixels.
as a weighted average of the point x and the next it-
erate f (x), obtains a better estimate than f(x) for an
appropriate value of λ.
4 EXPERIMENTAL RESULTS
AND DISCUSSION
To illustrate the efficiency of the proposed iterative al-
gorithm and the computational method, we treat two
examples: (i) the first image is made of four pixels
and six projection rays with the projection operator
p
1
=(1, 1, 0, 0), p
2
=(0, 0, 1, 1), p
3
=(1, 0, 0, 1),
p
4
=(0, 1, 0, 1), p
5
=(1, 0, 1, 0), and p
6
=
(0, 1, 1, 0), and the phantom image x
∗
=(5, 6, 7, 2)
T
;
and (ii) the image as the second example is made of
5 × 5 pixels and 56 projection rays with phantom im-
age shown in Fig.1.
Figure 2 shows an ICC forming a torus observed in
the map f for the first example (J =4) with γ =2,
consisting of 100,000 iterated points. The characteris-
tic multipliers of the fixed point satisfy the condition
of the double Neimark-Sacker bifurcation as a codi-
mension two bifurcation.
Figure 3 shows a phase transition diagram of fixed
points observed in the extended PMART of Eq.(9)
with J =4. In the figure, the parameter sets of equal
values of characteristic multipliers of fixed points are
indicated by solid and dashed curves with symbols
µ
∗
R and
ρ
∗
C, in the case of real multiplier µ
∗
and
the absolute value ρ
∗
of complex multiplier, respec-
tively, each of which is the maximum absolute value
among all characteristic multipliers. Period-doubling
and the Neimark-Sacker bifurcations are convention-
ally denoted by the symbols P and NS, which are
equivalent to
−1
R and
1
C, respectively.
In the diagram, there exists a unique stable fixed
point in the parameter regions without shading, and
with shading patterns
, and . Whereas,
in the regions with patterns
and surrounded
by the Neimark-Sacker and period-doubling bifurca-
tion curves, the fixed point is unstable and a solution
does not converge to the fixed point corresponding to
the phantom image. By increasing the value of λ for
fixed, e.g., γ =0.9, through the bifurcation curve P ,
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