A Julia package for fast wavelet transforms (1-D, 2-D, 3-D, by filtering or lifting). The package includes discrete wavelet transforms, column-wise discrete wavelet transforms, and wavelet packet transforms.

• 1st generation wavelets using filter banks (periodic and orthogonal). Filters are included for the following types: Haar, Daubechies, Coiflet, Symmlet, Battle-Lemarie, Beylkin, Vaidyanathan.

• 2nd generation wavelets by lifting (periodic and general type including orthogonal and biorthogonal). Included lifting schemes are currently only for Haar and Daubechies (under development). A new lifting scheme can be easily constructed by users. The current implementation of the lifting transforms is 2x faster than the filter transforms.

• Thresholding, best basis and denoising functions, e.g. TI denoising by cycle spinning, best basis for WPT, noise estimation, matching pursuit. See example code and image below.

• Wavelet utilities e.g. indexing and size calculation, scaling and wavelet functions computation, test functions, up and down sampling, filter mirrors, coefficient counting, inplace circshifts, and more.

• Plotting/visualization utilities for 1-D and 2-D signals.

See license (MIT) in LICENSE.md.

Usage

Install via the package manager and load with using

julia> Pkg.add("Wavelets")
julia> using Wavelets

API

Wavelet Transforms

The functions dwt and wpt (and their inverses) are linear operators. See wavelet below for construction of the type wt.

Discrete Wavelet Transform

# DWT
dwt(x, wt, L=maxtransformlevels(x))
idwt(x, wt, L=maxtransformlevels(x))
dwt!(y, x, filter, L=maxtransformlevels(x))
idwt!(y, scheme, L=maxtransformlevels(x))

Wavelet Packet Transform

# WPT (tree can also be an integer, equivalent to maketree(length(x), L, :full))
wpt(x, wt, tree::BitVector=maketree(x, :full))
iwpt(x, wt, tree::BitVector=maketree(x, :full))
wpt!(y, x, filter, tree::BitVector=maketree(x, :full))
iwpt!(y, scheme, tree::BitVector=maketree(y, :full))

Wavelet Types

The function wavelet is a type contructor for the transform functions. The transform type t can be either WT.Filter or WT.Lifting.

wavelet(c, t=WT.Filter, boundary=WT.Periodic)

Wavelet Classes

The module WT contains the types for wavelet classes. The module defines constants of the form e.g. WT.coif4 as shortcuts for WT.Coiflet{4}(). The numbers for orthogonal wavelets indicate the number vanishing moments of the wavelet function.

Class Type	Namebase	Supertype	Numbers
Haar	haar	OrthoWaveletClass
Coiflet	coif	OrthoWaveletClass	2:2:8
Daubechies	db	OrthoWaveletClass	1:Inf
Symlet	sym	OrthoWaveletClass	4:10
Battle	batt	OrthoWaveletClass	2:2:6
Beylkin	beyl	OrthoWaveletClass
Vaidyanathan	vaid	OrthoWaveletClass
CDF	cdf	BiOrthoWaveletClass	(9,7)

Class information

WT.class(::WaveletClass) ::String              # class full name
WT.name(::WaveletClass) ::String               # type short name
WT.vanishingmoments(::WaveletClass)            # vanishing moments of wavelet function

Transform type information

WT.name(wt)                                     # type short name
WT.length(f::OrthoFilter)                       # length of filter
WT.qmf(f::OrthoFilter)                          # quadrature mirror filter
WT.makeqmfpair(f::OrthoFilter, fw=true)
WT.makereverseqmfpair(f::OrthoFilter, fw=true)

Examples

The simplest way to transform a signal x is

xt = dwt(x, wavelet(WT.db2))

The transform type can be more explicitly specified (filter, Periodic, Orthogonal, 4 vanishing moments)

wt = wavelet(WT.Coiflet{4}(), WT.Filter, WT.Periodic)

For a periodic biorthogonal CDF 9/7 lifting scheme:

wt = wavelet(WT.cdf97, WT.Lifting)

Perform a transform of vector x

# 5 level transform
xt = dwt(x, wt, 5)
# inverse tranform
xti = idwt(xt, wt, 5)
# a full transform
xt = dwt(x, wt)

Other examples:

# scaling filters is easy
wt = wavelet(WT.haar)
wt = WT.scale(wt, 1/sqrt(2))
# signals can be transformed inplace with a low-level command
# requiring very little memory allocation (especially for L=1 for filters)
dwt!(x, wt, L)      # inplace (lifting)
dwt!(xt, x, wt, L)  # write to xt (filter)

# denoising with default parameters (VisuShrink hard thresholding)
x0 = testfunction(128, "HeaviSine")
x = x0 + 0.3*randn(128)
y = denoise(x)

# plotting utilities 1-d (see images and code in /example)
x = testfunction(128, "Bumps")
y = dwt(x, wavelet(WT.cdf97, WT.Lifting))
d,l = wplotdots(y, 0.1, 128)
A = wplotim(y)
# plotting utilities 2-d
img = imread("lena.png")
x = permutedims(img.data, [ndims(img.data):-1:1])
L = 2
xts = wplotim(x, L, wavelet(WT.db3))

See Bumps and Lena for plot images.

