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LOSSY TO LOSSLESS IMAGE COMPRESSION BASED ON REVERSIBLE INTEGER DCT

Lei Wang, Jiaji Wu, Licheng Jiao, Li Zhang and Guangming Shi

Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education of China,

Institute of Intelligent Information Processing, Xidian University, Xi'an 710071, P.R. China

This work is supported by the National Natural Science Foundation of China under Grant Nos.60607010, 60672125, 60672126,

60736043, 60776795; the Program for Cheung Kong Scholars and Innovative Research Team in University (PCSIRT, IRT0645); Hunan

Provincial Natural Science Foundation of China under Grant No.08JJ3123.

ABSTRACT

A progressive image compression scheme is investigated

using reversible integer discrete cosine transform (RDCT)

which is derived from the matrix factorization theory.

Previous techniques based on DCT suffer from bad

performance in lossy image compression compared with

wavelet image codec. And lossless compression methods

such as IntDCT, I2I-DCT and so on could not compare

with JPEG-LS or integer discrete wavelet transform (DWT)

based codec. In this paper, lossy to lossless image

compression can be implemented by our proposed scheme

which consists of RDCT, coefficients reorganization, bit

plane encoding, and reversible integer pre- and post-filters.Simulation results show that our method is competitive

against JPEG-LS and JPEG2000 in lossless compression.

Moreover, our method outperforms JPEG2000 (reversible

5/3 filter) for lossy compression, and the performance is

even comparable with JPEG2000 which adopted

irreversible 9/7 floating-point filter (9/7F filter).

Index TermsBlock transform, JPEG2000, lossless

compression, reversible integer DCT

1. INTRODUCTION

Image compression has been becoming increasinglyimportant with the development of aviation,

communications, internet and space techniques; especially

lossless compression becomes indispensable when there is

no loss of information is tolerable such as medical image,

remote sensing, image archiving, and satellite

communications and so on. Compression ratio and bit

distortion always contradict each other, so the techniques

pursuing for higher compression ratio with less distortion

even without information loss has been one of the popular

research issues in image compression. Fortunately a unified

representation for lossy to lossless image compression can

be provided by reversible coding, so both satisfying

recovered images at reasonable compression ratio and fully

reconstructed images can be obtained from a single

compressed file.

DCT has been applied in many international compression

standards such as JPEG, MPEG, and H.26X and so on for

its special advantages including: highly energy-compacting

capability, transforming block by block as a result of

parallel implementation and low memory requirement.

However, blocking DCT is less of considerations on the

correlations of inter-blocks and always results in blocking

artifacts at low bit rate, both of these defects effect the rate

distortion (RD) performance and visual quality of

reconstructed images. Furthermore, the DCT-based

reversible image compression was not well developed.

Hao et al. [1] have obtained very good results aboutlossless image compression using integer-DCT-based

methods; however, the simulation results are still scarce

about lossy compression. In fact, there is still a large

margin to improve the RD performance. As far as we know,

researches about DCT can be classified into two categories:

first, researches on the integer approximation of DCT

matrix for lossless compression; second, researches on

lossy compression. The first class includes: Chen et al.

proposed the low-cost 8-point Integer DCT (IntDCT) which

is based on the Walsh-Hadamard Transform (WHT) and

integer lifting [2]; Abhayaratne proposed N-point I2I-DCT

by applying recursive methods and lifting techniques,

where N is power of 2 [3]. The second class includes:Xiong presented a DCT-based embedded image coder

which was called EZDCT [4]; Tran et al.designed TDLT

(time domain lapped transform) by adding pre- and post-

filter to the DCT [5]. Although some of the above

algorithms obtained good results, but a progressive

reversible image compression scheme based on DCT with

high performance in both lossy and lossless compression

nearly doesnt exist.

In this paper, we developed a progressive reversible

image compression scheme which can realize lossy to

lossless compression. In our scheme, RDCT is used for

transforming combined with reversible integer pre-filter

which will make blocked source pixels more compatible for

block transform and post-filter for reducing blocking

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artifacts. RDCT coefficients are reorganized into sub-band

structure, and then coded by context-based block coding.

