3.7. Some concluding remarks.

In this chapter we have studied in some depth the reconstruction of f from the sequence , where ? _m,n( x) = a ^{? m/2} ₀ ?( a ^{? m} ₀ x ? nb ₀) (and variations thereof see 3.3.4). We have seen that numerically stable reconstruction is only possible if the ? _m,n constitute a frame, and we have derived a reconstruction formula if the ? _m,n are a frame. One can, however, use other reconstruction formulas (provided the ? _m,n do constitute a frame: the necessity of that condition remains!). To conclude this chapter, let me sketch the approach of S. Mallat, which addresses moreover the problem of shift-invariance.

The discrete wavelet transform, such as I have described it in this chapter, is highly non-invariant under translations. By this I mean that two functions may be shifted versions of each other, while their wavelet coefficients may be very different. This is already illustrated by the "hyperbolic lattice" ^[17] in Figure 1.4a, where the axis t = 0 plays a unique role. In practice one does not use an infinite number of scales, but cuts off very low and very high frequencies: only those m for which m ₁ ? m ? m ₀ are used. The resulting truncated lattice is then invariant under translations by b ₀2 ^{m ₀} (choose a ₀ = 2 for simplicity), which is, however,...