An effieient algorithm for fractal image coding using kick-out and zero contrast conditions

Abstract

In this paper, we propose a fast algorithm for fractal image coding based on a single kick-out condition and zero contrast prediction. The single kick-out condition can eliminate lots of unmatched domain blocks in the early encoding phase. An efficient method based on zero contrast prediction is also proposed, which can determine whether the contrast factor for a domain block is zero or not and compute the corresponding difference between the range block and the transformed domain block efficiently and exactly. The proposed algorithm can achieve the same reconstructed image quality as the exhaustive search, and can greatly reduce the required computational complexity. In addition, this algorithm does not need any preprocessing steps and additional memory for its implementation, and can combine with other fast fractal algorithms to further improve the speed. Experimental results show that the runtime is reduced by about 50% when compared to the exhaustive search method. The runtime can be reduced by about 75% when our algorithm is combined with the DCT inner product algorithm.

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AN EFFIEIENT ALGORITHM FOR FRACTAL IMAGE CODING
USING KICK-OUT AND ZERO CONTRAST CONDITIONS*
Cheung-Ming Lai, Kin-Man Lam" and Wan-Chi Siu
Centre for Multimedia Signal Processing
Department of Electronic and Information Engineering
The Hong Kong Polytechnic University, Hong Kong
ABSTRACT
In this paper, we propose a fast algorithm for fractal image
coding based-on a single kick-out condition and the zero
contrast prediction., The single ' kick-out condition can
eliminate lots of unmatched domain blocks in the early
encoding phase. An efficient method based on the zero
contrast prediction is also proposed, which can determine
whether the contrast factor for a domain block is zero or
not and compute the corresponding difference between the
range block and the transformed domain block efficiently
and exactly. The proposed algorithm can achieve the same
reconstructed image quality as the exhaustive search, and
can greatly reduce the required computational complexity.
In addition, this algorithm does not need any pre-
processing step and additional
implementation, and can combine with other fast fractal
algorithms to further improve the speed. Experimental
results show that the runtime is reduced by ahout 50%
when compared to the exhaustive search method. The
runtime can he reduced by about 75% when our algorithm
is combined with the DCT Inner Product algorithm.
memory for its
1. INTRODUCTION
A significant amount of research work has been done on
fractal image compression recently. Fractal image coding
can provide a highly reconstructed image quality with a
high compression ratio (CR), is independent of resolution,
and has a fast decoding process. Fractal theory was first
presented by Bamsley, and is based on a mathematical
theory called Iterated Function Systems (IFS). Jacquin [I]
proposed the first practical fractal image compression
scheme that relies on the assumption that image
redundancy could he efficiently exploited through self-
transformahility on a block-wise basis. Fractal image
compression is based on the representation of an image by
a set of iterated contractive transformations for which the
' This research work was supported by the HKF'olyU
research grant G-W088.
.. E-mail: enkmlam@polyu.edu.hk.
0-7803-7761-31031517.00 01003 IEEE
reconstructed image is an approximate fixed point and
close to the original image.
The exhaustive search algorithm can obtain the optimal
domain block to represent the range block by searching
exhaustively all the blocks within the domain pool, but this
process suffers from long encoding time and limits its
practical applications. To solve this problem, extensive
research on fast fractal image encoding algorithms [Z-41
has been camed out. However, these techniques reduce the
required computation at the expense of additional memory
and degradation of the reconstructed image quality.
In this paper, we propose an efficient algorithm based on a
single kick-out condition and the zero contrast prediction,
which can greatly reduce the required computation as
compared to the exhaustive search, while maintaining the
same reconstructed image quality. Our proposed kick-out
condition can determine efficiently whether a domain
block is a good representation of a range block, and so
excessive computation can be avoided in the early stage.
With zero contrast prediction, the computation involved is
further reduced. Moreover, the algorithm dues to not
require any pre-processing and extra memory for its
implementation. Our proposed approach can also be
combined with other fast encoding methods to further
speedup the encoding time. Experimental results show that
the runtime can be reduced by about 75% when our
algorithm is combined with the DCT Inner Product
algorithm [SI.