Simulation experiments have been taken on benchmark

images for both lossy and lossless image compression and

our algorithm outperforms JPEG-LS [6] and JPEG2000

(5/3 filter) [7] in lossless compression, and performs better

than JPEG2000 (5/3 filter) in reversible lossy compression,

comparable with irreversible JPEG2000 (9/7 F filter).The rest of this paper is organized as follows. A brief

review about DCT and the modified matrix factorization

method will be taken in Sec. 2. In Sec. 3, the realization of

RDCT will be introduced. In the following, RDCT-based

entropy coding will be described in Sec. 4. Finally, the last

section will present the experimental results and discussions.

2. DCT AND MODIFIED MATRIX

FACTORIZATION METHOD

2.1 DCT

The type-II DCT and its inverse in one dimension are givenby the following equations:1

0

2( ) ( ) co s ((2 1) )

2

N

C k

n

kX k x n n

N

(1)1

0

2( ) ( ) co s((2 1) )

2

N

k C

n

kx k X n n

N

(2)for k = 0 N-1, where

10,

2

1 .

k

if k

else

( )x n is the input sequence of length N,

cos((2 1) )2

knN

is the discrete cosine transform kernel.

2.2 Modified Matrix Factorization Method

We will have a brief introduction about the modified matrix

factorization method in this section. Hao has proved that a

nonsingular matrix can be factorized into a product of at

most three triangular elementary reversible matrices

(TERMs) [1]. Galli and Salzo modified the method by

proposing a procedure of quasi-complete pivoting which

leads to a better integer approximation to the original float-

point transform matrix [8].

Suppose a nonsingular matrix

N NA R without loss of

generality with determinant of module 1, the decomposing

formula can be defined as

A PLUS , (3)where L and S are lower triangular matrices, U is upper

matrix and Pis the permutation matrix.

Table I: TERMs Factorized from 4-point DCT Matrix

P L

0 1 0 0 1

1 0 0 0 0.2346 1

0 0 0 1 0.4142 -0.7654 1

0 0 1 0 0.2346 0 -0.6934 1

U S1 -0.2929 -0.0137 -0.6533 1

1 0.3066 0.6533 0 1

1 0.5000 0 0 1

1 0.5307 -0.8626 0.3933 1

3 REVERSIBLE INTEGER HIERARCHICAL DCT

3.1 Reversible Integer Discrete Cosine Transform

From the formula (1), the four-point discrete cosine

transform matrix can be easily got as follows.

0.5000 0.5000 0.5000 0.5000

0.6533 0.2706 -0.2706 -0.6533

0.5000 -0.5000 -0.5000 0.5000

0.2706 -0.6533 0.6533 -0.2706

A

As we can see, the elements of the matrix are values

between -1 and 1, and this will result in float-point

transform coefficients. By calculating we can proved that

the determinant of DCT kernel matrix is equal to 1, so the

matrix factorization theory can be applied to it. Table I

tabulates the factorization result.

Now we take upper TERM Uto illustrate how to realize

reversible integer to integer transform. Suppose ,m nuU ,

then Y = UX can be realized as follows:

, ,1

,

1, 2, , 1.

N

N

m m m m m n nn m

N N N

y u x u x

y u x

where m N

(4)

And its inverse transform is:

,

,1,

/

1

1, 2, ,1.

N N N N

N

m m m n nn mm m

x y u

x y u xu

where m N N

(5)

Where

denotes rounding to the nearest integer.

Obviously, lower TERM can realize reversible integer to

integer transform in the same way and arbitrary point

integer DCT could be realized using the property of TERM.

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3.2 Hierarchical RDCT with integer pre- and post-

filters

In our scheme, pre-filter is added to the input signals before

RDCT acting as a flattening operator to make pixels in one

block more homogeneous to improve the efficiency of

compacting energy; while post-filter is added to the

reconstructed signals from inverse RDCT with the function

of de-blocking [5]. The same matrix factorization method is

used to the pre- and post-filter to realize reversible integer

to integer transform. The general formula of pre-filter [5]

can be defined as:

1

2

I J I 0 I JF

J -I 0 V J -I, (6)

where Iand J are identity matrix and reversal identity

matrix respectively. Vis free control matrix.

