Wavelets

A julia package for a fast Discrete Wavelet Transform, and plotting and evaluation of wavelets

#Pkg.clone("https://github.com/vincentcp/WaveletsCopy.jl.git")
#Pkg.build("WaveletsCopy")

using WaveletsCopy
using Plots

A list of all available wavelets is given by

print_implemented_wavelets()

("db1", "db2", "db3", "db4", "db5", "db6", "db7", "db8", "db9", "db10", "cdf11", "cdf13", "cdf15", "cdf24", "cdf26", "cdf31", "cdf33", "cdf35", "cdf42", "cdf44", "cdf46", "cdf51", "cdf53", "cdf55", "cdf62", "cdf64", "cdf66")

Some examples are

db4, cdf53

(WaveletsCopy.DWT.DaubechiesWavelet{4,Float64}(), WaveletsCopy.DWT.CDFWavelet{5,3,Float64}())

1 The Discrete Wavelet Transform

The wavelet transform is performed using

a = rand(1<<8);
dwt(a, db3, perbound);

Only periodic boundaries boundaries and dyadic length are supported.

perbound

WaveletsCopy.DWT.PeriodicBoundary()

b = dwt(a,db3,perbound);
c = idwt(b,db3,perbound);
@assert a≈c

2 Evaluation of wavelets

Use

function evaluate_periodic_in_dyadic_points{T}(side::DWT.Side, kind::DWT.Scl, w::DWT.DiscreteWavelet{T}, j=0, k=0, d=10)

to evaluate the primal/dual wavelet function $\phi_{k,j}(x) = 2^{j/2}\phi(2^jx-k)$ of type w in equispaced grid with a separation of $2^{-d}$

Here an example plotting the scaling (red) and wavelet (blue) functions of the CDF wavelets cdf11, cdf13, ..., cdf44. In the left columns the primal functions and in the righ columns the dual functions.

plot(layout=(4,4),legend=false,ticks=nothing, border=nothing)
i=1;j=0;k=0;d=6
for p in 1:4
  qs = 2:2:4
  isodd(p) && (qs = 1:2:4 )
  for q in qs
    f, x = evaluate_in_dyadic_points(Primal, wavelet, CDFWavelet{p,q,Float64}(), j, k, d; points=true)
    plot!(x, -f, subplot=i)
    f, x = evaluate_in_dyadic_points(Primal, scaling, CDFWavelet{p,q,Float64}(), j, k, d; points=true)
    plot!(x, f, subplot=i)
    i += 1
  end
  for q in qs
    f, x = evaluate_in_dyadic_points(Dual, scaling, CDFWavelet{p,q,Float64}(), j, k, d; points=true)
    plot!(x, f, subplot=i)
    f, x = evaluate_in_dyadic_points(Dual, wavelet, CDFWavelet{p,q,Float64}(), j, k, d; points=true)
    plot!(x, f, subplot=i)  
    i += 1
  end
end
plot!()

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�[1m�[33mWARNING: �[39m�[22m�[33mRecursion algorithm is not convergent�[39m

Different translations and scales

plot(legend=false, layout=2,size=(900,300))
for wavelet_index in wavelet_indices(2)
    wavelet, j, k = wavelet_index
    f, x = evaluate_in_dyadic_points(Primal, wavelet, db3, j, k; points=true)
    plot!(x,f)
    f, x = evaluate_in_dyadic_points(Primal, wavelet, cdf24, j, k; points=true)
    plot!(x,f,subplot=2)
end
plot!()

Use

function evaluate_periodic_in_dyadic_points{T}(side::DWT.Side, kind::DWT.Scl, w::DWT.DiscreteWavelet{T}, j=0, k=0, d=10)

for wavelets periodized to the interval [0,1].

plot(legend=false, layout=2,size=(900,300))
for wavelet_index in wavelet_indices(2)
    wavelet, j, k = wavelet_index
    f, x = evaluate_periodic_in_dyadic_points(Primal, wavelet, db3, j, k; points=true)
    plot!(x,f)
    f, x = evaluate_periodic_in_dyadic_points(Primal, wavelet, cdf24, j, k; points=true)
    plot!(x,f,subplot=2)
end
plot!()

Also pointwise evaluation of scaling functions is possible up to some given precision. evaluate{T, S<:Real}(side::Side, kind::Scl, w::DiscreteWavelet{T}, j::Int, k::Int, x::S; xtol::S=1e-5)

evaluate_periodic{T, S<:Real}(side::Side, kind::Kind, w::DiscreteWavelet{T}, j::Int, k::Int, x::S; xtol::S=1e-5)

evaluate(Primal, scaling, db3, 0, 0, .123, xtol=1e-1), evaluate(Primal, scaling, db3, 0, 0, .123, xtol=1e-4)

(0.13394983521281625, 0.13129163397063862)

t = linspace(-1,1,1000)
plot(t,evaluate_periodic.(Primal, scaling, db3, 3, 0, t))
plot!(t,evaluate.(Primal, scaling, db3, 3, 0, t))

3. Plotting of Wavelets

Plot all in one figure

plot(cdf33)

Or over different figures

plot(cdf33,layout=(2,2))

Of orthogonal wavelets only primal functions are shown since the duals are equal to the primal functions

plot(db3)

Plot only the dual (or primal) ones

plot(Dual,cdf24)

plot(Dual, wavelet,cdf13)