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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 8, No. 8, November 2010
A fast fractal image encoding based on Haar wavelet
transform
Sofia Douda Abdallah Bagri Abdelhakim El Imrani
Département de Mathématiques et ENIC, Faculté des Sciences et LCS, Faculté des Sciences,
Informatique & ENIC, Faculté des Techniques, Université Hassan 1er, Université Mohammed V, Rabat,
Sciences et Techniques, Université Settat, Morocco. Morocco.
Hassan 1er, Settat, Morocco.
sofia_douda@yahoo.fr
Abstract—In order to improve the fractal image encoding, we blocks were defined based on the edge of the image. Jacobs et
propose a fast method based on the Haar wavelet transform. This al. uses skipping adjacent domain blocks [11] and Monro and
proposed method speed up the fractal image encoding by Dudbridge localizes the domain pool relative to a given range
reducing the size of the domain pool. This reduction uses the block based on the assumption that domain blocks close to this
Haar wavelet coefficients. The experimental results on the test range block are well suited to match the given range block [12].
images show that the proposed method reaches a high speedup Methods based on reduction of the domain pool are also
factor without decreasing the image quality. developed. Saupe’s Lean Domain Pool method discards a
fraction of domain blocks with the smallest variance [13] and
Keywords- Fractal image compression, PIFS, Haar wavelet
in Hassaballah et al., the domain blocks with high entropies are
transform, SSIM index.
removed from the domain pool [14]. Other approaches focused
on improvements of the FIC by tree structure search methods
I. INTRODUCTION [15, 16], parallel search methods [17, 18] or using two domain
Fractal image compression (FIC) is one of the recent pools in two steps of FIC [19]. The spatial correlation in both
methods of image compression firstly presented by Barnsley the domain pool and the range pool was added to improve the
and Jacquin [1-5]. This method is characterized by its high FIC as developed by Truong et al. [20]. Tong [21] proposes an
compression ratio which is achieved with an acceptable image adaptive search algorithm based on the standard deviation
quality [6], a fast decoding and a multi-resolution property. It is (STD). Other approaches based on genetic algorithms are also
based on the theory of Iterated Function System (IFS) and on applied to speed up the FIC [22-23]. In these methods, higher
the collage theorem. Jacquin [3-5] developed the first algorithm speedup factor are often associated with some loss of
of FIC by Local or Partitioned Iterated Function Systems reconstructed image quality. In the present work, a new method
(PIFS) which makes use of local self-similarity propriety in is proposed to reduce the encoding time of FIC using the Haar
real images. In FIC, the image is represented through a wavelet transform. It speeds up the time encoding by
contractive transformation defined by PIFS for which the discarding the smooth domain blocks from the domain pool.
decoded image is approximately its fixed point and close to an The type of these blocks is defined using the Haar wavelet
input image. transform. A high speedup factor is reached and the image
quality is still preserved.
In Jacquin’s algorithm, an input image is partitioned into
non-overlapping sub-blocks Ri called range blocks, the union
of which covers the whole image. Each range block Ri is put in II. THE PROPOSED METHOD BASED ON HAAR WAVELET
corresponding transformation with another part of a different TRANSFORM
scale, called domain block, looked for in the image. The
domain blocks can be obtained by sliding a window of the A. The Haar wavelet transform
same size around the input image to construct the domain pool. The Haar Wavelet Transform (HWT) [24] is one of the
The classical encoding method, i.e. full search, is time simplest and basic transformations from the space domain to a
consuming because for every range block the corresponding local frequency domain and it is a very useful tool for signal
block is looked for among all the domain blocks. Several analysis and image processing. The HWT decompose a signal
methods are proposed to reduce the time encoding. The most into different components in the frequency domain. One-
common approach is the classification scheme [6-10]. In this dimensional HWT decomposes an input sequence into two
scheme, the domain and the range blocks are grouped in a components (the average component and the detail component)
number of classes according to their common characteristics. by applying a low-pass filter and a high-pass filter. In the HWT
For each range block, comparison is made only for the domain of 2D image of size NxN, a pair of low-pass and high-pass
blocks falling into its class. Fisher’s classification method [6] filters is applied separately along the horizontal and vertical
constructed 72 classes for image blocks according to the direction to divide the image into four sub-bands of size
variance and intensity. In Wang et al. [10], four types of range N/2xN/2 (Fig. 1). After one level of decomposition, the low-
30 http://sites.google.com/site/ijcsis/
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Vol. 8, No. 8, November 2010
low-pass sub-band LL is the multiresolution approximation of Thus, an image block D can be determined as belonging to
the original image, and the other three are high frequency sub- smooth or heterogeneous type by using its vertical coefficient
bands representing horizontal, vertical and diagonals edges, WVD and its horizontal coefficient WHD obtained by a
respectively. The LL band is again subject to the same pyramidal HWT at the highest level.
procedure.
