1. Hao Jiang Computer Science Department Sept. 27, 2007
JPEG
Hao Jiang
Computer Science Department
Sept. 27, 2007

2. 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 ...

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

4. Color Space Conversion
JPEG first converts RGB to YUV or YCrCb.
Y is the luminance component (brightness).
Y = R G B
U and V are color components
U = B – Y
V = R - Y
Y U V

5. Color Subsampling
JPEG down-samples the “color channels” by half and partitions images into 8x8 small blocks. Y1 Y2 U V Y3 Y4
Intensity Component
Color components

6. 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

7. Image block DCT Coefficients

8. Definition of DCT
A linear transfrom F(u,v) = åx=07 åy=0 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.

9. For DCT, h(u,v,x,y) can be decomposed into g(u,x) * g(v,y)
F(u,v) = åx=07 åy=0 7 h(u,v,x,y) f(x,y)
= åy=07 g(v,y) [åx=0 7 g(u,x) f(x,y)]
followed by
Horizontal 1D DCT
Vertical 1D DCT

10. Given u and v, h can be viewed as an image of x and y.
DCT coefficients are projection of image block into these patterns.

11. 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.

12. Fq(u,v) = round(F(u,v)/Q(u,v))
Y quantization table
U, V quantization table
Quantization:
Fq(u,v) = round(F(u,v)/Q(u,v))
De-quantization:
Fr(u,v) = Fq(u,v) * Q(u,v)

13. quantization
dequantization

14. 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))

15. DC Coefficients Encoding
D(m,n) D(m,n+1) D(m,n+2) D(m,n+3)
We compute the difference of each two successive quantized
coefficients
D(m,n) – 0 => d(m,n)
D(m,n+1) – D(m,n) => d(m,n+1)
D(m,n+2) – D(m,n+1) => d(m,n+2)
D(m,n+3) – D(m,n+2) => d(m,n+3)

16. 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 …
,…,-1024,1024,…,2047
7 will be coded as (3, 7)
The first number 3 is Huffman
coded.
The second 7 is encoded as
1’s complement 111.
(-7 will be 000)
The Size Table
Can we encode the dc difference value directly?

17. 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

18. 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)
Huffman coded
1’s complement

19. JPEG File Format Start of image Frame End of image Tables Header Scan…
Scan
Tables
Header
Segment
Restart
Segment ….
Block
Block
Block

20. 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.

21. 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.

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