Local reconstruction for sampling in shift-invariant spaces
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Local reconstruction for sampling in shift-invariant spaces Qiyu Sun
Received: 18 April 2008 / Accepted: 10 October 2008 / Published online: 13 December 2008 © Springer Science + Business Media, LLC 2008
Abstract The local reconstruction from samples is one of most desirable properties for many applications in signal processing, but it has not been given as much attention. In this paper, we will consider the local reconstruction problem for signals in a shift-invariant space. In particular, we consider finding sampling sets X such that signals in a shift-invariant space can be locally reconstructed from their samples on X. For a locally finite-dimensional shiftinvariant space V we show that signals in V can be locally reconstructed from its samples on any sampling set with sufficiently large density. For a shiftinvariant space V(φ1 , . . . , φ N ) generated by finitely many compactly supported functions φ1 , . . . , φ N , we characterize all periodic nonuniform sampling sets X such that signals in that shift-invariant space V(φ1 , . . . , φ N ) can be locally reconstructed from the samples taken from X. For a refinable shift-invariant space V(φ) generated by a compactly supported refinable function φ, we prove that for almost all (x0 , x1 ) ∈ [0, 1]2 , any signal in V(φ) can be locally reconstructed from its samples from {x0 , x1 } + Z with oversampling rate 2. The proofs of our results on the local sampling and reconstruction in the refinable shift-invariant space V(φ) depend heavily on the linear independent shifts of a refinable function on measurable sets with positive Lebesgue measure and the almost ripplet property for a refinable function, which are new and interesting by themselves.
Communicated by R.Q. Jia. Q. Sun (B) Department of Mathematics, University of Central Florida, 32816 Orlando, FL, USA e-mail: [email protected]
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Q. Sun
Keywords Local reconstruction · Sampling · Shift-invariant space · Locally finitely-generated space · Refinable space · Ripplets · Spline · Wavelets Mathematics Subject Classifications (2000) 94A20 · 42C40 · 41A15 · 42A65 · 46E15
1 Introduction Given a discrete sampling set X ⊂ R, we say that signals (functions) in a linear space V can be locally determined from their samples on X if for any compact set K ⊂ R there exists another compact set K˜ ⊃ K such that the restriction of a signal (function) f ∈ V on K is uniquely determined by (and hence can be reconstructed from) its finite number of samples f (x j) taken from K˜ ∩ X. We call such a sampling set X as a locally determining sampling set for the space V. The local reconstruction from samples is one of most desirable properties for many applications in signal processing, e.g. for implementing real-time reconstruction numerically. However, the local reconstruction problem has not been given as much attention [33, 50]. Aldroubi and Gröchenig [2], and Sun and Zhou [45] discussed the local reconstruction of sampling for the spline space Bn :=
f ∈ Cn−2 The restriction of f on [k, k + 1) is a
polynomial of degr
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