MRA basic concepts

1
MRAbasic 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 VN 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

6
Orthogonal Projection and Least Square Error
7
For Orthonormal Basis (N3)
8
Interpretations

• Error (detail) vector in VN
• orthogonal to VN-1
• WN-1 the orthogonal complement to VN-1 in VN
• dimensionality of 1
• Similarly, WN-2 the orthogonal complement to
  VN-2 in VN-1

• Approximating vectors
• Sequence of orthogonal projection vectors of x in
  the subspaces
• Finest approximation at VN-1
• Coarsest approximation at V1

9
Interpretations (cont)

• Every vector in VN can be written in the form
  below (the sum of one vector apiece from the N
  subspaces WN-1, WN-2, , W1, V1)
• Orthogonality of subspaces

• The vectors eN-1, eN-2, , e1, X1 form an
  orthogonal set
• VN is the direct sum of these subspaces

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)

• V0 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 fk, a set of orthonormal basis
• We wish to find an approximation of u in V
• Remarks
• Approximation error u-up 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 f(t) (called scaling
  function) such that is
  a basis for V0
• If and
  vice versa
• Remarks
• Does not require the set of f(t) and its integer
  translates to be orthogonal (in general)
• No mention of wavelet