A NOVEL CHAOTIC CODING SYSTEM FOR LOSSY IMAGE

COMPRESSION

Sebastiano Battiato and Francesco Rundo

Image Processing Laboratory, Dipartimento di Matematica ed Informatica - University of Catania

Viale Andrea Doria 6, 95125 - Catania, Italy

Keywords: Chaos, lossy compression, control system, coding pipeline.

Abstract: In this paper a novel image compression pipeline, by making use of a controlled chaotic system, is

proposed. Chaos is a particular dynamic generated by nonlinear systems. Under certain conditions it is

possible to properly manage the chaotic dynamics obtaining very feasible and powerful working

instruments. In the proposed compression pipeline a linear feedback control strategy has been used to

stabilize chaotic dynamic used to track the 1D signal generated by the input image. The pipeline is closed by

an entropy encoder. Preliminary experiments and comparison with respect to standard JPEG engine confirm

the effectiveness of the proposed chaotic coding system both for natural and graphic images. Also the

overall performances in terms of rate-distortion capabilities are promising.

1 INTRODUCTION

Image coding is mainly achieved by making use of

well known de-facto standards both for lossless and

lossy compression. Coding redundancy is typically

attacked by making use of DCT or Wavelet

transform.

Main goal of the proposed compression

methods is to improve the coding frameworks

proposing an alternative compression engine

(Amerijckx et al., 1998). A novel algorithm to

compress (lossy) colour images by means of

controlled chaotic dynamics is discussed. Chaos is a

typical dynamic generated by nonlinear systems

with at least one positive Lyapunov exponent.

There is no well accepted rigorous

mathematical definition of chaos but its properties of

high sensitivity to small perturbations as well as the

capability to generate many complex dynamics such

as limit cycles, attractors and unstable orbits, are

well known (Arena et al., 2002; Chen et al., 1997;

Perrone, 1997). Due to this extremely sensitivity to

tiny perturbations the chaotic trajectories can be

controlled to follow a reference dynamic very

quickly (i.e. the NASA scientists have used a similar

approach in the control system of the spacecraft

ISEE-3/IEC). By taking into account some of the

mentioned properties, several algorithms have been

proposed using chaos for encrypting signals and

images (Dedieu et al., 1995; Guan et al., 2005).

Moreover, interesting approaches for coding

signal and images have been proposed in (Perrone,

1997, Belkhouche et al., 2003, Nien et al., 2007). An

alternative compression approach based on “fractal”

theory has been presented in (Li et al., 2000) but

without improvement with respect to standard

compression engine such as JPEG. In the proposed

compression pipeline a linear feedback control

strategy has been used to stabilize chaotic dynamic

properly tracking the 1D signal generated by a

classic raster visit of the input image. Classic

differential coding (Gonzalez et al., 2000; Sayood,

2003)) and Huffman coder are used to complete the

compression pipeline.

The paper is organized as follows: next section

presents some recall about chaos control techniques

whereas section III describes the proposed

algorithm. In section IV experimental results and

future works are briefly sketched.

2 THE CONTROL SYSTEM

There are several reasons for controlling chaos (i.e.,

nonlinear systems which shows chaotic dynamics

such as the Lorenz system, the Logistic map, the

Duffing oscillator, the Chua’s system and so on).

321

Battiato S. and Rundo F. (2008).

A NOVEL CHAOTIC CODING SYSTEM FOR LOSSY IMAGE COMPRESSION.

In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 321-326

DOI: 10.5220/0001071903210326

Copyright

c

SciTePress

Without loss of generality, we concentrate our

attention into the Chua’s system as dynamical

system with chaotic behaviour. The Chua’s system

was first proposed by L. O. Chua as autonomous

nonlinear circuit. There are many implementations

of the Chua’s system and a lot of complex nonlinear

dynamics that can be generated by tuning of the

Chua system’s parameters. We are referring to

canonical representation reported in (Chen et al.,

1997) in which the state equations can be re-written

in canonical dimensionless form as follows:

)11)((5.0)(

))((

010

−−+−+=

⎪

⎩

⎪

⎨

⎧

−=

+−=

−+−=

xxmmxmxf

qyz

zyxy

xfyxpx

&

&

&

(1)

By tuning the parameters (p, q, m

0

, m

1

) a vast

variety of dynamics can be generated (Chen et al.,

1997; Manganaro et al., 1999) also including chaos.

