Introduction to Finite Frame Theory - Frame Research Center

Introduction to Finite Frame Theory 31
The advantage of the conjugate gradient method, which we will present next, is
the fact that it does not require knowledge of the frame bounds. However, as before,
the rate of convergence certainly does depend on them.
Proposition 20 (Conjugate Gradient Method, [97]). Let (ϕ i ) M i=1 be a frame for
H N with frame operator S. Given a signal x ∈ H N , define three sequences (y j ) ∞ j=0 ,
(r j ) ∞ j=0 , and (p j) ∞ j=−1 in H N and corresponding scalars (λ j ) ∞ j=−1 by
and for j ≥ 0, set
and
y 0 = 0, r 0 = p 0 = Sx, and p −1 = 0,
λ j = 〈r j, p j 〉
〈p j ,Sp j 〉 , y j+1 = y j + λ j p j , r j+1 = r j − λ j Sp j ,
p j+1 = Sp j − 〈Sp j,Sp j 〉
〈p j ,Sp j 〉 p j − 〈Sp j,Sp j−1 〉
〈p j−1 ,Sp j−1 〉 p j−1.
Then (y j ) ∞ j=0 converges to x in H N and the rate of convergence is
|||x − y j ||| ≤
2σ j
1 + σ 2 j |||x||| with σ = √
B −
√
A
√
B +
√
A
,
and ||| · ||| is the norm on H N given by |||x||| = 〈x,Sx〉 1/2 = ‖S 1/2 x‖, x ∈ H N .
6 Construction of Frames
As diverse as the various desiderata are which applications require a frame to satisfy,
as diverse are also the methods to construct frames [37, 59]. In this section,
we will present a prominent selection. For further details and results, such as, for
example, the construction of frames through spectral tetris [31, 47, 44] and through
eigensteps [30], we refer to Chapter [155].
6.1 Tight and Parseval Frames
Tight frames are particularly desirable due to the fact that the reconstruction of
a signal from tight frame coefficients is numerically optimally stable as already
discussed in Section 5. Most of the constructions we will present utilize a given
frame, which is then modified to become a tight frame.
We start with the most basic result for generating a Parseval frame which is the
application of S −1/2 , S being the frame operator.