Introduction to Finite Frame Theory - Frame Research Center

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24 Peter G. Casazza, Gitta Kutyniok, and Friedrich Philipp is called the Grammian (operator) of the frame (ϕ i ) M i=1 . Note that the (canonical) matrix representation of the Grammian of a frame (ϕ i ) M i=1 for H N (which will also be called the Grammian matrix) is given by ⎡ ‖ϕ 1 ‖ 2 ⎤ 〈ϕ 2 ,ϕ 1 〉 ··· 〈ϕ M ,ϕ 1 〉〈ϕ 1 ,ϕ 2 〉 ‖ϕ 2 ‖ 2 ··· 〈ϕ M ,ϕ 2 〉 ⎢ ⎣ . . . .. ⎥ . ⎦ . 〈ϕ 1 ,ϕ M 〉〈ϕ 2 ,ϕ M 〉 ··· ‖ϕ M ‖ 2 One property of the Grammian is immediate. In fact, if the frame is unit-norm then the entries of the Grammian matrix are exactly the cosines of the angles between the frame vectors. Hence, for instance, if a frame is equi-angular then all off diagonal entries of the Grammian matrix have the same modulus. The fundamental properties of the Grammian operator are collected in the following result. Theorem 7. Let (ϕ i ) M i=1 be a frame for H N with analysis operator T , frame operator S, and Grammian operator G. Then the following statements hold. (i) An operator U on H N is unitary if and only if the Grammian of (Uϕ i ) M i=1 coincides with G. (ii) The non-zero eigenvalues of G and S coincide. (iii) (ϕ i ) M i=1 is a Parseval frame if and only if G is an orthogonal projection of rank N (namely onto the range of T ). (iv) G is invertible if and only if M = N. Proof. (i). This follows immediately from the fact that the entries of the Grammian matrix for (Uϕ i ) M i=1 are of the form 〈Uϕ i,Uϕ j 〉. (ii). Since T T ∗ and T ∗ T have the same non-zero eigenvalues (see Proposition 7), the same is true for G and S. (iii). It is immediate to prove that G is self-adjoint and has rank N. Since T is injective, T ∗ is surjective, and G 2 = (T T ∗ )(T T ∗ ) = T (T ∗ T )T ∗ , it follows that G is an orthogonal projection if and only if T ∗ T = Id, which is equivalent to the frame being Parseval. (iv). This is immediate by (ii). ⊓⊔ 5 Reconstruction from Frame Coefficients The analysis of a signal is typically performed by merely considering its frame coefficients. However, if the task is transmission of a signal, the ability to reconstruct the signal from its frame coefficients and also to do so efficiently becomes crucial.
Introduction to Finite Frame Theory 25 Reconstruction from coefficients with respect to an orthonormal basis was already discussed in Corollary 1. However, reconstruction from coefficients with respect to a redundant system is much more delicate and requires the utilization of another frame, called dual frame. If computing such a dual frame is computationally too complex, a circumvention of this problem is the so-called frame algorithm. 5.1 Exact Reconstruction We start with stating an exact reconstruction formula. Theorem 8. Let (ϕ i ) M i=1 be a frame for H N with frame operator S. Then, for every x ∈ H N , we have x = M ∑ i=1 〈x,ϕ i 〉S −1 ϕ i = M ∑ i=1 〈x,S −1 ϕ i 〉ϕ i . Proof. This follows directly from the definition of the frame operator in Definition 17 by writing x = S −1 Sx and x = SS −1 x. ⊓⊔ Notice that the first formula can be interpreted as a reconstruction strategy, whereas the second formula has the flavor of a decomposition. We further observe that the sequence (S −1 ϕ i ) M i=1 plays a crucial role in the formulas in Theorem 8. The next result shows that this sequence indeed also constitutes a frame. Proposition 13. Let (ϕ i ) M i=1 be a frame for H N with frame bounds A and B and with frame operator S. Then the sequence (S −1 ϕ i ) M i=1 is a frame for H N with frame bounds B −1 and A −1 and with frame operator S −1 . Proof. By Proposition 10, the sequence (S −1 ϕ i ) M i=1 forms a frame for H N with associated frame operator S −1 S(S −1 ) ∗ = S −1 . This in turn yields the frame bounds B −1 and A −1 . ⊓⊔ This new frame is called the canonical dual frame. In the sequel, we will discuss that also other dual frames may be utilized for reconstruction. Definition 19. Let (ϕ i ) M i=1 be a frame for H N with frame operator denoted by S. Then (S −1 ϕ i ) M i=1 is called the canonical dual frame for (ϕ i) M i=1 . The canonical dual frame of a Parseval frame is now easily determined by Proposition 13. Corollary 7. Let (ϕ i ) M i=1 be a Parseval frame for H N . Then its canonical dual frame is the frame (ϕ i ) M i=1 itself, and the reconstruction formula in Theorem 8 reads x = M ∑ i=1 〈x,ϕ i 〉ϕ i , x ∈ H N .
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