arXiv subjects

Martin Vetterli

Publications and source records attributed to Martin Vetterli.

At least 37 records · Page 2Linked to original sources

arXiv2019

Shapes from Echoes: Uniqueness from Point-to-Plane Distance Matrices

We study the problem of localizing a configuration of points and planes from the collection of point-to-plane distances. This problem models simultaneous localization and mapping from acoustic echoes as well as the notable "structure from sound" approach to microphone localization with unknown sources. In our earlier work we proposed computational methods for localization from point-to-plane distances and noted that such localization suffers from various ambiguities beyond the usual rigid body motions; in this paper we provide a complete characterization of uniqueness. We enumerate equivalence classes of configurations which lead to the same distance measurements as a function of the number of planes and points, and algebraically characterize the related transformations in both 2D and 3D. Here we only discuss uniqueness; computational tools and heuristics for practical localization from point-to-plane distances using sound will be addressed in a companion paper.

cs.CG↗

arXiv2018

Super Resolution Phase Retrieval for Sparse Signals

In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts to recover the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. Solving the phase retrieval problem is equivalent to recovering a signal from its auto-correlation function. In this paper, we assume the original signal to be sparse; this is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. We propose an algorithm that resolves the phase retrieval problem in three stages: i) we leverage the finite rate of innovation sampling theory to super-resolve the auto-correlation function from a limited number of samples, ii) we design a greedy algorithm that identifies the locations of a sparse solution given the super-resolved auto-correlation function, iii) we recover the amplitudes of the atoms given their locations and the measured auto-correlation function. Unlike traditional approaches that recover a discrete approximation of the underlying signal, our algorithm estimates the signal on a continuous domain, which makes it the first of its kind. Along with the algorithm, we derive its performance bound with a theoretical analysis and propose a set of enhancements to improve its computational complexity and noise resilience. Finally, we demonstrate the benefits of the proposed method via a comparison against Charge Flipping, a notable algorithm in crystallography.

cs.IT↗

arXiv2018

Bound and Conquer: Improving Triangulation by Enforcing Consistency

We study the accuracy of triangulation in multi-camera systems with respect to the number of cameras. We show that, under certain conditions, the optimal achievable reconstruction error decays quadratically as more cameras are added to the system. Furthermore, we analyse the error decay-rate of major state-of-the-art algorithms with respect to the number of cameras. To this end, we introduce the notion of consistency for triangulation, and show that consistent reconstruction algorithms achieve the optimal quadratic decay, which is asymptotically faster than some other methods. Finally, we present simulations results supporting our findings. Our simulations have been implemented in MATLAB and the resulting code is available in the supplementary material.

cs.CV↗

arXiv2016

FRIDA: FRI-Based DOA Estimation for Arbitrary Array Layouts

In this paper we present FRIDA---an algorithm for estimating directions of arrival of multiple wideband sound sources. FRIDA combines multi-band information coherently and achieves state-of-the-art resolution at extremely low signal-to-noise ratios. It works for arbitrary array layouts, but unlike the various steered response power and subspace methods, it does not require a grid search. FRIDA leverages recent advances in sampling signals with a finite rate of innovation. It is based on the insight that for any array layout, the entries of the spatial covariance matrix can be linearly transformed into a uniformly sampled sum of sinusoids.

cs.SD↗

arXiv2016

Omnidirectional Bats, Point-to-Plane Distances, and the Price of Uniqueness

We study simultaneous localization and mapping with a device that uses reflections to measure its distance from walls. Such a device can be realized acoustically with a synchronized collocated source and receiver; it behaves like a bat with no capacity for directional hearing or vocalizing. In this paper we generalize our previous work in 2D, and show that the 3D case is not just a simple extension, but rather a fundamentally different inverse problem. While generically the 2D problem has a unique solution, in 3D uniqueness is always absent in rooms with fewer than nine walls. In addition to the complete characterization of ambiguities which arise due to this non-uniqueness, we propose a robust solution for inexact measurements similar to analogous results for Euclidean Distance Matrices. Our theoretical results have important consequences for the design of collocated range-only SLAM systems, and we support them with an array of computer experiments.