The computation of WHD and WVD do not require the
calculation of other wavelets coefficients. Indeed, let D be an
LL HL image block of size 4x4 represented as follows:
LH HH
1 2 3 4
Figure 1. The result of 2D image HWT decomposition. 5 6 7 8
9 10 11 12
This wavelet decomposition can be repeatedly applied on 13 14 15 16
the low-low-pass sub-band at a coarser scale unless it only has
one component as shown in fig. 2.
Then, analyzing the expression of the coefficients WHD and
Let D be a given image block of size NxN. D can be WVD obtained after two pyramidal HWT, allow us to find the
decomposed into one component low-pass signal by a log following simplified formula:
N/log 2 pyramidal HWT. In the case of N=8, D can be
transformed to one component by 3 decomposition (Fig. 2). A−B
WH D = (2)
The LL3 band in level 3 is the multiresolution approximation of 16
LL2 bands in level 2. The coefficients HL3, LH3 and HH3 of the C−D
highest level denote the coarsest edges along horizontal, WVD = (3)
16
vertical and diagonal directions respectively in level 2.
where A = 1 + 2 + 5 + 6 + 3 + 4 + 7 + 8 , B = 9 + 10 + 13 + 14 + 11 + 12 + 15 + 16 ,
C = 1 + 2 + 5 + 6 + 10 + 13 + 14 , D = 3 + 4 + 7 + 8 + 11 + 12 + 15 + 16 .
LL3 HL3 2
HL
LH3 HH3 B. The proposed method
HL1
The proposed method is aimed to reduce the encoding time
LH2 HH2 by reducing the cardinal of the domain pool. As only a fraction
of the domain pool is used in fractal encoding and the set of the
used blocks is localized along edges and in the regions of high
contrast of the image (designed as heterogeneous blocks), it’s
LH1 HH1 possible to reduce the cardinal of the domain pool by
discarding the smooth domain blocks. Therefore, each range
bloc is compared only to the heterogeneous domain blocks.
Figure 2. The result of three level HWT pyramidal decomposition of an This method of reduction of the domain pool is simple since
image block of size 8x8. only few computations are required to calculate the coefficients
WHD and WVD of a domain block D to classify it as smooth or
We will refer to these coefficients obtained at the highest heterogeneous.
level hereafter as WHD for HL3, WVD for LH3 and HH3 for
WDD. The threshold TW can be fixed or chosen in an adaptive
way. Determining TW adaptively allow us to choose the
If both WHD and WVD are small, then the block D tends to speedup ratio. The main idea is to set the thresholds such that a
have less edge structure (smooth block). When a block has high fraction α of the domain pool can be discarded. The value of α
degree of edge structure, either WHD or WVD will be large. If can be one third, tow thirds,... of the domain pool. Due to the
WHD is larger, D will have horizontal edge properties. On the fact that the encoding time depends on the number of
other hand, if WVD is larger, then D will have vertical edge comparisons between range and domain blocks, the speedup
properties. Finally, those blocks with high magnitudes of WHD ratio can be estimated.
and/or WVD are designed as heterogeneous domain blocks. The
type of each block D is determined as follows: The determination of the threshold TW, which depends on
the fraction α of the domain pool to be eliminated, is
if WH D < TW and WVD < TW summarised as follows:
then D is a smooth domain block (1) • For each domain block D, calculate the Haar wavelet
else D is a heterogeneous domain block coefficient WHD and WVD. Set SD=max(|WHD|,
|WVD|).
where |.| denotes the absolute value of its variable and TW is a
threshold. • Sort all the values of SD in increasing order.
31 http://sites.google.com/site/ijcsis/
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• Find S* corresponding to the value of α. Set the
threshold TW =S*. 1
∑ (xi − µx ) )
1 1 1
where µ x = ∑ x i , µ y = ∑ yi , σx = ( N −1 2 2
,
Due to the fact that we apply our method in the case of a N N
quadtree partitioning, we choose different thresholds for every 1
size of the domain blocks. σy = (
1
N −1 ∑ (yi − µy ) )
2 2
, σxy = 1
N −1 ∑ (xi − µx )(yi − µ y )) .
The first steps of the proposed method are as follows:
• Choose a value of α.
