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ReverseBiorthogonalSplineWavelet

represents a reverse biorthogonal spline wavelet of order 4 and dual order 2.

ReverseBiorthogonalSplineWavelet[n,m]

represents a reverse biorthogonal spline wavelet of order n and dual order m.

Details

ReverseBiorthogonalSplineWavelet defines a family of biorthogonal wavelets.
ReverseBiorthogonalSplineWavelet[n,m] is defined for positive integers m and n where m+n is even.
The scaling function () and wavelet function () have compact support. The functions are symmetric.
ReverseBiorthogonalSplineWavelet can be used with such functions as DiscreteWaveletTransform and WaveletPhi, etc.

Examples

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Basic Examples (6)

Primal scaling function:

Wolfram Language code: Plot[WaveletPhi[ReverseBiorthogonalSplineWavelet[2, 4], x], {x, -2, 2}, PlotRange -> All]

Primal wavelet function:

Wolfram Language code: Plot[WaveletPsi[ReverseBiorthogonalSplineWavelet[2, 4], x], {x, -2, 3}, PlotRange -> All]

Dual scaling function:

Wolfram Language code: Plot[WaveletPhi[ReverseBiorthogonalSplineWavelet[2, 4], x, "Dual"], {x, -4, 4}, PlotRange -> All]

Dual wavelet function:

Wolfram Language code: Plot[WaveletPsi[ReverseBiorthogonalSplineWavelet[2, 4], x, "Dual"], {x, -2, 3}, PlotRange -> All]

Primal filter coefficients:

Wolfram Language code:

WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[2, 4], {"PrimalLowpass", "PrimalHighpass"}, WorkingPrecision -> ∞]

Dual filter coefficients:

Wolfram Language code:

WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[2, 4], {"DualLowpass", "DualHighpass"}, WorkingPrecision -> ∞]

Scope (17)

Basic Uses (10)

Compute primal lowpass filter coefficients:

Wolfram Language code: WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[2, 4], "PrimalLowpass"]

Dual lowpass filter coefficients:

Wolfram Language code: WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[2, 4], "DualLowpass"]

Primal highpass filter coefficients:

Wolfram Language code: WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[2, 4], "PrimalHighpass"]

Dual highpass filter coefficients:

Wolfram Language code: WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[2, 4], "DualHighpass"]

Lifting filter coefficients:

Wolfram Language code: WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[2, 4], "LiftingFilter", WorkingPrecision -> ∞]

Wolfram Language code: %[{"LiftingMatrixForm", z}]

Generate a function to compute lifting wavelet transform:

Wolfram Language code: lf = WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[2, 4], "LiftingFilter", WorkingPrecision -> ∞]

Wolfram Language code: lf["ForwardLiftingFunction"][Range[8]]

Wolfram Language code: lf["InverseLiftingFunction"][%]

Primal scaling function:

Wolfram Language code: Plot[WaveletPhi[ReverseBiorthogonalSplineWavelet[2, 4], x], {x, -1, 1}, PlotRange -> All]

Dual scaling function:

Wolfram Language code: Plot[WaveletPhi[ReverseBiorthogonalSplineWavelet[2, 4], x, "Dual"], {x, -4, 4}, PlotRange -> All]

Plot scaling function using different levels of recursion:

Wolfram Language code:

Table[Plot[WaveletPhi[ReverseBiorthogonalSplineWavelet[4, 4], x, MaxRecursion -> i], {x, -2, 2}, PlotLabel -> i, PlotRange -> All], {i, 1, 8, 2}]

Primal wavelet function:

Wolfram Language code: Plot[WaveletPsi[ReverseBiorthogonalSplineWavelet[2, 4], x], {x, -2, 3}, PlotRange -> All]

Dual wavelet function:

Wolfram Language code: Plot[WaveletPsi[ReverseBiorthogonalSplineWavelet[2, 4], x, "Dual"], {x, -2, 3}, PlotRange -> All]

Plot wavelet function at different refinement scales:

Wolfram Language code:

Table[Plot[WaveletPsi[ReverseBiorthogonalSplineWavelet[4, 4], x, MaxRecursion -> i], {x, -3, 4}, PlotLabel -> i, PlotRange -> All], {i, 1, 8, 2}]

Wavelet Transforms (5)

Compute a DiscreteWaveletTransform:

Wolfram Language code: data = Table[Sin[4π t] + 2Exp[-10 ^ 5(1 / 3 - t) ^ 2], {t, 0, 1, (1/1023)}];

Wolfram Language code: ListLinePlot[data, PlotRange -> All]

Wolfram Language code: dwt = DiscreteWaveletTransform[data, ReverseBiorthogonalSplineWavelet[4, 4], 2]

View the tree of wavelet coefficients:

Wolfram Language code: dwt["TreeView"]

Get the dimensions of wavelet coefficients:

Wolfram Language code: dwt["Dimensions"]

Plot the wavelet coefficients:

Wolfram Language code: WaveletListPlot[dwt, PlotLayout -> "CommonXAxis"]

Compute a DiscreteWaveletPacketTransform:

Wolfram Language code: data = Table[Sin[4π t] + 2Exp[-10 ^ 5(1 / 3 - t) ^ 2], {t, 0, 1, (1/1023)}];

Wolfram Language code: dwpt = DiscreteWaveletPacketTransform[data, ReverseBiorthogonalSplineWavelet[4, 4], 2]

View the tree of wavelet coefficients:

Wolfram Language code: dwpt["TreeView"]

Get the dimensions of wavelet coefficients:

Wolfram Language code: dwpt["Dimensions"]

Plot the wavelet coefficients:

Wolfram Language code: WaveletListPlot[dwpt, PlotLayout -> "CommonXAxis"]

Compute a StationaryWaveletTransform:

Wolfram Language code: data = Table[Sin[4π t] + 2Exp[-10 ^ 5(1 / 3 - t) ^ 2], {t, 0, 1, (1/1023)}];

Wolfram Language code: swt = StationaryWaveletTransform[data, ReverseBiorthogonalSplineWavelet[4, 4], 2];

View the tree of wavelet coefficients:

Wolfram Language code: swt["TreeView"]

Get the dimensions of wavelet coefficients:

Wolfram Language code: swt["Dimensions"]

Plot the wavelet coefficients:

Wolfram Language code: WaveletListPlot[swt, PlotLayout -> "CommonXAxis"]

Compute a StationaryWaveletPacketTransform:

Wolfram Language code: data = Table[Sin[4π t] + 2Exp[-10 ^ 5(1 / 3 - t) ^ 2], {t, 0, 1, (1/1023)}];

Wolfram Language code: swpt = StationaryWaveletPacketTransform[data, ReverseBiorthogonalSplineWavelet[4, 4], 2];

View the tree of wavelet coefficients:

Wolfram Language code: swpt["TreeView"]

Get the dimensions of wavelet coefficients:

Wolfram Language code: swpt["Dimensions"]

Plot the wavelet coefficients:

Wolfram Language code: WaveletListPlot[swpt, PlotLayout -> "CommonXAxis"]

Compute a LiftingWaveletTransform:

Wolfram Language code: data = Table[Sin[4π t] + 2Exp[-10 ^ 5(1 / 3 - t) ^ 2], {t, 0, 1, (1/1023)}];

Wolfram Language code: lwt = LiftingWaveletTransform[data, ReverseBiorthogonalSplineWavelet[4, 4], 2];

View the tree of wavelet coefficients:

Wolfram Language code: lwt["TreeView"]

Get the dimensions of wavelet coefficients:

Wolfram Language code: lwt["Dimensions"]

Plot the wavelet coefficients:

Wolfram Language code: WaveletListPlot[lwt, PlotLayout -> "CommonXAxis"]

Higher Dimensions (2)

Multivariate scaling and wavelet functions are products of univariate ones:

Wolfram Language code:

ϕ = WaveletPhi[ReverseBiorthogonalSplineWavelet[3, 3]];
ψ = WaveletPsi[ReverseBiorthogonalSplineWavelet[3, 3]];

Wolfram Language code:

Plot3D[Evaluate[ϕ[x]ϕ[y]], {x, -1, 2}, {y, -1, 2}, PlotRange -> All, ColorFunction -> "SolarColors", Mesh -> None, Axes -> None]

Wolfram Language code:

Plot3D[Evaluate[ϕ[x]ψ[y]], {x, -1, 2}, {y, -2, 3}, PlotRange -> All, ColorFunction -> "SolarColors", Mesh -> None, Axes -> None]

Wolfram Language code:

Plot3D[Evaluate[ψ[x]ϕ[y]], {x, -2, 3}, {y, -1, 2}, PlotRange -> All, ColorFunction -> "SolarColors", Mesh -> None, Axes -> None]

Wolfram Language code:

Plot3D[Evaluate[ψ[x]ψ[y]], {x, -2, 3}, {y, -2, 3}, PlotRange -> All, ColorFunction -> "SolarColors", Mesh -> None, Axes -> None]

Multivariate dual scaling and wavelet functions are products of univariate ones:

Wolfram Language code:

Overscript[ϕ, ~] = WaveletPhi[ReverseBiorthogonalSplineWavelet[3, 3], #, "Dual"]&;
Overscript[ψ, ~] = WaveletPsi[ReverseBiorthogonalSplineWavelet[3, 3], #, "Dual"]&;

Wolfram Language code:

Plot3D[Evaluate[Overscript[ϕ, ~][x]Overscript[ϕ, ~][y]], {x, -3, 4}, {y, -3, 4}, PlotRange -> All, ColorFunction -> "RustTones", Mesh -> None, Axes -> None]

Wolfram Language code:

Plot3D[Evaluate[Overscript[ϕ, ~][x]Overscript[ψ, ~][y]], {x, -3, 4}, {y, -2, 3}, PlotRange -> All, ColorFunction -> "RustTones", Mesh -> None, Axes -> None]

Wolfram Language code:

Plot3D[Evaluate[Overscript[ψ, ~][x]Overscript[ϕ, ~][y]], {x, -2, 3}, {y, -3, 4}, PlotRange -> All, ColorFunction -> "RustTones", Mesh -> None, Axes -> None]

Wolfram Language code:

Plot3D[Evaluate[Overscript[ψ, ~][x]Overscript[ψ, ~][y]], {x, -2, 3}, {y, -2, 3}, PlotRange -> All, ColorFunction -> "RustTones", Mesh -> None, Axes -> None]

Properties & Relations (19)

ReverseBiorthogonalSplineWavelet[1,1] is equivalent to HaarWavelet:

Wolfram Language code: WaveletFilterCoefficients[HaarWavelet[]] == WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[1, 1]]

Wolfram Language code:

WaveletFilterCoefficients[HaarWavelet[]] == WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[1, 1], "DualLowpass"]

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

Wolfram Language code:

{Plot[WaveletPhi[ReverseBiorthogonalSplineWavelet[2, 4], x], {x, -1, 1}, PlotRange -> All], Plot[WaveletPhi[BiorthogonalSplineWavelet[2, 4], x, "Dual"], {x, -1, 1}, PlotRange -> All]}

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

Wolfram Language code:

{Plot[WaveletPhi[ReverseBiorthogonalSplineWavelet[2, 4], x, "Dual"], {x, -4, 4}, PlotRange -> All], Plot[WaveletPhi[BiorthogonalSplineWavelet[2, 4], x], {x, -4, 4}, PlotRange -> All]}

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

Wolfram Language code:

{Plot[WaveletPsi[ReverseBiorthogonalSplineWavelet[2, 4], x], {x, -2, 3}, PlotRange -> All], Plot[WaveletPsi[BiorthogonalSplineWavelet[2, 4], x, "Dual"], {x, -2, 3}, PlotRange -> All]}

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

Wolfram Language code:

{Plot[WaveletPsi[ReverseBiorthogonalSplineWavelet[2, 4], x, "Dual"], {x, -2, 3}, PlotRange -> All], Plot[WaveletPsi[BiorthogonalSplineWavelet[2, 4], x], {x, -2, 3}, PlotRange -> All]}

Lowpass filter coefficients sum to unity; :

Wolfram Language code: Total@WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[4, 2]][[All, 2]]

Highpass filter coefficients sum to zero; :

Wolfram Language code: Total@WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[4, 2], "PrimalHighpass"][[All, 2]]

Dual filter coefficients sum to unity; :

Wolfram Language code: Total@WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[4, 2], "DualLowpass"][[All, 2]]

Dual highpass filter coefficients sum to zero; :

Wolfram Language code: Total@WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[4, 2], "DualHighpass"][[All, 2]]

Scaling function integrates to unity; :

Wolfram Language code: Integrate[WaveletPhi[ReverseBiorthogonalSplineWavelet[2, 2], x], {x, -∞, ∞}]

