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Example Wavelets

Module by: Kileen Cheng. E-mail the author

Summary: Many considerations go into the design of a wavelet system including properties such as orthogonality, compact support, symmetry, and smoothness. Here we will examine several popularly-used wavelet systems and their respective properties.

Design Considerations:

There are several design properties for the construction of a wavelet basis that one would want to be fulfilled.

  1. symmetry: If the wavelets are not symmetric, then the wavelet transform of the mirror of an image is not the mirror of the wavelet transform.
  2. smoothness: This property is determined by the number of vanishing moments. Recall that the primal vanishing moments determine the smoothness of the reconstruction. The dual vanishing moments determine the convergence rate of the multiresolution projections and are necessary for detecting singularities.
  3. orthogonality: This property can be too restrictive at times. Thus the need for biorthgonal wavelet systems.
  4. compact support: This property is a function of the filter length.

Haar Wavelet

The Haar wavelet is the most fundamental of the wavelet systems and is also known as the length-2 Daubechies filter (See Figure 1).

Figure 1: Haar Scaling and Wavelet Functions
Figure 1 (haar.png)

The Haar wavelet system properties are:

  1. Symmetric scaling function
  2. Anti-symmetric wavelet function
  3. One vanishing moment (a minimum)
  4. Orthogonal
  5. Compact support
It is important to note that the Haar wavelet system is the only one that is orthogonal, symmetric, and has compact support.
h⁢n=1212 h n 1 2 1 2
(1)
g⁢n=12−12 g n 1 2 1 2
(2)

Sinc Wavelet

The Sinc wavelet is the second fundamental of the wavelet systems (see Figure 2). Recall that the Fourier transform of the sinc is the brick-wall filter (or ideal low-pass filter).

Figure 2: Sinc Scaling and Wavelet Functions
Figure 2 (sinc.png)

The Sinc wavelet system properties are:

  1. Orthogonal
  2. Infinite number of vanishing moments
  3. Infinite support (IIR and non-causal)
Given our scaling function to be the sinc function
φ⁢t=sinc⁢t=sinπ⁢tπ⁢t φ t sinc t t t
(3)
we have the following wavelet expression
ψ⁢t=2⁢φ⁢2⁢t−φ⁢t ψ t 2 φ 2 t φ t
(4)
The filter coefficients can then be found as
h⁢n=sinc⁢π⁢n2 h n sinc n 2
(5)

Daubechies Wavelet

The Daubechies coefficients for the scaling and wavelet filters are unique in that they have a high degree of smoothness. All N2−1 N 2 1 degrees of freedom are used to maximize the number of vanishing moments:

∀i,i∈0…N2−1: H i ⁢π=0 i i 0 … N 2 1 H i 0
(6)
The Daubechies wavelet system properties are (see Figure 3:
  1. If length of filter N=2 N 2 , Daubechies = Haar
  2. Anti-symmetric if length of filter N>2 N 2
  3. Compact support
  4. Orthogonal
  5. N2−1 N 2 1 vanishing moments
  6. Increasing smoothness as NN increases

Figure 3: Length-4 Daubechies Scaling and Wavelet Functions
Figure 3 (daub4.png)

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