C1 and C2 are positive constants chosen to prevent unstable
• Construct the domain pool. measurement when (µ 2 + µ2 ) or (σ2 + σ2 ) is close to zero. They
x y x y
• Compute the Haar wavelet coefficients |WHD| and are defined in [25] as:
|WVD| for each domain block D. C1= (K1L)2 , C2= (K2L)2 (8)
• Determine the threshold TW for each size domain where L is the dynamic range of pixel values (L= 255 for 8-bit
block. gray scale images). K1 and K2 are the same as in [20]: K1=
• Remove from the domain pool the smooth domain 0.01 and K2= 0.03.
blocks. In the present work, we use a mean SSIM (MSSIM) index
to evaluate the overall image quality:
III. EXPERIMENTAL RESULTS M
∑SSIM(xi , yi )
1
The different tests are performed on three 256x256 images, MSSIM(X, Y) = (9)
represented in fig. 3, with 8 bpp on PC with Intel Pentium Dual M
i =1
2.16 Ghz processor and 2 GO of RAM. The quadtree
partitioning [6] is adopted for the FIC. The encoding time is where X and Y are the original and the distorted images
measured in seconds. The rate of compression is represented by respectively; xi and yi are the image contents at the ith local
the compression ratio (CR), i.e. the size of the original image window of size 8x8 and M is the number of local windows of
divided by the size of the compressed image. The speedup the image.
factor (SF) of a particular method can be defined as the ratio of
the time taken in full search to that of the said method, i.e.,
Time taken in full search
SF = (4)
Time taken in a particular method
The image quality is measured by the peak signal to noise
ratio (PSNR) and the structural similarity Measure (SSIM)
index [25].
The PSNR of two images X and Y of sizes N, is defined as
follows:
(a)
2552
PSNR = 10 x log (5)
MSE
where
N
N∑
1
MSE = (x i − yi )2 (6)
i =1
xi and yi are the gray levels of pixel of the original image
and the distorted image respectively.
The SSIM index is a method for measuring the similarity
between two images x and y defined by Wang [25] as follows:
(b)
(2µ x µ y + C )(2σ xy + C2 )
SSIM(x, y) = 1 (7)
(µ2 + µ2 + C1)(σ2 + σ2 + C2 )
x y x y
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TABLE II. THE RESULTS OF THE PROPOSED METHOD FOR BABOUN IMAGE.
Baboun
α
Time CR PSNR MSSIM SF
1 25.47 7.46 32.55 0.8802 1.00
0.9 22.29 7.45 32.55 0.8805 1.14
0.8 21.54 7.45 32.56 0.8805 1.18
0.7 17.07 7.40 32.57 0.8813 1.49
0.6 14.71 7.33 32.63 0.8829 1.73
0.5 12.95 7.33 32.64 0.8831 1.97
(c) 0.4 10.84 7.27 32.65 0.8828 2.35
0.3 7.74 7.22 32.66 0.8827 3.29
Figure 3. Images of size 256 x 256 : Lena (a), Peppers (b) and baboun (c).
0.2 5.40 7.11 32.65 0.8829 4.72
Table 1 and 2 gives the encoding time, the compression 0.1 2.87 6.90 32.57 0.8806 8.87
ratio, the image qualities and the speedup factor measured on
0.08 2.31 6.81 32.48 0.8785 11.03
the three test images for different values of (1-α). The full
search occurs when α=1 and there is no time reduction because 0.06 1.72 6.73 32.43 0.8769 14.81
no domain block is eliminated. The results show that the 0.04 1.39 6.66 32.24 0.8741 18.32
encoding time scales linearly with α as illustrated in fig. 4. For 0.02 0.84 6.42 31.95 0.8693 30.32
values of α between 0.9 and 0.3, there is no degradation in the
image quality. On contrary, the PSNR improves slightly for the 0.008 0.56 6.29 31.67 0.8632 45.48
test images. The SF of 10 causes a drop of PSNR of 0.6 dB, 0.006 0.47 6.21 31.22 0.8533 54.19
0.55 dB and 0.07 dB for Lena, Peppers and Baboun images 0.004 0.45 6.22 31.18 0.8523 56.60
respectively.
TABLE I. THE RESULTS OF THE PROPOSED METHOD FOR LENA AND
PEPPERS IMAGES.
Lena Peppers
α
Time CR PSNR MSSIM SF Time CR PSNR MSSIM SF
1 19.64 10.46 30.92 0.8909 1.00 19.64 10.98 31.91 0.8931 1.00
0.9 18.17 10.41 30.94 0.8915 1.08 16.71 10.98 31.91 0.8931 1.18
0.8 15.32 10.35 30.98 0.8933 1.28 15.18 10.94 31.93 0.8936 1.29
Figure 4. Effect of parameter α on encoding time.
