An Efficient Vector Quantization Method for Image Compression with Codebook generation using Modified K-Means

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

An Efficient Vector Quantization Method for
Image Compression with Codebook generation using
Modified K-Means
S. Sathappan, Associate Professor of Computer Science, Erode Arts and Science College, Erode-638 009.
Tamil Nadu. India. E-Mail : devisathappan@yahoo.co.in

Abstract— With the growth of internet and multimedia, the codebook and produces the quantized image, called as
compression techniques have become the thrust area in the fields of reconstructed image. In order to attain low bit rate, many VQ
computers. Image compression is a technique of efficiently coding schemes, have been used in the past literature such as side-
digital image to reduce the number of bits required in representing match VQ (SMVQ) [6], classified SMVQ (CSMVQ) [7] and
image. Many image compression techniques presently exist for the
Gradient based SMVQ (GSMVQ) [8].
compression of different types of images. In this paper Vector
Quantization based compression scheme is introduced. In this
Codebook
scheme a low bit rate still image compression is performed by
compressing the indices of Vector Quantization and residual C={c1,c2..cn}
codebook is generated. The indices of VQ are compressed by
exploiting correlation among image blocks, which reduces the bit per
index. A residual codebook similar to VQ codebook is generated that
LBG / Modified
represents the distortion produced in VQ. Using this residual K-Means
codebook the distortion in the reconstructed image is removed, VQ Indices
thereby increasing the image quality. The proposed technique Image Blocks
combines these two methods and by replacing the Modified k-means Encoder
algorithm for LBG in the codebook generation. Experimental results
on standard image Lena show that the proposed scheme can give a
reconstructed image with a higher PSNR value than all the existing (a)
image compression techniques.
Codebook
Keywords—Image compression, Vector Quantization, Residual
Codebook, Modified K-Means C={c0,c2..cN-1}

I. INTRODUCTION LBG / Modified
K-Means

I MAGE compression is a method of efficiently coding
digital image, to reduce the number of bits required in
representing image. Its purpose is to decrease the storage
Indices
VQ
Decoder
Reconstructed Image

space and transmission cost while maintaining good quality.
VECTOR Quantization [1] has been found to be an efficient (b)
technique for image compression in the past decade. VQ
compression system mainly contains two components: VQ Fig. 1 (a) VQ Encoder (b) VQ Decoder
encoder and decoder as shown in Fig.1.
In VQ technique [2] [3], the input image is partitioned into Even though, SMVQ, CSMVQ, GSMVQ and JSPVQ are
a set of non-overlapping image blocks the low bit rate schemes, they require high encoding time than
that of VQ method. In this paper, an efficient low bit rate
of size 4x4 pixels each and a clustering algorithm, for example
image compression scheme is proposed based on VQ that
Linde–Buzo–Gray (LGB) algorithm [5] and Modified K-
makes use of compression of indices of VQ and residual
Means [2]. The Modified K-Means algorithm is used in the
codebook with modified k-means clustering instead of LGB.
proposed technique, to generate a codebook
This scheme attains low bit rate and better image quality than
for the given set of image blocks. The
previous VQ methods.
codebook C comprises a set of representative image blocks
called codewords. The VQ encoder discovers a closest match The rest of the paper is presented as follows: in section II,
codeword in the codebook for each of the image block and the the literature survey is presented. In section III the proposed
index of the codeword is transmitted to VQ decoder. The compression scheme is described. Performance of the
decoder phase has the following functionalities. VQ decoder proposed system is evaluated in section IV and section V
replaces the index values with the respective codewords from concludes the paper.

