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					                  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
                     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.
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)
       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       R = 1 – 1/C                                           (1)
          D          R

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 nc<<n1 , CR large value and RD 1. Larger values of C
indicate better compression
     A general algorithm for data compression and
                 image reconstruction
Input image ( f(x,y)                                 Reconstructed image
                                                     f‟(x,y)


Source                                                        Source
                   Channel      Channel      Channel          decoder
encoder
                   encoder                   decoder          Reconstru
Data
                                                              ction
redundancy
reduction
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 .
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 .
                          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)
                                          M 1 N 1

The total error between two images is:       f ' ( x, y ) 
                                          x 0 y 0
                                                                      f ( x, y ) 

                                                                                         1
                                                 1 M 1 N 1                    2
                                                                                        2
The root –mean square error , erms is : erms         f ' ( x, y)  f ( x, y)   
                                                 MN x 0 y 0
                                                                                    
                                                                                     
              Types of redundancy

   Three basic types of redundancy can be identified in a
   single image:
1) Coding redundancy
2) Interpixel redundancy
3) Psychovisual redundancy
                     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.
                          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(r ) = h /n for k = 0,1……L-1                  (3)
                      k     k

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 :

                   L 1

                    l (r
                                                                (4)
  Lav g                         k   )P ( rk )
                                       r
                   k 0
               Coding redundancy - Example
Rk          Pr(rk)       Code 1          l1(rk)    Code 2   l2(rk)
r0 = 0      0.19         000             3           11     2
r1 = 1/7    0.25         001             3           01     2
r2 = 2/7    021          010             3           10     2
r3 = 3/7    0.16         011             3          001     3
r4 = 4/7    0.08         100             3         0001     4
r5 = 5/7    0.06         101             3         00001    5
r6 = 6/7    0.03         110             3         000001   6
r7=1        0.02         111             3         000000   6

                         7
           Lav g     l
                      k 0
                                  2   ( rk ) Pr ( rk )

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
                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).
             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 2:1.
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 eye‟s 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.
                          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.
                         Delta compression




FIGURE:Example 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
                      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
CTRL            COUNT              CHAR         used to indicate compression
                                                COUNT- number of counted
                                                characters in stream of the same
FIGURE:. Format of three byte code word         characters
                                                CHAR - repeating characters
RLE - flow chart
            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.
                                        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) Second
                   Definition
        byte

        0          End-of-line

        1          End-of-Rle(Bitmap)

                   Following two bytes defines offset in x and y direction (x is right,y is up). The
        2
                   skipped pixels get color zero.

                   when expanding following >=3 nibbles (4bits) are just copied from compressed
        >=3        file, file/memory pointer must be on 16bit boundary so adequate number of
                   zeros follows
              Examples for 4bit RLE:

Compressed
              Expanded data
data

06 52         525252

08 1B         1B1B1B1B
00 06 83 14
              831434
34

00 02 09 06   Move 9 positions right and 6 up

00 00         End-of –line

04 22         2222

00 01         End-of-RLE(Bitmap)
                      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 >=3 then when expanding the following >=3
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.
                  Examples for 8bit RLE
Compressed data   Expanded data

06 52             52 52 52 52 52 52

08 1B             1B 1B 1B 1B 1B 1B 1B 1B

00 03 83 14 34    83 14 34


00 02 09 06       Move 9 positions right and 6 up

00 00             End-of -line

04 2A             2A 2A 2A 2A

00 01             End-of-RLE(Bitmap)
                         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.
      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.
                           2b 1
                 H    P(i ) log 2 P(i )
                            i 0
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
                          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:
1. Different codewords must have different lengths ( number of bits0
2. Codewords that occurs infrequently ( low probability ) should use
   more bits. Codewords that occur frequently ( high probability )
   should use fewer bits.
3. It must not be possible to mistake a particular sequence of
   concatenated codewords for any other sequence.
                                                     2b 1
The average bit length of codewords is L
                                        avg          l (i
                                                      i 0
                                                                 )P (i )


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



     http://www.compressconsult.com/huffman/
     http://www.cs.duke.edu/csed/poop/huff/info/
       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/compres
 s/algorithms/fund/lz/lz77.html)
          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:

				
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