Phaseless Sampling and Linear Reconstruction of Functions in Spline Spaces

Abstract

We study phaseless sampling in spline spaces generated by Bsplines with arbitrary knots. For real spline spaces, we give a necessary and sufficient condition for a sequence of sampling points to admit a local phase retrieval of any nonseparable function. We also study phaseless sampling in complex spline spaces and illustrate that phase retrieval is impossible in this case. Nevertheless, we show that phaseless sampling is possible. For any function f in a complex spline space, no mater it is separable or not, we show that |f (x)| 2 is uniquely determined and can be recovered linearly from its sampled values at a well chosen sequence of sampling points. We give necessary and sufficient conditions for such sequences.generated by the m-degree B-spline ϕ m = χ [0,1] * . . . * χ [0,1] ( m + 1 terms), m ≥ 1,

Wavelet invariants for statistically robust multi-reference alignment

“…Each invariant captures successively more information in f . While f (0) carries limited information, the power spectrum recovers f up to its phase, which in some cases can be resolved; results along these lines are in the field of phase retrieval [47,48]. The power spectrum is invariant to translations since the Fourier modulus kills the phase factor induced by a translation t of f .…”

Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

“…A spatial signal f on a domain D is defined by its evaluations f (x), x ∈ D. In this paper, we consider the problem whether and how a real-valued signal f can be reconstructed, up to a global sign, from magnitude information |f (x)|, x ∈ D, or from its phaseless samples |f (γ)|, γ ∈ Γ, taken on a discrete set Γ ⊂ D in a stable way. The above problem has been discussed for bandlimited signals [39] and wavelet signals residing in a principal shiftinvariant space [13,14,38]. It is a nonlinear ill-posed problem which can be solved only if we have some extra information about the signal f .…”

“…An equivalent statement to the above question is whether a signal f is determined, up to a sign, from the magnitude information |f (x)|, x ∈ D. The above question is an infinite-dimensional phase retrieval problem, which has been discussed for bandlimited signals [39], wavelet signals in a principal shift-invariant space [13,14,38], and spatial signals in a linear space [13]. The reader may refer to [1,2,8,20,27,28,33] for historical remarks and additional references on phase retrieval in an infinite-dimensional linear space.…”