MRA (from subdivision viewpoint) Jyun-Ming Chen Spring 2001

Nested Spaces : record the combined effect of splitting and averaging done to the initial control points to achieve the limit f(x) The same limit curve can be defined from each iteration Using matrix notation We will show that every subdivision scheme gives rise to refinable scaling functions and, hence, to nested spaces : some yet undetermined functions; (later we’ll show they are the scaling functions)

Nested Spaces (cont) Subdivision (refinement) matrix P j –Represent the combined effect of splitting and averaging (both are linear operations) refinement relation for scaling functions –Observe the similar relation for Haar and Daub4

Nested Space (cont) The refinement relation states that each of the coarser scaling functions can be written as a linear combination of the finer scaling functions. This linear combination depends on the subdivision(refinement) scheme used.

Nested Space Define space V j that includes all linear combinations of scaling functions (of j), whose dimension denoted by v(j) Then From P j is a v(j) by v(j-1) matrix

Wavelet Space Define wavelet space W j to be the complement of V j in V j+1 ; implying –Any function in V j+1 can be written as the sum of a unique function in V j and a unique function in W j –The dimensions of these spaces are related The basis for W j are called wavelets The corresponding scaling function space

Also write wavelet space W j as linear combination using basis of next space V j+1 From before, Wavelet Space (cont) Combining, this is called the two-scale relation.

Splitting of MRA Subspaces VNVN V N-1 W N-1 V N-2 W N-2 V N-3 W N-3

Example: Haar

Two-Scale Relations (graphically)

Analysis Filters Consider the approxi- mation of a function in some subspace V j Assume the function is described in some scaling function basis Write these coefficients as a column matrix Suppose we wish to create a lower resolution version with a smaller number of coefficients v(j-1), this can be done by A j is a constant matrix of dimension v(j-1) by v(j)

Analysis/Decomposition To capture the lost details as another column matrix d j-1 B j is a constant matrix of dimension w(j-1) by v(j), relating to A j This process is called analysis or decomposition The pair of matrices A j and B j are called analysis filters

Synthesis/Reconstruction If analysis matrices are chosen appropriately, original signal can be recovered using subdivision matrices This process is called synthesis or reconstruction The pair of matrices P j and Q j are called synthesis filters

Closer Look at Synthesis Performing the splitting and averaging to bring c j-1 to a finer scale A perturbation by interpolating the wavelets

Filter Bank Doing the aforementioned task repeatedly Recall Haar (next page)

Haar (Analysis) A2 B2

Haar (Synthesis) P2 Q2

Relation Between Analysis and Synthesis Filters In general, analysis filters are not necessarily transposed multiples of the synthesis filters (as in the Haar case)

Analysis & Synthesis Filters Dimension of filter matrices –Aj: v(j-1) by v(j); –Bj: w(j-1) by v(j) –Pj: v(j) by v(j-1) –Qj: v(j) by w(j-1) Hence are both square … and should be invertible Combining: We get:

Orthogonal Wavelets

Scaling function orthogonal to one another in the same level Wavelets orthogonal to one another in the same level and in all scales Each wavelet orthogonal to every coarser scaling function Haar and Daubechies are both orthogonal wavelets

Implication of Orthogonality Two row matrices of functions Define matrix Has these properties:

Implication of Orthogonality Changing subscript to j-1

Orthonormal … Scaling functions –Scaling functions are orthonormal only w.r.t. translations in a given scale –Not w.r.t. the scale (because of the nested nature of MRA) Wavelets –The wavelets are orthonormal w.r.t. scale as well as w.r.t. translation in a given scale