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II. LITERATURE SURVEY et al. applied IVQ to image restoration [13], where the encoder
does not need a codebook except for some parameters. The
Somasundaram et al, [1] presented a novel vector codebook at decoder is learned on image pairs consisting of an
quantization approach and also explained various VQ original image and its diffraction-limited counterpart. Several
schemes, such as side-match VQ (SMVQ), classified SMVQ follow-up work is reported in [14][15].
(CSMVQ) and Gradient based SMVQ (GSMVQ). In which
SMVQ uses the high correlation existing between neighboring The goal of quantization is to encode the data from a
blocks to achieve low bit rate and master codebook C is used source, with some loss, so that the best reproduction is
to encode image blocks in the first column and first row in obtained. Vector quantization (VQ) achieves more
advance. The other image blocks are encoded, utilizing the compression then scalar quantization [14], making it useful for
correlation with the neighboring encoded image blocks. Let x band-limited channels. The algorithm for the design of optimal
be the input image block for the compression system, and u VQ is commonly referred to as the Linde-Buzo-Gray (LBG)
and l be the upper and left neighboring codewords algorithm, and it is based on minimization of the squared-error
respectively. Let the size of the given image block size be k = distortion measure. The LBG algorithm starts with an initial
m× n. The side-match distortion of a codeword Y can be codebook and iteratively partitions the training sequence into
defined as: the Voronoi regions to obtain a new codebook that produces a
(1) lower distortion. Once the final codebook is got, it can be used
on new data outside the training sequence with the optimum
nearest neighbor rule. If the training sequence is sufficiently
long, it yields good performance for future data produced by
According to their side-match distortions of all codewords
the source.
SMVQ sorts the codewords and then selects NS codewords
with smallest side-match distortions from the master book C
In the paper by M.Antonini, et al. [18], images have been
of size N to form the state codebook SC, where NS < N. A
coded using two-level wavelet decomposition with VQ of the
best-match codeword Yi is selected to encode an image block
resulting coefficients. A multiresolution codebook has been
x from NS codewords and the corresponding index is coded in
designed with noise-shaping bit allocation among the various
log2NS bits. Thus, the SMVQ reduces the bit rate of VQ. Since
subbands. The test image has been coded at a rate of 0.78bpp,
mean square error caused by state codebook is higher than that
achieving a PSNR of 32.1dB. In the paper by Gersho and
of master codebook, SMVQ degrades the image quality and
Ramamurthy [19], images have been compressed using
also it requires long encoding time. Classified side-match
unstructured VQ, achieving a bitrate of 0.5 – 1.5bpp. Ho and
vector quantization [7] (CSMVQ) is an efficient low bit rate
Gersho [19] have used multistage VQ for progressive image
image compression technique which produces relatively high
coding, with a PSNR of 30.93dB at 0.36bpp using 4 stages.
quality image. It is a variable rate SMVQ and makes use of
R.L.Claypoole et al. [21] have coded images using nonlinear
variable sized state codebooks to encode the current image
wavelet transform via lifting and obtained 30.5dB at 0.6bpp.
block. The size of the state codebook is decided based on the
An adaptive lifting scheme with perfect reconstruction has
variances of left codewords and upper codewords that predict
been used in [21].
the block activity of the input blocks. Also, CSMVQ uses two
master codebooks, one for low detail blocks and another for
high detail blocks. Another variant, gradient-based SMVQ [8]
III. METHODOLOGY
(GSMVQ) has been proposed, in which gradient values are
used instead of variance values to predict the input vector.
The compression scheme consists of two components,
Another low bit rate VQ, called Jigsaw-puzzle vector
compression of indices and generation of residual codebook.
quantization (JPVQ) [9] was proposed,in which an input block
These two are explained in this section.
can be coded by the super codebook, the dynamic codebook or
the jigsaw-puzzle block. The jigsawpuzzle block is
3.1. Compression of Indices
constructed dynamically using four-step side match prediction
The Index compression step has the following procedure.
technique.
When the image blocks are vector quantized, there likely to
exist high correlation among the neighboring blocks and hence
Interpolative vector quantization, first proposed explicitly
among the corresponding codeword indices. Therefore, if
by Gersho in [12], introduces dimension reduction to
indices are coded by comparing with the previous indices,
traditional VQ. The codebook in the encoder is learned on
further reduction in the bit rate can be achieved. In Search
downsampled vectors and the codebook in the decoder on
Order Coding (SOC) [11], a simple searching scheme is
high-dimension vectors. Except for the difference on
followed to find a match for the current index from the
dimension, the two codebooks have the same number of
previous indices. The search order SO is defined as the order
representative vectors and structure. VQ encoder maps down
in which the current index is compared with the previous
the sampled inputs to a set of scalar indices and VQ decoder
indices.
reproduces high-dimension inputs by received indices. David

