Study and implementation of various image compression algorithms.

1. MAJOR PROJECT
STUDY AND IMPLEMENTATION OF IMAGE
COMPRESSION ALGORITHMS
SHIVAM SHRIVASTAVA
Jaypee Institute of Information Technology
CSE
9910103565

2. SHORT DESCRIPTION
AIM
Study and implementation of image compression algorithms.
Need for compression
Image storage requirements are a function of dimensions and
color depth. Consider an image of dimensions 512x512, with 8 bit
color depth. The size of this image in bits multiplied to the color
depth in bits multiplied by the the image size:
Size=512x512x8 bits = 262144 bytes = 256 Kbytes
The objective of image compression is to reduce irrelevance and
redundancy of the image data in order to be able to store or
transmit data in an efficient form.

3. SIMPLE APPROACH
APPROACH
In order to reduce the size of video we need to convert video into
subsequent frames(images). Images are reduced by reducing the pixel
value of the image .
Need to apply such an algorithm which extracts pixel from image in
form of matrix coefficients of the pixel and reduce it in such a way that
it eliminates unwanted pixel values and reduce other such that the
important information is not lost of the image.
METHODS
1) Discrete Cosine Transform (DCT)
2) Discrete Wavelet Trasform (DWT)
3) Run Length Encoding

4. Run Length Coding
• Run-length encoding (RLE) is a very simple form of data compression in
which runs of data (that is, sequences in which the same data value occurs in
many consecutive data elements) are stored as a single data value and count,
rather than as the original run. This is most useful on data that contains many
such runs. Assume that our 15-character string now contains four
different character runs:
AAAAAAbbbXXXXXt
• Using run-length encoding this could be compressed into four 2-byte
packets:
6A3b5X1t
• Thus, after run-length encoding, the 15-byte string would require only
eight bytes of data to represent the string, as opposed to the original
15 bytes. In this case, run-length encoding yielded a compression
ratio of almost 2 to 1.
• Consider a four bit image and its binary representation:

6. RLE works by reducing the physical size of a repeating
string of characters. This repeating string, called a run,
is typically encoded into two bytes. The first byte
represents the number of characters in the run and is
called the run count. In practice, an encoded run may
contain 1 to 128 or 256 characters; the run count usually
contains as the number of characters minus one (a value
in the range of 0 to 127 or 255).

7. • The second byte is the value of the character in the run, which is
in the range of 0 to 255, and is called the run value. Run length
encoding (RLE) is used to encode strings of zeros and ones by
the number of repetitions in each string. RLE has become a
standard in transmission. For a binary image, there are many
different implementations of RLE; one method is to encode
each line separately, starting with the number of 0's.

8. DISCRETE WAVELET TRANSFORM
• The discrete wavelet transform is a very useful tool for signal analysis and image
processing, especially in multi-resolution representation. It can decompose signal into
different components in the frequency domain. One-dimensional discrete wavelet
transform (1-D DWT) decomposes an input sequence into two components (the
average component and the detail component) by calculations with a low-pass filter
and a high-pass filter. Two-dimensional discrete wavelet transform (2-D DWT)
decomposes an input image into four sub-bands, one average component (LL) and
three detail components (LH, HL, HH) as shown in Figure.

9. • Here in this DWT we are using Haar wavelet transform because The operation for
Haar DWT is simpler than that of any other wavelets. It has been applied to image
processing especially in multi-resolution representation. Harr DWT has the following
important features.
• Haar wavelets are real, orthogonal, and symmetric.
• The high-pass filter and the low-pass filter coefficient is simple.
The procedure goes like this. A low pass filter and a high pass filter are chosen, such
that they exactly halve the frequency range between themselves. This filter pair is
called the Analysis Filter pair. First, the low pass filter is applied for each row of data,
thereby getting the low frequency components of the row. But since the lpf is a half
band filter, the output data contains frequencies only in the first half of the original
frequency range. They can be subsampled by two, so that the output data now contains
only half the original number of samples. Now, the high pass filter is applied for the
same row of data, and similarly the high pass components are separated, and placed by
the side of the low pass components. This procedure is done for all rows..

10. • Next, the filtering is done for each column of the intermediate data. The
resulting two-dimensional array of coefficients contains four bands of data,
each labelled as LL (low-low), HL (high-low), LH (low-high) and HH (high-
high). The LL band can be decomposed once again in the same manner,
thereby producing even more subbands. This can be done upto any level,
thereby resulting in a pyramidal decomposition as shown below.

11. Approximation
Vertical
horizontal
diagonal

12. 1) Matrix (a) shows detail coefficients of original image
2) Matrix (b) show detail coefficients after row operation
3) Matrix (c) show detail coefficients after column operation

15. • Use of high pass and low pass filters include averaging and differences

16. The image block represented by 8x8 matrix

18. • Now we choose a small value of delta such that value below it will get zero.
This will not affect in much loosing of data of the image .
here -5<=delta<=5

19. INVERSE DWT
• The Inverse DWT of an image
Just as a forward transform to used to separate the image data into various
classes of importance, a reverse transform is used to reassemble the various
classes of data into a reconstructed image. A pair of high pass and low pass
filters are used here also. This filter pair is called the Synthesis Filter pair. The
filtering procedure is just the opposite - we start from the topmost level, apply
the filters column wise first and then row wise, and proceed to the next level,
till we reach the first level.