cs.RO↗

arXiv2016

Look, no Beacons! Optimal All-in-One EchoSLAM

We study the problem of simultaneously reconstructing a polygonal room and a trajectory of a device equipped with a (nearly) collocated omnidirectional source and receiver. The device measures arrival times of echoes of pulses emitted by the source and picked up by the receiver. No prior knowledge about the device's trajectory is required. Most existing approaches addressing this problem assume multiple sources or receivers, or they assume that some of these are static, serving as beacons. Unlike earlier approaches, we take into account the measurement noise and various constraints on the geometry by formulating the solution as a minimizer of a cost function similar to \emph{stress} in multidimensional scaling. We study uniqueness of the reconstruction from first-order echoes, and we show that in addition to the usual invariance to rigid motions, new ambiguities arise for important classes of rooms and trajectories. We support our theoretical developments with a number of numerical experiments.

cs.RO↗

arXiv2016

The regularized tau estimator: A robust and efficient solution to ill-posed linear inverse problems

Linear inverse problems are ubiquitous. Often the measurements do not follow a Gaussian distribution. Additionally, a model matrix with a large condition number can complicate the problem further by making it ill-posed. In this case, the performance of popular estimators may deteriorate significantly. We have developed a new estimator that is both nearly optimal in the presence of Gaussian errors while being also robust against outliers. Furthermore, it obtains meaningful estimates when the problem is ill-posed through the inclusion of $\ell_1$ and $\ell_2$ regularizations. The computation of our estimate involves minimizing a non-convex objective function. Hence, we are not guaranteed to find the global minimum in a reasonable amount of time. Thus, we propose two algorithms that converge to a good local minimum in a reasonable (and adjustable) amount of time, as an approximation of the global minimum. We also analyze how the introduction of the regularization term affects the statistical properties of our estimator. We confirm high robustness against outliers and asymptotic efficiency for Gaussian distributions by deriving measures of robustness such as the influence function, sensitivity curve, bias, asymptotic variance, and mean square error. We verify the theoretical results using numerical experiments and show that the proposed estimator outperforms recently proposed methods, especially for increasing amounts of outlier contamination. Python code for all of the algorithms are available online in the spirit of reproducible research.

stat.ME↗

arXiv2016

SHAPE: Linear-Time Camera Pose Estimation With Quadratic Error-Decay

We propose a novel camera pose estimation or perspective-n-point (PnP) algorithm, based on the idea of consistency regions and half-space intersections. Our algorithm has linear time-complexity and a squared reconstruction error that decreases at least quadratically, as the number of feature point correspondences increase. Inspired by ideas from triangulation and frame quantisation theory, we define consistent reconstruction and then present SHAPE, our proposed consistent pose estimation algorithm. We compare this algorithm with state-of-the-art pose estimation techniques in terms of accuracy and error decay rate. The experimental results verify our hypothesis on the optimal worst-case quadratic decay and demonstrate its promising performance compared to other approaches.

cs.CV↗

arXiv2016

On the Accuracy of Point Localisation in a Circular Camera-Array

Although many advances have been made in light-field and camera-array image processing, there is still a lack of thorough analysis of the localisation accuracy of different multi-camera systems. By considering the problem from a frame-quantisation perspective, we are able to quantify the point localisation error of circular camera configurations. Specifically, we obtain closed form expressions bounding the localisation error in terms of the parameters describing the acquisition setup. These theoretical results are independent of the localisation algorithm and thus provide fundamental limits on performance. Furthermore, the new frame-quantisation perspective is general enough to be extended to more complex camera configurations.

cs.CV↗

arXiv2015

Sampling and Reconstruction of Shapes with Algebraic Boundaries

We present a sampling theory for a class of binary images with finite rate of innovation (FRI). Every image in our model is the restriction of $\mathds{1}_{\{p\leq0\}}$ to the image plane, where $\mathds{1}$ denotes the indicator function and $p$ is some real bivariate polynomial. This particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial $p$. We show that the image parameters --i.e., the polynomial coefficients-- satisfy a set of linear annihilation equations with the coefficients being the image moments. The inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels. As a remedy to these problems, we replace conventional moments with more stable \emph{generalized moments} that are adjusted to the given sampling kernel. The benefits are threefold: (1) it relaxes the requirements on the sampling kernels, (2) produces annihilation equations that are robust at numerical precision, and (3) extends the results to images with unbounded boundaries. We further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points, and sequentially enforcing measurement consistency. We consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images, including low to moderate noise levels and a range of realistic sampling kernels.