0.7 14.13 10.30 30.99 0.8934 1.39 13.85 10.92 31.92 0.8936 1.42
0.6 11.72 10.19 31.05 0.8948 1.68 11.40 10.81 31.94 0.8938 1.72 The CR decreases slightly when SF≤0.2 (0.99 for Lena,
0.5 10.30 10.11 31.05 0.8947 1.91 9.86 10.65 31.98 0.8958 1.99
0.78 for Peppers and 0.35 for Baboun). When SF increases, the
CR decreases. A higher SF is accompanied with a high
0.4 7.99 9.90 31.06 0.8971 2.46 7.71 10.49 31.99 0.8956 2.55 decrease of CR. This could be explained by the fact that some
0.3 6.40 9.66 31.02 0.8957 3.07 6.05 10.45 31.88 0.8943 3.25 large range blocks could be covered well by some domain
0.2 4.31 9.47 30.88 0.8936 4.56 4.09 10.20 31.77 0.8927 4.80
blocks which were excluded from the domain pool. Therefore,
these large range blocks are subdivided in four quadrants
0.1 2.39 9.09 30.48 0.8891 8.22 2.29 9.88 31.47 0.8877 8.58 resulting in a decrease of CR. For example, when SF≈42 the
0.08 1.92 9.03 30.32 0.8872 10.23 1.94 9.68 31.36 0.8870 10.12 drops of CR are 1.83, 2.16 and 1.17 for Lena, Peppers and
0.06 1.56 9.02 30.13 0.8843 12.59 1.51 9.54 31.13 0.8839 13.01 Baboun respectively. For comparison, the full search reaches a
PSNR of 30.92 dB with a required time of 19.64 seconds for
0.04 1.11 8.81 29.89 0.8805 17.69 1.14 9.29 30.90 0.8811 17.23 Lena image. In the proposed method, the encoding time of
0.02 0.67 8.73 29.53 0.8737 29.31 0.70 9.05 30.43 0.8752 28.06 Lena image is 1.11 seconds while the PSNR is 29.89 dB when
0.008 0.45 8.63 29.24 0.8683 43.64 0.47 8.82 29.95 0.8690 41.79 α=0.04. The speedup factor attains 17.69 with a drop of PSNR
of 1.03.
0.006 0.39 8.50 29.13 0.8655 50.36 0.41 8.76 29.56 0.8631 47.90
0.004 0.36 8.33 29.03 0.8637 54.56 0.36 8.80 29.06 0.8567 54.56
For visual comparison, fig. 5, fig. 6 and fig. 7 shows
examples of reconstructed images encoded using full search
and the proposed method.
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Vol. 8, No. 8, November 2010
(a)
(a)
(b)
(b) Figure 7. Reconstructed image Baboun by full search (a) and by the
proposed method (b) when SF = 18.32.
Figure 5. Reconstructed image Lena by full search (a) and by the proposed
method (b) when SF = 17.69.
Fig. 8, fig. 9 and fig. 10 show that PSNR and MSSIM vary
in the same way according to the encoding time for the test
images.
(a)
(a)
(b)
(b) Figure 8. Encoding time versus PSNR (a) and MSSIM (b) for Lena.
Figure 6. Reconstructed image Peppers by full search (a) and by the
proposed method (b) when SF = 17.23.
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Also the comparison with AP2D-ENT [26] shows that the
proposed method reaches higher SFs with lesser drops of
PSNR and of MSSIM than AP2D-ENT. Indeed for Lena
image, a SF of 43.64 generates a PSNR of 29.24 dB, a CR of
8.63 and a MSSIM of 0.8683. While the SF of 24.76 obtained
by AP2D-ENT, generates a PSNR of 28.63 dB, a CR of 8.84
and MSSIM of 0.8507. Similar improvements are observed for
(a) Peppers and Baboun images.
IV. CONCLUSION
In this study, we propose to reduce the time of fractal image
encoding by using a new method based on the Haar wavelet
transform (HWT). The two HWT horizontal and vertical
coefficients obtained at the last level of pyramidal
decomposition are used to determine the smooth or
(b) heterogeneous type of domain blocks. The proposed method
Figure 9. Encoding time versus PSNR (a) and MSSIM (b) for Peppers. reduces the encoding time by removing the smooth domain
blocks from the domain pool. Experimental results show that
discarding a fraction of smooth blocks has little effect on the
image quality while a high speedup factor is reached.
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