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8 6 9 12 1 Get the first index generated by the VQ encoder and transmit
Current
index as such.
7 3 2 4 10
2. Get the next index generated by VQ Encoder. Compare this
5 1 index with the previous indices according SO
3. if SO = 1, code it as “00” and go to the step 2
else
if SO = 2, code it as “01” and go to the step 2
else go to the next step.
4 if index value ≤ (ind_val (SO = 1) + J) and
index value ≥ ─ (ind_val (SO = 1)+J)
Fig. 2 Searching Order { if ind_val (SO =1) = ind_val (SO=2)
code it as “10” followed by log (2 * ) 2 J bits
The SO used in [11] is given in Fig.2. The label “1” else
indicates the highest searching priority, “2” denotes the second code it as “100” followed by log (2 * ) 2 J
highest priority and so on. In order to limit comparisons of the bits. }
current index with previous indices, the searching range (SR) go to step 2.
is fixed. The SR is defined as the number of previous indices else
to be compared with current index. In this paper Search Order if index value ≤ (ind_val (SO = 2) + J) and
[11], SR is taken as 10, which gives the lower bit rate than index value ≥ ─ (ind_val (SO = 2)+J)
other SR. code it as “101” followed by log (2 * ) 2 J bits
In this method, the index of the codeword of a block is and go to step 2.
encoded exploiting the degree of the similarity of the block else
with previously encoded upper or left blocks. When the degree code it as “11” followed by its original index and
of similarity of the current block with one of the two goto step 2.
previously encoded blocks is high, the index of the codeword
of the block is encoded using the index of the neighboring Decoding of the compressed indices is done by reversing the
codeword. I.e. the codeword index of the current block and above coding steps.
that of the neighboring blocks are same. If the degree of
similarity of the block with the neighboring blocks is not high, 3.2. Construction of Residual Codebook (RC)
it is assumed that the closest match codeword of the current The residual codebook can be represented
block may be nearer to the codewords of the neighboring as . Residual codebook is
blocks. For example, if one of the two neighboring blocks
constructed using absolute error values caused by VQ method,
codeword’s index is ‘N’, the closest match codeword of the
in the residual codebook construction, the image blocks that
block to be encoded may lie between (N-J) th codeword and
are less similar to their closest match codewords found in the
(N+J) th codeword in the codebook, where J is any arbitrary
codebook are taken into account. Less similarity blocks will
number. So the index can be coded in log2( 2*J ) bits. This
increase distortion than high similarity blocks in the
idea is based on the property existing in the codebook design
reconstructed image. Residual codeword (RYi) for a less
using LBG algorithm with splitting technique. In the splitting
similarity image block is constructed by comparing it with its
technique, bigger size codebook is generated by splitting each
closest match codeword. The collection of residual codewords
codeword of the smaller codebook into two. The size of the
RYi, RYi+1… is called residual codebook. Similarity of an
codebook is always in powers of two (2M → 2(M+1)). Hence,
image block x with its closet match codeword Yi is determined
relatively similar two image blocks may have same closest
based on minimum distortion rule (α) between them. If the
match codeword in Jth position at codebook of size 2M and at
mean square error (α) of an image block is greater than a
codebook of size 2(M+1), one of the two image blocks may have
predefined threshold value (σ), then the block is taken as less
its closest match codeword at Jth place in the codebook and
similarity block.
other block’s codeword may be in (J+1)th place. The other
Let be a k-pixels image block and
non-similar blocks are encoded using their original index
value. In this scheme, examining the resemblance of a block be a k-pixels closest match codeword,
with its left and upper blocks is not required to encode the
then the α is defined as:
index of the block. The above description is the idea behind
(2)
our VQ indices compression scheme. In order to implement
this idea, the index to be coded is compared with previous
indices according to the SO given in Fig.2 and SR is fixed as 2
in this scheme. Let 1, 2,..,12 be the SO and ind_val (1), The steps used for constructing residual codebook are
ind_val (2),..ind_val(12) be the indices values of the SO = given below.
1,2,…12. The following steps is used to encode VQ index.

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Step 1: An image which is to be compressed is
decomposed into a set of non-overlapped image blocks of 4x4
pixels.
Step 2: A codebook is generated for the image blocks
using LBG algorithm.
Step 3: Pick up the next codeword Yt from the codebook
C and find its all closest match less similarity image blocks(X) where pv (*) is the pre assigned value of the corresponding
found out using (2) from the given set of image blocks and bit in the residual sign bit plane and rrv (*) is the respective
construct residual codeword RYt using the following equation. reconstructed residual value of the bit in the residual sign bit
(3) plane.

