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A Hierarchy of Detail in the Haar System

Module by: Phil Schniter. E-mail the author

Given a mother scaling function φt 2 φ t 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=2k2φ2ktn . 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 nZ . 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 nZ 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 = 2 V 2 (contains all signals).
Because φ k , n t nZ φ k , n t n is an orthonormal basis, the best (in 2 2 norm) approximation of xt 2 x t 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 aR a ), the only constant signal in 2 2 , the space of square-integrable signals, is the zero signal.

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