1. MRA basic concepts Jyun-Ming Chen Spring 2001

2. Introduction MRA (multi- resolution analysis) –Construct a hierarchy of approximations to functions in various subspaces of a linear vector space First explained in finite-dimensional linear vector space, There exist N linearly independent and orthogonal basis vectors Any vector in V N can be expressed as a unique linear combination of these basis vectors

3. Simple Illustration in Number Representation

4. Nested Vector Spaces : Subspace of a lower dimension by taking only N-1 of the N basis vectors, say Continuing, … Hence,

5. Approximate a Vector in Subspaces Best approximation: minimize discrepancy Let be its orthogonal projection of x in the subspace a1a1 a2a2 a3a3 e

6. Orthogonal Projection and Least Square Error

7. For Orthonormal Basis (N=3) a1a1 a2a2 a3a3 e2e2 a1a1 a2a2 a3a3 e1e1

8. Interpretations Approximating vectors: –Sequence of orthogonal projection vectors of x in the subspaces –Finest approximation at V N-1 –Coarsest approximation at V 1 Error (detail) vector in V N –orthogonal to V N-1 W N-1 : the orthogonal complement to V N-1 in V N – dimensionality of 1 Similarly, W N-2 : the orthogonal complement to V N-2 in V N-1

9. Interpretations (cont) Every vector in V N can be written in the form below (the sum of one vector apiece from the N subspaces W N-1, W N-2, …, W 1, V 1 ) Orthogonality of subspaces The vectors e N-1, e N-2, …, e 1, X 1 form an orthogonal set V N is the direct sum of these subspaces V3V3 V2V2 W2W2 V1V1 W1W1

10. From Vector Space to Function Space

11. Example of an MRA Let f(t) be a continuous, real-valued, finite energy signal Approximate f(t) as follows:

12. MRA Example (cont) V 0 : linear vector space, formed by the set of functions that are piecewise constant over unit interval Nested subspaces

13. Approximating Function by Orthogonal Projection Assume u is not a member of the space V spanned by {  k }, a set of orthonormal basis We wish to find an approximation of u in V Remarks: –Approximation error u-u p is orthogonal to space V –Mean square error of such an approximation is minimum

14. Formal Definition of an MRA An MRA consists of the nested linear vector space such that There exists a function  (t) (called scaling function) such that is a basis for V 0 If and vice versa ; Remarks: –Does not require the set of  (t) and its integer translates to be orthogonal (in general) –No mention of wavelet