Dual scaling function integrates to unity; :

Wolfram Language code: Integrate[WaveletPhi[ReverseBiorthogonalSplineWavelet[2, 2], x, "Dual"], {x, -∞, ∞}]

Wavelet function integrates to zero; :

Wolfram Language code: Integrate[WaveletPsi[ReverseBiorthogonalSplineWavelet[2, 2], x], {x, -∞, ∞}]

Dual wavelet function integrates to zero; :

Wolfram Language code: Integrate[WaveletPsi[ReverseBiorthogonalSplineWavelet[2, 2], x, "Dual"], {x, -∞, ∞}]//Chop

Scaling function has compact support {n₁,n₂}:

Wolfram Language code: ϕ = WaveletPhi[ReverseBiorthogonalSplineWavelet[3, 3], x]

Wolfram Language code: {n1, n2} = {-1, 2};

Dual scaling function has compact support {nd₁,nd₂}:

Wolfram Language code: Overscript[ϕ, ~] = WaveletPhi[ReverseBiorthogonalSplineWavelet[3, 3], x, "Dual"]

Wolfram Language code: {nd1, nd2} = {-3, 4};

Corresponding wavelet function has support {(n₁– nd₂+1)/2, (n₂– nd₁+1)/2}:

Wolfram Language code: ψ = WaveletPsi[ReverseBiorthogonalSplineWavelet[3, 3], x]

Wolfram Language code: {(n1 - nd2 + 1/2), (n2 - nd1 + 1/2)}

Dual wavelet function has support {(nd₁– n₂+1)/2, (nd₂– n₁+1)/2}:

Wolfram Language code: Overscript[ψ, ~] = WaveletPsi[ReverseBiorthogonalSplineWavelet[3, 3], x, "Dual"]

Wolfram Language code: {(nd1 - n2 + 1/2), (nd2 - n1 + 1/2)}

satisfies the recursion equation :

Wolfram Language code: ϕ = WaveletPhi[ReverseBiorthogonalSplineWavelet[4, 2]];

Wolfram Language code: a = WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[4, 2], "PrimalLowpass"];

Plot the components and the sum of the recursion:

Wolfram Language code: scalet[x_, ϕ_, a_] := 2Table[a[[i, 2]]ϕ[2x - a[[i, 1]]], {i, Length[a]}]

Wolfram Language code:

{Plot[Evaluate@scalet[x, ϕ, a], {x, -2, 2}, PlotRange -> All], 
	Plot[Total@scalet[x, ϕ, a], {x, -2, 2}, PlotRange -> All]}

satisfies the recursion equation :

Wolfram Language code: ϕ = WaveletPhi[ReverseBiorthogonalSplineWavelet[4, 2]];

Wolfram Language code: b = WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[4, 2], "PrimalHighpass"];

Plot the components and the sum of the recursion:

Wolfram Language code: wavelet[x_, ϕ_, b_] := 2Table[b[[i, 2]]ϕ[2x - b[[i, 1]]], {i, Length[b]}]

Wolfram Language code:

{Plot[Evaluate@wavelet[x, ϕ, b], {x, -2, 3}, PlotRange -> All], Plot[Total@wavelet[x, ϕ, b], {x, -2, 3}, PlotRange -> All]}

satisfies the recursion equation :

Wolfram Language code: Overscript[ϕ, ~] = ( Evaluate[WaveletPhi[ReverseBiorthogonalSplineWavelet[3, 5], #, "Dual"]]&);

Wolfram Language code: Overscript[a, ~] = WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[3, 5], "DualLowpass"];

Plot the components and the sum of the recursion:

Wolfram Language code: scalet[x_, ϕ_, a_] := 2Table[a[[i, 2]]ϕ[2x - a[[i, 1]]], {i, Length[a]}]

Wolfram Language code:

{Plot[Evaluate@scalet[x, Overscript[ϕ, ~], Overscript[a, ~]], {x, -3, 4}, PlotRange -> All], 
	Plot[Total@scalet[x, Overscript[ϕ, ~], Overscript[a, ~]], {x, -3, 4}, PlotRange -> All]}

satisfies the recursion equation :

Wolfram Language code: Overscript[ϕ, ~] = ( Evaluate[WaveletPhi[ReverseBiorthogonalSplineWavelet[3, 5], #, "Dual"]]&);

