Bg044357364

Er. Abhishek Kaushik et al Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 4( Version 1), April 2014, pp.357-364
www.ijera.com 357 | P a g e
Image Compression Algorithms Using Dct
Er. Abhishek Kaushik*, Er. Deepti Nain**
*(Department of Electronics and communication, Shri Krishan Institute of Engineering and Technology,
Kurukshetra)
** (Department of Electronics and communication, Shri Krishan Institute of Engineering and Technology,
Kurukshetra)
ABSTRACT
Image compression is the application of Data compression on digital images. The discrete cosine transform
(DCT) is a technique for converting a signal into elementary frequency components. It is widely used in image
compression. Here we develop some simple functions to compute the DCT and to compress images. An image
compression algorithm was comprehended using Matlab code, and modified to perform better when
implemented in hardware description language. The IMAP block and IMAQ block of MATLAB was used to
analyse and study the results of Image Compression using DCT and varying co-efficients for compression were
developed to show the resulting image and error image from the original images. Image Compression is studied
using 2-D discrete Cosine Transform. The original image is transformed in 8-by-8 blocks and then inverse
transformed in 8-by-8 blocks to create the reconstructed image. The inverse DCT would be performed using
the subset of DCT coefficients. The error image (the difference between the original and reconstructed image)
would be displayed. Error value for every image would be calculated over various values of DCT co-efficients
as selected by the user and would be displayed in the end to detect the accuracy and compression in the resulting
image and resulting performance parameter would be indicated in terms of MSE , i.e. Mean Square Error.
Keywords- Discrete Cosine Transform, Pixels, Inverse -DCT, Encoding, Decoding, Quantization, Entropy,
MSE.
I. INTRODUCTION
Data compression is the technique to reduce
the redundancies in data representation in order to
decrease data storage requirements and hence
communication costs. Reducing the storage
requirement is equivalent to increasing the capacity
of the storage medium and hence communication
bandwidth. Thus the development of efficient
compression techniques will continue to be a design
challenge for future communication systems and
advanced multimedia applications. Data is
represented as a combination of information and
redundancy. Information is the portion of data that
must be preserved permanently in its original form in
order to correctly interpret the meaning or purpose of
the data. Redundancy is that portion of data that can
be removed when it is not needed or can be reinserted
to interpret the data when needed. Most often, the
redundancy is reinserted in order to generate the
original data in its original form. A technique to
reduce the redundancy of data is defined as Data
compression. The redundancy in data representation
is reduced such a way that it can be subsequently
reinserted to recover the original data, which is called
decompression of the data.
Data compression can be understood as a
method that takes an input data D and generates a
shorter representation of the data c(D) with less
number of bits compared to that of D. The reverse
process is called decompression, which takes the
compressed data c(D) and generates or reconstructs
the data D‟ as shown in Figure 1. Sometimes the
compression (coding) and decompression (decoding)
systems together are called a “CODEC”. The
reconstructed data D‟ could be identical to the
original data d or it could be an approxima-
figure.1 Block Diagram of CODEC
-tion of the original data D, depending on the
reconstruction requirements. If the reconstructed data
D‟ is an exact replica of the original data D, the
algorithm applied to compress D and decompress
c(D) is lossless. On the other hand, the algorithms are
lossy when D‟is not an exact replica of D. Hence as
far as the reversibility of the original data is
concerned, the data compression algorithms can be
broadly classified in two categories – lossless and
lossy. Usually loseless data compression techniques
are applied on text data or scientific data. Sometimes
data compression is referred as coding, and the terms
noiseless or noisy coding, usually refer to loseless
RESEARCH ARTICLE OPEN ACCESS

and lossy compression techniques respectively. The
term “noise” here is the “error of reconstruction” in
the lossy compression techniques because the
reconstructed data item is not identical to the original
one. Data compression schemes could be static or
dynamic. In static methods, the mapping from a set of
messages (data or signal) to the corresponding set of
compressed codes is always fixed. In dynamic
methods, the mapping from the set of messages to the
set of compressed codes changes over time. A
dynamic method is called adaptive if the codes adapt
to changes in ensemble characteristics over time. For
example, if the probabilities of occurrences of the
symbols from the source are not fixed over time, an
adaptive formulation of the binary codewords of the
symbols is suitable, so that the compressed file size
can adaptively change for better compression
efficiency. More recently, the wavelet transform has
emerged as a cutting edge technology, within the
field of image compression. Wavelet-based coding
provides substantial improvements in picture quality
at higher compression ratios.
