Image Compression

1
Image Compression
Image compression address the problem of reducing
the amount of data required to represent a
digital image with no significant loss of
information. Interest in image compression dates
back more than 25 years. The field is now poised
significant growth through the practical
application of the theoretic work that began in
1940s , when C.E. Shannon and others first
formulated the probabilistic view of information
and its representation , transmission and
compression.
Claude Elwood Shannon http//www-gap.dcs.st-and.ac
.uk/history/Mathematicians/Shannon.html
2
Image Compression
Images take a lot of storage space -         -
1024 x 1024 x 32 x bits images requires 4
MB -         - suppose you have some video that
is 640 x 480 x 24 bits x 30 frames per second , 1
minute of video would require 1.54 GB
Many bytes take a long time to transfer slow
connections suppose we have 56,000 bps
-       - 4MB will take almost 10
minutes -         - 1.54 GB will take almost 66
hours
Storage problems, plus the desire to exchange
images over the Internet, have lead to a large
interest in image compression algorithms.
3
 The same information can be represented many
ways.
Data are the means by which information is
conveyed. Various amounts of data can be used to
convey the same amount of information. Example
Four different representation of the same
information ( number five) 1) a picture (
1001,632 bits ) 2) a word five spelled in
English using the ASCII character set ( 32 bits)
3) a single ASCII digit ( 8bits) 4) binary
integer ( 3bits)
4
Compression algorithms remove redundancy
If more data are used than is strictly necessary,
then we say that there is redundancy in the
dataset. Data redundancy is not abstract concept
but a mathematically quantifiable entity . If n1
and nc denote the number of information carrying
units in two data sets that represent the same
information, the relative data redundancy RD of
the first data set ( n1 ) can be defined as
RD 1 1/CR (1)
Where CR is compression ration, defined as
CR n1/nc (2)
Where n1 is the number of information carrying
units used in the uncompressed dataset and nc is
the number of units in the compressed dataset.
The same units should be used for n1 and nc bits
or bytes are typically used.
When ncltltn1 , CR? large value and RD? 1. Larger
values of C indicate better compression
5
A general algorithm for data compression and
image reconstruction
Input image ( f(x,y)
Reconstructed image f(x,y)
Source decoder Reconstruction
Source encoder Data redundancy reduction
Channel
Channel encoder
Channel decoder
An input image is fed into the encoder which
creates a set of symbols from the input data.
After transmission over the channel, the
encoded representation is fed to the decoder,
where a reconstructed output image f(x,y) is
generated . In general , f(x,y) may or may not
an exact replica of f(x,y). If it is , the system
is error free or information preserving, if not,
some level of distortion is present in the
reconstructed image .
6
Data compression algorithms can be divided into
two groups
1 Lossless algorithms remove only redundancy
present in the data . The reconstructed image is
identical to the original , i.e., all af the
information present in the input image has been
preserved by compression . 2. Higher compression
is possible using lossy algorithms which create
redundancy (by discarding some information ) and
then remove it .
7
Fidelity criteria
When lossy compression techniques are employed,
the decompressed image will not be identical to
the original image. In such cases , we can define
fidelity criteria that measure the difference
between this two images. Two general classes of
criteria are used (1) objective fidelity
criteria and (2) subjective fidelity criteria A
good example for (1) objective fidelity criteria
is root-mean square ( RMS ) error between on
input and output image For any value of x,and y ,
the error e(x,y) can be defined as
e(x,y) f(x,y) f(x,y)
The total error between two images is
The root mean square error , erms is
8
 Types of redundancy

• Three basic types of redundancy can be
  identified in a single image
• Coding redundancy
• Interpixel redundancy
• Psychovisual redundancy

9
Coding redundancy

• our quantized data is represented using
  codewords
• The codewords are ordered in the same way as the
  intensities that they represent thus the bit
  pattern 00000000, corresponding to the value 0,
  represents the darkest points in an image and the
  bit pattern 11111111, corresponding to the value
  255, represents the brightest points.
• if the size of the codeword is larger than is
  necessary to represent all quantization levels,
  then we have coding redundancy
• An 8-bit coding scheme has the capacity to
  represent 256 distinct levels of intensity in an
  image . But if there are only 16 different grey
  levels in a image , the image exhibits coding
  redundancy because it could be represented using
  a 4-bit coding scheme. Coding redundancy can also
  arise due to the use of fixed-length codewords.

