data compression (dc)
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8/8/2019 Data Compression (Dc)
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DATA
COMP
RESSI
ON
(DC)NAME: JAGARNATH PASWANEmail:paswan.jagarnath@gmail.com
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What is Data Compression
• Data Compression refers to the process of reducing
the amount of data required to represent a given
quantity of Information.
• In fact, Data are means by which information isconveyed.
• Data that either provide no relevant information or
simply restate that which is already known, is said to
contain data redundancy.
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Why Data Compression
• In terms of communications, the bandwidth of adigital communication link can be effectively increased
by compressing data at the sending end and
decompressing data at the receiving end.
• In terms of storage, the capacity of a storage device
can be effectively increased with methods that
compresses a body of data on its way to a storagedevice and decompresses it when it is retrieved.
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Compression Techniques Basis Techniques
Entropy Coding
Run-Length Coding
Huffman Coding
Arithmetic Coding
Liv-Zempel-Welch (LZW) Coding
Source Coding
Prediction DPCM
Transformation DCT
Layer Coding Subband Coding
Vector Quantization
Hybrid Coding
JPEG
MPEG
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Data Compression MethodsLossless Compression
Text Entropy Sannon-fano, Huffman, Arithmetic
Dictionary LZ, LZW
Lossy Compression
Audio Audio Codec part DPCM, Sub-band
Image Method RLE, DCT
Video Video Codec part DCT, Vector Quantization
Hybrid Compression
Video Confrence Method JPEG, MPEG
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Lossless Data Compression
1838Samuel Finley Breeze Morse
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Information Theory
1948-Prof. Dr. Claude Elwood Shannon
“ A Mathematical Theory of Communication ”
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Information Theory
Information is quantifiable as:Average Information = - log2 (prob. of occurrence)
For English: -log(1/26) = 4.7
Entropy (in our context) - smallest number of bitsneeded, on the average, to represent a symbol(the average on all the symbols code lengths).
Note: log2(pi ) is the uncertainty in symbol (or the
“surprise” when we see this symbol). Entropy –
average “surprise”.
Assumption: there are no dependencies betweenthe symbols’ appearances
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Shannon-Fano Data Compression
1949-Prof. Dr. Claude Elwood Shannon
-Prof. Dr. Robert Mario Fano
"The transmission of information". Technical Report No. 65
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Shannon-Fano Data Compression1. Line up the symbols
by falling probability
of incidence.2. Divide the symbols in
two groups, so thatboth groups haveequal or almost equalsum of theprobabilities.
3. Assign value 0 to thefirst group, and value 1to the second.
4. For each of the bothgroups go to step 2.
Symbol A B C D E
Count 15 7 6 6 5
Probabilities 0.38461538 0.17948718 0.15384615 0.15384615 0.12820513
Symbol A B C D E
Code0
0
0
1
1
0
11
0
11
1
H(s)=2Bit *(15+7+6)+3Bit * (6+5)
/ 39 symbols= 2.28 Bit PerSymbol
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Huffman Data Compression
1952Dr. David Albert Huffman
“ A Method for the Construction of Minimum-Redundancy Codes ”
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Huffman Data Compression
Symbol A B C D E
Count 15 7 6 6 5
Probabilities 0.38461538 0.17948718 0.15384615 0.15384615 0.12820513
Symbol A B C D E
Code 0
1
0
0
1
0
1
1
1
0
1
1
1
1. Line up the symbols by fallingprobabilities
2. Link two symbols with least
probabilities into one newsymbol which probability is asum of probabilities of twosymbols
3. Go to step 2. until yougenerate a single symbolwhich probability is 1
4. Trace the coding tree from aroot (the generated symbolwith probability 1) to originsymbols, and assign to eachlower branch 1, and to eachupper branch 0
1Bit * 15+3 Bit * (7+6+6+5)
/ 39 Symbols= 2.23 BPS
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Arithmetic Coding
1976
- Prof. Peter Elias-Prof. Jorma Rissanen
-Prof. Richard Clark Pasco
"Generalized Kraft Inequality and Arithmetic Coding"
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Arithmetic Coding
1.0
0.8
0.4
0.2
0.8
0.72
0.56
0.48
0.40.0
0.72
0.688
0.624
0.592
0.592
0.5856
0.5728
0.5664
0.5728
0.57152
056896
0.56768
0.56 0.56 0.5664
Source
Symbol
Probability Initial subInterval
a1 0.2 [0.0, 0.2]
a2 0.2 [0.2, 0.4]
a3 0.4 [0.4, 0.8]a4 0.2 [0.8, 1.0]
Let the message to be
encoded be a3a
3a
1a
2a
4
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Arithmetic Coding
Therefore, the
message is
a3a3a1a2a4
Decoding:
Decode 0.572.