II. DISCRETE COSINE TRANSFORM
The discrete cosine transform (DCT) helps
separate the image into parts (or spectral sub-bands)
of differing importance (with respect to the image's
visual quality). The DCT is similar to the discrete
Fourier transform: it transforms a signal or image
from the spatial domain to the frequency domain
(Fig 2)
figure.2 Transformation of function into DCT
A discrete cosine transform (DCT)
expresses a sequence of finitely many data points in
terms of a sum of cosine functions oscillating at
different frequencies. DCTs are important to
numerous applications in science and engineering,
from lossy compression of audio (e.g. MP3)
and images (e.g. JPEG) (where small high-frequency
components can be discarded), to spectral for the
numerical solution of partial differential equations.
The use of cosine rather than sine functions is critical
in these applications: for compression, it turns out
that cosine functions are much more efficient (as
described below, fewer are needed to approximate a
typical signal), whereas for differential equations the
cosines express a particular choice of boundary
conditions.
In particular, a DCT is a Fourier-related
transform similar to the discrete Fourier
transform (DFT), but using only real numbers. DCTs
are equivalent to DFTs of roughly twice the length,
operating on real data with even symmetry (since the
Fourier transform of a real and even function is real
and even), where in some variants the input and/or
output data are shifted by half a sample. There are
eight standard DCT variants, of which four are
common.
The most common variant of discrete cosine
transform is the type-II DCT, which is often called
simply "the DCT"; its inverse, the type-III DCT, is
correspondingly often called simply "the inverse
DCT" or "the IDCT". Two related transforms are
the discrete sine transforms (DST), which is
equivalent to a DFT of real and odd functions, and
the modified discrete cosine transforms (MDCT),
which is based on a DCT of overlapping data.
III. RELATED WORK, ISSUES AND
POSSIBLE SOLUTIONS
Apart of so many advantages of Discrete
Wavelet Transform, Discrete wavelet transform
donot justify wide replacement of DCT to avoid
blocking artifacts. Discrete Cosine Transform is a
technique for converting a signal into elementary
frequency components, widely used in image
compression. The rapid growth of digital imaging
applications, including desktop publishing,
multimedia, teleconferencing, and high-definition
television (HDTV) has increased the need for
effective and standardized image compression
techniques. Among the emerging standards are JPEG,
for compression of still images [Wallace 1991];
MPEG, for compression of motion video [Puri 1992];
and CCITT H.261 (also known as Px64), for
compression of video telephony and
teleconferencing. There are many research papers
that propose 1D Discrete cosine transform technique
to modify an image so as to get the compressed data
or image model. [1] shows a discrete wavelet
technique to compress the images using wavelet
theory in VHDL, Verilog. [2] shows FFT approach
for data compression.that its histogram has a desired
shape. [3] shows the lossless image compression
algorithm using FPGA technology. [4] has shown an
image compression algorithm using verilog with area,
time and power constraints. [5] has shown a simple
DCT technique to for converting signals into
elementary frequency components using mathematica
toolbox. [6] shows comparative analysis of various
compression methods for medical images depicting
lossless and lossy image compression. [7] shows
Fourier analysis and Image processing technique. [8]
shows Image compression Implementation using Fast
Fourier Transform. [9] depicts a comparative study of
Image Compression using Curvelet, Ridgilet and
Wavelet Transform Techniques.

