9.3 Scaling and Wavelet Functions

Scaling and wavelet functions

Properties of scaling and wavelet functions

Scaling functions and wavelet functions divide the labor of representing a signal. The scaling function handles the smooth, low-frequency content, while the wavelet function picks up the detail and rapid changes. Together, they let you analyze a signal at multiple resolutions without losing information.

Scaling functions are denoted $\phi(t)$. They approximate a signal at a given resolution level. For a scaling function to work properly in wavelet analysis, it needs to satisfy several properties:

• Normalization: $\int \phi(t)\, dt = 1$, so the function has unit area.
• Orthogonality: Integer-shifted copies of $\phi(t)$ are orthogonal to each other, meaning $\int \phi(t)\phi(t-k)\,dt = 0$ for all nonzero integers $k$. This prevents redundancy in the representation.
• Compact support: $\phi(t)$ is nonzero only over a finite interval, which keeps computations local and efficient.

Depending on the application, you may also want smoothness (for image compression, where blocky artifacts are undesirable) or symmetry (for linear-phase filtering).

Wavelet functions, denoted $\psi(t)$, are derived from the scaling function and capture the high-frequency detail that the scaling function misses. Two key properties distinguish them:

• Zero mean: $\int \psi(t)\, dt = 0$. This is what makes wavelets sensitive to changes and fluctuations rather than constant offsets.
• Vanishing moments: If $\psi(t)$ has $N$ vanishing moments, it is orthogonal to all polynomials up to degree $N-1$. More vanishing moments means the wavelet can ignore smooth polynomial trends and focus on genuine detail, which directly improves compression performance.

Multiresolution analysis (MRA) framework

Multiresolution analysis (MRA) is the mathematical framework that ties scaling and wavelet functions together into a hierarchy of approximation spaces.

The core idea: at each resolution level $j$, the scaling function $\phi(t)$ spans a subspace $V_j$ that contains all the information about the signal up to that resolution. When you move from a coarser level $V_{j+1}$ to a finer level $V_j$, the extra detail lives in a "wavelet subspace" $W_{j+1}$, spanned by the wavelet function. So:

$V_j = V_{j+1} \oplus W_{j+1}$

This means the finer approximation space equals the coarser approximation space plus the detail space. You can keep splitting recursively, peeling off detail at each level.

Within this framework:

• The scaling function is associated with a low-pass filter, extracting the smooth approximation.
• The wavelet function is associated with a high-pass filter, extracting the detail.

This filter-bank interpretation is what makes wavelet transforms computationally fast in practice.

Relationship between scaling and wavelet functions

Two-scale equation and filter coefficients

The two-scale equation (also called the refinement equation) is the key link between the scaling function and itself at a finer scale. It states that $\phi(t)$ can be written as a weighted sum of compressed, shifted copies of itself:

$\phi(t) = \sum_{k} h_k \, \phi(2t - k)$

The coefficients $h_k$ are the low-pass filter coefficients (also called scaling coefficients). They completely determine the shape and properties of $\phi(t)$. For example:

• The number of nonzero $h_k$ values determines the support size of $\phi(t)$. The Daubechies-4 wavelet has 4 nonzero coefficients, giving a support of length 3.
• The specific values of $h_k$ control smoothness and the number of vanishing moments.
• The coefficients must satisfy $\sum_k h_k = \sqrt{2}$ for proper normalization.

Obtaining the wavelet function from the scaling function

Once you have the low-pass coefficients $h_k$, you can construct the wavelet function using a parallel equation with high-pass filter coefficients $g_k$:

$\psi(t) = \sum_{k} g_k \, \phi(2t - k)$

The relationship between the two sets of coefficients is given by the quadrature mirror filter (QMF) condition:

$g_k = (-1)^k \, h_{1-k}$

This alternating sign-flip ensures that $\psi(t)$ is orthogonal to $\phi(t)$ and its shifts. In plain terms, the high-pass filter is just a sign-alternated, reversed version of the low-pass filter.

To summarize the roles:

• $h_k$ (low-pass) → scaling function → coarse approximation of the signal
• $g_k$ (high-pass) → wavelet function → detail/high-frequency content of the signal

Constructing wavelet basis functions

Dilation and translation operations

A single wavelet function $\psi(t)$ can only capture detail at one scale and one location. To cover an entire signal at all scales, you apply two operations:

1. Dilation (scaling): Stretch or compress $\psi(t)$ by a factor of $2^j$, where $j$ is an integer. Larger $j$ values stretch the wavelet, making it sensitive to coarser features; smaller $j$ values compress it, making it sensitive to fine detail.

   • $j = 0$: original scale
   • $j = 1$: wavelet is twice as wide, captures features at half the frequency
   • $j = -1$: wavelet is half as wide, captures features at double the frequency

2. Translation (shifting): Slide the dilated wavelet along the time axis by $k$ units (at the appropriate scale) so that every part of the signal is covered.

   • $k = 0$: centered at the origin
   • $k = 1$: shifted by one unit at scale $j$
   • $k = 2$: shifted by two units at scale $j$

Wavelet basis functions and signal representation

Combining dilation and translation produces the family of wavelet basis functions:

$\psi_{j,k}(t) = 2^{-j/2} \, \psi(2^{-j}t - k)$

The factor $2^{-j/2}$ is a normalization term that keeps the energy (squared integral) of each basis function equal to 1, regardless of scale.

These basis functions have two critical properties:

• Completeness: Any finite-energy signal can be represented as a combination of these functions. Nothing is lost.
• Orthonormality: $\int \psi_{j,k}(t)\,\psi_{j',k'}(t)\,dt = 0$ whenever $j \neq j'$ or $k \neq k'$. This means the wavelet coefficients are independent of each other, which simplifies both computation and interpretation.

A signal $f(t)$ is then decomposed as:

$f(t) = \sum_{j} \sum_{k} d_{j,k} \, \psi_{j,k}(t)$

where the wavelet coefficients $d_{j,k} = \int f(t)\,\psi_{j,k}(t)\,dt$ measure how much of basis function $\psi_{j,k}$ is present in the signal. In practice, most of these coefficients are small or zero for natural signals, which is exactly why wavelet-based compression (like JPEG 2000) works so well: you keep the large coefficients and discard the rest.