cs.CG↗

arXiv2015

Where You Are Is Who You Are: User Identification by Matching Statistics

Most users of online services have unique behavioral or usage patterns. These behavioral patterns can be exploited to identify and track users by using only the observed patterns in the behavior. We study the task of identifying users from statistics of their behavioral patterns. Specifically, we focus on the setting in which we are given histograms of users' data collected during two different experiments. We assume that, in the first dataset, the users' identities are anonymized or hidden and that, in the second dataset, their identities are known. We study the task of identifying the users by matching the histograms of their data in the first dataset with the histograms from the second dataset. In recent works, the optimal algorithm for this user identification task is introduced. In this paper, we evaluate the effectiveness of this method on three different types of datasets and in multiple scenarios. Using datasets such as call data records, web browsing histories, and GPS trajectories, we show that a large fraction of users can be easily identified given only histograms of their data; hence these histograms can act as users' fingerprints. We also verify that simultaneous identification of users achieves better performance compared to one-by-one user identification. We show that using the optimal method for identification gives higher identification accuracy than heuristics-based approaches in practical scenarios. The accuracy obtained under this optimal method can thus be used to quantify the maximum level of user identification that is possible in such settings. We show that the key factors affecting the accuracy of the optimal identification algorithm are the duration of the data collection, the number of users in the anonymized dataset, and the resolution of the dataset. We analyze the effectiveness of k-anonymization in resisting user identification attacks on these datasets.

cs.LG↗

arXiv2015

Unlabeled Sensing with Random Linear Measurements

We study the problem of solving a linear sensing system when the observations are unlabeled. Specifically we seek a solution to a linear system of equations y = Ax when the order of the observations in the vector y is unknown. Focusing on the setting in which A is a random matrix with i.i.d. entries, we show that if the sensing matrix A admits an oversampling ratio of 2 or higher, then with probability 1 it is possible to recover x exactly without the knowledge of the order of the observations in y. Furthermore, if x is of dimension K, then any 2K entries of y are sufficient to recover x. This result implies the existence of deterministic unlabeled sensing matrices with an oversampling factor of 2 that admit perfect reconstruction. The result is universal in that recovery is guaranteed for all possible choices of x. While the proof is constructive, it uses a combinatorial algorithm which is not practical, leaving the question of complexity open. We also analyze a noisy version of the problem and show that local stability is guaranteed by the solution. In particular, for every x, the recovery error tends to zero as the signal-to-noise-ratio tends to infinity. The question of universal stability is unclear. We also obtain a converse of the result in the noiseless case: If the number of observations in y is less than 2K, then with probability 1, universal recovery fails, i.e., with probability 1, there exists distinct choices of x which lead to the same unordered list of observations in y. In terms of applications, the unlabeled sensing problem is related to data association problems encountered in different domains including robotics where it is appears in a method called "simultaneous localization and mapping" (SLAM), multi-target tracking applications, and in sampling signals in the presence of jitter.

cs.IT↗

arXiv2015

Shapes From Pixels

Continuous-domain visual signals are usually captured as discrete (digital) images. This operation is not invertible in general, in the sense that the continuous-domain signal cannot be exactly reconstructed based on the discrete image, unless it satisfies certain constraints (\emph{e.g.}, bandlimitedness). In this paper, we study the problem of recovering shape images with smooth boundaries from a set of samples. Thus, the reconstructed image is constrained to regenerate the same samples (consistency), as well as forming a shape (bilevel) image. We initially formulate the reconstruction technique by minimizing the shape perimeter over the set of consistent binary shapes. Next, we relax the non-convex shape constraint to transform the problem into minimizing the total variation over consistent non-negative-valued images. We also introduce a requirement (called reducibility) that guarantees equivalence between the two problems. We illustrate that the reducibility property effectively sets a requirement on the minimum sampling density. One can draw analogy between the reducibility property and the so-called restricted isometry property (RIP) in compressed sensing which establishes the equivalence of the $\ell_0$ minimization with the relaxed $\ell_1$ minimization. We also evaluate the performance of the relaxed alternative in various numerical experiments.

cs.IT↗

arXiv2015

Euclidean Distance Matrices: Essential Theory, Algorithms and Applications

Euclidean distance matrices (EDM) are matrices of squared distances between points. The definition is deceivingly simple: thanks to their many useful properties they have found applications in psychometrics, crystallography, machine learning, wireless sensor networks, acoustics, and more. Despite the usefulness of EDMs, they seem to be insufficiently known in the signal processing community. Our goal is to rectify this mishap in a concise tutorial. We review the fundamental properties of EDMs, such as rank or (non)definiteness. We show how various EDM properties can be used to design algorithms for completing and denoising distance data. Along the way, we demonstrate applications to microphone position calibration, ultrasound tomography, room reconstruction from echoes and phase retrieval. By spelling out the essential algorithms, we hope to fast-track the readers in applying EDMs to their own problems. Matlab code for all the described algorithms, and to generate the figures in the paper, is available online. Finally, we suggest directions for further research.

