Group Testing with Probabilistic Tests: Theory, Design and Application

Identification of defective members of large populations has been widely studied in the statistics community under the name of group testing. It involves grouping subsets of items into different pools and detecting defective members based on the set of test results obtained for each pool. In a classical noiseless group testing setup, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in the sense that the existence of a defective member in a pool results in the test outcome of that pool to be positive. However, this may not be always a valid assumption in some cases of interest. In particular, we consider the case where the defective items in a pool can become independently inactive with a certain probability. Hence, one may obtain a negative test result in a pool despite containing some defective items. As a result, any sampling and reconstruction method should be able to cope with two different types of uncertainty, i.e., the unknown set of defective items and the partially unknown, probabilistic testing procedure. In this work, motivated by the application of detecting infected people in viral epidemics, we design non-adaptive sampling procedures that allow successful identification of the defective items through a set of probabilistic tests. Our design requires only a small number of tests to single out the defective items. In particular, for a population of size $N$ and at most $K$ defective items with activation probability $p$, our results show that $M = O(K^2\log{(N/K)}/p^3)$ tests is sufficient if the sampling procedure should work for all possible sets of defective items, while $M = O(K\log{(N)}/p^3)$ tests is enough to be successful for any single set of defective items. Moreover, we show that the defective members can be recovered using a simple reconstruction algorithm with complexity of $O(MN)$.Comment: Full version of the conference paper "Compressed Sensing with Probabilistic Measurements: A Group Testing Solution" appearing in proceedings of the 47th Annual Allerton Conference on Communication, Control, and Computing, 2009 (arXiv:0909.3508). To appear in IEEE Transactions on Information Theor

Estimation of Sparse MIMO Channels with Common Support

We consider the problem of estimating sparse communication channels in the MIMO context. In small to medium bandwidth communications, as in the current standards for OFDM and CDMA communication systems (with bandwidth up to 20 MHz), such channels are individually sparse and at the same time share a common support set. Since the underlying physical channels are inherently continuous-time, we propose a parametric sparse estimation technique based on finite rate of innovation (FRI) principles. Parametric estimation is especially relevant to MIMO communications as it allows for a robust estimation and concise description of the channels. The core of the algorithm is a generalization of conventional spectral estimation methods to multiple input signals with common support. We show the application of our technique for channel estimation in OFDM (uniformly/contiguous DFT pilots) and CDMA downlink (Walsh-Hadamard coded schemes). In the presence of additive white Gaussian noise, theoretical lower bounds on the estimation of SCS channel parameters in Rayleigh fading conditions are derived. Finally, an analytical spatial channel model is derived, and simulations on this model in the OFDM setting show the symbol error rate (SER) is reduced by a factor 2 (0 dB of SNR) to 5 (high SNR) compared to standard non-parametric methods - e.g. lowpass interpolation.Comment: 12 pages / 7 figures. Submitted to IEEE Transactions on Communicatio

Efficient and Stable Acoustic Tomography Using Sparse Reconstruction Methods

We study an acoustic tomography problem and propose a new inversion technique based on sparsity. Acoustic tomography observes the parameters of the medium that influence the speed of sound propagation. In the human body, the parameters that mostly influence the sound speed are temperature and density, in the ocean - temperature and current, in the atmosphere - temperature and wind. In this study, we focus on estimating temperature in the atmosphere using the information on the average sound speed along the propagation path. The latter is practically obtained from travel time measurements. We propose a reconstruction algorithm that exploits the concept of sparsity. Namely, the temperature is assumed to be a linear combination of some functions (e.g. bases or set of different bases) where many of the coefficients are known to be zero. The goal is to find the non-zero coefficients. To this end, we apply an algorithm based on linear programming that under some constrains finds the solution with minimum l0 norm. This is actually equivalent to the fact that many of the unknown coefficients are zeros. Finally, we perform numerical simulations to assess the effectiveness of our approach. The simulation results confirm the applicability of the method and demonstrate high reconstruction quality and robustness to noise