where k represents the number of elements in the
codeword Yt and the image block Xi respectively and m
denotes the number of less similarity image blocks that are
closer to the codeword Yt.
Repeat the step 3 until no more codeword exists in the
codebook. Since residual codeword RYi is constructed only Fig. 3 Bits encircled are used for prediction
for less similarity image blocks, some of the codewords Yi
may not have their respective residual codewords, i.e; these After reconstructing the residual codeword, each value of
codewords may not have less similarity image blocks. In the residual codeword is added to respective value of the
residual codebook construction, only absolute values of the closest match codeword of the block Since the residual sign
residuals of the less similarity image blocks are used. The sign bit plane for each image block has only eight bits, alternate
information for each less similarity image block is preserved residual values in the residual codeword RYt are dropped and
and is called residual sign bit plane. In encoding phase, for it also reduces the cost of storing residual codebook. The
each less similarity image block, pixels of the block are dropped residual values are predicted from the neighboring
subtracted from the corresponding pixel values of the residual values as given above.
codeword Yi, then sign values (positive or negative) of the
residual values of that block, called residual sign bit plane, are 3.3. Modified K-Means Algorithm to Replace LBG
preserved. To reduce the bits needed for residual sign bit Initial Cluster Centers Deriving from Data Partitioning
plane, only alternate bits are stored and others are dropped The algorithm follows a novel approach that performs data
based on the assumption that there exists correlation among partitioning along the data axis with the highest variance. The
neighboring bits. The bits used for prediction is shown in approach has been used successfully for color quantization [9].
Fig.3. In the decoding process, the bits of the residual sign bit The data partitioning tries to divide data space into small cells
plane of a block are replaced with the respective residual or clusters where intercluster distances are large as possible
values of the residual codeword from the residual codebook and intracluster distances are small as possible.
(RC) with appropriate sign. The residual values of the dropped For instance, Suppose ten data points in 2D data space are
bits are predicted from neighboring residual values using given. The goal is to partition the ten data points into two
following steps. disjoint cells where sum of the total clustering errors of the
two cells is minimal. Suppose a cutting plane perpendicular to
X-axis will be used to partition the data. Let C1 and C2 be the
first cell and the second cell respectively and and be the
cell centroids of the first cell and the second cell, respectively.
The total clustering error of the first cell is thus computed by:
(4)

and the total clustering error of the second cell is thus
computed by:
(5)

Where ci is the ith data in a cell. As a result, the sum of total
clustering errors of both cells is minimal.
The partition could be done using a cutting plane that passes
through m. Thus

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Fig. 4 Graphs depict the total clustering error, lines 1 and 2 represent
the total clustering error of the first cell and second cell, respectively,
Line 3 represents a summation of the total clustering errors of the
(6) first and the second cells

A parabola curve shown in Fig. 4 represents a summation of
Thus
the total clustering error of the first cell and the second cell,
represented by the dash line 2. Note that the lowest point of
the parabola curve is the optimal clustering point (m). At this
(7) point, the summation of total clustering error of the first cell
and the second cell are minimum.
Since time complexity of locating the optimal point m is
O(n2), the distances between adjacent data is used along the X-
m is called as the partitioning data point where |C1| and |C2| axis to find the approximated point of n but with time of O(n).
are the numbers of data points in cluster C1 and C2 Let be the squared Euclidean distance of
respectively. The total clustering error of the first cell can be adjacent data points along the X-axis.
minimized by reducing the total discrepancies between all data If i is in the first cell then . On the one
in first cell to m, which is computed by:
(8) hand, if i is in the second cell then
The task of approximating the optimal point (m) in 2D is
The same argument is also true for the second cell. The total thus replaced by finding m in one-dimensional line.
clustering error of second cell can be minimized by reducing
the total discrepancies between all data in second cell to m,
which is computed by:

(9) Fig. 5 Illustration of the ten data points on a one-dimensional
line and the relevant Dj

The point (m) is therefore a centroid on the one dimensional
where d(ci,cm) is the distance between m and each data in line (as shown in Fig. 5), which yields
each cell. Therefore the problem to minimize the sum of total (10)
clustering errors of both cells can be transformed into the
problem to minimize the sum of total clustering error of all
data in the two cells to m.
Let and a centroidDist can be computed
The relationship between the total clustering error and the
clustering point is illustrated in Fig. 4, where the horizontal-
axis represents the partitioning point that runs from 1 to n (11)
where n is the total number of data points and the vertical-axis
represents the total clustering error. When m=0, the total
clustering error of second cell equals to the total clustering
error of all data points while the total clustering error of first It is probable to choose either the X-axis or Y-axis as the
cell is zero. On the other hand, when m=n, the total clustering principal axis for data partitioning. However, data axis with
error of the first cell equals to the total clustering error of all the highest variance will be chosen as the principal axis for
data points, while the total clustering error of the second cell is data partitioning. The reason is to make the inter distance
zero. between the centers of the two cells as large as possible while
the sum of total clustering errors of the two cells are reduced
from that of the original cell. To partition the given data into k
cells, it is started with a cell containing all given data and
partition the cell into two cells. Later on the next cell is
selected to be partitioned that yields the largest reduction of
total clustering errors (or Delta clustering error). This can be
described as Total clustering error of the original cell – the
sum of Total clustering errors of the two sub cells of the
original cell. This is done so that every time a partition on a
cell is performed, the partition will help reduce the sum of
total clustering errors for all cells, as much as possible.

