JPEG

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JPEG

• Hao Jiang
• Computer Science Department
• Sept. 27, 2007

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What is JPEG?

• JPEG Joint Photographic Expert Group an
  international standard in 1992.
• Works for both color and grayscale images.
• Targets at natural images.
• Applications include satellite, medical imaging,
  general photography ...

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The System of JPEG
8x8 block
DCT
Quantization
Quantization tables
Header
DC
Entropy coding
Tables
Difference coding
Data
Run length coding
AC
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Color Space Conversion

• JPEG first converts RGB to YUV or YCrCb.
• Y is the luminance component (brightness).
• Y 0.299 R 0.587 G 0.144 B
• U and V are color components
• U B Y
• V R - Y

Y U V
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Color Subsampling

• JPEG down-samples the color channels by half
  and partitions images into 8x8 small blocks.

U
V
Y1
Y2
Y3
Y4
Intensity Component
Color components
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DCT (Discrete Cosine Transform)

• DCT converts each image 8x8 block into another
  8x8 block.
• The energy in DCT domain is concentrated into
  very few coefficients.

Image block
DCT
DCT coefficients
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Image block
DCT Coefficients
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Definition of DCT
A linear transfrom F(u,v) åx07 åy0 7
h(u,v,x,y) f(x,y)
where h(u,v,x,y) is the linear weighting
function.
DCT
The inverse transform IDCT recovers the original
data from DCT coefficients.
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• For DCT, h(u,v,x,y) can be decomposed into g(u,x)
  g(v,y)

F(u,v) åx07 åy0 7 h(u,v,x,y) f(x,y)
åy07 g(v,y) åx0 7 g(u,x) f(x,y)
followed by
Horizontal 1D DCT
Vertical 1D DCT
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Given u and v, h can be viewed as an image of x
and y.
u
v
DCT coefficients are projection of image block
into these patterns.
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Quantization

• Recall that quantization can be used to collapse
  the input into smaller number of values.
• For DCT coefficients, we want to quantize
  different coefficients in different details
• DC coefficient should have the most quantization
  levels.
• Quantization for AC coefficients can be coarser
  and coarser as the frequency increases.

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Y quantization table
U, V quantization table
17 18 24 99 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
Fq(u,v) round(F(u,v)/Q(u,v))
Quantization
Fr(u,v) Fq(u,v) Q(u,v)
De-quantization
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quantization
dequantization
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The Quality Factor

• For most current JPEG encoders, we can choose a
  quality factor from 1 to 100.
• The method of controlling the compression quality
  is by scaling the quantization table. For
  example,
• Fq(u,v) round(F(u,v) (quality)
  / Q(u,v))

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DC Coefficients Encoding
D(m,n) D(m,n1) D(m,n2)
D(m,n3)
We compute the difference of each two successive
quantized coefficients
D(m,n) 0 gt d(m,n) D(m,n1)
D(m,n) gt d(m,n1) D(m,n2) D(m,n1) gt
d(m,n2) D(m,n3) D(m,n2) gt d(m,n3)
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DC Coefficients Encoding

• Then, we encode each DC difference value by
  (size, coefficient)

• 1 -1, 1
• 2 -3, -2, 2, 3
• -7 -4, 4, ,7
• 4
• 11 -2047,,-1024,1024,,2047

• 7 will be coded as (3, 7)
• The first number 3 is Huffman
• coded.
• The second 7 is encoded as
• 1s complement 111.
• (-7 will be 000)

The Size Table
Can we encode the dc difference value directly?
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AC Coefficients Encoding

• AC coefficients are not differentially encoded.
• Instead, we first do run-length coding.

The run length code
(0,11), (0,-12), (1, -12), (0, 10), (0,16), (0,
0)
Indicates all are 0 from here
Zig-zag scanning
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AC Coefficients Encoding (cont)

• For AC coefficients, we now have a bunch of
  symbols like
• The value has large number of possible values
  in -1023, 1023. Direct Huffman coding is
  infeasible.
• Instead, we generate symbols like

(zero-run length, value)
( zero-run length, size, value)
1s complement
Huffman coded
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JPEG File Format
Start of image
Frame
End of image

Tables
Header
Scan
Scan
Scan
Tables
Header
Segment
Restart
Segment
.
Block
Block
Block
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JPEG Extensions

• The basic mode of JPEG supports sequential coding
  (the order is from top to bottom and left to
  right).
• JPEG extensions support progressive modes
• Spectrum selection.
• Successive approximation.
• Hierarchical mode.

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JPEG 2000

• JPEG 2000 is a new standard based on wavelet
  transform.
• JPEG2000 does not partition an image into small
  blocks.
• Wavelet decomposes the whole image into different
  bands and then encodes the coefficients in
  different bands smartly.

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LH
LL
HL
HH
The wavelet decomposition of Lena image using
Haar Wavelet.