Fast Algorithms for Discrete Wavelet Transform

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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

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  • 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)

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    #1: Arithmetic Complexity

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    #2: Various Structures

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    #1: Arithmetic Complexity

    MULTSADDS

    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

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    000

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    Tran.Filter (FFT)

    Tran.Filter(Short-length)

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    Tran.Filter(Short-length)

    Binomial QMF filters

    Classical Lattice

    CORDIC Lattice

    Sheet1

    #1: Arithmetic Complexity

    MULTSADDS

    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

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    Tran.Filter (FFT)

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    Classical Lattice

    CORDIC Lattice

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    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

    Comments, Implementations, etc.

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