# Fast Algorithms for Discrete Wavelet Transform

• Published on
25-Feb-2016

• View
22

4

DESCRIPTION

Fast Algorithms for Discrete Wavelet Transform. Review and Implementation. by Dan Li F2000. DWT and FWT: Significance. DWT Multi-resolution mode to access the information Extensively (and intensively) used in information processing Advantage over other transforms - PowerPoint PPT Presentation

Transcript

• Fast Algorithms for Discrete Wavelet Transform

Review and Implementationby Dan LiF2000

• DWT and FWT: Significance DWTMulti-resolution mode to access the information Extensively (and intensively) used in information processingAdvantage over other transformse.g. (in JPEG 2000), DWT provides 20-30% improvement in compression efficiency as oppose to DCT.

FWTMulti-resolution mode to access the informationDWT: intensive computation and large memory requirement.FWT makes DWT practicable in real applicationsMain factors controlling the speed of DWT:Filter lengthFloating point operation vs. integer operation

• FWT Algorithm: An OutlineRegular StructureMallat Straightfoward Filter Bank Polyphase Transversal Filters#Polyphase Short-length Filters*Classical Lattice Filters CORDIC Lattice Filters Lifting Scheme and Integer WTBinomial QMF Filters (* also known as fast-running FIR algo)(* based on FFT for fast filtering)Irregular StructureRegular Structure

• FWT Algorithm: An Overview (I)Mallat Filter BankTransversal FiltersShort-length Filters

• FWT Algorithm: An Overview (II)Binomial QMFClassical LatticeCORDIC LatticeLifting Ladder

• Comput. Complexity: A Comparison# of multsFilter length L# of addsFilter length LWord length w (bits)# of addersneededArithmetic Complexity (per input point & per decomposition cell)Computational Structure Complexity (per filter coefficient)

Chart2

443

64.674

85.234.5

105.674.8

126.186

166.569

186.838

207.137.2

247.3212

307.769.6

327.9218

Mallat FB

Tran.Filter (FFT)

Tran.Filter(Short-length)

Sheet1

#1: Arithmetic Complexity

Mallat FBTran.Filter (FFT)Tran.Filter(Short-length)filter lengthMallat FBTran.Filter (FFT)Tran.Filter(Short-length)

4443439.334

664.67465126.3

885.234.58714.158.5

10105.674.810915.3314.2

12126.186121116.7312

16166.569161518.2413

18186.83818171917

20207.137.2201919.8321.4

24247.3212242320.6818

30307.769.6302921.9227

32327.9218323122.3722

#2: Various Structures

word lengthTran.Filter (FFT)Tran.Filter(Short-length)Binomial QMF filtersClassical LatticeCORDIC Lattice

820151688

1022016161410

122618182010

143425202210

163828222410

184432222812

Sheet1

000

000

000

000

000

000

000

000

000

000

000

Mallat FB

Tran.Filter (FFT)

Tran.Filter(Short-length)

Sheet2

Sheet3

Chart3

39.334

5126.3

714.158.5

915.3314.2

1116.7312

1517.713

171917

1919.8321.4

2320.6818

2921.9227

3122.3722

Mallat FB

Tran.Filter (FFT)

Tran.Filter(Short-length)

Sheet1

#1: Arithmetic Complexity

Mallat FBTran.Filter (FFT)Tran.Filter(Short-length)Mallat FBTran.Filter (FFT)Tran.Filter(Short-length)

4443439.334

664.67465126.3

885.234.58714.158.5

10105.674.810915.3314.2

12126.186121116.7312

16166.569161517.713

18186.83818171917

20207.137.2201919.8321.4

24247.3212242320.6818

30307.769.6302921.9227

32327.9218323122.3722

#2: Various Structures

word lengthTran.Filter (FFT)Tran.Filter(Short-length)Binomial QMF filtersClassical LatticeCORDIC Lattice

820151688

1022016161410

122618182010

143425202210

163828222410

184432222812

Sheet1

000

000

000

000

000

000

000

000

000

000

000

Mallat FB

Tran.Filter (FFT)

Tran.Filter(Short-length)

Sheet2

Sheet3

Chart4

20151688

2216161410

2618182010

3425202210

3828222410

4432222812

Tran.Filter (FFT)

Tran.Filter(Short-length)

Binomial QMF filters

Classical Lattice

CORDIC Lattice

Sheet1

#1: Arithmetic Complexity

Mallat FBTran.Filter (FFT)Tran.Filter(Short-length)Mallat FBTran.Filter (FFT)Tran.Filter(Short-length)

4443439.334

664.67465126.3

885.234.58714.158.5

10105.674.810915.3314.2

12126.186121116.7312

16166.569161518.2413

18186.83818171917

20207.137.2201919.8321.4

24247.3212242320.6818

30307.769.6302921.9227

32327.9218323122.3722

#2: Various Structures

Tran.Filter (FFT)Tran.Filter(Short-length)Binomial QMF filtersClassical LatticeCORDIC Lattice

820151688

102216161410

122618182010

143425202210

163828222410

184432222812

Sheet1

00000

00000

00000

00000

00000

00000

Tran.Filter (FFT)

Tran.Filter(Short-length)

Binomial QMF filters

Classical Lattice

CORDIC Lattice

Sheet2

Sheet3

• Efficiency in the sense of arithmetic complexity and computational structureStraightforward filter bank: classical and used in many commercial s/wPolyphase structure: more efficient than direct FB. (Worthy of further exploration!)FFT based filtering: efficient for medium or long filtersFast running FIR filter: good for short filtersBinomial QMF: reduces the # of mults with expense of additional adds Lattice: easier to implement with each relatively simpler stagesCORDIC: most suitable fore efficient VLSI implementation since only addition and shifts involved and least possible adders requiredLifting scheme: lead to IWT which is faster than floating-point DWT and ideal for lossless coding/compression.

Implementation focused on the following:Fast filtering for short and long filters Various formats of polyphase structures Reformulation of polyphase transversal filter with the consideration of reduced inter-channel communicationInteger filter and IWT implementationsSimulations