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A Hierarchy of Detail in the Haar System

Module by: Phil Schniter

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Given a mother scaling function φt∈ ℒ 2 φ t ℒ 2 — the choice of which will be discussed later — let us construct scaling functions at "coarseness-level-k"k" and "shift-nn" as follows: φ k , n t=2-k2φ2-kt−n . φ k , n t 2 k 2 φ 2 k t n . Let us then use V k V k to denote the subspace defined by linear combinations of scaling functions at the k th k th level: V k =span{ φ k , n t|n∈ℤ} . V k span φ k , n t n . In the Haar system, for example, V 0 V 0 and V 1 V 1 consist of signals with the characteristics of x 0 t x 0 t and x 1 t x 1 t illustrated in Figure 1.

Figure 1
Figure 1 (sq_scalingfn.png)

We will be careful to choose a scaling function φt φ t which ensures that the following nesting property is satisfied: …⊂ V 2 ⊂ V 1 ⊂ V 0 ⊂ V -1 ⊂ V -2 ⊂… … V 2 V 1 V 0 V -1 V -2 … coarse detailed coarse detailed In other words, any signal in V k V k can be constructed as a linear combination of more detailed signals in V k − 1 V k − 1 . (The Haar system gives proof that at least one such φt φ t exists.)

The nesting property can be depicted using the set-theoretic diagram, Figure 2, where V − 1 V − 1 is represented by the contents of the largest egg (which includes the smaller two eggs), V 0 V 0 is represented by the contents of the medium-sized egg (which includes the smallest egg), and V 1 V 1 is represented by the contents of the smallest egg.

Figure 2
Figure 2 (vspaces.png)

Going further, we will assume that φt φ t is designed to yield the following three important properties:

  1. { φ k , n t|n∈ℤ} φ k , n t n constitutes an orthonormal basis for V k V k ,
  2. V ∞ =0 V ∞ 0 (contains no signals). 1
  3. V − ∞ = ℒ 2 V − ∞ ℒ 2 (contains all signals).
Because { φ k , n t|n∈ℤ} φ k , n t n is an orthonormal basis, the best (in ℒ 2 ℒ 2 norm) approximation of xt∈ ℒ 2 x t ℒ 2 at coarseness-level-kk is given by the orthogonal projection, Figure 3
x k t=∑n=-∞∞ c k , n φ k , n t x k t n c k , n φ k , n t (1)
c k , n =< φ k , n t,xt> c k , n φ k , n t x t (2)

Figure 3
Figure 3 (projection.png)

We will soon derive conditions on the scaling function φt φ t which ensure that the properties above are satisfied.

Footnotes

  1. While at first glance it might seem that V ∞ V ∞ should contain non-zero constant signals (e.g., xt=a x t a for a∈ℝ a ), the only constant signal in ℒ 2 ℒ 2 , the space of square-integrable signals, is the zero signal.

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