Calibration Using Matrix Completion with Application to Ultrasound Tomography

We study the calibration process in circular ultrasound tomography devices where the sensor positions deviate from the circumference of a perfect circle. This problem arises in a variety of applications in signal processing ranging from breast imaging to sensor network localization. We introduce a novel method of calibration/localization based on the time-of-flight (ToF) measurements between sensors when the enclosed medium is homogeneous. In the presence of all the pairwise ToFs, one can easily estimate the sensor positions using multi-dimensional scaling (MDS) method. In practice however, due to the transitional behaviour of the sensors and the beam form of the transducers, the ToF measurements for close-by sensors are unavailable. Further, random malfunctioning of the sensors leads to random missing ToF measurements. On top of the missing entries, in practice an unknown time delay is also added to the measurements. In this work, we incorporate the fact that a matrix defined from all the ToF measurements is of rank at most four. In order to estimate the missing ToFs, we apply a state-of-the-art low-rank matrix completion algorithm, OPTSPACE . To find the correct positions of the sensors (our ultimate goal) we then apply MDS. We show analytic bounds on the overall error of the whole process in the presence of noise and hence deduce its robustness. Finally, we confirm the functionality of our method in practice by simulations mimicking the measurements of a circular ultrasound tomography device.Comment: submitted to IEEE Transaction on Signal Processin

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.Comment: - 17 pages, 12 figures, to appear in IEEE Signal Processing Magazine - change of title in the last revisio

Spatial Error Concealment in Ad-hoc Audio Conferencing Systems

In this work we consider an ad-hoc audio conferensing system based on VoIP services in which the participants connect to the conference using mobile communication devices with wireless connectivity. To overcome possible quality problems in the wireless link, we propose improvements to the existing conferencing systems. Some networking modifications are suggested to increase the channel capacity and robustness from the conference server to multiple clients. On the other hand, for the improvement of the uplink quality, we suggest a new spatial error concealment method, where a backup device captures and sends the audio signals to the server together with the primary device. In the server the lost frames from the primary channel are estimated based on the backup signal. Several methods for estimating the primary signal based on the backup signal are studied. The results of the methods are evaluated by a psychoacoustic error metric based on Zwicker’s loudness model. An informal subjective test is also performed to compare the results of these methods in order to chose one for implementing on the real- time conferencing setup. Both objective and subjective tests show consistent results and confirm that usage of spatial error concealment improves significantly the audio quality in the primary signal