In the theory of chaos control, several goals can be

achieved (Boccaletti et al., 2000). The Fig. 1 shows

a typical double-scroll attractor generated by the

Chua’s system described in (1). Typical approaches

to chaos control are: open loop strategies, feedback

control systems, adaptive control systems

(Boccaletti et al., 2000). A typical open loop control

strategy is the so called entrainment in which the

controlled chaotic dynamic is forced to follow a

reference trajectory (Chen et al., 1997).

Figure 1: The Double Scroll attractor.

A closed loop (feedback) version of the

entrainment control strategy was proposed in

(Jackson,1998). The general entrainment strategy

can be defined as follows. Let a dynamical system

which shows chaotic behaviour:

n

tutxtuttxftx ℜ∈+= )(),()()),(()(

&

(2)

The goal of the above control scheme is to find

a controller “u(t)” to force the dynamical system in

(2) to follow (asymptotically stability) a target

trajectory r(t):

n

t

trtrtx ℜ∈∀=−

∞→

)(0)()(lim

(3)

In realistic applications, the target to be

achieved when a chaotic system is controlled is the

so called “near” control goal (Chen et al., 1997):

∞<≥∀<−

o

ttttrtx

0

)()(

ε

(4)

In (4) the term ε represents a prefixed allowable

tolerance ad t

0

a terminal time. There are several

approaches to determine the controller u(t) (Mascolo

et al., 1999). The controller u(t) can be implemented

by means of a careful analysis of the dynamics to be

controlled or by using adaptive algorithms.

Recently, “intelligent” controllers have been widely

used in several applications for controlling nonlinear

dynamics such as chaos (Arena et al., 2002). In

(Chen et al., 1997) a linear feedback control system

has been used for controlling Chua’s system

successfully. With this approach, the state equations

of the controlled Chua’s system can be written as

follows:

)11)((5.0)(

)(

)(

)())((

010

−−+−+=

⎪

⎪

⎩

⎪

⎪

⎨

⎧

−−−=

−−+−=

−−−+−=

xxmmxmxf

zzkqyz

yykzyxy

xxkxfyxpx

refz

refy

refx

&

&

&

(5)

where the control gains (k

x

,k

y

,k

z

) have to be

computed during the design phase whereas (x

ref

, y

ref

,

z

ref

) are the reference trajectories.

The design of the above control gains can be

made by using several techniques (Arena et al.,

2002; Boccaletti et al., 2000).

In (Chen et al., 1997) a Lyapunov’s theorem

based algorithm has been used for getting the

following conditions:

0,0,

1

≥≥

−

≥

zyx

kkpmk

(6)

The reference trajectories can be generated by a

Chua’s circuit with different initial conditions or

parameters or from another dynamical system.

Clearly, the control action can be also applied to a

subset of variables of the Chua’s system as showed

in (Arena et al., 2002). They were able to control the

Chua’s system acting only to its x and y state

variables by means of a linear feedback control

scheme with adaptive gains. The experimental

results (tested with many kind of complex target

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

322

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

323

coding, codewords length generated by the Huffman

coder for each mage plane (Y,C

b

,C

r

). In Fig. 2 the

proposed encoding pipeline is showed. The signal

e

e

(k) represents the encoded image.

3.2 The Decoder

In the decoding phase all previous steps are simply

inverted. Firstly, the included header is parsed to

extract the basic information. The compressed

encoded signal e

e

(k) is processed by an Huffman

decoder and then by an inverse differential coding.