JPEG is primarily a lossy method of
compression.JPEG was designed specifically to
discard information that the human eye cannot easily
see. Slight changes in color are not perceived well by
the human eye, while slight changes in intensity
(light and dark) are. Therefore JPEG's lossy encoding
tends to be more frugal with the gray-scale part of an
image and to be more frivolous with the color .DCT
separates images into parts of different frequencies
where less important frequencies are discarded
through quantization and important frequencies are
used to retrieve the image during decompression.
IV. PROBLEM FORMULATION
Wavelet based techniques for image
compression have been increasingly used for image
compression.
The wavelet uses subband coding to
selectively extract different subbands from the given
image.These subbands can then be quantized with
different quantizers to give better compression.The
wavelet filters are specifically designed to satisfy
certain constraints called the smoothness constraints.
The wavelet filters are designed so that the
coefficients in each subband are almost uncorrelated
from the coefficients in other subbands [3]. The
wavelet transform achieves better energy compaction
than the DCT and hence can help in providing better
compression for the same Peak Signal to Noise Ratio
(PSNR). A lot of research has been done on the
performance comparison of the DWT and DCT for
image compression which is less than 1 dB. This
proposed work is to analyse the image compression
algorithm using 2-dimension DCT. According to the
DCT properties, a DC is transformed to discrete
delta-function at zero frequency. Hence, the
transform image contains only the DC component.
To transformed an image into 8 x 8 subsets by
applying DCT in 2 dimension. Also, a subset of DCT
co-efficients have been prepared in order to perform
inverse DCT to get the reconstructed image. The
work to be done is to perform the inverse transform
of the transformed image and also to generate the
error image in order to give the results in terms of
MSE (Mean Square Error), as MSE increases, the
image quality degrades and as the MSE would
decrease, image quality would be enhanced with the
help of changing the co-efficients for DCT Blocks.
V. WORK METHODOLOGY
Image compression is a technique used to
reduce the storage and transmission costs. The
existing techniques used for compressing image files
are broadly classified into two categories, namely
lossless and lossy compression techniques. In lossy
compression techniques, the original digital image is
usually transformed through an invertible linear
transform into another domain, where it is highly de-
correlated by the transform. This de-correlation
concentrates the important image information into a
more compact form. The transformed coefficients are
then quantized yielding bit-streams containing long
stretches of zeros. Such bit-streams can be coded
efficiently to remove the redundancy and store it into
a compressed file. The decompression reverses this
process to produce the recovered image. The 2-D
discrete cosine transform (DCT) is an invertible
linear transform and is widely used in many practical
image compression systems because of its
compression performance and computational
efficiency [10-12]. DCT converts data (image pixels)
into sets of frequencies. The first frequencies in the
set are the most meaningful; the latter, the least. The
least meaningful frequencies can be stripped away
based on allowable resolution loss. DCT-based image
compression relies on two techniques to reduce data
required to represent the image. The first is
quantization of the image‟s DCT coefficients; the
second is entropy coding of the quantized coefficients
[13]. Quantization is the process of reducing the
number of possible values of a quantity, thereby
reducing the number of bits needed to represent it.
Quantization is a lossy process and implies in a
reduction of the color information associated with
each pixel in the image. Entropy coding is a
technique for representing the quantized coefficients
as compactly as possible.
figure.3: Block Diagram For Compression/
Decompression Algorithm
In this paper, a simple entropy encoder
algorithm will be proposed and implemented. It
works on quantized coefficients of the discrete cosine
transform. The basic idea of the new approach is to
divide the image into 8x8 blocks and then extract the
consecutive non-zero coefficients preceding the zero
coefficients in each block. In contrast to the
Run_Length decoder, the output of this encoder
consists of the number of the nono-zero coefficients
followed by the coefficients themselves for each
block. The decompression process can be performed
systematically and the number of zero coefficients
can be computed by subtracting the number of non-
zero coefficients from 64 for each block.
After the reconstructed image is obtained,
the error image is also generated using MATLAB
image processing commands by subtracting the

reconstructed image from the original image and
MSE (main square error) is computed to indicate the
quality of the reconstructed image that is best when
MSE = 0 , that happens when all the 64 co-efficients
are selected in the simulations.
