Download - 4C8 Dr. David Corrigan
4C8
Dr. David Corrigan
Jpeg and the DCT
2D DCT
DCT Basis Functions
2D DCT
Qstep = 15
Each band is the same size and there are 64 bands in total so the entropy is
bits/pixel 36.164
entropies band
H
Optimum Block Size is 8!
Slow DCT
• Sledgehammer implementation for 8 point DCT
• Each row multiply requires 8 MADDs (approx)• So for all 8 rows requires 64 MADDs (approx)
Fast DCT
• Exploit Symmetry
Fast DCT
• So split Matrix T into two parts...
)8()7()6()5()4()3()2()1(
yyyyyyyy
Fast DCT
• split Matrix T into two parts, change y...
)8()7()6()5()4()3()2()1(
yyyyyyyy
)5()4()6()3()7()2()8()1(
yyyyyyyy
)5()4()6()3()7()2()8()1(
yyyyyyyy
Fast DCT
)8()7()6()5()4()3()2()1(
yyyyyyyy
)5()4()6()3()7()2()8()1(
yyyyyyyy
)5()4()6()3()7()2()8()1(
yyyyyyyy
4 “adds”, 16 MADDS for each operation = 8 adds and 32 MADDS = 40 opsCompare with 64 MADDS from before .
Fast DCT
)5()4()6()3()7()2(
)8()1(
yyyyyyyy
This sub-matrix can be simplified with symmetry again!
4 “adds”, 8 MADDS in total = 12 ops (down from 20)
So now we are at 20 (for the first sub matrix) + 12 (for these two) = 32 ops
So we have saved about x2!
JPEG and Colour Images
• JPEG uses YCBCR colourspace.• The chrominance channels are usually
downsampled. • There are 3 commonly used modes– 4:4:4 – no chrominance subsampling– 4:2:2 – Every 2nd column in the chrominance
channels are dropped.– 4:2:0 – Every 2nd column and row is dropped.
Subjectively Weighted Quantisation
• In JPEG it is standard to apply different thresholds to different bands
9910310011298959272101120121103877864499211310481645535247710310968563722186280875129221714566957402416131455605826191412126151402416101116
lumQ
99999999999999999999999999999999999999999999999999999999999999999999999999996647999999999956262499999999662621189999999947241817
chrQ
Subjectively Weighted Quantisation
• These values are obtained by perceptual tests.
• A user is asked to view an image of a particular size on at specified distance from the screen.– Usually a multiple of the screen height.
• User is presented with an image and is asked to increase the gain of a given band until he/she just notices a difference in the image.
– Note typically a flat grey image is used to avoid masking effects caused by edges and texture
• The set of form the quantisation matrix.
),(),(),( yxyxIyxI klklorigvis
Subjectively Weighted Quantisation
• Lower Frequency Bands are assigned lower step sizes.
• There is a slight drop of in step size from the DC coefficient to low frequency coefficients.
• The step sizes for the chrominance channels increase faster than for luminance.
9910310011298959272101120121103877864499211310481645535247710310968563722186280875129221714566957402416131455605826191412126151402416101116
lumQ
99999999999999999999999999999999999999999999999999999999999999999999999999996647999999999956262499999999662621189999999947241817
chrQ
We have seen this before
Comparing Different Quantisations
Qstep = Qlum
Uncompressed JPEG
Comparing Different Quantisations
Qstep = QlumPSNR = 32.9 dB
Comparing Different Quantisations
Qstep = 2 * QlumPSNR = 30.6 dB
Uncompressed JPEG
Comparing Different Quantisations
Qstep = 15 PSNR = 37.6 dB
Qstep = QlumQstep = 15
Comparing Different Quantisations
Qstep = 30 PSNR = 33.4 dBQstep = Qlum
Qstep = 30
Comparing Different Quantisations
Qstep = 30 PSNR = 33.4 dBQstep = Qlum
Qstep = 30PSNR indicates better quality for Qstep = 30 over Qstep = Qlum but this clearly is not true from a subjective analysis.
