MODIFIED HORIZONTAL VERTICAL PARTITION SCHEME FOR FRACTAL IMAGE COMPRESSION Nikolay Ponomarenko1, Vladimir Lukin1, Karen Egiazarian2, Jaakko T. Astola 2 1 Dept 504, National Aerospace University, 17 Chkalova St, 61070, Kharkov, Ukraine; tel/fax + 38 0572 441186, e-mail
[email protected],
[email protected] 2Inst. of Information Processing, Tampere University of Technology, P.O.Box FIN-553, 33101, Tampere, Finland, tel. +358 3 3115 3860, Fax +358 3 3115 3857; e-mails (karen, jta)@cs.tut.fi ABSTRACT A horizontal-vertical partition scheme modification for fractal and hybrid image compression and an algorithm of its optimization are proposed. The proposed partition scheme efficiency analysis for two test images is carried out. The advantages of the proposed partition scheme in comparison to conventional horizontal vertical partition scheme and to Quadtree partition scheme are shown. 1. INTRODUCTION Nowadays lossy compression of still images using fractal techniques is of rather great interest [1-4]. Despite many theoretical aspects of fractal compression have been intensively studied, some practical aspects, in particular, those ones dealing with partition scheme (PS) type selection and its appropriately fast optimization still draw attention of many researchers [2, 4]. This is explained by the fact that it is impractical to perform full search of all possible variants of an image division into range blocks of different size and shapes. This is why, the basic task to be solved is to put forward such methods of image division into range blocks that in appropriate manner satisfy the following two contradictory requirements: 1) to perform PS optimization quite quickly, i.e. using some simple operations instead of full search of all possible partitioning variants; 2) to provide the decoded image quality (for image compression based on the obtained PS) comparable to the quality of decompressed image that have been coded using "optimal" partition schemes. By “optimal” PS we mean such partition scheme that ensures the best PSNR for given compression ratio (CR). However, in fractal image compression it is difficult or practically impossible to give theoretical background of optimality of one or another PS [1]. This is why the efficiency of PS can be evaluated using practical results of its application for many test images. For example, the horizontal vertical PS [4, 5] and the region based PS [6] are considered to be the most efficient ones. However, these types of PS require tremendous computations to optimize them. In practice the requirement to obtain the PS quite quickly is also important. This explains the fact why the investigations having the aim to design new efficient and less time consuming fractal methods of PS-
based image compression continue. Besides, the design of new PS types and methods of their optimization is stimulated by rapid development of hybrid methods of image compression [7,8]. There are many different approaches to selection of the block shape that can be considered as one restriction imposed on the PS structure [1, 2]. Below we consider only the case of rectangular shape blocks since this variant is one of the simplest and it offers favorable preconditions for fast optimization of PS. Our intention is to design an effective method of PS optimization providing minimization of losses at compression stage, to ensure possibility of its integration to hybrid methods of image compression [7,8]. The particular task is to get high speed of image coding and decoding. For these purposes below we design the modified horizontal vertical PS that, to our opinion, satisfies the aforementioned requirements. Its performance characteristics are compared to one of fractal compression methods that imply conventional horizontal vertical and Quadtree partition schemes. 2. CONVENTIONAL HORIZONTAL VERTICAL PARTITION SCHEME One rather simple PS is Quadtree proposed by Jacquin [3]. At the very beginning the PS consists of squares of equal size. Then, during optimization, some squares are recursively divided into four smaller size squares until the required compression quality is achieved. The selection of such square is performed so that its division provides the largest improvement of image compression quality. An example demonstrating the Quadtree PS appearance for image Lenna is presented in Fig.1. Another approach is to divide the blocks horizontally or vertically into two rectangular shape sub-blocks, thus, getting the horizontal vertical (HV) PS [4, 5]. It is then not necessary that these sub-blocks have equal size. The data remembered for each coded block contain the sign whether or not this block has been divided into smaller blocks (one bit). If the block has been divided into two sub-blocks the sign denoting what was the division (horizontal or vertical, one bit) and the index of the corresponding row or column are remembered. For these
indices data the number of required bits depends upon the divided block size. An example that shows the HV PS for image Lenna is represented in Fig.2.
M
lated as
E = ∑ σ i2 N i where M is the number of i =1
blocks in PS, th block, σ
2 i
N i denotes the number of pixels in the iis the variance of pixel values for the i-th
block. Due to the use of the above given criterion E the PS is optimized quite quickly although the image compression quality occurs to be considerably worse than for conventional Quadtree (see numerical simulation results). This obstacle restricts practical application of SHV PS. The SHV partition scheme example is given in Fig.3.