/ 2 / 2( )II T IV

M S MV J C D C J , (7)

where/ 2

II

MC and

/ 2

IV

MC stand for / 2M point type-II and

type-IV DCT matrix respectively; ,1, ,1S diag sD is a

diagonal matrix where s is a scaling factor. It should be

noticed that, the determinant of the filter matrix Fdoes not

equal to 1 and should be modified to satisfy det 1F

before factorizing.

In order to combine RDCT with wavelet-based codec,

the transform coefficients should be reorganized into tree

structure before coding [4]. With the transformed

coefficients of sub-band structure in hand, a simple way to

improve the efficiency of transform is to apply another

RDCT in the DC sub-band for further de-correlation, and

this can be defined as hierarchical RDCT. Simulation

results show that two-level RDCT could improve the PSNR

by 0.2~0.5 dB compared with one-level RDCT at the same

rate of the same image.

4. RDCT-BASED LOSSY AND LOSSLESS

ENTROPY ENCODING

Once the DCT coefficients have been organized into tree-

structure, wavelet-based codec can be used here.

An improved SPECK algorithm is designed to encode

transform coefficients. The original SPECK [9] has not

adopted high order arithmetic coding. Our algorithm

improves the SPECK by adopting context model likesJPEG2000 on its arithmetic coding for further reducing

correlations of transform coefficients. In our algorithm,

refinement pass coding adopts 3 contexts, sign coding

adopts 5 contexts, and significant coefficients and blocks

coding adopt 20 contexts.

5. RESULTS AND DISCUSSIONS

We perform experiments on still images using DWT and

RDCT combined with several progressive codec, such as

Said and Pearlmans SPIHT [10] and our coding algorithm.

The DWT in SPIHT codec is replaced by RDCT combined

with integer pre- and post-filters. Also, JPEG2000 which

adopts single layer coding stream has been included in ourexperiments for comparison. The lossy and lossless

compression performances are evaluated by PSNR (peak

signal to noise ratio) and bpp (bits per pixel) respectively.

In our simulation experiments, testing images include Lena,

Barbara, Goldhill, Baboon and Finger, all of which are

gray-scale (8bpp) images with size of 512512.

Table II: Lossy compression, PSNR comparison (in dB)

Reversible Irreversible

RDCT (8 point) RDCT (16 point)bpp

JPEG2000

(5/3) SPIHT OURS SPIHT OURS

JPEG2000

(5/3)

JPEG2000

(9/7F)

Lena 512512,8bpp

1 39.31 39.20 39.48 39.22 39.51 39.86 40.35

0.5 36.32 36.55 36.82 36.73 37.00 36.61 37.28

0.25 33.26 33.52 33.81 33.77 34.08 33.44 34.14

Barbara 512512,8bpp

1 35.81 37.05 37.44 37.53 37.95 36.11 37.17

0.5 30.86 32.55 33.03 33.47 33.87 31.03 32.29

0.25 27.36 28.75 29.20 29.85 30.26 27.41 28.39

Goldhill 512512,8bpp

1 35.88 35.98 36.33 35.99 36.39 36.21 36.59

0.5 32.74 32.93 33.21 32.95 33.26 32.91 33.25

0.25 30.09 30.44 30.69 30.46 30.79 30.31 30.54

Baboon 512512,8bpp

1 28.61 28.93 29.27 29.07 29.40 28.63 29.11

0.5 25.06 25.52 25.76 25.66 25.93 25.20 25.59

0.25 22.81 23.21 23.45 23.30 23.52 22.88 23.15

Finger 512512,8bpp

1 30.54 31.67 32.13 32.19 32.66 30.66 31.64

0.5 26.86 27.84 28.20 28.18 28.58 27.09 27.86

0.25 23.68 24.39 24.76 24.80 25.12 23.82 24.37

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5.1 Evaluation of lossy compression

Table II summarizes the PSNR of benchmark images with

different methods. Both 88 and 1616 RDCT have been

performed in our experiments. Reversible and irreversible

JPEG2000 have been taken for comparison. Images at

different bit rate can be recovered from a single codestream

and one can see that the results of our method are better

than JPEG2000 for most images at most bit rates. The

PSNR of our method based on RDCT on Barbara are about

2.14~3.01 dB and 0.78~1.86 dB higher than the

performances of JPEG2000 using reversible 5/3 filter and

irreversible 9/7F filter respectively. And 16 point DCTperforms about 0.1~0.4 dB better than 8 point DCT with a

little cost of the complexity increment.

