# Discrete Wavelet Transform¶

Discrete Wavelet Transform

Wraps GSL's gsl_wavelet_transform_forward(), and gsl_wavelet_transform_inverse() and creates plot methods.

AUTHOR:

• Josh Kantor (2006-10-07) - initial version
• David Joyner (2006-10-09) - minor changes to docstrings and examples.
sage.gsl.dwt.DWT(n, wavelet_type, wavelet_k)

This function initializes an GSLDoubleArray of length n which can perform a discrete wavelet transform.

INPUT:

• n – a power of 2
• T – the data in the GSLDoubleArray must be real
• wavelet_type – the name of the type of wavelet, valid choices are:
• 'daubechies'
• 'daubechies_centered'
• 'haar'
• 'haar_centered'
• 'bspline'
• 'bspline_centered'

For daubechies wavelets, wavelet_k specifies a daubechie wavelet with $$k/2$$ vanishing moments. $$k = 4,6,...,20$$ for $$k$$ even are the only ones implemented.

For Haar wavelets, wavelet_k must be 2.

For bspline wavelets, wavelet_k of $$103,105,202,204,206,208,301, 305,307,309$$ will give biorthogonal B-spline wavelets of order $$(i,j)$$ where wavelet_k is $$100*i+j$$. The wavelet transform uses $$J = \log_2(n)$$ levels.

OUTPUT:

An array of the form $$(s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0}, \ldots, d_{J-1,2^{J-1}-1})$$ for $$d_{j,k}$$ the detail coefficients of level $$j$$. The centered forms align the coefficients of the sub-bands on edges.

EXAMPLES:

sage: a = WaveletTransform(128,'daubechies',4)
sage: for i in range(1, 11):
...    a[i] = 1
...    a[128-i] = 1
sage: a.plot().show(ymin=0)
sage: a.forward_transform()
sage: a.plot().show()
sage: a = WaveletTransform(128,'haar',2)
sage: for i in range(1, 11): a[i] = 1; a[128-i] = 1
sage: a.forward_transform()
sage: a.plot().show(ymin=0)
sage: a = WaveletTransform(128,'bspline_centered',103)
sage: for i in range(1, 11): a[i] = 1; a[100+i] = 1
sage: a.forward_transform()
sage: a.plot().show(ymin=0)


This example gives a simple example of wavelet compression:

sage: a = DWT(2048,'daubechies',6)
sage: for i in range(2048): a[i]=float(sin((i*5/2048)**2))
sage: a.plot().show()  # long time (7s on sage.math, 2011)
sage: a.forward_transform()
sage: for i in range(1800): a[2048-i-1] = 0
sage: a.backward_transform()
sage: a.plot().show()  # long time (7s on sage.math, 2011)

class sage.gsl.dwt.DiscreteWaveletTransform

Bases: sage.gsl.gsl_array.GSLDoubleArray

Discrete wavelet transform class.

backward_transform()
forward_transform()
plot(xmin=None, xmax=None, **args)
sage.gsl.dwt.WaveletTransform(n, wavelet_type, wavelet_k)

This function initializes an GSLDoubleArray of length n which can perform a discrete wavelet transform.

INPUT:

• n – a power of 2
• T – the data in the GSLDoubleArray must be real
• wavelet_type – the name of the type of wavelet, valid choices are:
• 'daubechies'
• 'daubechies_centered'
• 'haar'
• 'haar_centered'
• 'bspline'
• 'bspline_centered'

For daubechies wavelets, wavelet_k specifies a daubechie wavelet with $$k/2$$ vanishing moments. $$k = 4,6,...,20$$ for $$k$$ even are the only ones implemented.

For Haar wavelets, wavelet_k must be 2.

For bspline wavelets, wavelet_k of $$103,105,202,204,206,208,301, 305,307,309$$ will give biorthogonal B-spline wavelets of order $$(i,j)$$ where wavelet_k is $$100*i+j$$. The wavelet transform uses $$J = \log_2(n)$$ levels.

OUTPUT:

An array of the form $$(s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0}, \ldots, d_{J-1,2^{J-1}-1})$$ for $$d_{j,k}$$ the detail coefficients of level $$j$$. The centered forms align the coefficients of the sub-bands on edges.

EXAMPLES:

sage: a = WaveletTransform(128,'daubechies',4)
sage: for i in range(1, 11):
...    a[i] = 1
...    a[128-i] = 1
sage: a.plot().show(ymin=0)
sage: a.forward_transform()
sage: a.plot().show()
sage: a = WaveletTransform(128,'haar',2)
sage: for i in range(1, 11): a[i] = 1; a[128-i] = 1
sage: a.forward_transform()
sage: a.plot().show(ymin=0)
sage: a = WaveletTransform(128,'bspline_centered',103)
sage: for i in range(1, 11): a[i] = 1; a[100+i] = 1
sage: a.forward_transform()
sage: a.plot().show(ymin=0)


This example gives a simple example of wavelet compression:

sage: a = DWT(2048,'daubechies',6)
sage: for i in range(2048): a[i]=float(sin((i*5/2048)**2))
sage: a.plot().show()  # long time (7s on sage.math, 2011)
sage: a.forward_transform()
sage: for i in range(1800): a[2048-i-1] = 0
sage: a.backward_transform()
sage: a.plot().show()  # long time (7s on sage.math, 2011)

sage.gsl.dwt.is2pow(n)

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