Comparing Different Quantisations
Quantisation PSNR (dB) Subjective Ranking
Entropy (bits/pel)
15 37.6 2 1.36
30 33.4 4 0.820.5 * Qlum 35.6 1 1.28
Qlum 32.9 3 0.862*Qlum 30.6 5 0.55
Using the subjectively weighted Quantisation achieves much higher levels of compression for equivalents levels of quality.
JPEG Coding• The most obvious way might seem to code each band
separately– ie. Huffman with RLC like we suggested with the Haar Transform.– We could get close to the entropy
• This is not the way it is coded because– It would require 64 different codes. High cost in computation and
storage of codebooks.– It ignores the fact that the zero coefficients occur at the same
positions in multiple bands.
JPEG Coding• Instead we code each block separately
– A block contains 64 coefficients, one from each band.
• Each block contains 1 DC coefficient (from the top left band) and 63 AC coefficients
• Two codebooks are used in total for all the blocks, one for the DC coefficients and the other for the AC coefficients.
• At the end of each Block we insert an End Of Block (EOB) symbol in the datastream
Data Ordering
• Each block covers is a 8x8 grid of coeffs– A Zig-Zag scan converts them into
a 1D stream.– As most non-zero values occur in
the top left corner using a Zig-Zag scan maximises the lengths zero runs so improves efficiency of RLC
Zig-Zag Scan Example
000000000000000000000000000000010000000000000006010002313
-13, -3, 6, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 36 more zeros, the end
Typical DCT Block Coefficients
Non-Zero values are at the top left corner of the block Zig-Zag scan concentrates
the non-zero coefficients at the start of the stream
Coding the DC Coefficients
Differential Coding
000000000000000000000000000000010000000000000006010002313
Coding the DC Coefficients
This value is actually the difference between the dc coefficient of the current and previous blocks
Typical DCT Block Coefficients
Coding DC Coefficients• There is potentially a large number of levels to encode.
– Up to 4096 depending on the quantization step size.
• We break down the symbol value into a size index pair
Coding DC Coefficients
• So if the DC value is -13– The size is 4– The index is 0010
• In JPEG only the size is encoded using Huffman– The index is uncoded, efficiency is not dramatically
affected.– Only 12 codes required in huffman table– Table size is 16 + 12 = 28 bytes
Value Size Index
-7 3 000
-6 3 001
-5 3 010
-4 3 011
-3 2 00
-2 2 01
-1 1 0
0 0 -
1 1 1
2 2 10
3 2 11
4 3 100
5 3 101
6 3 110
7 3 111
More examples of Coefficient to size/index pair conversions
Coding the AC Coefficients
000000000000000000000000000000010000000000000006010002313
4 0010, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end
Typical DCT Block Coefficients
The block usually ends with a long run of zeros
The length of the run and the value of the coeff after it are strongly correlated
Size/Index Pair for DC coefficient
Coding the AC Coefficients
• Code/Size Correlations– High coeffs follow short runs and low coeffs follow
long runs
• Final run of zeros– These don’t need to be coded– Just tell the encoder that there are no more non-
zero coefficients and move onto the next block.
SymbolsRun/Coefficient Symbols
eg. 0, 0, 9 is a run of 2 zeros followed by a 9
However we represent 9 using the size/index format from the dc coeffs
9 has a size of 4 and an index 1001
So we code the run/size pair (2,4) and the index 1001 is appended to the stream
Symbols
• Run/Size Symbols– All possible combinations of runs from 0->15 and
size from 1->10– 160 total symbols– Huffman Codes are used for each symbol– Index values are not coded further
Special Symbols• ZRL
– Used to represent a run of 16 zeros– Used when the run of zeros is greater than 15– Eg. 17 zeros, 14 - is coded as (ZRL) (1,4) 1110
• EOB– Inserted when a block ends with a run of zeros
In total there are 160 run/size symbols and 2 special symbols 162 symbols to 2 encodecodetable is 16 + 162 = 178 bytes
Coding Example
000000000000000000000000000000010000000000000006010002313
Typical DCT Block Coefficients
-13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end
DC Coefficient is -13. The size is 4 and the index is 0010
Current Stream State: 4 0010
Coding Example
-13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end
The first ac value is -3. That is a run of 0 zeros followed by -3.