This paper is organized as follows. We briefly describe the
fundamentals of the fractal coding algorithm and review
the DCT Inner Product approaches in Section 2. Section 3
presents our new fractal image compression algorithm
based on a kick-out condition and the zero contrast
prediction. In Section 4, we compare the performance of
our proposed fast algorithm, the full search, the DCT Inner
Product algorithm, and our algorithm combined with the
DCT algorithm. Finally, conclusions are given in Section
5.
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2. REVIEW
COMPRESSION
In the fractal image compression scheme, an image,f;,,,, of
size mr is partitioned into two basic block units: the range
block and the domain block. The range blocks are a set of
non-overlapping image blocks of size knxn, which are
denoted as {R,]3 ={<,,r,2,...,qk]2.
range blocks is N , = ($)x($), and image& is a union of
the range blocks, (R, 1 2 :
OF FRACTAL IMAGE
The number of
Overlapping image blocks off,,
size larger than that of the range blocks are called domain
blocks. A collection of all these domain blocks form a
domain pool. These domain blocks can be obtained by
sliding a window of size I = mxm, where m > n,
throughout the image to construct the domain pool. The
size of a domain block is usually four times that of a range
block, i.e. I = 2nx2n. To encode a range block R, each of
the blocks in the domain pool is scaled to the size of the
range block, and is then compared to R with respect to
intensity offset and contrast parameters, as well as the
isometry transformations. The set of contracted domain
blocks is denoted as {Dip = {d,, ,d,, ..., d,, , where ND is
the number of domain blocks in the domain pool. The
corresponding parameters for the affine transformation ‘T
are determined by minimizing the following equation:
in a domain pool with
E(R,D,)=BR-(s.Dj+oI~J1
,whereo,sE R (2)
In this equation, 0, is the contracted domain block under
an isometry transformation, I denotes a unity vector of
dimension k, sand o are the contrast and offset parameters,
respectively. For a given range block and the
corresponding domain block, these two parameters are
given as follows:
o =+((R,I)-s(D,I))
(3)
The /I 1) is the two-norm and (.,.) is inner product. The
contrast factor should be -I <s< 1
contractivity of the transformation. The domain block
which results in the smallest difference with equation (2)
is then chosen as the hest matched block, and the
corresponding parameters for the transformations { r, 1 i=
1,2 ,..., N , } are encoded and stored. At the decoding phase,
the transformation parameters are recursively applied to an
arbitrary initial image, which wrill then converge to the
fractal image after fewer than 10 iterations.
to ensure the
Truong et al [SI proposed an efficient image coding
algorithm which can produce the same image quality as
exhaustive search. In this method, the image blocks are
first demeaned, and the error function (2) between a range
block and a transformed domain block can be simplified as
follows:
where F and aare the means of the range block and
domain block, respectively, while I is a vector with all 1’s
and of the same dimension as R and D. The most
computational part of (4)
(R - j q , ~
-21). To determine the best matched domain
block for a range block, the isometry transformation
consists four orientations and four reflections of each
domain block. In other words, the error function has to be
computed eight times; once for each of the transformed
domain blocks. Most of the computational cost of this
method comes from the overhead for calculating the inner
product of the range block and the transformed domain
block in computing the error function. in order to reduce
computation involved, [5] proposed using Discrete Cosine
Transform (DCT) to convert the image block to the
frequency domain, which can reduce the number of
computations of the inner product from eight to two. The
other inner products can be obtained by a proper
arrangement of these two inner products.
is the inner product
3. PROPOSED ALGORITHM
Our algorithm uses a kick-out condition in searching for
the best matched domain block to represent a range block.
Those domain blocks in the domain pool satisfying kick-
out condition will be bypassed, so no further computations
will .he needed. For this kick-out condition, we first
convert the full search equation (2) from two parameters,
i.e. the contrasts and the offset 0, to a function which only
contains the contrasts. Based on this formulation, we can
successively eliminate the search space in the domain pool
and thus decrease the computation required to compare a
range block and a transformed domain block. To further
reduce the computation, we propose a simple method to
determine the zero contrast condition. When this condition
occurs, the range block can be coded without performing
any range-domain block matching.