Thesholding

The Wavelets.Threshold module includes the following utilities

• denoising (VisuShrink, translation invariant (TI))
• best basis for WPT
• noise estimator
• matching pursuit
• threshold functions (see table for types)

API

# threshold types with parameter
threshold!(x::AbstractArray, TH::THType, t::Real)
threshold(x::AbstractArray, TH::THType, t::Real)
# without parameter (PosTH, NegTH)
threshold!(x::AbstractArray, TH::THType)
threshold(x::AbstractArray, TH::THType)
# denoising
denoise(x::AbstractArray,
        wt=DEFAULT_WAVELET;
        L::Int=min(maxtransformlevels(x),6),
        dnt=VisuShrink(size(x,1)),
        estnoise::Function=noisest,
        TI=false,
        nspin=tuple([8 for i=1:ndims(x)]...) )
# entropy
coefentropy(x, et::Entropy, nrm)
# best basis for WPT limited to active inital tree nodes
bestbasistree(y::AbstractVector, wt::DiscreteWavelet,
        L::Integer=nscales(y), et::Entropy=ShannonEntropy() )
bestbasistree(y::AbstractVector, wt::DiscreteWavelet,
        tree::BitVector, et::Entropy=ShannonEntropy() )

Type	Details
Thresholding	<: THType
HardTH	hard thresholding
SoftTH	soft threshold
SemiSoftTH	semisoft thresholding
SteinTH	stein thresholding
PosTH	positive thresholding
NegTH	negative thresholding
BiggestTH	biggest m-term (best m-term) approx.
Denoising	<: DNFT
VisuShrink	VisuShrink denoising
Entropy	<: Entropy
ShannonEntropy	-v^2*log(v^2) (Coifman-Wickerhauser)
LogEnergyEntropy	-log(v^2)

Examples

Find best basis tree for wpt, and compare to dwt using biggest m-term approximations.

wt = wavelet(WT.db4)
x = sin(4*linspace(0,2*pi-eps(),1024))
tree = bestbasistree(x, wt)
xtb = wpt(x, wt, tree)
xt = dwt(x, wt)
# get biggest m-term approximations
m = 50
threshold!(xtb, BiggestTH(), m)
threshold!(xt, BiggestTH(), m)
# compare sparse approximations in ell_2 norm
vecnorm(x - iwpt(xtb, wt, tree), 2) # best basis wpt
vecnorm(x - idwt(xt, wt), 2)        # regular dwt

julia> vecnorm(x - iwpt(xtb, wt, tree), 2)
0.008941070750964843
julia> vecnorm(x - idwt(xt, wt), 2)
0.05964431178940861

Denoising:

n = 2^11;
x0 = testfunction(n,"Doppler")
x = x0 + 0.05*randn(n)
y = denoise(x,TI=true)

Benchmarks

Timing of dwt (using db2 filter of length 4) and fft. The lifting wavelet transforms are faster and use less memory than fft in 1-D and 2-D. dwt by lifting is currently 2x faster than by filtering.

# 10 iterations
dwt by filtering (N=1048576), 20 levels
elapsed time: 0.247907616 seconds (125861504 bytes allocated, 8.81% gc time)
dwt by lifting (N=1048576), 20 levels
elapsed time: 0.131240966 seconds (104898544 bytes allocated, 17.48% gc time)
fft (N=1048576), (FFTW)
elapsed time: 0.487693289 seconds (167805296 bytes allocated, 8.67% gc time)

For 2-D transforms (using a db4 filter and CDF 9/7 lifting scheme):

# 10 iterations
dwt by filtering (N=1024x1024), 10 levels
elapsed time: 0.773281141 seconds (85813504 bytes allocated, 2.87% gc time)
dwt by lifting (N=1024x1024), 10 levels
elapsed time: 0.317705928 seconds (88765424 bytes allocated, 3.44% gc time)
fft (N=1024x1024), (FFTW)
elapsed time: 0.577537263 seconds (167805888 bytes allocated, 5.53% gc time)

By using the low-level function dwt! and pre-allocating temporary arrays, significant gains can be made in terms of memory usage (and some speedup). This is useful when transforming multiple times.

To-do list

• Transforms for non-square 2-D signals
• Boundary extensions (other than periodic)
• Boundary orthogonal wavelets
• Define more lifting schemes
• WPT in 2-D
• Stationary transform
• Continuous wavelets
• Wavelet scalogram