Since 0.8>code word > 0.4, the first symbol should be a3.
1.0
0.8
0.4
0.2
0.8
0.72
0.56
0.48
0.40.0
0.72
0.688
0.624
0.592
0.592
0.5856
0.5728
0.5664
0.5728
0.57152
056896
0.56768
0.56 0.56 0.5664
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LZ Data Compression
1977-Prof. Abraham Lempel
-Prof. Dr. Jacob Ziv
“ A Universal Algorithm for Sequential Data Compression “
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LZ Data Compression
Total no of bit = 23 After comptession = 13
23
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ComparisonHuffman Arithmetic Lempel-Ziv
Probabilities Known in advance Known in advance Not known in advance
Alphabet Known in advance Known in advance Not known in advance
Data loss None None None
Symbols
dependency
Not used Not used Used – better
compression
Preprocessing Tree building None First pass on data
(can be eliminated)
Entropy If probabilities are
negative powers of 2
Very close Best results when
alphabet not known
Codewords One codeword for
each symbol
One codeword for
all data
Codewords for set
of alphabet
Intuition Intuitive Not intuitive Not intuitive
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LZW Data Compression
“ A Technique for High Performance Data Compression ”
1984.-Prof. Abraham Lempel
-Prof. Dr. Jacob Ziv-Dr. Terry A. Welch
C l Pi l E d d Di i Di i
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39 39 126 126
39 39 126 126
39 39 126 126
Currently
Recognized
Sequence
Pixel
Being
Processed
Encoded
Output
Dictionary
Location
(Code Word)
Dictionary
Entry
39
39 39 39 256 39-39
39 126 39 257 39-126
126 126 126 258 126-126
126 39 126 259 126-39
39 39
39-39 126 256 260 39-39-126
126 126
126-126 39 258 261 126-126-39
39 3939-39 126
39-39-126 126 260 262 39-39-126-126
126 eof
Total No. Of bit =12Now Coded bit =7
LZWCompression
Algorithm
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Rate-Distortion Theory
1948-Prof. Dr. Claude Elwood Shannon
“ A Mathematical Theory of Communication ”
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Rate-Distortion Theory
– Rate –distortion theory is a major
branch of information theory which
provides the theoretical foundations for
lossy data compression. it addresses
the problem of determining the
minimal amount of entropy (orinformation) R that should be
communicated over a channel, so that
the source (input signal) can be
approximately reconstructed at the
receiver (output signal) without
exceeding a given distortion D.
WhereR(D) = Rate Distortion FunctionH = Trade off rateD = Distortion
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Distortion Measures
• A distortion measure is a mathematical quality that specifieshow close an approximation is to its original
– The average pixel difference is given by the Mean Square
Error (MSE)
• The size of the error relative to the signal is given by the
signal-to-noise ratio (SNR)
•Another common measure is the peak-signal-to-noise ratio
(PSNR)
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DPCM Data Compression“Differential Quantization of Communication Signals”
1950C.Chapin Cutler
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DPCM Data Compression
Schematic DiagramAn Audio Signal
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DPCM Data Compression
(fn - fn’) = e = 0 20 -2 56 63
fn = 130 150 140 200 230
fn’ = 130 130 142 144 167
e’ = 0 24 -8 56 56
e’= Q[en]= 16* trunc ((255+ en)/16) – 256 +8
fn”= 130 154 134 200 223
Prediction Error = fn – fn’
Reconstruction Error = Quantization Errorfn” – fn = e’ – e = q
fn’= (fn”+ fn-1”)/2 e.g. (154+134)/2=144
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DCT Data Compression
“Discrete Cosine Transform”
1974
Dr. Nasir AhmedDr.T. Natarajan
Dr. Kamisetty R. Rao
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DCT Data Compression
The most common DCT definition of a 1-D sequence of length N(8) is
for u = 0,1,2,…,N −1. C(u)=Transform Coefficient, f(x)= 1D Matrix Pixel value
Similarly, the inverse transformation is defined as
for x = 0,1,2,…,N −1. f(x)= 1D Matrix Pixel value, C(u)=Transform Coefficient
The One-Dimensional DCT
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DCT Data Compression
The Two-Dimensional DCT
The 2-D DCT is a direct extension of the 1-D case and is given by
for x,y = 0,1,2,…,N −1.
for u,v = 0,1,2,…,N −1 .