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The partitioning algorithm can be used now to partition a 5. if (α ≤σ ), the current block is encoded as “0”.
given set of data into k cells. The centers of the cells can then else
be used as good initial cluster centers for the K-means the current block is encoded as “1” followed by
algorithm. Following are the steps of the initial centroid interpolated residual sign bitplane which is computed as
predicting algorithm. described in section 3.2.
1. Let cell c contain the entire data set. 6. Repeat the step 4 until no more blocks exist in the
2. Sort all data in the cell c in ascending order on each image.
attribute value and links data by a linked list for each attribute. The decoding of the compressed images is done by reversing
3. Compute variance of each attribute of cell c. Choose an the above said steps and residual block to be added is
attribute axis with the highest variance as the principal axis for reconstructed for each less similarity block as described in
partitioning. section 3.2.
4. Compute squared Euclidean distances between adjacent
data along the data axis with the highest variance IV. EXPERIMENTAL RESULTS
and compute the To evaluate the proposed technique experiments are carried
5. Compute centroid distance of cell c: out on standard gray scale images using a Pentium-IV
computer running at 1.60 GHz under Windows XP. Three
images of 512 x 512 pixels in size are used. Codebook is
generated using Modified K-Means algorithm for all the
Where dsumi is the summation of distances between the methods. Codebook is also generated with LBG [5] for
adjacent data. comparison. For this scheme, a codebook of size 64 is used.
6. Divide cell c into two smaller cells. The partition Performances of the above algorithms are evaluated in terms
boundary is the plane perpendicular to the principal axis and of bit rate (bits per pixel) and peak signal-to-noise ratio
passes through a point m whose dsumi approximately equals (PSNR) given by:
to centroidDist. The sorted linked lists of cell c are scanned
and divided into two for the two smaller cells accordingly
7. Calculate Delta clustering error for c as the total
clustering error before partition minus total clustering error of
its two sub cells and insert the cell into an empty Max heap where MSE (mean squared error) is defined as:
with Delta clustering error as a key.
8. Delete a max cell from Max heap and assign it as a
current cell.
9. For each of the two sub cells of c, which is not empty,
perform step 3 - 7 on the sub cell. where xi and yi denote the original and the encoded pixel
10. Repeat steps 8 - 9. Until the number of cells (Size of values and n is the total number of pixels in an image. Bit rate
heap) reaches K. including overhead bits (i.e bits need to store codebook) for
11. Use centroids of cells in max heap as the initial cluster different threshold values ranging from 50 to 2000 for Lena,
centers for K-means clustering Camera man and Pepper.

3.4. The Proposed Algorithm The performance of proposed scheme is evaluated with
The proposed scheme combines compression of VQ the existing techniques for different gray- scale images of size
indices and residual codebook. The steps used in this 512x512 and is given in the table I. J is set to 4 for the
compressor are as follows proposed scheme. From table I, it can be observed note that
1. An image which is to be compressed is decomposed proposed method with the modified k-means instead of LBG
into a set of non-overlapped image blocks of size 4x4 pixels. has an improvement in coding the VQ indices.
2. A codebook is generated for the image blocks using
Modified K-Means algorithm. TABLE I
3. Construct a Residual Codebook (as described in section PERFORMANCE OF PROPOSED METHOD VQ WITH CODEBOOK
3.4) for those image blocks (less similarity blocks) whose α is SIZE 64 USING LBG, K-MEANS AND MODIFIED K-MEANS IN
greater than σ . CODING STANDARD GRAY SCALE IMAGES OF SIZE 512 X 512
4. Pick the next image block (current block) and find its EACH
closest match codeword in the codebook. Calculate mean Modified
VQ LBG K-Means
square error α for the image block using equation (2) and Images K-Means
bits/index (bits/index) (bits/index)
index of the codeword is encoded using VQ indices (bits/index)
compression scheme presented in section 3.1. Lena 6 3.92 3.88 3.47
Cameraman 6 4.08 3.99 3.32
Peppers 6 3.72 3.66 3.20

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