22. YCbCr
The YCbCr colour space and its variations (sometimes referred
to as YUV) is a popular way of efficiently representing colour
images. Y is the luminance (luma) component and can be
calculated as a weighted average of R, G and B:
Y = kr R + kgG + kbB
where k are weighting factors.
In the YCbCr colour space, only the luma (Y ) and blue and red
chroma (Cb, Cr) are transmitted. YCbCr has an important
advantage over RGB, that is the Cr and Cb components may be
represented with a lower resolution than Y because the HVS is
less sensitive to colour than luminance.

23. YCbCr Sampling Formats:
4:4:4 sampling means that the three components (Y, Cb and
Cr) have the same resolution and hence a sample of each
component exists at every pixel position.
4:2:0 sampling is widely used for consumer applications such
as video conferencing, digital television and digital versatile
disk (DVD) storage.
Each colour difference component contains one quarter of the
number of samples in the Y component, 4:2:0 YCbCr video
requires exactly half as many samples as 4:4:4 (or R:G:B)
video.

24. Discrete Cousine Transform(DCT)
-Organize information by order of importance to the human visual system
-Used to compress small blocks of an image (8 x 8 pixels in our case)
-We will exploit the fact that the DCT matrix is based on our visual system for
the purpose of image compression.
-This means we can delete the least significant values without our eyes noticing
the difference
The Discrete Cosine Transform (DCT) operates on X, a block of N × N
samples (typically image samples or residual values after prediction) and
creates Y, an N × N block of coefficients. The action of the DCT (and its
inverse, the IDCT) can be described in terms of a transform matrix A. The
forward DCT (FDCT) of an N × N sample block is given by: Y = AXA’ and
the inverse DCT (IDCT) by: X = A’YA where X is a matrix of samples, Y is
a matrix of coefficients and A is an N × N transform matrix.
The output of a two-dimensional FDCT is a set of N × N coefficients
representing the image block data in the DCT domain and these coefficients
can be considered as ‘weights’ of a set of standard basis patterns. The basis
patterns for the 4×4 and 8×8 DCTs are shown in respectively and are
composed of combinations of horizontal and vertical cosine functions. Any
image block may be reconstructed by combining all N × N basis patterns,
with each basis multiplied by the appropriate weighting factor (coefficient).

25. 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.
In the image compression standard, each DCT coefficient is
quantized using a weight that depends on the frequencies for that
coefficient. The coefficients in each 8 x 8 block are divided by a
corresponding entry of an 8 x 8 quantization matrix, and the result
is rounded to the nearest integer.
Quantisation may be used to reduce the precision of image data
after applying a transform such as the DCT or wavelet transform
removing remove insignificant values such as near-zero DCT or
wavelet coefficients. The forward quantiser in an image or video
encoder is designed to map insignificant coefficient values to zero
whilst retaining a reduced number of significant, nonzero
coefficients.

26. Quantization cont..
Use Quantization Matrix (Q)
qkl = 8p(k + l + 1) for 0 < k, l < 7
Q = p * 8 16 24 32 40 48 56 64
16 24 32 40 48 56 64 72
24 32 40 48 56 64 72 80
32 40 48 56 64 72 80 88
40 48 56 64 72 80 88 96
48 56 64 72 80 88 96 104
56 64 72 80 88 95 104 112
64 72 80 88 96 104 112 120
-p is called the loss parameter. It acts like a “knob” to
control compression.
The greater p is the more you compress the image.

27. Luminance: Chrominance:
QY =
p { 16 11 10 16 24 40
51 61
12 12 14 19 26 58
60 55
14 13 16 24 40 57
69 56
14 17 22 29 51 87
80 62
18 22 37 56 68 109
103 77
24 35 55 64 81 104
113 92
49 64 78 87 103 121
120 101
QC =
p{ 17 18 24 47 99 99
99 99
18 21 26 66 99 99
99 99
24 26 56 99 99 99
99 99
47 66 99 99 99 99
99 99
99 99 99 99 99 99
99 99
99 99 99 99 99 99
99 99
99 99 99 99 99 99
99 99

28. ERROR IMAGE
At last find the error image by subractind the original image
from the reconstructed image. We can laso find out the mean
square error of the output image.

29. Comparison between algorithm
• For DCT technique we can achieve the Cr=1.6 compression
ratio. For DWT technique we can achieve the Cr=1.9 to 2.3
compression ratio. Both techniques have its’ own advantage and
disadvantage. But, both techniques are quite efficient for image
compression. We can get quite reasonable compression ratio
without loss of much important information.

30. • Though our experiments show that DWT technique is much
efficient than DCT technique in quality and efficiency wise. But
in performance time wise DCT is better than DWT .When we
deal with compression, the use of the DWT have shown the
ability to be more robust than the DCT. The introduction of the
DWT is made to deal with the issue related to non-stationnary
signals. In this case, the use of the DWT is more appropriate
and one can extract the existing frequencies .

33. • In RLE there is a problem with bit planes ie. that small changes
of gray value may cause significant changes in bits. For
example, the change from value 7 to 8 causes the change of all
four bits, since we are changing the binary strings 0111 to 1000.
For RLE to be effective, we should hope that long runs of very
similar grey values would result in very good compression rates
for the code.

34. • To overcome this difficulty, we may encode the grey values
with their binary Gray codes. A Gray code is an ordering of all
binary strings of a given length so that there is only one bit
change between a string and the next. Hence in RLE the
complexity increases and there is risk of loosing data . RLE can
be used in DCT algorithm after Quantization process to
represent the quantized coefficients .

35. CONCLUSION
• DWT and DCT algorithm can be successfully applied for image
compression.
• We can decompose upto certain level in DWT.
• Haar wavelet can be applied to image processing especially in
multi-resolution representation.
• Concept of RLE is successfully studied.
• DWT is the best algorithm.