Figure 1: Example of Quadtree PS, Lenna, bpp=0.5
Figure 3: Example of SHV PS, Lenna, bpp=0.5 3. PROPOSED MODIFIED HORIZONTAL VERTICAL PARTITION SCHEME
Figure 2: Example of HV PS, Lenna, bpp=0.5 However, in this case one needs to consider a very large number of possible variants of block division to ensure the best quality of image compression although due to this the HV PS is more flexible and it provides better compression results than Quadtree. The necessity to analyze a great number of possible variants for each block division leads to high computations required. This restricts the practical application of HV PS. One way to accelerate HV PS optimisation (this method has been called simplified horizontal vertical partition scheme - SHV PS) is to use some simple quality criteria [4]. For example while optimizing the PS it is possible to minimize the aggregate estimate of block variance calcu-
An idea we put forward is to design a trade-off variant that requires computation load for PS optimization comparable to Quadtree but provides image compression quality quite close to the reachable limit for conventional HV PS. This means that the number of possible variants of block division should be drastically reduced in comparison to conventional HV PS while the adaptation facilities of this PS have to be remained. To get this compromise we propose a new method that we called modified horizontal vertical (MHV) PS. The restriction imposed on the manner of block division is that at each step the block can be divided horizontally or vertically into only two rectangular shape equal size subblocks. Thus, at each step, it is necessary to consider only two possible variants. Remind, that for Quadtree method one has to analyze only one variant while for conventional HV method the number of variants is equal to (M-1)(N-1) where M and N denote the horizontal and vertical size of the block to be divided. This means that for the proposed method the computational load can increase, at least, by two times in comparison to Quadtree, but it is anyway by tens times less
than for conventional HV method (more detailed results are in section 4). The memory required for remembering the PS for the proposed method is also much smaller than for basic HV PS. In opposite to conventional HV PS, for the proposed MHV PS there is no need to remember the indices of rows or columns the block has been divided. This permits to use the saved memory for additional division of some number of blocks. In turn, this partially compensate the loss of PS flexibility and makes closer the quality of image compression provided by the proposed method and conventional HV PS. One advantage of the proposed method is the following. Suppose that the size of initial square blocks is the power of two (say, 16x16 or 32x32). Then, at all stages of PS optimization the obtained sub-blocks size (horizontal and vertical) are also the powers of two (like 16x8 or 32x16). This provides favorable facilities for integration of the PS obtained by the proposed method in hybrid image compression techniques, in particular, for those that presume the use of methods based on discrete cosine transform (DCT) for which the fast transform algorithms can be applied. Fig. 4 gives an example of the obtained MHV partition scheme.
Table 1. Efficiency of fractal image compression for Quadtree, HV, SHV, and MHV PS Image
Lenna
Barbara
PSNR, dB
bpp Quadtree
HV
SHV
MHV
0,02
22.11
22.85
19,27
22.55
0,1
27.43
28.50
25,19
28.19
0,5
34.38
35.44
32,24
35.13
0,02
20.21
21.01
20,35
20.62
0,1
23.21
23.55
21,05
23.42
0,5
27.41
28.39
25,44
28,37
As seen, the use of the proposed MHV PS for all considered cases ensures PSNR which is by 0.2...1.0 dB better than for the conventional Quadtree. At the same time, the difference in PSNR for the MHV and conventional HV partition schemes is not larger than 0.4 dB. This rather small difference is explained by the following. The memory required for storing the PS data for MHV is considerably smaller than for HV PS. To demonstrate this, Table 2 gives the average number of bits for coding the information for block of PS. This permits to use a larger number of blocks in MHV PS than for HV PS (for the same CR) and, thus, to additionally improve image compression PSNR. Table 2. Average number of bits required for saving PS data in the compressed image for one block of PS (different PS, CR and test images) Image
bpp 0,02
Lenna
Barbara
d) Figure 4: Example of MHV PS, Lenna, bpp=0.5 4. PERFORMANCE ANALYSIS OF IMAGE FRACTAL CONPRESSION BASED ON THE PROPOSED AND CONVENTIONAL HV PARTITION SCHEMES The comparison of the efficiency of fractal image compression methods for the four considered methods of PS optimization has been performed for two test images (Lenna and Barbara, 512x512 pixels both). Three different compression ratios - 400, 80 and 16 corresponding to bit rates 0.02, 0.1 and 0.5 bpp have been considered. The obtained results are presented in Table 1. The decompressed image quality is characterized by PSNR.
