MA5242 Wavelets Lecture 3 Discrete Wavelet Transform

1. MA5242 Wavelets Lecture 3 Discrete Wavelet Transform Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 6874-2749

2. Riesz Representation Theorem. If is a finite dimensional unitary space there exists an antilinear isomorphism such that Proof. Let be an ONB for and define Then

3. Adjoint Transformations Theorem: Given unitary spaces and a linear transformation there exists a unique linear transformation (adjoint of T) with Proof. Define by composition let be the Riesz Rep. transformations and define

4. Problem Set 1 1. Assume that are ONB for unitary spaces and that is linear. Derive the relationship between the matrices that represent with respect to these bases. 2. Prove that a transformation is unitary iff 3. Derive the Riesz Representation, Adjoint and matrix representations, and characterization for orthogonal transformations for Euclidean spaces.

5. General Discrete Wavelet Transform

6. Convolution Representation where a,b,c,dare infinite sequences that extend the finite sequences

7. Orthogonality Conditions Theorem. The wavelet transform matrix is unitary iff for all

8. Laurent Polynomials Definition: A Laurent polynomial is a function that admits a representation where c is a finitely supported sequence. Definition: For a sequence c let and define the unit circle Theorem: For seq. a, b,

9. Conjugate Quadrature Filters Definition: A sequence c that satisfies the quadratic equations necessary for a wavelet transform matrix to me unitary is called a Conjugate Quadrature Filter Theorem. A sequence c is a CQF iff it satisfies Theorem: Prove that if c is a CQF and if d is related to c by the equation on the previous page then d is also a CQF and the WT is unitary Theorem: If c is a CQF then the WT is unitary if

10. Problem Set 2 1. Derive the conditions for a WT to be unitary. 2. Prove the theorems about Laurent polynomials and the two theorems on the preceding page. 3. Prove that c, d form a unitary WT iff (the modulation matrix) is unitary for all 4. Prove that d on the previous page is the same as

11. Moment Conditions Definition. d has -1< p vanishing moments if Theorem. If c,d gives a unitary WT then d has -1< p vanishing moments iff has a factor has a factor iff

12. Moment Consequences Theorem. If d has -1< p vanishing moments and is supported on the set {0,1,…,2N-1} then if the finite sequence d(k),d(k-1),…,d(k-2N+1) can be represented by a polynomial having degree < N Proof. by the binomial theorem and vanishing moments.

13. Riesz-Fejer Spectral Factorization Theorem. A Laurent polynomial N is on iff there exists a LP P such that Proof. Let be the set of roots of where is the multiplicity of Since N is real-valued furthermore, since N is non-negative the are even hence paired, now choose P to contain one root from each pair and the result easily follows.

14. Daubechies Wavelets Theorem. If c is a CQF supported on 0,1,…,2N-1 and then has a factor satisfies and is uniquely determined by the equations Furthermore, and c can be chosen by the R.-F. Theorem.

15. Problem Set 3 1. Prove all of the Theorems after Problem Set 2.