Wolfram Language code: Overscript[b, ~] = WaveletFilterCoefficients[ReverseBiorthogonalSplineWavelet[3, 5], "DualHighpass"];

Plot the components and the sum of the recursion:

Wolfram Language code: wavelet[x_, ϕ_, b_] := 2Table[b[[i, 2]]ϕ[2x - b[[i, 1]]], {i, Length[b]}]

Wolfram Language code:

{Plot[Evaluate@wavelet[x, Overscript[ϕ, ~], Overscript[b, ~]], {x, -2, 3}, PlotRange -> All], 
	Plot[Total@wavelet[x, Overscript[ϕ, ~], Overscript[b, ~]], {x, -2, 3}, PlotRange -> All]}

Frequency response for is given by :

Wolfram Language code: h[wav_, ω_] := With[{a = WaveletFilterCoefficients[wav]}, Sum[a[[i, 2]]Exp[-I a[[i, 1]]ω], {i, Length[a]}]]

The filter is a lowpass filter:

Wolfram Language code:

Plot[Abs[h[ReverseBiorthogonalSplineWavelet[3, 3], ω]], {ω, -π, π}, Ticks -> {{-π, -(π/2), 0, (π/2), π}, Automatic}, AxesLabel -> {ω, Abs[H[ω]]}]

Fourier transform of is given by :

Wolfram Language code: h[wav_, ω_] := With[{a = WaveletFilterCoefficients[wav]}, Sum[a[[i, 2]]Exp[-I a[[i, 1]]ω], {i, Length[a]}]]

Wolfram Language code: fh[wav_, ω_, j_] := Abs[Product[h[wav, (ω/2^i)], {i, j}]]

Wolfram Language code:

Plot[fh[ReverseBiorthogonalSplineWavelet[3, 3], ω, 10], {ω, -10Pi, 10 Pi}, PlotRange -> All, 
AxesLabel -> {ω, Abs[ Overscript[ϕ,  ^ ][ω]]}]

Frequency response for is given by :

Wolfram Language code:

Overscript[h, ~][wav_, ω_] := With[{a  = WaveletFilterCoefficients[wav, "DualLowpass"]}, Sum[a[[i, 2]]Exp[-I a[[i, 1]]ω], {i, Length[a]}]]

The filter is a dual lowpass filter:

Wolfram Language code:

Plot[Abs[Overscript[h, ~][ReverseBiorthogonalSplineWavelet[3, 3], ω]], {ω, -π, π}, Ticks -> {{-π, -(π/2), 0, (π/2), π}, Automatic}, AxesLabel -> {ω, Abs[Overscript[H, ~][ω]]}]

Fourier transform of is given by :

Wolfram Language code:

Overscript[h, ~][wav_, ω_] := With[{a  = WaveletFilterCoefficients[wav, "DualLowpass"]}, Sum[a[[i, 2]]Exp[-I a[[i, 1]]ω], {i, Length[a]}]]

Wolfram Language code: fh[wav_, ω_, j_] := Abs[Product[Overscript[h, ~][wav, (ω/2^i)], {i, j}]]

Wolfram Language code:

Plot[fh[ReverseBiorthogonalSplineWavelet[3, 3], ω, 10], {ω, -10Pi, 10Pi}, PlotRange -> All, 
AxesLabel -> {ω, Abs[ Overscript[Overscript[ϕ, ~],  ^ ][ω]]}]

Frequency response for is given by :

Wolfram Language code:

g[wav_, ω_] := With[{b  = WaveletFilterCoefficients[wav, "PrimalHighpass"]}, Sum[b[[i, 2]]Exp[-I b[[i, 1]]ω], {i, Length[b]}]
	]

The filter is a lowpass filter:

Wolfram Language code:

Plot[Abs[g[ReverseBiorthogonalSplineWavelet[3, 3], ω]], {ω, -π, π}, Ticks -> {{-π, -(π/2), 0, (π/2), π}, Automatic}, AxesLabel -> {ω, Abs[G[ω]]}]

Fourier transform of is given by :

Wolfram Language code: h[wav_, ω_] := With[{a = WaveletFilterCoefficients[wav]}, Sum[a[[i, 2]]Exp[-I a[[i, 1]]ω], {i, Length[a]}]]

