Module 25 Lossy Data Compression

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Module 25Lossy Data Compression
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• Textbook sections
• Topics
• JPEG
• Discrete Cosine Transform (DCT)
• Quantization
• Lossless Compression

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1. JPEG

• JPEG
• JPEG stands for the Joint Photographic Experts
  Group. It is referred as a joint group
  because this committee is sanctioned by the CCITT
  and the ISO, two prominent international
  standards groups.
• JPEG refers both to the committee and their work
  in progress a compression standard that will
  define a method for compressing photographic
  images. Images compressed with the JPEG
  algorithm undergo a lossy compression.

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JPEG Lossy Compression
8x8 block
Lossless Compression
DCT
Quantization

• The JPEG compression process is a three-step
  procedure.
• Discrete Cosine Transformation (DCT) DCT is a
  lossless transformation that does not actually
  perform compression. It prepares for the lossy
  or quantization, step of the process.
• Quantization Quantization is the process of
  reducing the number of bits needed to store an
  integer value by reducing the precision of the
  integer.
• The JPEG algorithm outputs the elements using
  an entry encoding mechanism. The output has
  RLE built into it as an integral part of the
  coding mechanism.

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1. JPEG - DCT

• Why use DCT to transform from spatial
  representation to spectral representation?
• After the transformation, there are still just as
  many points as before. It would be much more
  impressive if the DCT took an N-by-N matrix of
  data and transformed it to an N/2 by N/2 matrix.
• The DCT transformation identifies pieces of
  information in the signal that can be effectively
  thrown away without seriously compromising the
  quality of the image.
• Most graphical images on our computer screens are
  composed of low-frequency information. The
  components found in row and column 0 (the DC
  components) carry more useful information about
  the image than the higher-frequency components.
  As we move farther away form the DC components in
  the picture, we find that the coefficients not
  only tend to have lower values, but they become
  far less important for describing the picture.
• It is hard to imagine how we would do this with a
  picture that hadnt been transformed. With the
  image still described in spatial terms, using
  pixels, a program would have a difficult time
  figuring out which pixels are important to the
  overall look of the picture and which arent.

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1. JPEG - DCT

• DCT takes an 8 x 8 pixel values as input and
  outputs an 8 x 8 matrix of frequency
  coefficients.
• DC coefficient
• The first frequency coefficients, at location
  (0,0) in the output matrix, is called the DC
  coefficient. Intuitively, we can see that the DC
  coefficients is a measure of the average value of
  the 64 input pixels.
• AC coefficients
• The other 63 elements of the output matrix are
  called the AC coefficients. They add the
  higher-spatial-frequency information to this
  average value.
• Thus, as you go from the first frequency
  coefficient toward the 64th frequency
  coefficient, you are moving from the broad
  strokes of the image to finer and finer detail.
• These higher-frequency coefficients are
  increasingly unimportant to the perceived quality
  of the image.

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1. JPEG - DCT

• Why choose blocks of 8x8 pixel values?
• The calculation time required for each element in
  the DCT is heavily dependent on the size of the
  matrix
• One of the consequence of this is that it is
  virtually impossible to perform a DCT on an
  entire image. The amount of calculation needed
  to perform a DCT transformation on even a
  256-by-256 gray-scale block is prohibitively
  large. To get around this, DCT implementations
  typically break the image down into small, more
  manageable blocks. The JPEG selected an 8-by-8
  block for the size of their DCT calculation

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LG Figure 12.26 One-dimensional and
two-dimensional DCTs
1-D DCT
X(f)
(a)
x(t)
(time)
(frequency)
(b)
2-D DCT
80 10 3 2 0 8 2 1 0
.. 2 0 0 .. 0 0 ... 0
...
100 95 85 .. 102 99 70.. 101 80 70.. 95 77 65..
(n,m) Space
(u,v) Frequency
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1. JPEG - DCT
Input Pixel matrix
DCT
Output DCT matrix
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1. JPEG - DCT

• The Discrete Cosine Transformation (DCT)
• DCT(i,j) (1 / (2N)1/2) C(i)C(j) S S pixel(x,y)
  COS(2x1)ip)/2NCOS(2y1)jp)/2N
• The Inverse Discrete Cosine Transformation (IDCT)
• pixel(i,j) (1 / (2N)1/2) S S C(i)C(j) DCT(i,j)
  COS(2x1)ip)/2NCOS(2y1)jp)/2N

N-1
N-1
X0
y0
C(x) (1 / (21/2) if x is 0, else 1 if xgt0
N-1
N-1
j0
i0
C(x) (1 / (21/2) if x is 0, else 1 if xgt0
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1. JPEG - Quantization

• Quantization
• The process of reducing the number of bits needed
  to store an integer value by reducing the
  precision of the integer
• The JPEG algorithm implements quantization using
  a quantization matrix
• For every element position in the DCT matrix, a
  corresponding value in the quantization matrix
  gives a quantum value.
• The quantum value indicates what the step size is
  going to be for that element in the compressed
  rendition of the picture
• The quantum values range from one to 255.

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1. JPEG - Quantization

• The basic quantization equation is
• Quantized Value(i,j) (DCT(i,j) / Quantum(i,j)
  Rounded to nearest integer
• The decompression is defined as
• DCT(i,j) Quantized Value(i,j) x Quantum (i,j)

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1. JPEG - Quantization
Quantization matrix
DCT Matrix before Quantization
DCT Matrix after Quantization
DCT Matrix after Dequantization
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1. JPEG - Quantization

• Selecting a Quantization Matrix
• An enormous number of schemes could be used to
  define values in the quantization matrix.
• ISO has developed a standard set of quantization
  values supplied for use by implementers of JPEG
  code
• By choosing extraordinarily high step sizes for
  most DCT coefficients (values of the elements in
  the quantization matrix), we get excellent
  compression ratios and poor picture quality.
• By choosing low step sizes, compression ratios
  will begin to reduce, but picture quality should
  be excellent.

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1. JPEG Lossless Compression
LG Figure 12.28 Zigzag scanning process
In image and video coding, the picture array is
divided into 8x8 pixel blocks which are coded
separately.
180 150 115 100 100 100 100 100 250 180 128 100
100 100 100 100 190 170 120 100
100 100 100 100 160 130 110 100
100 100 100 100 110 100 100 100 100 100 100 100
100 100 100 100 100 100 100 100 100 100
100 100 100 100 100 100 100 100
100 100 100 100 100 100
111 22 15 5 1 0 0 0 14 17 10 4 1 0 0 0 2 2
1 0 0 0 0 0 -4 -4 -2 -1 0 0 0 0 -3 -3 -1
0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
DCT
8x8 block of 8-bit pixel values
Quantized DCT Coefficients
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1. JPEG Lossless Compression

• Instead of relying of Huffman to compress the
  zero values, they are coded using a Run-Length
  Encoding (RLE)
• One way to increase the length of runs is to
  recording the coefficients in the zig-zag
  sequence.
• Instead of compressing the coefficients in
  row-major order, as a programmer would probably
  do, the JPEG algorithm moves through the block
  along diagonal paths, selecting what should be
  the highest value elements first, and working its
  way toward the values likely to be lowest.