10
Coding redundancy
Grey level histogram of an image also can provide
a great deal of insight into the construction of
codes to reduce the amount of data used to
represent it . Let us assume, that a discrete
random variable rk in the interval (0,1)
represents the grey levels of an image and that
each rk occurs with probability Pr(rk).
Probability can be estimated from the histogram
of an image using
Pr(rk) hk/n for k 0,1L-1 (3)
Where L is the number of grey levels and hk is
the frequency of occurrence of grey level k (the
number of times that the kth grey level appears
in the image) and n is the total number of the
pixels in the image. If the number of the bits
used to represent each value of rk is l(rk), the
average number of bits required to represent each
pixel is
(4)
11
Coding redundancy - Example
Using eq. (2) the resulting compression ratio Cn
is 3/2.7 or 1.11 Thus approximately 10 percent
of the data resulting from the use of code 1 is
redundant. The exact level of redundancy is RD
1 1/1.11 0.099
12
Interpixel redundancy
The intensity at a pixel may correlate strongly
with the intensity value of its
neighbors. Because the value of any given pixel
can be reasonably predicted from the value of its
neighbors Much of the visual contribution of a
single pixel to an image is redundant it could
have been guessed on the bases of its neighbors
values. We can remove redundancy by representing
changes in intensity rather than absolute
intensity values .For example , the differences
between adjacent pixels can be used to represent
an image . Transformation of this type are
referred to as mappings. They are called
reversible if the original image elements can be
reconstructed from the transformed data set.
For example the sequence (50,50, ..50) becomes
(50, 4).
13
Psychovisual redundancy
Example First we have a image with 256 possible
gray levels . We can apply uniform quantization
to four bits or 16 possible levels The resulting
compression ratio is 21. Note , that false
contouring is present in the previously smooth
regions of the original image. The significant
improvements possible with quantization that
takes advantage of the peculiarities of the human
visual system . The method used to produce this
result is known as improved gray-scale ( IGS)
quantization. It recognizes the eyes inherent
sensitivity to edges and breaks them up by adding
to each pixel a pseudo-random number, which is
generated from the order bits of neighboring
pixels, before quantizing the result.
14
Error free compression
Delta compression ( differential coding ) is a
very simple, lossless techniques in which we
recode an image in terms of the difference in
gray level between each pixel and the previous
pixel in the row. The first pixel must be
represented as an absolute value, but subsequent
values can be represented as differences , or
deltas. For example
FIGURE Example of delta encoding. The first
value in the encoded file is the same as the
first value in the original file. Thereafter,
each sample in the encoded file is the difference
between the current and last sample in the
original file.
15
Delta compression
FIGUREExample of delta encoding. Figure (a) is
an audio signal digitized to 8 bits. Figure (b)
shows the delta encoded version of this signal.
Delta encoding is useful for data compression if
the signal being encoded varies slowly from
sample-to-sample.
Takes advantage of interpixel redundancy in a
scan line
16
Run length encoding
  Also take advantage of interpixel redundancy .
A run of consecutive pixels whose gray levels
are identical is replaced with two values the
length of the run and the gray level of all
pixels in the run. Exampe ( 50, 50,50,50) becomes
(4,50) Especially suited for synthetic images
containing large homogeneous regions . The
encoding process is effective only if there are
sequences of 4 or more repeating
characters Applications compression of binary
images to be faxed.
CTRL - control character which is used to
indicate compressionCOUNT- number of counted
characters in stream of the same charactersCHAR
- repeating characters
FIGURE. Format of three byte code word
17
RLE - flow chart
18
Examples of RLE implementations
RLE algorithms are parts of various image
compression techniques like BMP, PCX, TIFF, and
is also used in PDF file format, but RLE also
exists as separate compression technique and file
format.
MS Windows standard for RLE have the same file
format as well-known BMP file format, but it's
RLE format is defined only for 4-bit and 8-bit
color images.Two types of RLE compression is
used 4bit RLE and 8bit RLE as expected the first
type is used for 4-bit images, second for 8-bit
images.
19
4bit RLE
Compression sequence consists of two bytes, first
byte (if not zero) determines number of pixels
which will be drawn. The second byte specifies
two colors, high-order 4 bits (upper 4 bits)
specifies the first color, low-order 4bits
specifies the second color this means that after
expansion 1st, 3rd and other odd pixels will be
in color specified by high-order bits, while even
2nd, 4th and other even pixels will be in color
specified by low-order bits. If first byte is
zero then the second byte specifies escape code.
(See table below)
20
Examples for 4bit RLE
21
8bit RLE
Sequence when compressing is also formed from 2
bytes, the first byte (if not zero) is a number
of consecutive pixels which are in color
specified by the second byte.Same as 4bit RLE if
the first byte is zero the second byte defines
escape code, escape codes 0, 1, 2, have same
meaning as described in Table 1. while if escape
code is gt3 then when expanding the following gt3
bytes will be just copied from the compressed
file, if escape code is 3 or other greater odd
number then zero follows to ensure 16bit boundary.
22
Examples for 8bit RLE
23
Statistical coding
Statistical coding techniques remove the coding
redundancy in an image. Information theory tells
us that the amount of information conveyed by a
codeword relates to its probability of
occurrence. Codeword that occur rarely convey
more information that codeword that occur
frequently in the data. A random event i that
occurs with probability P(i) is said to contain
I(i) -logP(i) units of information ( self
information ) If P(i) 1 ( that is, the event
always occurs) I(i) 0 and no information is
attributed to it . Let us assume that information
source generates a random sequence of symbols (
grey level). The probability of occurrence for a
grey level i is P(i) . If we have 2b-1 gray level
( symbols ) the average self-information
obtained from i outputs is called entropy.
24
Entropy in information theory is the measure of
the information content of a message.
Entropy gives a average bits per pixel required
to encode an image.
Probabilities are computed by normalizing the
histogram of the image P(i)hi/n Where hi is
the frequency of occurrence of grey level i and
n is the total number of pixels in the image. If
b is the smallest number of bits needed to
generate a number of quatisation levels observed
in an image, then the information redundancy of
that image is defined as R b-H  The compression
ratio is Cmax b/H
25
Statistical coding