Memory, bit/block Quadtree
HV
MHV
1.33
9.32
2.89
0,1
1.33
8.11
2.90
0,5
1.33
6.79
2.90
0,02
1.33
9.43
2.89
0,1
1.33
8.07
2.90
0,5
1.33
6.74
2.90
It is worth noting that the quality of decompressed images in case of MHV PS use in fractal compression occurs to be comparable to the cases when one uses such complicated PS like Region Based [6] for which the blocks (regions) might have arbitrary shapes. To get imagination about computational load of the considered methods we have evaluated the time required for getting the optimized PS. Since the PS optimization time is the smallest for Quadtree PS, in Table 3 we give the ratio of time required for HV and MHV methods with respect to Quadtree method. While estimating the computational load for different PSs we took into account the number of comparison operations needed for partition scheme optimization for fractal image compression techniques. As seen, MHV PS optimization requires 2.6-3.9 times larger time (comparison operations) than Quadtree PS.
The HV PS optimization needs hundreds of times larger computational expenses than Quadtree PS. Table 3. Comparison of time consumption for optimisation of different PSs Image
Lenna
Barbara
bpp
Time ratio with respect to Quadtree
The decompressed image Lenna for the cases of using the HV and MHV PSs is shown in Fig.5. The visual quality of decompressed images is approximately the same although the local distortion peculiarities slightly differ. 5. CONCLUSIONS
HV
MHV
0,02
255.6
3.0
0,1
222.5
2.6
0,5
198.7
2.6
0,02
238.7
3.9
Taking into account the aforementioned advantages of the proposed MHV PS like fast optimization, high quality of decompressed images, simple integration to hybrid methods it can be recommended for practical application in fractal and/or hybrid compression of images based on PS use.
0,1
207.2
2.9
REFERENCES
0,5
183.6
2.5
[1] Fisher Y., Ed., "Fractal Image Compression: Theory and Application". Berlin, Germ.: Springer-Verlag, 1995. [2] Wohlberg B., Jager G., "A Rewiev of the Fractal Image Coding Literature" // IEEE Transactions on image processing,Vol.8, N12, Dec. 1999, pp.17161729. [3] Jacquin A.E., "Fractal image coding based on a theory of iterated contractive image transformations", Proc. SPIE: Vis. Commun. Image Process., M.Kunt, Ed., Lausanne, Switz., vol.1360, pp.227-239, Oct. 1990. [4] Saupe D., Hamzaoui R., Hartenstein H., "Fractal image compression - An introductory overview", in: Fractal Models for Image Synthesis, Compression, and Analysis, D. Saupe, J. Hart (eds.), ACM SIGGRAPH'96 Course Notes. [5] Saupe D. et al., Optimal hierarchical partitions for fractal image compression, in Proc. IEEE International Conference Image Processing, Chicago, IL, October, 1998, Vol. 1, pp. 737-741. [6] Hartenstein H., Ruhl M., Saupe D., "Region-based fractal image compression". IEEE Transactions on Image Processing, 9(7):1171-1184, 2000. [7] R. Hamzaoui, D. Saupe., "Combining fractal image compression and vector quantization", IEEE Trans. on Image Processing, Vol. 9(2), pp.197-208, 2000. [8] N. T. Thao, "A hybrid fractal-DCT coding scheme for image compression", In Proc. ICIP-96 IEEE International Conference on Image Processing, Lausanne, Sept. 1996. [9] Wavelet Image and Video Compression, Edited by Pankaj N. Topiwala, Boston, USA: Kluwer Academic Publishers, 1998. [10] Christopoulos, C.; Skodras, A.; Ebrahimi, T., "The JPEG2000 still image coding system: an overview", IEEE Transactions on Consumer Electronics, Volume: 46 Issue: 4 , Nov. 2000, pp. 1103-1127. [11] Saupe D., "A new view of fractal image compression as convulution transform coding", IEEE Signal Processing Letters 3, 1996.
a)
b) Figure 5: Decompressed image Lenna, bpp=0.1, for HV PS (a), PSNR=28.50 dB, and MHV PS (b), PSNR=28.19 dB
