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How to contact
Prof. Dr. Paul Müller
Raum: 34/312
Telefon: 0631/205−2263
Email: mueller@uni−kl.de
Bernd Reuther
Raum: 32/344
Telefon: 0631/205−2161
Email: reuther@informatik.uni−kl.de
Dirk Henrici
Raum: 32/414
Telefon: 0631/205−2891
Email: henrici@informatik.uni−kl.de
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JPEG as an application of the discrete cosine transform
How to contact
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Overview
Seminar
Basics of Compression
• Motivation
• Derivation of the DCT out of
the DFT
• JPEG overview
• JPEG baseline codec
• JPEG summary
• Summary and Perspectives
Ulrich Dauenhauer
Student of the University of Kaiserslautern, MatNr.: 309 941
email: dauenha@rhrk.uni−kl.de
JPEG as an application of the discrete cosine transform
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Overview
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Motivation
From DFT to DCT (1)
• Periodic extension of the DFT leads to
leaps at the boundaries of the analyzed
section
• Compression is important despite of higher computing performance and faster networks.
• The concept of the DFT to transform pixels from a viewable plane of spacial coordinates (pixel
representation) into an abstract plain of frequency representation is excellent but has weaknesses.
• Definition of a platform−spanning standard for picture exchange is reasonable
Motivation
• Concept of the DCT: Symmetric extension
♦ DFT−analysis over signal part of
length 2N
♦ Symmetry about the center n=−1/2,
thus DFT needs phase shift
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From DFT to DCT (2)
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From DFT to DCT (3)
• This 1−D DCT is invertible like the DFT,
but
• FDCT (Forward DCT) equals its IDCT
• More appropriate for image processing
is the 2−D DCT
• Deductible from the 1−D DCT since
♦ linear transform
♦ separately applicable to rows and
columns
• For the DFT, symmetry results in an even
function and real coefficients
• Splitting into a sum, each term with
(existing) positive n−coordinates
• Euler's identity leads to the DCT
From DFT to DCT (2)
From DFT to DCT (1)
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From DFT to DCT (3)
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JPEG overview
JPEG baseline codec (1)
• Joint Photographic Experts Group (JPEG)
• Developer consortium on the field of image compression
• Goal: generic standard for all kinds of images (true−color 24bit or
grayscale)
• portability
• JPEG “standard”: collection of algorithms
• smallest common denominator: JPEG baseline codec
JPEG overview
• JPEG compression program using baseline
codec
• select color space
• color−subsampling
• DCT
• quantization
• coding
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JPEG baseline codec (2)
• select color space
♦ grayscale
♦ RGB
♦ YCbCr
JPEG baseline codec (1)
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JPEG baseline codec (3)
• select color space
• color−subsampling
• DCT
• quantization
• coding
• The following example shows an 8x8
pixel image section
• 8−bit numbers represent grayscales
• Assuming a given 16x16 block
with Cb or Cr
• The eye can sense local
differences only poorly
• Averaged values of 2x2 sections
• color−subsampling
• DCT
• quantization
• coding
JPEG baseline codec (2)
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JPEG baseline codec (3)
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JPEG baseline codec (4)
• select color space
• color−subsampling
• DCT
♦ left−shift 2P−1
♦ enforce NxN
blocks
♦ apply 2−D DCT
• quantization
• coding
JPEG baseline codec (5)
• 8x8 block with 8−bit values
• thus P=8; left shift of all items
corresponds to −128
JPEG baseline codec (4)
• select color space
• color−subsampling
• DCT
♦ left−shift 2P−1
♦ enforce NxN blocks
♦ apply 2−D DCT
• quantization
• coding
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JPEG baseline codec (6)
• select color space
• color−subsampling
• DCT
• quantization
♦ select table Q
♦ quantify DCT−coefficients
• Der 8x8 Block enthält nun Werte zwischen −128 und
127
• Application of the 2−D DCT leads to the transform
matrix ¸ with the items Æi,j
JPEG baseline codec (5)
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JPEG baseline codec (7)
• select color space
• color−subsampling
• DCT
• quantization
• coding
♦ DC−coefficient
♦ AC−coefficients
• select a quantization table Q with items qij
according to JPEG recommendation
• quantify according to stated formula and obtain
table L with quantified DCT−coefficients
• DC−coefficient
• coding table: 16 categories with increasing
number of entries
• code difference to preceding
DC−coefficient:
♦ here: 3
♦ 3 is in group 2 and is on position 4
there (thus bitword 11)
♦ assuming Huffman−Code for group 2
is 2−bit long (e.g. 01)
♦ resulting sequence for DC: 0111
♦ store results in L
• coding
JPEG baseline codec (6)
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JPEG baseline codec (7)
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JPEG baseline codec (8)
JPEG baseline codec (9)
• AC−coefficients are traversed in zick−zack and
coded absolutely
• select color space
• color−subsampling
• DCT
• quantization
• coding
♦ DC−coefficient
♦ AC−coefficients
• select color space
• color−subsampling
• DCT
• quantization
• coding
• result
• 8x8 section coded with 23 bit
0111 0011 0110 1100 1111 010
• i.e. in average 23/64 instead of 8 bit
per pixel
• the reconstructed block has only small
errors [values in %]
• The (Z,C) table contains Huffman codes for
AC−sequences
♦ Z: number of preceding zeros
♦ C: category of the AC coefficient
♦ For the example the results are:
◊ 1 −> (0,1) −> 00 1
(2nd position in 1st category)
◊ −9 −> (0,4) −> 1011 0110
◊ 3 −> (0,2) −> 01 11
◊ 0…0 −> (0,0) −> 1010
JPEG baseline codec (8)
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JPEG summary
• speed
♦ fast compression and
decompression
♦ images can be viewed without long
waiting even on slow PCs
• image quality
♦ using low compression very good
image quality
♦ depending on image type
♦ quantization can lead to visible
color distortion
♦ sharp edges problematic
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Summary (1)
• Summary
♦ DCT has better characteristics
than DFT
♦ DCT is used in JPEG
• Outlook on JPEG 2000
♦ JPEG 2000 addresses the
stated problems
• Fault liability
♦ blocks are interconnected
systematically
♦ errors in bitstream affect whole
image
• Implementation
♦ no provision for metadata
♦ provision for a ROI (Region of
Interest)
JPEG summary
JPEG baseline codec (9)
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Summary (1)
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Summary (2)
References
• Demonstration of JPEG vs. JPEG2000
• Link to seminar of Matthias Klapperstück, Stefanie
Kreide
♦ satellite images
♦ radiograms
♦ TV still images
♦ forrest
♦ buildings
♦ test image
♦ test image 2
♦ chessboard
♦ text
Summary (2)
• Khalid Sayood: Introduction to Data Compression
• Prof. Dr. A. Schmidt: Seminar Multimedia Datenformate; WS 94/95; University of
Karlsruhe
• Internet
♦ (see above)
♦ FAQ located at www.jpeg.org
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References
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