Wavelet Transforms in One Dimension

1. Wavelet Series Expansion

A wavelet series expansion represents a function as a combination of scaling functions and wavelet functions. The scaling functions capture the low-frequency or smooth part of the signal, while the wavelet functions encode the high-frequency or detailed parts.

Consider a function $f(x) \in L^2(\mathbb{R})$ (the space of square-integrable functions). We can represent this function using scaling and wavelet functions. The wavelet series expansion of a function is given by:

Where:

• $\phi_{j_0,k}(x)$ is the scaling function at scale $j_0$.
• $\psi_{j,k}(x)$ are the wavelet functions at different scales $j$ and translations $k$.
• $c_{j_0}(k)$ are the approximation coefficients that capture the low-frequency content.
• $d_j(k)$ are the detail coefficients that capture the high-frequency details.

The scaling function, $\phi_{j_0,k}(x)$, provides a coarse approximation of the function, while the wavelet functions, $\psi_{j,k}(x)$, add finer details as we go to higher scales $j$. The first sum represents the low-frequency approximation, and the second sum adds detail at various scales to improve accuracy.

Calculation of Coefficients

To compute the coefficients, we use the inner product of the function with the scaling and wavelet functions:

This allows us to decompose the function $f(x)$ into components at different scales, where the approximation coefficients $c_{j_0}(k)$ provide the smooth representation, and the detail coefficients $d_j(k)$ add higher-resolution information.

2. Discrete Wavelet Transform (DWT)

The Discrete Wavelet Transform (DWT) is a computationally efficient method that maps a discrete signal (a sequence of values) into wavelet coefficients, representing both the low- and high-frequency components of the signal.

The DWT operates by applying discrete filters to a signal $f(n)$, which is sampled at discrete intervals. The filters decompose the signal into a set of approximation coefficients (low-frequency components) and detail coefficients (high-frequency components). This decomposition can be applied iteratively to obtain finer details.

Formulation of the DWT

For a discrete signal $f(n)$, where $n = 0, 1, \dots, M-1$, the discrete wavelet transform is given by:

Where:

• $W_\phi(j_0, k)$ are the approximation coefficients at the coarsest scale $j_0$.
• $W_\psi(j, k)$ are the detail coefficients at finer scales $j$.

The discrete wavelet transform uses a pair of filters:

• Lowpass filter: Captures the approximation (scaling) coefficients.
• Highpass filter: Captures the detail (wavelet) coefficients.

Inverse Discrete Wavelet Transform

Once the signal is decomposed, it can be reconstructed using the inverse discrete wavelet transform (IDWT). The IDWT reconstructs the original signal by summing up the contributions from both the approximation and detail coefficients:

This allows us to fully reconstruct the original signal $f(n)$ from its wavelet coefficients.

3. Continuous Wavelet Transform (CWT)

The Continuous Wavelet Transform (CWT) is similar to the DWT but is defined for continuous functions and produces a redundant representation of the signal, capturing both time and frequency information. The CWT transforms a continuous function into a two-dimensional representation over scale and translation.

The CWT of a function $f(x)$ is defined as:

Where:

• $\psi_{s,t}(x) = \frac{1}{\sqrt{s}} \psi\left(\frac{x – t}{s}\right)$ is the mother wavelet, scaled by $s$ and translated by $t$.
• $s$ is the scale parameter that controls the dilation or compression of the wavelet.
• $t$ is the translation parameter that shifts the wavelet along the $x$-axis.

The inverse CWT allows us to reconstruct the original function by integrating over the scales and translations:

Where $C_\psi$ is a constant that depends on the wavelet used.

4. Example: Haar Wavelets

The Haar wavelet is one of the simplest and most commonly used wavelets. It is defined as:

Consider the function $f(x) = x^2$ over the interval $[0, 1]$. Using Haar wavelets, we can approximate this function by applying the wavelet series expansion.

Step-by-Step Approximation

1. Coarse Approximation: Start by approximating $f(x)$ using the scaling function at the coarsest level. The approximation captures the average behavior of $f(x)$ over each interval.

2. Add Detail: At the next level, we add the first wavelet function, which captures the difference between the original function and the coarse approximation. This adds finer details to the approximation.

3. Higher-Order Approximation: As we go to higher scales, we add more wavelets, capturing even finer details of $f(x)$. The more levels we include, the closer the wavelet series expansion gets to the original function.

After applying several levels of the wavelet expansion, the approximation will become closer to the original function $f(x) = x^2$, and the error will decrease as more wavelets are added.

5. Fast Wavelet Transform (FWT)

The Fast Wavelet Transform (FWT) is an efficient algorithm for computing the DWT. It exploits the relationship between wavelet coefficients at successive scales to reduce the computational complexity. The FWT reduces the number of calculations required by successively applying lowpass and highpass filters to the signal and downsampling the result.

This approach is similar to subband coding, where the signal is decomposed into different frequency bands. By iterating this process, the FWT can efficiently compute the wavelet coefficients for large datasets, making it suitable for real-time signal processing and image compression.