3.1 The Kick-out Condition
From equation (2). the error function can be further
simplified as follows:
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can be written as follows:
E(R,D) = A - szB
(6)
where
The coefficients is limited to the range (-1,l) to ensure
convergence in the decoding process. If A is greater than
B, the maximum error occurs when s=O while minimum
emor occurs when s=l. Therefore, we have:
IfA-B L 0, then
I ) The maximum error occurs when s=O,
A - ~3
= A
2) The minimum error occurs when s-l ,
E,,=A - s 2 ~
= A - B
A=IIRII‘ -+(R,IY
and B=\IDI/’ -+(D,I)’.
E,,=
(7)
(8)
This means that, in finding the hest matched domain
block, the search is performed only if the minimum error
E,,for the domain block under consideration is less than
the current minimum error d , . .
kick-out condition as follows:
Thus, we propose the
E,.= A - B 2 d , .
(9)
Based on ( 9 ) , we propose a fast search algorithm which
can reject dissimilar domain blocks efficiently for a given
range block. In our algorithm, we select the domain blocks
from left to right and top to bottom. The first domain block
D, is considered to be the initial best matched domain
block. The current minimum distance d,. is set to the
distortion E(R.D,) and the search proceeds in the raster
scan order. To determine whether the next candidate
domain block Dz is closer to R than the current best match
D,,
we compute E,,(R,Dz) and compare it to d,,,;”. If
E,,,(R,DZ) is larger than or equal to d,,, it also means that
the condition E(R,D,) 2 d , ,
domain block D2 is therefore rejected. Otherwise, the
actual distortion E(R,D2) is calculated and compared to
dmiv If E(R,D2) L d , , ,
D2 is rejected for the same reason
mentioned above. Otherwise, d , .
and the current best matched domain block is set to 4.
This process is repeated for all the domain blocks Di in the
domain pool to find the best matched one for an input
range block. Based on this kick-out condition, the required
computation for searching the best matched domain block
will be greatly reduced.
is always guaranteed. The
is replaced by E(R,Dz)
3.2 Fast Error Calculation using Zero Contrast
Prediction
I
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[n the implementation, the contrast factor s is encoded
using 5 bits. Therefore, any value of the contrast s falling
within (-0.03125, 0.03125) will be set to zero after
quantization. With (6), as /s/<l, this means that the zero
contrast condition will happen only when A<B:
-
or - 2 ( S I
E
The contrast factors is quantized to 0 if the absolute value
ofs is less than 0.03125, i.e.:
E(R,D) = A - s’B 2 0
0.03125 >
2 (SI
(I 1)
When s is set to zero, the corresponding error is given as
follows:
E(R,D)=A
In this case, the range block can be encoded without
performing any range-domain matching, and their error
can be represented by the constant value A.
(12)
3.3 Combining Other Approaches
Our proposed algorithm can combine with other fast
fractal algorithms to further improve their speed. One
example is the DCT Inner Product [5] approach, which
allows the computation of two inner products only for the
four orientations and four reflections of a domain block.
However, the whole domain pool still has to be considered
in order to obtain the best matched domain block. This
DCT approach can combine with our algorithm to further
improve its speed. In encoding an image, the single kick-
out condition (9) will be checked to reject those dissimilar
domain blocks. Then, zero contrast prediction (1 1) is used
to determine whether the contrast factor is zero or not, and
the corresponding error function can be computed without
performing the range-domain block matching. Therefore,
the required runtime for the algorithm can be further
reduced.
4. EXPERIMENTAL RESULTS
In the experiment, a three-level quadtree partition scheme
with range block sizes of4x4, 8x8 and 16x16 pixels, and a
search grid of one are used. Three popular 512x512
images, Lena, Boat and Goldhill, are used to evaluate the
performance of the proposed algorithm and other
algorithms. The computer used is a Pentium Ill SOOMHz.
