# Frames

Frames are a stable, possibly redundant,
representation of signals.

It is a generalization of the concept of basis in
a linear space.

A frame is a family of vectors which can represent
any finite energy signal by the sequence of its inner products with
the vectors of the family. However, it may be possible that not all
sequences of reals represent an finite energy signal. Oversampling is
an example of a represention of signals in a frame. One can see that
not all sequences of values may represent a sequence of samples. In
general, frames are a stable and redundant representation of
signals.

## Definition

A family of vectors in a
Hilbert space H is a frame of H if there are two constants A>0 and
B>0 such that, for any f in H,

If A=B, the frame is said to be *tight*.

A **Riesz
basis** is a frame whose vectors are
linearly independant.

Example: consider a family of three vectors in the plane which are
obtained by succesive rotations of a third of turn of one vector.
This family is a tight frame of the plane, with A=B=3/2.

## Properties

The frame vectors are supposed to be of unit
norm.

If the frame vectors are independent, then
A<=1<=B. The frame is then an orthonormal basis if and only if
A=B=1. If A>1, then the frame is redundant. A finite family is
always a frame of the linear space that it generates.

## Pseudo Inverse

U denotes the operateur which transforms a signal
f into the sequence of its frame inner products.

**U has one or an infinity of left
inverses.**

The *pseudo-inverse* of U is the left inverse of U
which is zero on the orthogonal complement of the image of U. It is
the minimum norm left inverse.

It is used to build a signal approximation from
any sequence of real numbers. The computation of (U^{*}U)^{-1}f can be performed by a
conjugate gradient algorithm.

## Dual Frame

The image of the frame through (U^{*}U)^{-1} is a frame called
the* dual frame*.
For any f in H,

and

If the original frame is a Riesz basis, then the
two frames form a **biorthogonal basis
system**, that is

Windowed Fourier Frames and
Wavelet Frames