Sampling at Unknown Locations: Uniqueness and Reconstruction Under Constraints

Abstract

Traditional sampling results assume that the sample locations are known. Motivated by simultaneous localization and mapping (SLAM) and structure from motion (SfM), we investigate sampling at unknown locations. We show that, without further constraints, the problem is often hopeless. More precisely, we show that, for polynomial and bandlimited signals, it is possible to find two signals, arbitrarily far from each other, that fit the measurements. To overcome this, we propose to add constraints to the sample positions. As we show, this leads to a uniform sampling of a composite of functions. Our formulation retains the key aspects of the SLAM and SfM problems, whilst providing uniqueness, in many cases. We demonstrate this by studying two simple examples of constrained sampling at unknown locations. In the first, we consider sampling a periodic bandlimited signal composite with an unknown linear function. We derive the sampling requirements for uniqueness and present an algorithm that recovers both the bandlimited signal and the linear warping. Furthermore, we prove that, when the requirements for uniqueness are not met, the cases of multiple solutions have measure zero. For our second example, we consider polynomials sampled such that the sampling positions are constrained by a rational function. We prove that, if a specific sampling requirement is met, uniqueness is achieved. In addition, we present an alternate minimization scheme for solving the resulting nonconvex optimization problem. Finally, simulation results are provided to support our theoretical analysis.

Sampling and Reconstruction of Bandlimited Signals With Multi-Channel Time Encoding

“…Proof: Spike times have a maximal separation between them defined by (6). According to this bound, every machine produces a sampling set t (i) k , k ∈ Z where two spike times have a separation of at most 2κδ/(b − c).…”

“…On the other hand, to help us prove that the Nyquist-like condition is satisfied, the following lemma provides us with a lower bound on the average sampling rate of our spike times tk , k ∈ Z . Proof: Spike times have a maximal separation between them defined by (6). According to this bound, every machine produces a sampling set t (i) k , k ∈ Z where two spike times have a separation of at most 2κδ/(b − c).…”

“…Classical sampling and reconstruction has been revisited frequently with extensions proposed along two axes: the sampling setup and the signal class. For example, the traditional Shannon-Nyquist-Kotelnikov sampling setup [2,3] has been extended to setups where samples are irregularly spaced in time [4,5], and where samples are taken at unknown locations [6,7], among others. On the other hand, reconstructibility results have been established for multiple signal classes, from bandlimited signals [2], to signals in general shift-invariant subspaces [8], and signals of finite rate of innovation [9,10].…”

Multi-channel Time Encoding for Improved Reconstruction of Bandlimited Signals

“…More recently, sampling results have been developed for other signal classes such as general shift-invariant subspaces [2], unions of subspaces [3], and signals of finite rate of innovation [4]. The extension beyond uniform sampling includes results for irregular samples [5], jittered samples [6], samples at unknown locations [7,8] and incomplete frequency samples [9]. Whilst these results are of both theoretical importance and practical interest, they diverge from the example set by biology.…”