A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ironing method) and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James L. Crowley.
{0}⊂⋯⊂V1⊂V0⊂V−1⊂⋯⊂V−n⊂V−(n+1)⊂⋯⊂L2(ℝ)
that satisfies certain self-similarity relations in time-space and scale-frequency, as well as completeness and regularity relations.
- Self-similarity in time demands that each subspace Vk is invariant under shifts by integer multiples of 2. That is, for each Vk.
- Self-similarity in scale demands that all subspaces l,"}}'>Vk⊂Vl,k>l, are time-scaled versions of each other, with scaling respectively dilation factor 2. I.e., for each ∀x∈ℝ:g(x)=f(2k−lx).
- In the sequence of subspaces, for k>l the space resolution 2 of the l-th subspace is higher than the resolution 2 of the k-th subspace.
- Regularity demands that the model subspace V0 be generated as the linear hull (algebraically or even topologically closed) of the integer shifts of one or a finite number of generating functions V0⊂L2(ℝ), which imposes certain conditions on the decay at infinity. The generating functions are also known as scaling functions or father wavelets. In most cases one demands of those functions to be piecewise continuous with compact support.
- Completeness demands that those nested subspaces fill the whole space, i.e., their union should be dense in ak=2⟨ϕ(x),ϕ(2x−k)⟩ for |k|>N, such that
- W0⊂V−1, which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to V−1 is the orthogonal sum (denoted by V0. By self-similarity, there are scaled versions L2(ℝ)=closure of ⨁k∈ℤWk,
thus the set
- ↑ Mallat, S.G. "A Wavelet Tour of Signal Processing". www.di.ens.fr. Retrieved 2019-12-30.