![](bg3.png)
trajectories) shows, also in this case, the good
performance of the linear feedback control approach
in controlling chaotic dynamics. For the proposed
compression pipeline, we use the same linear
feedback control strategy used in (Arena et al.,
2002) applied to the x variable of the Chua’s system
showed in (1). The so controlled Chua’s system,
integrated with a classic Euler algorithm, can be re-
written as follows:
27.1,68.0,87.14,10
1,..1,0
)())(()()(()(
))(()()1(
))()()(()()1(
)()()1(
)()()(
10
−=−===
−=∀
−−+−=
⎪
⎩
⎪
⎨
⎧
−⋅+=+
+−⋅+=+
⋅+=+
−=
mmqp
Nk
kekkxfkykxpk
kqyhkzkz
kzkykxhkyky
khkxkx
kxkxke
xxx
x
refx
ψ
ψ
(7)
In (7) the term h represents the integration step
while the reported parameters (p, q, m
0
, m
1
) are
suitable to generate a double scroll chaotic attractor
(Chen et al., 1997). The described control theory is
the main core of the proposed lossy compression
pipeline. The key idea is based on the specific
property of chaotic dynamic: high sensitivity to
small perturbations. As mentioned in section 1, the
previous property leads a controlled chaotic system
to follow a desired trajectory very quickly.
3 THE PROPOSED PIPELINE
In the proposed compression pipeline we force the
controlled Chua’s system showed in (7) to track the
1D representation of bi-dimensional source image
(the target trajectory x
ref
). Due to the above
considerations about main chaos properties, we
make sure that at least near goal feedback control
(showed in (4)) can be achieved (Arena et al., 2002).
Both for the encoder and decoder sides a Chua’s
system as showed in (7) is used. The initial
conditions are x(0)=0.1; y(0)=0.2; z(0)=0.3 for both
encoder/decoder side. The used integration step is
h=0.01.
3.1 The Encoder
The encoding pipeline starts converting the source
colour image I(x,y) of size (m x n) from RGB to
YC
b
C
r
colour space (Gonzales et al., 2000). After
that, the chrominance components, C
b
and C
r
, are
down-sampled by a factor 2. In the next step, the
encoding scheme is applied for each plane (Y, C
b
and C
r
) separately. We refer in the next paragraphs
to Y plane of the source image but the same
consideration may be applied to the chrominance
components (down-sampled by factor 2) of the same
image. The 2D image plane is than translated into
1D by a classical raster visit. Finally, a
normalization in the range [0,1] of the 1D image
vector is applied. Let i(k) the obtained vector
corresponds to the reference trajectory x
ref
showed in
(7). At this point the tracking error can be defined as
follows:
1),..(1,0),()()( −×=
nmkkikxke
N
(8)
Each of the Chua’s system variables (x(k), y(k),
z(k)) are normalized in order to define the precision
of the non-integer values involved in the proposed
algorithm:
1),..(1,0
/))(()(
/))(()(
/))(()(
−×=
⋅=
⋅=
⋅=
nmk
RFRFkzroundkz
RFRFkyroundky
RFRFkxroundkx
N
N
N
(9)
where RF is ad hoc heuristically defined round-off
factor. Finally, in order to re-map the non-integer
values of the tracking error to an integer range,
before to the Huffman encoding, the following re-
mapping equation is used:
By tuning this RF factor we are able to change
the compression rate of the proposed algorithm. The
encoder defines an RF parameter both for luminance
(RF
y
) and chrominance (RF
c
) quantization. After
that, we proceed to compress the error e(k) as
showed in (8); really we compress the quantized
version of e(k) described in (10) instead of i(k). The
linear feedback control system, leads the chaotic
dynamic of he x-variable of the Chua’s system to
follow the target (i.e. the vector i(k)) very quickly).
The residual entropy of the error signal e(k) is, of
course, more achievable than original i(k) signal,
allowing to obtain near-optimal rate-distortion
performances. A classical differential coding
(Gonzales et al. 2000; Sayood, 2003) followed by an
Huffman encoder is used to complete the
compression pipeline. In the proposed algorithm we
have defined ad hoc header (just a few bytes)
included together with the data stream. This header
contains the size of the original image, the round-off
factors RF
y
and RF
c
(both luminance and
chrominance) three parameters used by differential
1),..(1,0
])/))(([()(
−×=
⋅⋅
nmk
RFRFRFkeroundroundke
(10)
A NOVEL CHAOTIC CODING SYSTEM FOR LOSSY IMAGE COMPRESSION
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