The runtimes (in second) for our proposed algorithm and
full search are listed in Table I . We measured the runtimes
of our algorithm based on (i) the single kick-out condition
(i.e. case I), (ii) the zero contrast prediction (i.e. case 2),

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and (iii) the combination of both conditions (i.e. case1 and
2). Experimental results show that our proposed algorithm
considering both conditions can reduce the required
computation by about 50% as compared to the exhaustive
search method. In other words, a large number of domain
blocks are rejected for performing the range-domain block
matching by the kick-out condition, and a number of the
error functions are obtained based on the zero contrast
prediction. Experimental results show that about 52 % of
the domain blocks are rejected by the kick-out condition,
while 6% of the remaining range-domain block matching
can use the zero contrast prediction to compute the
corresponding error functions. This means that the
computational complexity can be reduced by more than
half using our proposed algorithms. In (6), the kick-out
condition has not considered the effect of quantizing the
luminance offset, so we cannot guarantee that the optimal
domain block will he obtained. Therefore, in order to
obtain the hest domain block for representing a range
block, we set a tolerance of 10% more when comparing
the minimum error between a range block and a domain
block of the current minimum error d , , .
we found that the reconstructed image quality will be
equal to the exhaustive search.
With this setting,
The performance of our algorithm combined with the DCT
Inner Product approach was also investigated. The size of
the range blocks is set to 8x8 only. We combined the DCT
approach with the single kick-out condition and the zero
contrast prediction. These combined algorithms were
compared with the baseline method and the DCT Inner
Product method [5] in terms of the encoding time and
PSNR. The experimental results are tabulated in Table 2,
which shows that the runtime of the combined algorithm is
about 25% of the baseline approach and 50% of the DCT
approach. Furthermore, the PSNR based on our algorithm
is the same as that of the baseline method.
4. CONCLUSIONS
In this paper, we propose a single kick-out condition and
the use of zero contrast prediction to speedup the encoding
process. The efficiency of the kick-out condition depends
on how quickly the global minimum error is detected.
Once this global error is found, most of the remaining
domain blocks will be rejected and range-domain block
matching will not be performed. Experimental results
show that the runtime of our algorithm is about 50% of the
exhaustive search. Our algorithm can also be combined
with other fast fractal coding algorithms, such as the DCT
Inner Product, to further improve the speed. The combined
algorithm can reduce the required computation by about
75% as compared to the baseline approach.
Peppers
5. REFERENCES
Time(s)
5340 2413
1521
1319
PSNR
I (dB)
(34.11 134.11 134.11 134.11
[I]
A. E. Jacquin, “Fractal image coding: A review,” Proc.
IEEE,vol. 8 1 . ~ ~ .
1451-1465,Ocf. 1993.
[2] Y. Fisher, Ed.; Fractal Image Compression: Theory and
Application. Berlin, Gemany: Springer-Verlag, 1995.
[3] C. K. Lee and W. K. Lee, “Fast fractal image block coding
based on local variances,” IEEE Trans. Image Processing,
vol. 7, pp. 888-891, lune 1998.
D. Saupe,”Accelerating fractal image compression by multi-
dimensional nearest neighbor search“, Data Compression
Conference, pp. 222-231, March 1995.
(51 T. K TNong, J. H Jeng, I. S. Reed, P. C. Lee, and A. Q. Li,
“A Fast Encoding Algorithm for Fractal Image Compression
Using the DCT Inner product”. IEEE Trans. Image
Processing, vol. 9,No. 4, pp. 529-535, April 2000.
[4]
Table 1. Comparison of the coding results using domain
grid of one and three level quadtree partitioning (16x16,
8x8 and 4x4).
Algonrhms
I
DCT +
1 .... I ~“,
P ~ N R 131.14
I
131.14 131.14
I
I
131.14
I
Table 2. Comparison of the coding results using domain
grid of one and 8x8 range block
I
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