Discrete Cosine Transform

1

• Discrete Cosine Transform
• (DCT)
• Eric Rust
• ESS 522
• Spring/2007
• University of Washington

2
References

• International Telecommunication Union,
  Information Technology - Digital Compression and
  Coding of Continuous-Tone Still Images -
  Requirements and Guidelines, T.81 ed. , New York
  n.p., 1992.
• I. Richardson, Video CODEC Design, New Jersey
  John Wiley and Sons Inc., 2002
• Khalid Sayood, Introduction to Data Compression,
  Second ed., San Francisco Morgan Kaufmann, 2000.
  
• W. Pennebaker and J. Mitchell, JPEG Still Image
  Data Compression Standard, New York Van Nostrand
  Reinhold, 1993.
• D. Taubman and M. Marcellin, JPEG2000 Image
  Compression Fundamentals, Standards and Practice,
  Boston Kluwer Academic Publishers, 2002.
• Multiple Wikipedia searches

3
Objectives

• Learn where Discrete Cosine Transform (DCT) is
  used.
• Learn how DCT works
• Experiment in using DCT

4
Why Did I Choose This?

• We learned Fourier Transform.
• Wanted to show that there are other domains that
  may be more useful.

5
Where DCT is used

• Data compression
• Image Compression

6
Image Compression (why?)

• Common Cameras
• 8 megapixels gt 24MB
• Yahoo!, Hotmail, AOL 10MB
• GMAIL 20MB
• Uploading pictures
• Myspace 5MB

7
JPEG

• JPEG - current image compression standard. The
  group was organized in 1986, issuing a standard
  in 1992 1.
• lossy and lossless
• Gif - CompuServe in 1987 2
• Lossless
• png came about because of patent issues with
  LZW algorithm
• 1 - http//en.wikipedia.org/wiki/JPEG - last
  visited on 06/02/2007
• 2 - http//en.wikipedia.org/wiki/GIF - last
  visited on 06/02/2007

8
JPEG Overview
Current Image
JPEG
Compressed File
9
Baseline JPEG CODEC
Level Shift
FDCT
Quant.
Zigzag
DC Pred.
Entropy Encode
Output Buffer
Markers
10
The image is divided into 8x8 matrices
Image Division
Matrix image taken from Wikipedia.com
11

Level Shift

• Each pixel has a numeric value representing its
  color.

Level Shift
(0,255)
(-128,127)
-Recall that this is done on each individual 8x8
block for each color plane
12
DCT II
13
Plot of DCT Coefficients
High value coefficients Low frequency
We can eliminate low value coefficients without
losing much image quality
Low value coefficients High frequency
14
FDCT Coefficients
Shifting
8X8 block of pixels
FDCT
Matrix images taken from Wikipedia.com
15
Moving into Quant.
Completed
FDCT
Quant.
Zigzag
DC Pred.
Entropy Encode
Level Shift
Output Buffer
Markers
16
FDCT Quantization
Common JPEG quantization matrix.
Note that these matrices will change depending
upon what quality you want your compressed image
to be.
17
Quantization
FDCT Coefficients
Note that the numbers are rounded which leads to
many zeros
Resulting quantized matrix
Common JPEG quantization matrix.
18
Zigzag
Zigzag arranges the DCT coefficients in an array
such that the larger coefficients are grouped
together at the beginning.
19
Update on JPEG
FDCT
Quant.
Zigzag
DC Pred.
Entropy Encode
Level Shift
Output Buffer
Markers
20
Entropy Encoding

• After quantization, it is not unusual for more
  than half of the DCT coefficients to equal
  zero1.
• JPEG uses Run-Length Encoding to reduce the size
  of the file by taking advantage of repeating
  characters2 (the zeros).
• Huffman coding could be used as well

• ece.perdue.edu
• http//www.eee.bham.ac.uk/WoolleySI/All7/run_1.htm

21
DCT

• Why DCT in JPEG?

22
FT versus DCT

• DFT
• Sines and cosines
• Periodic extension

• DCT
• Cosines
• Periodic and even extension

• DCT
• 4 types of DCT
• (Martucci 1994) shows four of them
• The other four are

23
(No Transcript)
24
WHY DCT?

• Different boundary conditions strongly affect the
  applications of the transform.
• Discontinuities in a function reduce the rate of
  convergence of the Fourier series, so that more
  sinusoids are needed to represent the function
  with a given accuracy

25
WHY DCT cont.

• The periodicity of the DFT results in
  discontinuities usually occur at boundaries
• DCT where both boundaries are even always yields
  a continuous extension at the boundaries
• DCT II is preferred for applications, in part for
  reasons of computational convenience.

26
Example
0.9ncos(0.1pin), n0,1,2...31
27
(No Transcript)
28
High-Pass
High value coefficients Low frequency
We can eliminate low value coefficients without
losing much image quality
Low value coefficients High frequency
29
Background on Images

• Color image has three color planes
• Black and white has one color plane
• Each color plane is represented by pixels with
  numerical values that correspond to a color.

30
Matrix image taken from Wikipedia.com
31
Background on Images Cont.

• Each pixel has 8-bits (1 or 0) that gives the
  pixel a value from 0 to 255.
• 0000 0001 1
• 0001 0010 18
• 1000 0000 128
• Goal is to find out haw many bits per pixel it
  takes to represent an image (uncompressed image
  would be 8 bits per pixel).

32
Finding Bits Per Pixel

• Download files from website.
• Review code to understand what it is doing
• Run code (look at images that are displayed)
• Complete two Exercises
• QualityJPEG1 2 5 80 100