VI. LOSSLESS AND LOSSY IMAGE
COMPRESSION
When hearing that image data are reduced,
one could expect that automatically also the image
quality will be reduced. A loss of information is,
however, totally avoided in lossless compression,
where image data are reduced while image
information is totally preserved. It uses the predictive
encoding which uses the gray level of each pixel to
predict the gray value of its right neighbor. Only the
small deviation from this prediction is stored. This is
a first step of lossless data reduction. Its effect is to
change the statistics of the image signal drastically.
Statistical encoding is another important approach to
lossless data reduction. Statistical encoding can be
especially successful if the gray level statistics of the
images has already been changed by predictive
coding. The overall result is redundancy reduction,
that is reduction of the reiteration of the same bit
patterns in the data. Of course, when reading the
reduced image data, these processes can be
performed in reverse order without any error and thus
the original image is recovered. Lossless compression
is therefore also called reversible compression. Lossy
data compression has of course a strong negative
connotation and sometimes it is doubted quite
emotionally that it is at all applicable in medical
imaging. In transform encoding one performs for
each image run a mathematical transformation that is
similar to the Fourier transform thus separating image
information on gradual spatial variation of brightness
(regions of essentially constant brightness) from
information with faster variation of brightness at
edges of the image (compare: the grouping by the
editor of news according to the classes of contents).
In the next step, the information on slower changes is
transmitted essentially lossless (compare: careful
reading of highly relevant pages in the newspaper),
but information on faster local changes is
communicated with lower accuracy (compare:
looking only at the large headings on the less relevant
pages). In image data reduction, this second step is
called quantization. Since this quantization step
cannot be reversed when decompressing the data, the
overall compression is „lossy‟ or „irreversible‟.
VII. JPEG
The DCT is used in JPEG image
compression, MJPEG, MPEG, DV, and Theora video
compression. There, the two-dimensional DCT-II of
N x N blocks are computed and the results
are quantized and entropy coded. In this case, N is
typically 8 and the DCT-II formula is applied to each
row and column of the block. The result is an 8 × 8
transform coefficient array in which the (0,0) element
(top-left) is the DC (zero-frequency) component and
entries with increasing vertical and horizontal index
values represent higher vertical and horizontal spatial
frequencies.
It incorporated the last advances in the
image compression to provide a unified optimized
tool to accomplish both lossless and lossy
compression and decomposition using the same
algorithm and the bit stream syntax. The systems
architecture is not only optimized for compression
efficiency at even very low bit-rates, it is also
optimized for scalability and interoperability in the
networks and noisy mobile environments. The JPEG
standard will be effective in wide application areas
such as internet, digital photography, digital library,
image archival, compound documents, image
databases, colour reprography (photocopying,
printing, scanning, facsimile), graphics, medical
imagery, mobile multimedia communication, 3G
cellular telephony, client-server networking, e-
commerce, etc.
VIII. QUANTIZATION
DCT-based image compression relies on
two techniques to reduce the data required to
represent the image. The first is quantization of the
image's DCT coefficients; the second is entropy
coding of the quantized coefficients. Quantization is
the process of reducing the number of possible values
of a quantity, thereby reducing the number of bits
needed to represent it. Entropy coding is a technique
for representing the quantized data as compactly as
possible. I have developed functions to quantize
images and to calculate the level of compression
provided by different degrees of quantization.
IX. ENTROPY ENCODING
After quantization, most of the high
frequency coefficients (lower right corner) are zeros.
To exploit the number of zeros, a zig-zag scan of the
matrix is used yielding to Long strings of zeros. The
current coder acts as filter to pass only the string of
non-zero coefficients. By the end of this process we
will have a list of non-zero tokens for each block
preceded by their count. DCT based image
compression using blocks of size 8x8 is considered.
After this, the quantization of DCT coefficients of
image blocks is carried out. After that, entropy
encoding technique is applied to the resulting image.