cs.OH↗

arXiv2015

Raking the Cocktail Party

We present the concept of an acoustic rake receiver---a microphone beamformer that uses echoes to improve the noise and interference suppression. The rake idea is well-known in wireless communications; it involves constructively combining different multipath components that arrive at the receiver antennas. Unlike spread-spectrum signals used in wireless communications, speech signals are not orthogonal to their shifts. Therefore, we focus on the spatial structure, rather than temporal. Instead of explicitly estimating the channel, we create correspondences between early echoes in time and image sources in space. These multiple sources of the desired and the interfering signal offer additional spatial diversity that we can exploit in the beamformer design. We present several "intuitive" and optimal formulations of acoustic rake receivers, and show theoretically and numerically that the rake formulation of the maximum signal-to-interference-and-noise beamformer offers significant performance boosts in terms of noise and interference suppression. Beyond signal-to-noise ratio, we observe gains in terms of the \emph{perceptual evaluation of speech quality} (PESQ) metric for the speech quality. We accompany the paper by the complete simulation and processing chain written in Python. The code and the sound samples are available online at \url{http://lcav.github.io/AcousticRakeReceiver/}.

cs.SD↗

arXiv2014

On Minimal Trajectories for Mobile Sampling of Bandlimited Fields

We study the design of sampling trajectories for stable sampling and the reconstruction of bandlimited spatial fields using mobile sensors. The spectrum is assumed to be a symmetric convex set. As a performance metric we use the path density of the set of sampling trajectories that is defined as the total distance traveled by the moving sensors per unit spatial volume of the spatial region being monitored. Focussing first on parallel lines, we identify the set of parallel lines with minimal path density that contains a set of stable sampling for fields bandlimited to a known set. We then show that the problem becomes ill-posed when the optimization is performed over all trajectories by demonstrating a feasible trajectory set with arbitrarily low path density. However, the problem becomes well-posed if we explicitly specify the stability margins. We demonstrate this by obtaining a non-trivial lower bound on the path density of an arbitrary set of trajectories that contain a sampling set with explicitly specified stability bounds.

cs.IT↗

arXiv2014

The Beauty of the Commons: Optimal Load Sharing by Base Station Hopping in Wireless Sensor Networks

In wireless sensor networks (WSNs), the base station (BS) is a critical sensor node whose failure causes severe data losses. Deploying multiple fixed BSs improves the robustness, yet requires all BSs to be installed with large batteries and large energy-harvesting devices due to the high energy consumption of BSs. In this paper, we propose a scheme to coordinate the multiple deployed BSs such that the energy supplies required by individual BSs can be substantially reduced. In this scheme, only one BS is selected to be active at a time and the other BSs act as regular sensor nodes. We first present the basic architecture of our system, including how we keep the network running with only one active BS and how we manage the handover of the role of the active BS. Then, we propose an algorithm for adaptively selecting the active BS under the spatial and temporal variations of energy resources. This algorithm is simple to implement but is also asymptotically optimal under mild conditions. Finally, by running simulations and real experiments on an outdoor testbed, we verify that the proposed scheme is energy-efficient, has low communication overhead and reacts rapidly to network changes.

cs.NI↗

arXiv2014

Sequences with Minimal Time-Frequency Uncertainty

A central problem in signal processing and communications is to design signals that are compact both in time and frequency. Heisenberg's uncertainty principle states that a given function cannot be arbitrarily compact both in time and frequency, defining an "uncertainty" lower bound. Taking the variance as a measure of localization in time and frequency, Gaussian functions reach this bound for continuous-time signals. For sequences, however, this is not true; it is known that Heisenberg's bound is generally unachievable. For a chosen frequency variance, we formulate the search for "maximally compact sequences" as an exactly and efficiently solved convex optimization problem, thus providing a sharp uncertainty principle for sequences. Interestingly, the optimization formulation also reveals that maximally compact sequences are derived from Mathieu's harmonic cosine function of order zero. We further provide rational asymptotic expansions of this sharp uncertainty bound. We use the derived bounds as a benchmark to compare the compactness of well-known window functions with that of the optimal Mathieu's functions.

cs.IT↗