JPEG

1
JPEG

• Compresses real images
• Standard set by the Joint Photographic Experts
  Group in 1991

2
Stages of JPEG

• We will consider (monochrome) black and white.
  Images.
• Colour images are treated similarly, but the
  colour information is extracted and processed
  separately using a reduced sample set.
• The whole image is divided into 8 x 8 blocks.

3
Stages of JPEG

• Transform each 8 x 8 block into an 8 x 8 Discrete
  cosine transform.
• Quantise DCT co-efficients (values) giving finer
  quantisation to certain values.
• Run length code zero values.
• Apply Huffman coding to the run length coded
  sequence.

4
Take 8x8 blocks and produce 2 dimensional (8 x 8)
DCT
The image is split up into 8 x 8 pixel blocks
giving 64 values. The following (scary) formula
is applied. Which gives an 8 x 8 DCT
block.
5
The 2 dimensional (8 x 8) DCT

• However, as before, all the equation does, is
  calculate what amount of certain cosine patterns
  make up the 8 x 8 pixel block.
• This time the patterns are in two dimensions and
  are shown in a following slide.

6
The 2 dimensional (8 x 8) DCT

• They are called basis functions or pictures,
  because they form the basis (building blocks) of
  any image. That is any picture can be completely
  reconstructed from a combination of these
  patterns.

7
The 2 dimensional (8 x 8) DCT

• Each co-efficient (value) in the DCT represents
  a spatial frequency in the image.
• That is how many times it goes from light to dark
  (or from dark to light) in the block.

8
The 2 dimensional (8 x 8) DCT

• The block can be see to be separated into
  horizontal and vertical spatial frequencies.
• The value in the upper left corner represents
  zero frequency, dc or the average pixel
  brightness in the original 8x8 block.

9
The 2 dimensional (8 x 8) DCT

• It is processed separately to the other (called
  ac co-efficients) and is not subject to the lossy
  compression process which follows.

10
DCT basis functions (Basis pictures).
11
Why DCT?

• We have produced this complex transformation,
  but it still contains 8 X 8 or 64 values.

12
Why DCT?

• However, while any the original 8 x 8 pixel
  values could take on any value between 0 255,
  it turns out that for most images, that the DCT
  values are high and significant towards the top
  left hand corner and low (and less significant)
  towards the lower right hand corner of the block.
• We use this statistical information to our
  advantage.

13
Typical DCT co-efficients.
14
Re-quantisation

• The eye is less sensitive to high frequency noise
  (error).
• We can afford to code the high frequencies less
  accurately.
• We use larger quantisation steps.
• Since the high frequency values are usually small
  anyway, large quantisation steps reduce these
  values to zero.
• That is they are not counted in the coded
  representation and we have achieved (lossy)
  compression.

15
Re-quantisation

• To re- quantise to larger steps we divide the
  actual value by a values larger than 1 and round
  the result.
• The following table shows a typical set of 8x8
  quantisation values which we divide the dct
  values by to achieve re-quantisation.
• Note that the lower values in this case receive
  no re-quantisation.
• Compression may be increased by increasing the
  re-quantisation values.

16
Typical re-quantisation values
17
DCT coefficients after re-quantisation
18
The effect of loosing the high frequency
co-efficients completely

• Matlab program metdct1.m
• Syntax
• Metdct(image.bmp, mask)
• image is an image to be compressed.
• Mask is an 8x8 matrix 1s and 0s which
  determines which co-efficients are used and not
  used (unlike quantising which allows higher
  co-efficients to play a limited part)..

19
The effect of loosing the high frequency
co-efficients completely

• Use zeros(8, 8), ones(8,8)
• Ones(8, 8) gives no deterioration (and no
  compression)
• Observe the result of using the dc component
  only.
• Experiment by adding ones to the mask to see the
  minimum number of co-efficients required to give
  a reasonable compressed image.
• Estimate the degree of compression.

20
Zig-Zag scan

• The dc component and the re-quantised
  coefficients are placed in a string using a
  zig-zag scan.
• This cunning stunt leaves long runs of zeroes.

21
Zig-zag scan
22
Run-length encoding

• The string of co-efficients has many zeroes at
  the end
• 168, 45, 67, 12, 32, 7, 3, 3, 5, 3, 1, 1, 5, 3,
  2, 0, 1, 3, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
  0, 0
• Long runs of zeros can be run-length coded.
• However, a special (single) code is allocated to
  a run of 15 zeros and another end of block (EOB)
  code is used if there are no more non-zero values
  in the block..

23
Variable length coding

• As we approach the end of the the zig-zag scan,
  it is more likely that any non-zero value will
  have a run of zeros before it. Also the non-zero
  value is more likely to have a low value than a
  high one, particularly if the run of zeroes is
  large.
• These combinations of the non-zero value and the
  preceding run of zeros can therefore be Huffman
  coded.

24
Further reading
www.snellwilcox..com/reference/pdfs/ecomp.pdf Art
of Digital Video, Watkinson, Focal Press
Digital image processing, Gonzalez and Woods,
Prentice Hall