Portions of Goldhill and Barbara, reconstructed from

JPEG2000 and our method based on 16-point RDCT at

0.25 bpp are illustrated in Fig.1. Block artifacts have been

reduced due to the pre- and post-filters adopted in our

method and the detailed textures are preserved better than

JPEG2000.

5.2 Evaluation of lossless compression

Table III illustrates the lossless compression results of our

method, JPEG2000, and JPEG-LS on testing images. It can

be seen that our method based on RDCT outperforms

JPEG2000 (5/3 filter) and JPEG-LS for most images.

6. CONCLUSIONS

In this paper, we present a progressive lossy to lossless

image compression scheme based on hierarchical RDCT.Simulation results show that the new scheme performs well

in both reversible lossy and lossless compression. Besides,

block transform can be implemented parallel as a result of

fast computing compatibility compared with DWT.

7. REFERENCES

[1] P. Hao and Q. Shi, Matrix factorizations for reversible

integer mapping, IEEE Trans. Signal Processing, vol.49,

pp.2314-2324, Oct. 2001.

[2] Y. Chen, S. Oraintara, and T. Nguyen, Integer discrete

cosine transform (IntDCT), inProc. 2nd Int. Conf. Inform.,

Commun. Signa. Process, Dec. 1999.

[3] G.C.K. Abhayaratne, Reversible integer-to-integermapping of N-point orthonormal block transforms. Signal

Processing,v.87 n.5 pp. 950969,2007

[4] Z. Xiong, O. Guleryuz, and M. T. Orchard, A DCT-based

embedded image coder, IEEE Signal Processing Lett., vol.

3, pp. 289290, Nov. 1996.

[5] T. D. Tran, J. Liang, and C. Tu, Lapped transform via time-

domain pre- and post-processing, IEEE Trans. Signal

Process, vol. 51, no. 6, pp. 15571571, Jun. 2003.

[6] ISO / IEC JTC1 SC29 WG1 (JPEG / JBIG), FCD 14495,

Lossless and near-lossless coding of continuous tone still

images (JPEG-LS).

[7] ISO/IEC JTC1/SC 29/WG 1 (ITU-T SG8), The JPEG 2000

Still Image Compression Standard.

[8] L. Galli and S. Salzo, Lossless hyperspectral compression

using KLT,IEEE IGARSS, vol.1, pp.313-316, Sept. 2004.

[9] Asad Islam, William A. Pearlman, Embedded and efficient

low-complexity hierarchical image coder, inProc. Visual

Communications and Image Processing'99, San Jose, CA,

USA, vol.3653, pp. 294-305, 1998.

[10] A. Said, W. A. Pearlman, A new, fast and efficient image

codec based on set partitioning in hierarchical trees, IEEE

Trans. Circuits and Systems for Video Technology, vol.6,

no.3, pp. 243-250, 1996.

Table III: Lossless Performance (in bpp)

OURSJPEG2000

(5/3)JPEG-LS

8 point 16 point

Lena 4.316 4.243 4.341 4.317

Barbara 4.786 4.863 4.689 4.558

Goldhill 4.837 4.712 4.854 4.816

Baboon 6.111 6.038 5.968 5.919

Finger 5.665 5.663 5.535 5.415

(a) Goldhill, 30.79dB (b) Barbara, 30.26 dB

(c) Goldhill, 30.09 dB (d) Barbara, 27.36 dB

(e) Goldhill, 30.54 dB (f) Barbara, 28.39 dB

Fig.1. Portion of Reconstructed Images at 0.25 bpp, our method (top)

versus JPEG2000 5/3 filter (middle) based on reversible transform and

9/7F filter (bottom).

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