• After computing the histogram and normalizing
  the task is to construct a set of codewords to
  represent each pixel value . These codewords must
  have the following properties
• Different codewords must have different lengths (
  number of bits0
• Codewords that occurs infrequently ( low
  probability ) should use more bits. Codewords
  that occur frequently ( high probability ) should
  use fewer bits.
• It must not be possible to mistake a particular
  sequence of concatenated codewords for any other
  sequence.
• The average bit length of codewords is

where l(i) is the length of the codeword used to
represent the grey level i. From Shannon first
coding theorem the upper limit for Lavg is b and
the lower limit for Lavg is the entropy.
26
Huffman coding

• Ranking pixel values in decreasing order of their
  probability
• Pair the two values with the lowest
  probabilities, labeling one of them with 0 and
  other with 1.
• Link two symbols with lowest probabilities .
• Go to step 2 until you generate a single symbol
  which probability is 1.
• Trace the coding tree from a root.
•  

http//www.compressconsult.com/huffman/ http//www
.cs.duke.edu/csed/poop/huff/info/
27
Dictionary based coding
The methods of the first group try to find if the
character sequence currently being compressed has
already occurred earlier in the input data and
then, instead of repeating it, output only a
pointer to the earlier occurrence.
http//www.rasip.fer.hr/research/compress/algorith
ms/fund/lz/lz77.html)
28
Dictionary based coding
The algorithms of the second group create a
dictionary of the phrases that occur in the input
data. When they encounter a phrase already
present in the dictionary, they just output the
index number of the phrase in the dictionary.
This is explained in the diagram below