Data compression strategies for exascale CFD [email protected] 5. Dezember 2016 MotivationImage Compression OverviewExtended JP2000 FormatSummaryAppendix

Motivation Image Compression Overview Extended JP2000 Format Summary Appendix

Data compression strategies for exascaleCFD simulations

Patrick VoglerInstitut für Aerodynamik und Gasdynamik - Universität Stuttgart

[email protected]

5. Dezember 2016


Outline

1. Motivation

2. Image Compression OverviewReduction of Spacial RedundancyEntropy of a Random VariableReduction of EntropyEntropy Encoding

3. Extended JP2000 FormatTime-Frequency TransformDeadzone QuantizationEmbedded Block Coding

4. Summary

IAGUniversität Stuttgart P. Vogler Data Compression Strategies 1 / 26


Outline

1. Motivation



4. Summary



Motivation

Move towards exascale computing due to increasingly complexdemands on scientific and numerical modelling.

Growing mismatch between the ability to produce and store/analyzedata.

Alleviating the I/O bottleneck in exascale computing by considerabledata-reduction before I/O.

Image and video compression algorithms offer robust and portablesource codes with a wide range of compression ratios (lossy/lossless).



Outline

1. Motivation



4. Summary



Reduction of Spacial RedundancyMotivation




2418m 3047m 2466m 2545m 2779m 3611m 3388m 3497m 4052m 3282m 2590m 3038m 2546m

n∑i=0

bhi = 12bit+ 12bit+ 12bit+ 12bit . . . = 324bit.




+241m +341m -239m -7m +207m +1462m -279m +271m +193m -493m -361m +329m -430m

n∑i=0

bhi = 12bit+ 9bit+ 10bit+ 10bit . . . = 249bit.



Entropy of a Random Variable

H(x) = −∑x∈Ax

fx(x) log2 fx(x)



Entropy of a Random Variable

P

0 0.2 0.4 0.6 0.8 1

H(P

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X ={

1, with probability p,0, with probability 1− p.

H (X) = −p log p− (1− p) log (1− p) := H (P )



Reduction of EntropyQuantization

I1 I2 I3 I4 I5 I6

y[k1, k2]

q[k1, k2] = 3

Q (y) ={i, if y[k1, k2] ∈ Ii0, otherwise



Entropy EncodingWilliam Shakespeare 1 Year Old Baby

O, full of scorpions is my mind, dearwife! Thou know’st that Banquo, andhis Fleance, lives.

Baa ba babaaababa ba ba ba babaabab babbababa bababaaaa baabababaaaaa babaaababa baaabababa



Entropy EncodingWilliam Shakespeare 1 Year Old Baby

f() = 0.172; f(!) = 0.011; f(‘) = 0.011;f(, ) = 0.043; f(.) = 0.011; f(B) = 0.011;f(F ) = 0.011; f(O) = 0.011; f(T ) = 0.011;f(a) = 0.054; f(c) = 0.022; f(d) = 0.032;...

H(x) = −∑x∈Ax

fx(x) log2 fx(x) = 4.42053

f() = 0.14; f(B) = 0.011; f(a) = 0.505;f(b) = 0.344;

H(x) = −∑

x∈Ax

fx(x) log2 fx(x) = 1.49431



Entropy Encoding

{B, , b}

P = 495100

{a}

P = 505100

{b}

P = 344100

{B, }

P = 151100

{B}

P = 11100

{ }

P = 1410

root“0“

“0“

“1“

“1“

“1“

“0“



Outline

1. Motivation



4. Summary



Time-Frequency Transform

t

f(t)

f̂ (ω) = (2π)−1/2∫Rf(x)e−iωx dx




time resolution

freq

uency

reso

luti

on Fourier Transform

time resolutionfr

eq

uency

reso

luti

on Windowed Fourier Transform

time resolution

freq

uency

reso

luti

on

low frequency(high scale)

high frequency(low scale)

Wavelet Transform

f̂ (ω) = (2π)−1/2∫Rf(x)e−iωx dx




time resolution

freq

uency

reso

luti


time resolutionfr

eq

uency

reso

luti


time resolution

freq

uency

reso

luti

on



Wavelet Transform

Ggf(ω, t) = (2π)−1/2∫Rf(x)g(x− t)e−iωx dx




time resolution

freq

uency

reso

luti


time resolutionfr

eq

uency

reso

luti


time resolution

freq

uency

reso

luti

on



Wavelet Transform

Wψf(a, b) =∫Rf(x) 1√

|a|ψ

(x− ba

)dx



Time-Frequency TransformDiscrete 1D-Wavelet Transform

LeGall-5/3-Wavelet

y(2n+ 1) = x(2n+ 1)− bx(2n)+x(2n+2)2 c,

y(2n) = x(2n)− bx(2n−1)+x(2n+1)+24 c.