Wolfram Language code:

g[wav_, ω_] := With[{b  = WaveletFilterCoefficients[wav, "PrimalHighpass"]}, Sum[b[[i, 2]]Exp[-I b[[i, 1]]ω], {i, Length[b]}]
	]

Wolfram Language code: fg[wav_, ω_, j_] := Abs[g[wav, (ω/2)]Product[h[wav, (ω/2^i)], {i, 2, j}]]

Wolfram Language code:

Plot[fg[ReverseBiorthogonalSplineWavelet[3, 3], ω, 10], {ω, -10Pi, 10Pi}, PlotRange -> All, 
AxesLabel -> {ω, Abs[ Overscript[ψ,  ^ ][ω]]}]

Frequency response for is given by :

Wolfram Language code:

Overscript[g, ~][wav_, ω_] := With[{b  = WaveletFilterCoefficients[wav, "DualHighpass"]}, Sum[b[[i, 2]]Exp[-I b[[i, 1]]ω], {i, Length[b]}]
	]

The filter is a lowpass filter:

Wolfram Language code:

Plot[Abs[Overscript[g, ~][ReverseBiorthogonalSplineWavelet[3, 3], ω]], {ω, -π, π}, Ticks -> {{-π, -(π/2), 0, (π/2), π}, Automatic}, AxesLabel -> {ω, Abs[Overscript[G, ~][ω]]}]

Fourier transform of is given by :

Wolfram Language code:

Overscript[h, ~][wav_, ω_] := With[{a  = WaveletFilterCoefficients[wav, "DualLowpass"]}, Sum[a[[i, 2]]Exp[-I a[[i, 1]]ω], {i, Length[a]}]]

Wolfram Language code:

Overscript[g, ~][wav_, ω_] := With[{b  = WaveletFilterCoefficients[wav, "DualHighpass"]}, Sum[b[[i, 2]]Exp[-I b[[i, 1]]ω], {i, Length[b]}]
	]

Wolfram Language code:

Overscript[fg, ~][wav_, ω_, j_] := Abs[Overscript[g, ~][wav, (ω/2)]Product[Overscript[h, ~][wav, (ω/2^i)], {i, 2, j}]]

Wolfram Language code:

Plot[Overscript[fg, ~][ReverseBiorthogonalSplineWavelet[3, 3], ω, 10], {ω, -10Pi, 10Pi}, PlotRange -> All, 
AxesLabel -> {ω, Abs[ Overscript[Overscript[ψ, ~],  ^ ][ω]]}]

Neat Examples (2)

Plot translates and dilations of scaling function:

Wolfram Language code: ϕ[x_, j_, k_] := 2^j / 2WaveletPhi[ReverseBiorthogonalSplineWavelet[3, 3], 2^jx - k]

Wolfram Language code: Plot[Evaluate@Table[ϕ[x, j, 0], {j, 0, 4}], {x, -1, 2}, Filling -> Axis, PlotRange -> All, Axes -> {True, False}]

Wolfram Language code: Plot[Evaluate@Table[ϕ[x, 2, k], {k, 0, 2^2 - 1}], {x, -0.5, 2}, Filling -> Axis, PlotRange -> All]

Plot translates and dilations of wavelet function:

Wolfram Language code: ψ[x_, j_, k_] := 2^j / 2WaveletPsi[ReverseBiorthogonalSplineWavelet[3, 3], 2^jx - k]

Wolfram Language code: Plot[Evaluate@Table[ψ[x, j, 0], {j, 0, 4}], {x, -0.5, 1}, Filling -> Axis, PlotRange -> All, Axes -> {True, False}]

Wolfram Language code: Plot[Evaluate@Table[ψ[x, 2, k], {k, 0, 2^2 - 1}], {x, -0.5, 1.5}, Filling -> Axis, PlotRange -> All]

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ReverseBiorthogonalSplineWavelet

Details

Examples

Basic Examples (6)

Scope (17)

Basic Uses (10)

Wavelet Transforms (5)

Higher Dimensions (2)

Properties & Relations (19)

Neat Examples (2)

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ReverseBiorthogonalSplineWavelet

Details

Examples

Basic Examples (6)

Scope (17)

Basic Uses (10)

Wavelet Transforms (5)

Higher Dimensions (2)

Properties & Relations (19)

Neat Examples (2)

See Also

Tech Notes

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CMS

APA

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