The inverse transform is defined as
α( u) = α( v) = u=0 or v=0
u>0 or v>0
u=0 or v=0
u>0 or v>0α( u) = α( v) =
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DCT Data Compression
The Matrix form of equestionOne dimensional cosine basis function
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DCT Data Compression
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DCT Data CompressionStep 1: Sample the Image into 8*8
Block
Step 2: The original image is Leveled off
by subreacting 128 from each entry.
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DCT Data CompressionStep 3: DCT Transform by Matrix
Manipulation D=T M T’
Step 4: Now DCT Matrix is Divided
by Quantization Table.
162.3
= DC coefficient
DCT Transform MatrixQuantization Table Quality Level 50
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DCT Data CompressionStep 5: Dividing D by Q and rounding
to nearest integer value.
Step 6: Now Zig- Zag Scan to
compress AC coefficent .
Quantized Matrix Zig- Zag Scan
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DCT Data Compression
Decompression:
N=
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DCT Data Compression
Compresion between Original and Decompressed image
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DCT Data Compression
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DCT Data Compression
Baboon Original Image DCT Baboon Decompress Image
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JPEG (Joint Photographic Experts Group)
JPEG (pronounced jay-peg) is a most commonly used standard method of lossy
compression for photographic images.
JPEG itself specifies only how an image is transformed into a stream of bytes,but not how those bytes are encapsulated in any particular storage medium.
A further standard, created by the Independent JPEG Group, called JFIF (JPEGFile Interchange Format) specifies how to produce a file suitable for computerstorage and transmission from a JPEG stream.
In common usage, when one speaks of a "JPEG file" one generally means a JFIFfile, or sometimes an Exif JPEG file.
JPEG/JFIF is the format most used for storing and transmitting photographs on
the web.. It is not as well suited for line drawings and other textual or iconicgraphics because its compression method performs badly on these types of images
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Baseline JPEG compression
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Y = 0.299R + 0.587G + 0.114B
U =Cb= 0.492(B − Y )= − 0.147R − 0.289G + 0.436B
V =Cr= 0.877(R − Y )= 0.615R − 0.515G − 0.100B
Y = luminanceCr, Cb = chrominance
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JPEG File Interchange Format (JFIF)
The encoded data is written into the JPEG File Interchange Format (JFIF), which,
as the name suggests, is a simplified format allowing JPEG-compressed imagesto be shared across multiple platforms and applications.
JFIF includes embedded image and coding parameters, framed by appropriateheader information.
Specifically, aside from the encoded data, a JFIF file must store all coding andquantization tables that are necessary for the JPEG decoder to do its job properly.
MPEG D t C i
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MPEG Data Compression
“Motion Picture Experts Group”
1980Motion Picture Experts Group (MPEG)
MPEG 1 D t C i
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MPEG-1 Data Compression
“Motion Picture Experts Group”
MPEG 1 D t C i
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MPEG-1 Data Compression“Motion Picture Experts Group”
MPEG 1 D t C i
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MPEG-1 Data Compression“Motion Picture Experts Group”
MPEG 1 D t C i
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MPEG-1 Data Compression“Motion Picture Experts Group”
MPEG 1 D t C i
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MPEG-1 Data Compression“Motion Picture Experts Group”
MPEG 1 Data Compression
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MPEG-1 Data Compression“Motion Picture Experts Group”
MPEG 1 Data Compression
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MPEG-1 Data Compression“Motion Picture Experts Group”
MPEG 2 Data Compression
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MPEG-2 Data Compression“Motion Picture Experts Group”
MPEG 2 Data Compression
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MPEG-2 Data Compression“Motion Picture Experts Group”
MPEG 4 Data Compression
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MPEG-4 Data Compression“Motion Picture Experts Group”
MPEG 4 Data Compression
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MPEG-4 Data Compression“Motion Picture Experts Group”
MPEG 7 Data Compression
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MPEG- 7 Data Compression“Motion Picture Experts Group”
H 261 Data Compression
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H.261 Data Compression“Motion Picture Experts Group”
H 261 Data Compression
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H.261 Data Compression“Motion Picture Experts Group”
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Refrences
Multimedia Fundamentals Vol 1
-by Ralf Steinmetz and Klara Nahrstedt
http://en.wikipedia.org/wiki/Data_compression
http://navatrump.de/Technology/Datacompression/compression.html
Digital Image Processing 2nd Edition
-by Rafael C. Gonzalez and Richard E. Woods
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