Let p 2 ( t), (2 t k), we have the scaling relation,

Transcription

1 Multiresolution Analysis and Daubechies N Wavelet We have discussed decomposing a signal into its Haar wavelet components of varying frequencies. The Haar wavelet scheme relied on two functions: the Haar scaling function and the Haar wavelet. Both are simple to describe and lead to an easy decomposition algorithm. The disadvantage with the Haar decomposition algorithm is that both of these functions are discontinuous. As a result, the Haar decomposition algorithm provides only crude approximations to a continuously varying signal. It is desirable to have continuous versions of our building blocs, and. Due to Daubechies, we do have these functions, called Daubechies wavelets. We first briefly discuss the framewor for creating more general and is called a multiresolution analysis due to Stephane Mallat. Definition. Let V,,,,,,, be a sequence of subspaces of functions in L ( R ). The collection of spaces V, Z is called a multiresolution analysis with scaling function if the following conditions hold:. (nested) V V. (density) V L ( R). (separation) V {} 4. (scaling) f ( x) V ( f x) V 5. (orthonormal basis) The function V (using the L inner product) for V. and the set ( x ), Z is an orthonormal basis Remar: From the definition, we have two immediate consequences of this definition. ) the set / ( ), x Z is an orthonormal basis for V. ) Scaling relation. Since ( x) V V we have and ( x ), Z ( x) ( t), ( t ) ( x ) Let p ( t), ( t ), we have the scaling relation, () ( x) p ( x ) is a orthonormal basis for V,

2 Here, to eep things simple, we will assume that the p s are all real, and that only p, p, p, p are different from. In the end, this corresponds to the simplest Daubechies wavelet, after the Haar wavelet. In that case, the scaling relation becomes The ( x) p ( x ) p ( x) p (x ) p (x ) p (x ) p s are not random numbers. The set ( x ), Z is an orthonormal basis for the space V. In terms of inner products, this means that ( t m), ( t l) lm, l m l m where lm, is the Kronecer. Since, if we set ( x l) p (x l) m above, we have ( t), ( t l) ( t), p (x l) p ( t), (x l p p l, l because the (real) inner product is linear in each variable. Since only p, p, p, p are nonzero, so there are only two values of l give nontrivial equations, namely l,, so we have p p p p p p p p 4 The wavelet. The formula for the wavelet is constructed from the scaling relation. Once we now ( x), we now an orthonormal basis for all of the spaces V. In particular, we now that ( x ), Z is an orthonormal basis for V. The wavelet space W is defined to be all functions in V that are orthogonal to the entire space V. In other words, W wv : w, f for all f V, our obective is to construct a function ( x) such that ( x l), l Z is an orthonormal basis for the space W. Since ( x) V the basis of V, we have ( x) q ( x ) Z Notice that: ( x l) p (x l) p ( x ) l l, so we have l ( x), ( x l) p q p q p q p q () l l l l and ( x ), Z are

3 Let l, we have () pq pq pq pq We can choose q p, q p, q p, q p and q otherwise. This choice maes () is. For this choice, to see whether ( x), ( x l) for all other integer l. For l, we have For l, we have ( x), ( x ) p q p q p q p q 4 5 p p p ( p ) p () p () ( x), ( x ) pq pq pq pq p () p () p p p ( p ) For other l, we have ( x), ( x l) because the q s involved are all. Thus, if has the form ( x) p ( x) p (x ) p(x ) p (x ) (4) Then we now that ( x) is in W. It is not hard to show that ( x m), m Z is in W, and ( x m), mz is an orthonormal set for W, and actually this is a basis for W. Similarly, we have / ( ) : x ( x ) Z is an orthonormal basis for the th level wavelet space, V V W, V W W. We have the following decomposition: So, we have Thus, L ( R) W W W W W W / ( ) : x ( x ) is an orthonormal basis for the whole space of signals,, Z L ( R ).

4 N Daubechies Wavelet Theorem. Suppose P() z p z is a polynomial that satisfies the following conditions: a. P() b. P P z (z) ( z) for it c. p( e ) for t Let be the Haar scaling function and let n ( x) pn ( x ) for n. Then, the sequence n converges pointwise and in (), the orthonormality condition ( x n), ( x m) ( x n) ( x m) dx, and the scaling function, ( x) p ( x ) L to a function which satisfies the normalization condition, mn Example: Daubechies Example, let P() z p z, where p p p p Then Pz () satisfies the conditions in the above Theorem. Therefore, the iterative scheme stated in the Theorem converges to a scaling function whose construction is due to Daubechies. Plots of the first four iterations,,,, 4 of this process are given below.

5 The graph of The graph of

6 The graph of The graph of 4

7 The limit of n is the Daubechies scaling function, and the graph is below. The graph of Daubechies scaling function From (), we have the wavelet associated with the Daubechies scaling function is ( x) p ( x) p (x ) p(x ) p (x ) where p p p p The graph of Daubechies wavelet is below

8 The graph of Daubechies wavelet function Daubechies N Wavelet is the simplest wavelet after Haar. The scaling and wavelet relations are ( x) p ( x) p(x ) p (x )+ p (x ) ( x) p ( x) p (x ) p (x )- p (x ) where p p p p

9 Decomposition and Reconstruction Formulas for Daubechies N Wavelet The following formulas can be derived similarly to those in the Haar case. Daubechies N decomposition formulas a p a p a p a p a l l l l l b p a p a p a p a l l l l l / / Daubechies N reconstruction formulas a p a p a p b p b l even l l l l l l l l a p a p a p b p b l odd