The reconstructed error signal e

rec

(k) is used for

controlling the x-variable of the Chua’s system as

showed in (7) (with the same initial conditions,

system parameters and control gains used in the

encoder).

At this point, the source vector i(k) can be

reconstructed as follows:

1),..(1,0

)()()(

−×=

−=

−

nmk

kekxki

rec

DecoderChua

Nrec

(11)

Figure 2: The proposed Encoder.

We reconstruct the 2D image from the 1D

i

rec

(k) vector (using also the information included in

the header). For the chrominance components, an

up-sampling operation (2x bi-cubic interpolation) is

also applied. Finally, the so reconstructed 2D image

(in YC

b

C

r

)

is converted into RGB colour image

I’(x,y). In Fig. 3 the proposed decoding pipeline is

showed.

4 EXPERIMENTAL RESULTS

The proposed lossy image compression pipeline has

been tested with different images. The value of the

used control gain during the tests execution is k

x

=10.

The tests have been run under MATLAB framework

(rel. 7.0.1).

The reported results show good performances

in terms of quality (PSNR) and compression ratio

(Original image filesize/compressed image filesize)

of the proposed pipeline. We report in Fig. 4 and

Fig. 5 some comparisons with JPEG standard

applied to the synthetic image named “Benjerry”.

Moreover, in Fig. 6 we have tracked the PSNR

dynamic versus RF

y

for the “Benjerry” image. For

some synthetic images (466x60) we report the full

rate-distortion curve in Fig. 7 while some visual

results are showed in Fig. 8. Further results on

graphic and natural images are briefly reported in

Table 1. Finally in Fig. 9 and Fig. 10 we have

tracked respectively the Bpps versus RFy and PSNR

versus RFy for a single natural image.

Preliminary comparisons with JPEG standard

are also reported by measuring the improvement

obtained in terms of compression ratio by

considering the same amount of visual quality

(measured by PSNR values). We are planning to

make further experiments to provide a full detailed

rate-distortion repository on large images dataset

including a JPEG2000 (lossy pipeline) comparisons

as well. Future works aims to apply the proposed

pipeline for the encryption of digital images.

Figure 3: The proposed Decoder.

(Original)

(JPEG, PSNR = 27.11 dB, Bits/pixel = 4.14)

(CCS, PSNR = 27.23 dB, Bits/pixel = 2.23 )

Figure 4: The comparison between the proposed algorithm

(CCS – Chaotic Coding System) with respect to JPEG.

Original JPEG CCS

Figure 5: A detail of the “Benjerry” image compressed

with the proposed pipeline (CCS) and with JPEG codec.

The above detail shows the absence of artefacts (ringing)

typically showed by JPEG engine.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

324

Figure 6: PSNR versus RF

y

plot (for Y plane of the

synthetic image “Benjerry”).

Figure 7: The Rate-Distortion plot of the proposed

pipeline with respect to the JPEG standard (for synthetic

images).

(a)

(b)

(c)

Figure 8: The above images show a further comparison

between the proposed pipeline with respect to JPEG

standard. In (a) the original image is reported

(gimp_temp.115971). In (b) the JPEG image is reported

(Compression ratio=5.54, PSNR=37.4 dB). Finally in (c)

the image compressed with the proposed pipeline is

reported (Compression ratio=6.24, PSNR=37.71 dB). All

the above images contain a superimposed detail to show

the absence of typical ringing artefacts (normally present

in JPEG coding) in the proposed compression pipeline.

Table 1: Comparison between the proposed pipeline

(CCS) with respect to JPEG standard in terms of

compression ratio (CR) and PSNR (in dB). The used

quality factor for JPEG coding is in the range 90 ÷ 100.

The CCS coding has been performed tuning the parameter

RF

y

in the range 35 ÷ 80.

