0.4 Bases, orthogonal bases, biorthogonal bases, frames, tight  (Page 4/5)

Frames are an over-complete version of a basis set, and tight frames are an over-complete version of an orthogonal basis set. If one is using aframe that is neither a basis nor a tight frame, a dual frame set can be specified so that analysis and synthesis can be done as for anon-orthogonal basis. If a tight frame is being used, the mathematics is very similar to using an orthogonal basis. The Fourier type system in [link] is essentially the same as [link] , and [link] is essentially a Parseval's theorem.

The use of frames and tight frames rather than bases and orthogonal bases means a certain amount of redundancy exists. In some cases,redundancy is desirable in giving a robustness to the representation so that errors or faults are less destructive. In other cases, redundancy is aninefficiency and, therefore, undesirable. The concept of a frame originates with Duffin and Schaeffer [link] and is discussed in [link] , [link] , [link] . In finite dimensions, vectors can always be removed from a frame to get a basis, but in infinite dimensions, that isnot always possible.

An example of a frame in finite dimensions is a matrix with more columns than rows but with independent rows. An example of a tight frame is asimilar matrix with orthogonal rows. An example of a tight frame in infinite dimensions would be an over-sampled Shannon expansion. It isinformative to examine this example.

Matrix examples

An example of a frame of four expansion vectors $f_k$ in a three-dimensional space would be

which corresponds to the basis shown in the square matrix in [link] . The corresponding analysis equation is

which corresponds to [link] . One can calculate a set of dual frame vectors by temporarily appending an arbitrary independent row to [link] , making the matrix square, then using the first three columns of the inverse as the dual frame vectors. This clearly illustrates the dualframe is not unique. Daubechies [link] shows how to calculate an “economical" unique dual frame.

The tight frame system occurs in wavelet infinite expansions as well as other finite and infinite dimensional systems. A numerical example of aframe which is a normalized tight frame with four vectors in three dimensions is

which includes the redundancy factor from [link] . Note the rows are orthogonal and the columns are normalized, which gives

or

which is the matrix form of [link] . The factor of $A=4/3$ is the measure of redundancy in this tight frame using four expansion vectors ina three-dimensional space.

The identity for the expansion coefficients is

which for the numerical example gives

Although this is not a general identity operator, it is an identity operator over the three-dimensional subspace that $\mathbf{a}$ is in and it illustrates the unity norm of the rows of ${\mathbf{F}}^{\mathbf{T}}$ and columns of $\mathbf{F}$ .