Multiresolution Expansions

1. Series Expansions and Function Spaces

A multiresolution analysis (MRA) constructs a signal through a sequence of function spaces, each representing a different resolution. Consider a function $f(x)$, which can be approximated as a linear combination of basis functions:

where $\phi_k(x)$ are the basis functions and $a_k$ are the expansion coefficients. This expansion enables the representation of the function in various resolutions, typically from coarser to finer.

The space of functions that can be expressed this way is referred to as a function space, typically denoted by $V_j$, where $j$ corresponds to the level of resolution. For a finer resolution, you get smaller details, while for coarser resolutions, only the broad features of the function are captured.

2. Nested Subspaces and Multiresolution Property

MRE uses nested subspaces where each finer resolution contains the coarser one:

This nesting property means that functions in a higher-resolution space can approximate finer details. At each level $j$, the approximation space $V_j$ contains the approximation at the previous level $V_{j-1}$, along with additional detail captured in a wavelet space $W_j$.

The relationship between these spaces is fundamental:

where $\oplus$ denotes the direct sum, meaning that $V_{j+1}$ contains both the approximation at resolution $j$ and the details at resolution $j+1$.

3. Scaling Functions and Wavelet Functions

In MRA, two types of functions are essential: scaling functions $\phi(x)$ and wavelet functions $\psi(x)$.

• The scaling function $\phi(x)$ generates the approximation spaces $V_j$ through integer translations and dyadic scalings:

  $$\phi_{j,k}(x) = 2^{j/2} \phi(2^j x – k)$$

  where $k$ translates the function, and $j$ controls the resolution.

• The wavelet function $\psi(x)$ is used to represent the detail spaces $W_j$. It captures the differences between consecutive levels of resolution:

  $$\psi_{j,k}(x) = 2^{j/2} \psi(2^j x – k)$$

The scaling function $\phi(x)$ approximates the signal at a coarse level, while the wavelet function $\psi(x)$ refines this approximation by adding details.

4. Orthonormal and Biorthogonal Bases

In many cases, the basis functions $\phi(x)$ and $\psi(x)$ form an orthonormal basis, meaning they are orthogonal and normalized:

where $\delta_{kl}$ is the Kronecker delta, equal to 1 if $k = l$, and 0 otherwise. This property simplifies the computation of coefficients in the series expansion.

In other cases, wavelets can form biorthogonal bases, where the basis functions are not orthonormal, but they still allow signal reconstruction. The dual functions, denoted $\tilde{\phi}(x)$ and $\tilde{\psi}(x)$, form a basis for reconstruction, while the original functions are used for decomposition.

5. Refinement Equation

A critical part of MRA is the refinement equation, which defines the scaling function in terms of a sum of scaled and translated versions of itself:

where $h_k$ are the scaling coefficients, which act as low-pass filters. This equation reflects how a function at one resolution can be built from finer resolution components. Similarly, the wavelet function can be defined as:

where $g_k$ are the wavelet coefficients, often corresponding to high-pass filtering.

6. Wavelet Transform

The Discrete Wavelet Transform (DWT) is a key application of multiresolution analysis, used to decompose a signal into approximation and detail components at various scales. The DWT recursively applies the scaling and wavelet functions, resulting in a hierarchical, multiscale representation.

A signal $f(x)$ can be expressed using both scaling and wavelet coefficients:

where $a_{j,k}$ are the approximation coefficients at scale $j$, and $d_{j,k}$ are the detail coefficients. The higher the resolution, the more detailed the information that wavelets capture.

7. Wavelet Decomposition and Reconstruction

MRE provides a framework for wavelet decomposition, where a signal is broken down into its approximation and detail components at various scales. This process can be inverted through reconstruction, allowing the original signal to be recovered from its multiresolution components.

In image processing, for instance, this decomposition allows images to be stored and transmitted efficiently by keeping only the essential approximation and detail coefficients.

Example: Haar Wavelet

The Haar wavelet is one of the simplest wavelets used in MRA. The Haar scaling function $\phi(x)$ and wavelet function $\psi(x)$ are defined as:

These functions decompose a signal into coarse and fine components, making them suitable for fast computation, though more advanced wavelets (like Daubechies wavelets) provide smoother decompositions.