Entropy encoding is a compression method
based on the idea of allotting fewer bits to represent
colors that occur frequently in an image and more
bits to those that occur infrequently. Shannons

entropy equation, allows us to compare an encoding
to a theoretical optimum.Processes with this principle
are called entropy encoding. There are several of this
kind. Entropy encoding is used regardless of the
media‟s specific characteristics. The data stream to
be compressed is considered to be a simple digital
sequence, and the semantic of the data is ignored.
Entropy encoding is an example of lossless encoding
as the decompression process regenerates the data
completely. The raw data and the decompressed data
are identical, no information is lost.
figure.4 Zigzag Sequence
The general equation for a 1D (N data items)
DCT is defined by the following equation:
(1)
and the corresponding inverse 1D DCT transform is
simple F-1
(u), i.e.:where
(2)
The general equation for a 2D
(N by M image) DCT is defined by the following
equation:
(3)
and the corresponding inverse 2D DCT transform is
simple F-1
(u,v), i.e.:
where
(4)
The basic operation of the DCT is as follows:
 The input image is N by M;
 f(i,j) is the intensity of the pixel in row i and
column j;
 F(u,v) is the DCT coefficient in row k1 and
column k2 of the DCT matrix.
 For most images, much of the signal energy lies
at low frequencies; these appear in the upper left
corner of the DCT.
 Compression is achieved since the lower right
values represent higher frequencies, and are
often small - small enough to be neglected with
little visible distortion.
 The DCT input is an 8 by 8 array of integers.
This array contains each pixel's gray scale level;
 8 bit pixels have levels from 0 to 255.
 Therefore an 8 point DCT would be:
where,
(5)
X. INVERSE DCT
Using the normalization conventions above,
the inverse of DCT-I is DCT-I multiplied by 2/(N-1)
The inverse of DCT-IV is DCT-IV multiplied by 2/N.
The inverse of DCT-II is DCT-III multiplied by
2/N and vice versa.
Like for the DFT, the normalization factor in
front of these transform definitions is merely a
convention and differs between treatments. For
example, some authors multiply the transforms
by √2/N so that the inverse does not require any
additional multiplicative factor. Combined with
appropriate factors of √2 (see above), this can be used
to make the transform matrix orthogonal.
10.1. MULTIDIMENSIONAL DCT’S
Multidimensional variants of the various
DCT types follow straightforwardly from the one-
dimensional definitions: they are simply a separable
product (equivalently, a composition) of DCTs along
each dimension.
For example, a two-dimensional DCT-II of an
image or a matrix is simply the one-dimensional
DCT-II, from above, performed along the rows and
then along the columns (or vice versa). Technically,
computing a two- (or multi-) dimensional DCT by
sequences of one-dimensional DCTs along each
dimension is known as arow-column algorithm (after
the two-dimensional case). That is, the 2d DCT-II is
given by the formula (omitting normalization and
other scale factors, as above):
(6)
figure 5: Two-dimensional DCT frequencies from the
JPEG DCT

As with multidimensional FFT algorithms,
however, there exist other methods to compute the
same thing while performing the computations in a
different order (i.e. interleaving/combining the
algorithms for the different dimensions). The inverse
of a multi-dimensional DCT is just a separable
product of the inverse(s) of the corresponding one-
dimensional DCT(s) (see above), e.g. the one-
dimensional inverses applied along one dimension at
a time in a row-column algorithm.The image to the
right shows combination of horizontal and vertical
frequencies for an 8 x 8 (N1 = N2 = 8) two
dimensional DCT. Each step from left to right and
top to bottom is an increase in frequency by 1/2
cycle. For example, moving right one from the top-
left square yields a half-cycle increase in the
horizontal frequency. Another move to the right
yields two half-cycles. A move down yields two half-
cycles horizontally and a half-cycle vertically. The
source data (8x8) is transformed to a linear
combination of these 64 frequency squares
XI. MSE: MEAN SQUARE ERROR
The phrase Mean Square Error, often
abbreviated MSE (also called PSNR, Peak Signal to
Noise Ratio) is an engineering term for the ratio
between the maximum possible power of a signal and
the power of corrupting noise that affects the fidelity
of its representation. Because many signals have a
very wide dynamic range, MSE is usually expressed
in terms of the logarithmic decibel scale.
