It often happens that we are interested in reconstructing an unknown signal from partial measurements. Also, it is typically assumed that the location (temporal or spatial) of each sample is known and that the only distortion present in the observations is due to additive measurement noise. However, there are some applications where such location information is lost. In this paper, we consider the situation in which the order of noisy samples, taken from a linear measurement system, is missing. Previous work on this topic has only considered the noiseless case and exhaustive search combinatorial algorithms. We propose a much more efficient algorithm based on a geometrical viewpoint of the problem. We also study the uniqueness of the solution under different choices of the sampling matrix and its robustness to noise for the case of two-dimensional signals. Finally we provide simulation results to confirm the theoretical findings of the paper.

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.

No abstract

Self-localization of nodes in a sensor network is typically achieved using either range or direction measurements; in this paper, we show that a constructive combination of both improves the estimation. We propose two localization algorithms that make use of the differences between the sensors' coordinates, or edge vectors; these can be calculated from measured distances and angles. Our first method improves the existing edge-multidimensional scaling algorithm (E-MDS) by introducing additional constraints that enforce geometric consistency between the edge vectors. On the other hand, our second method decomposes the edge vectors onto 1-dimensional spaces and introduces the concept of coordinate difference matrices (CDMs) to independently regularize each projection. This solution is optimal when Gaussian noise is added to the edge vectors. We demonstrate in numerical simulations that both algorithms outperform state-of-the-art solutions.

No abstract