-3 has size 2 and index 0000
Therefore the run/size pair is (0,2)
Current Stream State: 4 0010 (0,2) 00
Coding Example
-13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end
The next ac value is 6. That is a run of 0 zeros followed by 6.
6 has size 3 and index 110
Therefore the run/size pair is (0,3)
Current Stream State: 4 0010 (0,2) 00 (0,3) 110
Coding Example
-13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end
The next ac value to encode is a run of 2 zeros followed by a ac coefficient 2.
2 has size 2 and index 10
Therefore the run/size pair is (2,2)
Current Stream State: 4 0010 (0,2) 00 (0,3) 110 (2,2) 10
Coding Example
The next ac value to encode is a run of 3 zeros followed by a ac coefficient -1.
-1 has size 1 and index 0
Therefore the run/size pair is (3,1)
Current Stream State: 4 0010 (0,2) 00 (0,3) 110 (2,2) 10 (3,1) 0
-13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end
Coding Example
The next ac value to encode is a run of 17 zeros followed by a ac coefficient 1.
As the run is > 15 zeros we have to use the ZRL symbol to code the first 16 zeros. The remaining run length consists of (17 - 16) = 1 zero.
An ac coefficient of 1 has size 1 and index 1
Therefore we insert the run/size pair (1,1) after the ZRL marker
Current Stream State: 4 0010 (0,2) 00 (0,3) 110 (2,2) 10 (3,1) 0 ZRL (1,1) 1
-13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end
Coding Example
The remaining coeffs are all 0. Therefore the EOB marker is used.
If the last ac coeff is non-zero, then the EOB marker is not used.
Current Stream State: 4 0010 (0,2) 00 (0,3) 110 (2,2) 10 (3,1) 0 ZRL (1,1) 1 EOB
-13, -3, 6, 2 zeros, 2, 3 zeros, -1, 17 zeros, 1, 36 more zeros, the end
Huffman Coding
• Best Solution is to define the 2 Huffman codes for each image during compression
• However a default Huffman codetable is defined in the JPEG standard.
Final Stream: 4 0010 (0,2) 00 (0,3) 110 (2,2) 10 (3,1) 0 ZRL (1,1) 1 EOB
Encoded using dc codetable
Encoded using ac codetable
No further encoding
Default CodetablesAC tableDC table
Final Stream: 4 0110 (0,2) 0000 (0,3) 110 (2,2) 10 (3,1) 0 ZRL (1,1) 1 EOB
Fully Encoded Stream:
101 0110 01 0000 100 110 11111001 10 111010 0 11111111001 1100 1 1010
56 bits to encode 64 coefficients = 0.875 bits/coefficient
How good is this scheme?
Should we use default codetables?
Even though doubling the quantisation sizes reduces the number of events the distribution of those events doesn’t change much. Only the EOB probability changes significantly.
Therefore using the same codetable for both cases is reasonable
How good is this scheme?
In fact using the same codetable for multiple images doesn’t reduce the efficiency of the code much.
Efficiency when the default codetable is used
97.35%
95.74%
Special Markers
Synchronisation Markers
• There are 8 synch markersFFD0 ->FFD7
They can be placed at intervals which can be specified by using the DRI (FFDD) marker
Each marker is sent sequentially so if any marker is corrupted its absence can be easily detected.
Summary
• We have covered the basics of JPEG standard
• The standard specifies a syntax rather than specifying exactly how it is implemented
• Most implementations use the recommended settings provided by the JPEG community.