The MSE / PSNR is most commonly used as
a measure of quality of reconstruction of lossy
compression codecs (e.g., for image compression).
The signal in this case is the original data, and the
noise is the error introduced by compression. When
comparing compression codecs it is used as
an approximation to human perception of
reconstruction quality, therefore in some cases one
reconstruction may appear to be closer to the original
than another, even though it has a lower PSNR (a
higher PSNR would normally indicate that the
reconstruction is of higher quality). One has to be
extremely careful with the range of validity of this
metric; it is only conclusively valid when it is used to
compare results from the same codec (or codec type)
and same content.
It is most easily defined via the mean
squared error (MSE) which for
two m×n monochrome images I and K where one of
the images is considered a noisy approximation of the
other is defined as:
(7)
The PSNR is defined as:
(8)
Here, MAXI is the maximum possible pixel
value of the image. When the pixels are represented
using 8 bits per sample, this is 255. More generally,
when samples are represented using
linear PCM with B bits per sample, MAXI is 2B−1.
For color images with three RGB values per pixel,
the definition of PSNR is the same except the MSE is
the sum over all squared value differences divided by
image size and by three.
figure.6: Example luma PSNR values for
a cjpeg compressed image at various quality levels
XII.SIMULATIONS AND RESULTS
The window shows the original image,
reconstructed image after applying 2D – DCT and the
computed error image. On the right hand side, top is
shown a slider option to vary the number of
coefficients of 8 x 8 blocks image input for DCT
matrix that could have a value in between 1 to 64.
Apply button is present to finally apply the DCT on
the image selected after setting the number of co-
efficients for the image compression. Also, MSE is
calculated and displayed on the bottom of the images
indicating the quality of reconstructed image.In the
above picture, all the 64 co-efficients have been
selected and error image is obtained zero with mean
square error value i.e. MSE = 0 indicating that best
reconstruction is attained at highest number of co-
efficients has been selected. The window for showing
the results that appear on the screen when running the
code is shown below:

Analysis of Image Compression Algorithm
using DCT is conducted on Six natural images out of
which Pepper Image has been shown before and
after reconstruction along with their obtained MSE.
a) Original Image
After DCT (25%) : DCT Coefficients selected = 16
b) Reconstructed Image c) Error Image
Mean Square Error = 0.000241
After DCT (50%) : Coefficients selected = 32
d) Reconstructed Image e) Error Image
Mean Square Error = 7.62e-005
After DCT (100%) : Coefficients selected = 64
f) Reconstructed Image g) Error Image
Mean Square Error = 0.0
In the above picture, all the 64 co-efficients
have been selected and error image is obtained zero
with MSE = 0 indicating that best reconstruction is
attained at highest number of co-efficients been
selected
XIII. FUTURE SCOPE AND CONCLUSIONS
The results presented in this document show
that the DCT exploits interpixel redundancies to
render excellent decorrelation for most natural
images. Thus, all (uncorrelated) transform
coefficients can be encoded independently without
compromising coding efficiency. In addition, the
DCT packs energy in the low frequency regions.
Therefore, some of the high frequency content can be
discarded without significant quality degradation.
Such a (course) quantization scheme causes further
reduction in the entropy (or average number of bits
per pixel). Lastly, it is concluded that successive
frames in a video transmission exhibit high temporal
correlation (mutual information). This correlation can
be employed to improve coding efficiency.
The aforementioned attributes of the DCT have
led to its widespread deployment in virtually every
image/video processing standard of the last decade,
for example, JPEG (classical), MPEG- 1, MPEG-2,
MPEG-4, MPEG-4 FGS, H.261, H.263 and JVT
(H.26L). Nevertheless, the DCT still offers new
research directions that are being explored in the
current and upcoming image/video coding standards.

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