Nonuniform sampling and reconstruction with Hilbert frames. Taylor Hines,
advisor: Dr. Anne Gelb. School of Mathematical and Statistical Sciences,. Arizona
...
Nonuniform sampling and reconstruction with Hilbert frames Taylor Hines, advisor: Dr. Anne Gelb School of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ.
[email protected]
Motivation
Induced Hilbert Spaces
Localization and Re-projection for Frames
In magnetic resonance (MR) imaging, image data is collected as Fourier coefficients, usually in non-Cartesian coordinates. For this reason, standard reconstruction methods cannot be implemented.
As described by S. Saitoh, the given sampling sequence {fn}n∈N naturally generates a reproducing kernel Hilbert space in the following way:
Introduction. For f ∈ H, say that we are given a finite sequence of frame coefficients N {hf, fni}n=1, and a corresponding approximation SN f for f . We now would like to reproject SN f onto a ‘better’ family {gm}m∈N, whereby we recover f pointwise with high accuracy.
Let Z be any set, and Φ : Z → H be a function. Denote by F(Z) the set of scalar-valued functions on Z, and define the map L : H → F(Z) by (LF )(p) = hF, Φ(p)i.
(3)
Note. Here, we take Z = N, and Φ(n) = fn, so that L : F 7→ {hF, fniH}.
Fig.: Fourier transform (nonuniformly sampled).
Fig.: Physical signal.
Example. Consider the following 1D signal f ∈ L2[−π, π]. Say that we are given Fourier samples of f as the linear functionals −iλnx ˆ ˆ {f (λn)}n∈Z, where f (λn) = f (x)e dx
Theorem. (Saitoh, 1983) The range of L, denoted K, is a Hilbert space under the inner product h·, ·iK defined by kf kK = inf{kF kH | LF = f }. (4) Furthermore, K admits the reproducing kernel K : Z × Z → H defined by K(p, q) = hΦ(q), Φ(p)iH.
(5)
Z
for some family {λn}n∈Z in R. If we in fact have {λn} = Z, then we can reconstruct f exactly. However, in many instances this is not the case. Therefore, non-standard methods of reconstructing a signal from nonuniform samples are necessary.
Let H be a separable Hilbert space of functions on a domain E. The problem we have studied has two forms: 1 Reconstruct (exactly, if possible) the signal g ∈ H given only information in the form of the linear functionals {hg, fni}n∈N for a fixed family {fn}n∈N in H. 2 Construct a family {fn}n∈N in H so that every g ∈ H can be reconstructed given the samples {hg, fni}n∈N. In addition to accuracy and uniqueness, “good” solutions to both of these questions also gives numerically efficient and practically implementable reconstruction algorithms.
Hilbert Frames We are particularly interested in the case when sampling functions {fn}n∈N form a frame in H, that is, there exist positive constants A and B such that 2
2
Akf k ≤ Σ |hf, fni| ≤ Bkf k n∈N
2
(1)
for all f ∈ H. Theorem. When {fn}n∈N is a frame for H, then the frame operator S : H → H, defined by Sf = Σ hf, fnifn (2) n∈N
is positive, self-adjoint, and invertible. As a result, a function can be reconstructed (or represented) exactly by its frame samples.
kf − TM (SN f )k ≤ kf − TM f k + kTM f − TM (SN f )k |
Lemma. Associated to every element f ∈ K there is a unique F ∗ ∈ K with LF ∗ = f ∗ and which satisfies kF kH = kf kK.
∗
F = Σ hf, LEniKEn n∈N
(6)
where {En}n∈N is an orthonormal basis for H. Problem. Computation of the norm k · kK, is a costly minimization process. For this reason, we hope to design the sampling scheme {Φ(p)}p∈Z so that k · kK can be defined as an integral formula. Fact. The family {ϕp}p∈Z = {LΦ(p)}p∈Z is a frame in K if and only if it is a frame sequence (a frame for its closed span). Corollary. If {Φ(p)}p∈Z is a frame sequence, then the function ν : K → R defined by 2 2 ν(f ) = Σ |f (p)| (7 )
|
{z
re-projection error
}
We hope to generate good re-projections using the theory of localized frames. Definition. (Gröchenig, 2004) Given s > 0, the frame {fn}n∈N is s-localized with respect to the Riesz basis {gm}m∈N if there exists a positive constant C such that |hfn, gmi|2 ≤ C(1 + |n − m|)−s and |hfn, g˜mi|2 ≤ C(1 + |n − m|)−s
(9) (10)
for all n, m ∈ N, where {˜ gm} is the dual basis to {gm}. Similarly, the frame {fn}n∈N is self-localized if |hfn, g˜mi|2 ≤ C(1 + |n − m|)−s. (11) Question. If SN f is the approximation of f with respect to the frame {fn}n∈N, and {fn}n∈N is localized with respect to the Riesz basis {gm}m∈N, how accurate is the reprojection of SN f onto this Riesz basis? Similarly, given a frame {fn}n∈N, is it possible to find a Riez basis which localizes {fn}, and if so, to what degree? Example. When {fn}n∈Z is the standard Fourier basis, it has been shown that a finitedimensional approximation SN f can be re-projected onto Gegenbauer polynomials to recover f pointwise with exponential accuracy. We hope to generalize this to more general families using the theory of localized frames. Of particular interest are nonuniform frames of complex exponentials reprojected onto orthogonal polynomials (e.g. Gegenbauer, Hermite, Freud).
References and Acknowledgement
is a norm on K that is equivalent to the induced norm k · kK.
Example. This is particularly applicable when the set {Φ(p)}p∈Z is a family of translations of a single function, since these are a commonly-studied type of frame sequence in L2(R). Furthermore, the set {ϕp} = {LΦ(p)} then corresponds to a frame of translations on K which also form a reproducing kernel, and hence we expect K to behave much like a shift-invariant space. Investigating this question is one of our current research projects.
}
(8)
where TM f is the finite-dimensional approximation of f with respect to {gm}.
p∈Z
Result. If {Φ(p)}p∈Z is a Parseval frame sequence (a frame sequence with frame bounds A = B = 1), then the function ν above is in fact an integral formula for the norm k · kK.
{z
truncation error
Fact. If {Φ(p)} is complete in H, then L is an isometry (otherwise, L will not be injective.)
Provided that the sampled function f ∈ K satisfies some regularization conditions, there is an explicit reconstruction algorithm given by
Problem Statement
Technique. If SN f is the finite-dimensional approximation of f with respect to the family {fn}n∈N, then the error with respect to the family {gm}m∈N is
Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator, J. Fourier Anal. Appl. 10 (2004), is. 2, 105–132. 2 A. Aldroubi, K. Gröchenig, Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces, Soc. Ind. App. Math. 43 (2001), no. 4, 585–620. 3 S. Saitoh, Hilbert Spaced Induced by Hilbert Space Valued Functions, Proc. Amer. Math. Soc 89 (1983), no. 2, 74–78. 1
K. Gröchenig,
This work was supported in part by National Science Foundation grant FRG 0652833. This was presented at the International Conference on Advances in Scientific Computing on December 6-8, 2009 at Brown University in Providence, RI.