Euclidean Distance Matrices:Properties, Algorithms and Applications

Euclidean distance matrices (EDMs) are central players in many diverse fields including psychometrics, NMR spectroscopy, machine learning and sensor networks. However, they are not often exploited in signal processing. In this thesis, we analyze attributes of EDMs and derive new key properties of them. These analyses allow us to propose algorithms to approximate EDMs and provide analytic bounds on the performance of our methods. We use these techniques to suggest new solutions for several practical problems in signal processing. Together with these properties, algorithms and applications, EDMs can thus be considered as a fundamental toolbox to be used in signal processing. In more detail, we start by introducing the structure and properties of EDMs. In particular, we focus on their rank property; the rank of an EDM is at most the dimension of the set of points generating it plus 2. Using this property, we introduce the use of low rank matrix completion methods for approximating and completing noisy and partially revealed EDMs. We apply this algorithm to the problem of sensor position calibration in ultrasound tomography devices. By adapting the matrix completion framework, in addition to proposing a self calibration process for these devices, we also provide analytic bounds for the calibration error. We then study the problem of sensor localization using distance information by minimizing a non-linear cost function known as the s-stress function in the multidimensional scaling (MDS) community. We derive key properties of this cost function that can be used to reduce the search domain for finding its global minimum. We provide an efficient, low cost and distributed algorithm for minimizing this cost function for incomplete networks and noisy measurements. In randomized experiments, the proposed method converges to the global minimum of the s-stress in more than 99% of the cases. We also address the open problem of existence of non-global minimizers of the s-stress and reduce this problem to a hypothesis. If the hypothesis is true then the cost function has only global minimizers, otherwise, it has non-global minimizers. Using the rank property of EDMs and the proposed minimization algorithm for approximating them, we address an interesting and practical problem in acoustics. We show that using five microphones and one loudspeaker, we can hear the shape of a room. We reformulate this problem as finding the locations of the image sources of the loudspeaker with respect to the walls. We propose an algorithm to find these positions only using first-order echoes. We prove that the reconstruction of the room is almost surely unique. We further introduce a new algorithm for locating a microphone inside a known room using only one loudspeaker. Our experimental evaluations conducted on the EPFL campus and also in the Lausanne cathedral, confirm the robustness and accuracy of the proposed methods. By integrating further properties of EDMs into the matrix completion framework, we propose a new method for calibrating microphone arrays in a diffuse noise field. We use a specific characterization of diffuse noise fields to relate the coherence of recorded signals by two microphones to their mutual distance. As this model is not reliable for large distances between microphones, we use matrix completion coupled with other properties of EDMs to estimate these distances and calibrate the microphone array. Evaluation of our algorithm using real data measurements demonstrates, for the first time, the possibility of accurately calibrating large ad-hoc microphone arrays in a diffuse noise field. The last part of the thesis addresses a central problem in signal processing; the design of discrete-time filters (equivalently window functions) that are compact both in time and frequency. By properly adapting the definitions of compactness in the continuous time to discrete time, we formulate the search for maximally compact sequences as solving a semi-definite program. We show that the spectra of maximally compact sequences are a special class of Mathieu’s cosine functions. Using the asymptotic behavior of these functions, we provide a tight bound for the time-frequency spread of discrete-time sequences. Our analysis shows that the Heisenberg uncertainty bound on the time-frequency spread of sequences is not tight and the lower bound depends on the frequency spread, unlike in the continuous time case

Ad Hoc Microphone Array Calibration: Euclidean Distance Matrix Completion Algorithm and Theoretical Guarantees

This paper addresses the problem of ad hoc microphone array calibration where only partial information about the distances between microphones is available. We construct a matrix consisting of the pairwise distances and propose to estimate the missing entries based on a novel Euclidean distance matrix completion algorithm by alternative low-rank matrix completion and projection onto the Euclidean distance space. This approach confines the recovered matrix to the EDM cone at each iteration of the matrix completion algorithm. The theoretical guarantees of the calibration performance are obtained considering the random and locally structured missing entries as well as the measurement noise on the known distances. This study elucidates the links between the calibration error and the number of microphones along with the noise level and the ratio of missing distances. Thorough experiments on real data recordings and simulated setups are conducted to demonstrate these theoretical insights. A significant improvement is achieved by the proposed Euclidean distance matrix completion algorithm over the state-of-the-art techniques for ad hoc microphone array calibration.Comment: In Press, available online, August 1, 2014. http://www.sciencedirect.com/science/article/pii/S0165168414003508, Signal Processing, 201

Single-Channel Indoor Microphone Localization

We propose a novel method for single-channel microphone localization inside a known room. Unlike other approaches, we take advantage of the room reverberation, which enables us to use only a single fixed loudspeaker to localize the microphone. Our method uses an echo labeling approach that associates the echoes to the correct walls. Echo labeling leverages the properties of the Euclidean distance matrices formed from the distances between the virtual sources and the microphone. Experiments performed in a real lecture room verify the effectiveness of the proposed localization algorithm

Euclidean Distance Matrix Completion for Ad-hoc Microphone Array Calibration

This paper addresses the application of missing data recovery via matrix completion for audio sensor networks. We propose a method based on Euclidean distance matrix completion for ad-hoc microphone array location calibration. This method can calibrate a full network from partial connectivity informa- tion. The pairwise distances of microphones in close proximity are estimated using the coherence model of the diffuse noise field. The distance matrix of the ad-hoc network is constructed where the distances of the microphones above a threshold are missing. We exploit the low-rank property of the squared distance matrix and apply a matrix completion method to recover the missing entries. In order to constrain the Euclidean space geometry, we propose the additional use of the Cadzow algorithm for matrix completion. The applicability of the proposed method is evaluated on real data recordings where a significant improvement over the state-of-the-art is achieved