A Fast Fractal Image Encoding Based On Haar Wavelet Transform

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A Fast Fractal Image Encoding Based On Haar Wavelet Transform Powered By Docstoc
					                                                          (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-




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                                                                                                      ISSN 1947-5500
                                                                   (IJCSIS) International Journal of Computer Science and Information Security,
                                                                   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.




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                                                                                                                           ISSN 1947-5500
                                                                (IJCSIS) International Journal of Computer Science and Information Security,
                                                                Vol. 8, No. 8, November 2010




   •    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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                                                                                                                      ISSN 1947-5500
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                                                                   Vol. 8, No. 8, November 2010



                                                                                   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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                                                                                                                     (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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                                                                       35                                     http://sites.google.com/site/ijcsis/
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                                                                    (IJCSIS) International Journal of Computer Science and Information Security,
                                                                    Vol. 8, No. 8, November 2010



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