Images JPEG CCS

CR PSNR CR PSNR

Benjerry 3.72 28.31 5.61 28.99

Netscape 3.01 34.47 7.83 34.53

Book 2.87 26.60 3.40 25.34

gimp_temp.115971 5.54 37.41 6.24 37.71

gimp_temp.115977 5.87 37.15 6.98 37.42

gimp_temp.115979 3.11 35.21 4.02 35.31

gimp_temp.1159715 4.53 35.53 4.82 35.24

gimp_temp.1159717

7.98 38.38 4.74 39.25

gimp_temp.1159719

3.94 39.59 4.25 40.93

gimp_temp.26473

5.27 39.93 7.23 39.35

gimp_temp.26475

3.23 37.42 3.87 37.43

gimp_temp.264711

5.14 39.19 5.72 39.23

gimp_temp.264715

3.84 37.10 4.02 37.02

gimp_temp.264723

4.26 39.01 4.15 38.43

A NOVEL CHAOTIC CODING SYSTEM FOR LOSSY IMAGE COMPRESSION

325

Figure 9: Bit per pixels versus RF

y

(computed for the

natural image “gimp_test.115979”).

Figure 10: PSNR (in dB) versus RF

y

(computed for the

natural image “gimp_test.1159715”).

REFERENCES

Arena P., Fortuna L., Frasca M., 2002. Chaos Control by

using Motor Maps. In Chaos, vol. 12, no 3, September

2002.

Amerijckx C., Verleysen V., Thissen P., Legat J., 1998.

Image Compression by Self-Organized Kohonen Map.

In IEEE Trans. on Neural Networks 1998, vol. 9, pag.

503-507.

Belkhouche F., Qidwai U., 2003. Binary image encoding

using 1D chaotic maps. In IEEE Region 5, A. T.

Conference, 2003, pag. 39-43.

Boccaletti S., Grebogi C., Lai Y.C., Mancini H., Maza D.,

2000. The control of chaos: Theory and applications.

In Phys. Rep. 329, pag. 103-197, 2000.

Chen G., Dong X., 1997. From Chaos To Order

Methodologies, Perspectives and Applications, World

Scientific Series on Nonlinear Science, USA, 1997.

Dedieu H., Ogorzalek M., 1995. Signal Coding and

Compression Based on Chaos Control Techinique. In

ISCAS’9, vol. 2, pag. 1191-1194.

Gonzalez R., Woods R. E., 2000. Digital Image

Processing –Second Edition, Prentice Hall, 2000.

Guan Z., Huang F., Guan W., 2005. Chaos based image

encryption algorithm. In Elsevier Physics Letters A ,

pag. 153-157.

Jackson E. A., 1998. The OPCL control method for

entrainment, model-resonance, and migration actions

on multiple-attractors systems. In Chaos vol. 7, pag.

550-559.

Li Z., Zhao L., Soma N.Y., 2000. Fractal Color Image

Compression. In SIBGRA 2000, pag. 185-192.

Manganaro G., Arena P., Fortuna L., 1999. Cellular

Neural Networks. Chaos, Complexity and VLSI

Processing,. Springer edition, 1999.

Mascolo S., Grassi G., 1999. Controlling Chaotic

Dynamics Using backstepping Design with

Application to the Lorenz System and Chua’s Circuit.

In Int. Journal of Bifurcation and Chaos, vol. 9, no. 7.

Nien H.H, Huang C.K., Changchien S.K., Shieh H.W.,

Chen C.T., Tuan Y.Y, 2007. Digital color image

encoding and decoding using a novel chaotic random

generator. In Elsevier, Chaos, Solitons and Fractals

vol. 32, pag. 1070-1080.

Perrone A.L., 1997. Applications of chaos theory to lossy

image compression. In Elsevier, Nuclear Instruments

& Methods in Physics Research A 389 (1997) 221-

225.

Sayood K., 2003. Lossless Compression Handbook. The

Academic Press Series in Communications,

Networking and Multimedia, 2003.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

326