Some Brief Mathematical Preliminaries

Wavelet Analysis

Some Brief Mathematical Preliminaries

The discrete wavelet transform decomposes a function as a sum of basis functions called wavelets. These basis functions have the property that they can be obtained by dilating and translating two basic types of wavelets known as the scaling function or father wavelet $\phi$ , and the mother wavelet $\psi$ . These translates and dilations are defined as follows:

$\phi_{j,k}(x) &=& 2^{j/2} \phi( 2^j x - k) \ \psi_{j,k}(x) &=& 2^{j/2} \psi( 2^j x - k) \$

The index j defines the dilation or level while the index k defines the translate. Loosely speaking, sums of the $\phi_{j,k}(x)$ capture low frequencies and sums of the $\psi_{j,k}(x)$ represent high frequencies in the data. More precisely, for any suitable function f(x) and for any j₀,

$f(x)=\sum_k c^{j_0}_k \phi_{j_0,k}(x) + \sum_{j \geq j_0} \sum_k d^j_k \psi_{j,k}(x)$

where the c^j_k and d^j_k are known as the scaling coefficients and the detail coefficients respectively. For orthonormal wavelet families these coefficients can be computed by

$c^j_k &=& \int f(x) \phi_{j,k}(x) \,dx \ d^j_k &=& \int f(x) \psi_{j,k}(x) \,dx$

The key to obtaining fast numerical algorithms for computing the detail and scaling coefficients for a given function f(x) is that there are simple recurrence relationships that enable you to compute the coefficients at level j-1 from the values of the scaling coefficients at level j. These formulae are

$c^{j-1}_k &=& \sum_i h_{i-2k} c^j_i \ d^{j-1}_k &=& \sum_i g_{i-2k} c^j_i$

The coefficients h_k and g_k that appear in these formulae are called filter coefficients. The h_k are determined by the father wavelet and they form a low-pass filter; g_k=(-1)^k h_1-k and form a high-pass filter. The preceding sums are formally over the entire (infinite) range of integers. However, for wavelets that are zero except on a finite interval, only finitely many of the filter coefficients are non-zero and so in this case the sums in the recurrence relationships for the detail and scaling coefficients are finite.

Conversely, if you know the detail and scaling coefficients at level j-1 then you can obtain the scaling coefficients at level j using the relationship

$c^j_k=\sum_i h_{k-2i} c^{j-1}_i + \sum_i g_{k-2i}d^{j-1}_i$

Suppose that you have data values

y_k = f(x_k), k = 0,1,2, ... ,N-1

at N=2^J equally spaced points x_k. It turns out that the values 2^-J/2 y_k are good approximations of the scaling coefficients c^J_k. Then using the recurrence formula you can find c^J-1_k and d^J-1_k, k = 0,1,2, ... ,N/2-1. The discrete wavelet transform of the y_k at level J-1 consists of the N/2 scaling and N/2 detail coefficients at level J-1. A technical point that arises is that in applying the recurrence relationships to finite data, a few values of the c^J_k for k<0 or $k\geq N$ may be needed. One way to cope with this difficulty is to extend the sequence c^J_k to the left and right using some specified boundary treatment.

Continuing by replacing the scaling coefficients at any level j by the scaling and detail coefficients at level j-1 yields a sequence of N coefficients

{c⁰₀,d⁰₀,d¹₀,d¹₁,d²₀,d²₁,d²₂,d²₃,d³₁, ... ,d³₇, ... ,d^J-1₀, ... ,d^J-1_N/2-1}

This sequence is the finite discrete wavelet transform of the input data {y_k}. At any level j₀ the finite dimensional approximation of the function f(x) is

$f(x)\approx \sum_k c^{j_0}_k \phi_{j_0,k}(x) + \sum_{j=j_0}^{J-1} \sum_k d^j_k \psi_{j,k}(x)$

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