Cohen-Daubechies-Feauveau-9/7-Wavelet

y(2n+ 1) ← x(2n+ 1) + (α× |x(2n) + x(2n+ 2)|),y(2n) ← x(2n) + (β × |y(2n− 1) + y(2n+ 1)|),y(2n+ 1) ← x(2n+ 1) + (γ × |y(2n) + y(2n+ 2)|),y(2n) ← x(2n) + (δ × |y(2n− 1) + y(2n+ 1)|),y(2n+ 1) ← −K × y(2n+ 1),y(2n) ← (1/K)× y(2n).




95... 128 255 128 95 ... ... 77 89 122 89 77 ...

symmetric point symmetric point

Leading boundary Trailing boundary










H1L1L1

H1H1L1L 1H1L1

H2H2L2L 2H2L2

L 2L 2L 2 H2L2L 2

L 1L 1H 1 H 1L 1H 1

L 1H1L1 H1L1H1

H1 H

1 H1



Time-Frequency TransformFloating Point Datasets

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

0

20

40

60

80

100

120

140

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

(−1)sign(

1 +52∑i=1

b52−i 2−i)× 2e−1023




Velocity Field

Sign Mantissa Exponent



Time-Frequency TransformShape Adaptive Wavelet Transform [2]

95... 128 255 128 95 ... ... 77 89 122 89 77 ...

symmetric point symmetric point

Leading boundary Trailing boundary









Deadzone Quantization

∆b ∆b 2∆b ∆b ∆b

−2 −1 0 1 2qb(u, v)

y(u, v)

∆b = 2Rb−εb(1 + µb

211

),

qb(u, v) = sign(yb(u, v))b |yb(u,v)|∆b

c.



Embedded Block CodingCoefficient Bit Modeling

1

0

0

0MSB

LSB



Embedded Block CodingCoefficient Bit Modeling

××××



Embedded Block CodingRate Allocation for IEEE 754 Data [3]

1 2 3 4 5 6 7Code Block

123456

LSB

MSB

BitPlanes

Quality Layer 4Quality Layer 3Quality Layer 2Quality Layer 1



Embedded Block CodingRate Allocation for IEEE 754 Data [3]

D =∑iGi

∑j d

ij ,

di =(∆aij

)2 +(∆bij

)2 +(∆sij

)2,

∆aij = 223 · ln(2) · xij ·{

∆yij}a,

∆bij = 223 · sij · 2aij ·{

∆yij}b,

∆sij = 223 · |xij | ·{

∆yij}s,



Outline

1. Motivation



4. Summary



Summary

JPEG 2000 can be extended to support IEEE754 floating pointnumbers.

Provides lossless and lossy compression in one codestream.

Superior compression performance (compared to JPEG).

Resolution and quality scalability.

High dynamic range support.

Robust to bit errors.



Thank you for your attention



References ISakai, R., Onda, H., Sasaki, D. and Nakahashi, K.Data Compression of Large-scale Flow Computation forAerodynamic/Aeroacoustic Analysis49th AIAA Aerospace Sciences Meeting, 2011

Li , S. and Li, W.Shape Adaptive Discrete Wavelet Transforms for Arbitrarily ShpaedVisual Object Coding.IEEE Transactions on Circuits and Systems for Video Technology10(5), 2000

Gamito, M. N. and Dias, M. S.Lossless coding of floating point data with JPEG 2000 Part 10Proceedings of SPIE, 2004



References IIBruylants, T., Munteanu, A. and Schelkens, P.Wavelet based volumetric medical image compressionSignal Processing: Image Communication 31, p. 112-133, 2015

Do, M. N. and Vetterli, M.The Contourlet Transform: An Efficient Directional MultiresolutionImage RepresentationIEEE Transactions on Image Processing Volume:14 Issue:12, p.2091-2106, 2005ExaFlow Project (2015): ExaFlow FlyerURL: http: // exaflow-project. eu/ index. php/ objectives(visited on 30.03.2016)


http://exaflow-project.eu/index.php/objectives


References IIIExaFlow Project (2015): ObjectivesURL: http: // exaflow-project. eu/ images/DisseminationMaterials/ flyer_ exaflow_ printing. pdf(visited on 30.03.2016)

HPCwire (2014): Burst Buffers Flash Exascale PotentialURL: http: // www. hpcwire. com/ 2014/ 05/ 01/burst-buffers-flash-exascale-potential/(visited on 30.03.2016)

AXIS Communications: Video compressionURL: http: // www. axis. com/ de/ de/ learning/web-articles/ technical-guide-to-network-video/video-compression-guide(visited on 30.03.2016)


http://exaflow-project.eu/images/DisseminationMaterials/flyer_exaflow_printing.pdf

http://exaflow-project.eu/images/DisseminationMaterials/flyer_exaflow_printing.pdf

http://www.hpcwire.com/2014/05/01/burst-buffers-flash-exascale-potential/

http://www.hpcwire.com/2014/05/01/burst-buffers-flash-exascale-potential/

http://www.axis.com/de/de/learning/web-articles/technical-guide-to-network-video/video-compression-guide



Documents

Data compression strategies for exascale CFD [email protected] 5. Dezember 2016 MotivationImage Compression OverviewExtended JP2000 FormatSummaryAppendix