Annihilating filter-based decoding in the compressed sensing framework

Recent results in compressed sensing or compressive sampling suggest that a relatively small set of measurements taken as the inner product with universal random measurement vectors can well represent a source that is sparse in some fixed basis. By adapting a deterministic, non-universal and structured sensing device, this paper presents results on using the annihilating filter to decode the information taken in this new compressed sensing environment. The information is the minimum amount of nonadaptive knowledge that makes it possible to go back to the original object. We will show that for a k-sparse signal of dimension n, the proposed decoder needs 2k measurements and its complexity is of O(k(2)) whereas for the decoding based on the l(1) minimization, the number of measurements needs to be of O(k log(n)) and the complexity is of O(n(3)). In the case of noisy measurements, we first denoise the signal using an iterative algorithm that finds the closest rank k and Toeplitz matrix to the measurements matrix (in Frobenius norm) before applying the annihilating filter method. Furthermore, for a k-sparse vector with known equal coefficients, we propose art algebraic decoder which needs only k measurements for the signal reconstruction. Finally, we provide simulation results that demonstrate the performance of our algorithm

Compressed Sensing with Probabilistic Measurements: A Group Testing Solution

Detection of defective members of large populations has been widely studied in the statistics community under the name "group testing", a problem which dates back to World War II when it was suggested for syphilis screening. There the main interest is to identify a small number of infected people among a large population using collective samples. In viral epidemics, one way to acquire collective samples is by sending agents inside the population. While in classical group testing, it is assumed that the sampling procedure is fully known to the reconstruction algorithm, in this work we assume that the decoder possesses only partial knowledge about the sampling process. This assumption is justified by observing the fact that in a viral sickness, there is a chance that an agent remains healthy despite having contact with an infected person. Therefore, the reconstruction method has to cope with two different types of uncertainty; namely, identification of the infected population and the partially unknown sampling procedure. In this work, by using a natural probabilistic model for "viral infections", we design non-adaptive sampling procedures that allow successful identification of the infected population with overwhelming probability 1-o(1). We propose both probabilistic and explicit design procedures that require a "small" number of agents to single out the infected individuals. More precisely, for a contamination probability p, the number of agents required by the probabilistic and explicit designs for identification of up to k infected members is bounded by m = O(k^2 (log n)/p^2) and m = O(k^2 (log n)^2 /p^2), respectively. In both cases, a simple decoder is able to successfully identify the infected population in time O(mn).Comment: In Proceedings of the Forty-Seventh Annual Allerton Conference on Communication, Control, and Computin

Methods and apparatus for estimating a sparse channel

Embodiments include a method for sending a selected number of pilots (20) to a sparse channel having a channel impulse response limited in time comprising sending the selected number of the pilots (20). The pilots (20) are equally spaced in the frequency domain the number is selected based on the finite rate of innovation of the channel impulse response. Once received the pilots (20), such a channel is estimated by: low-pass filtering (100) the received pilots, sampling (200) the filtered pilots with a rate below the Nyquist rate of the pilots, applying a FFT (300) on the sampled pilots, verifying (500) the level of noise of the transformed pilots, if the level of noise is below to a determined threshold, applying an annihilating filter method (600) to the transformed pilots, and dividing the temporal parameters by the distance (D) between two consecutive pilots

Fast and robust estimation of jointly sparse channels

A device and method for estimating multipath jointly sparse channels. The method comprises receiving a number K of signal components by a number P of receiving antennas, where P≧2. The method further comprises estimating the sparsity condition of the multipath jointly sparse channels. The method further comprises, if the sparsity condition is not satisfied, estimating the channels by using a non-sparse technique. The method further comprises, if the sparsity condition is satisfied, estimating the channels by using a sparse technique

Sampling of Sparse Channels with Common Support

The present paper proposes and studies an algorithm to estimate chan- nels with a sparse common support (SCS). It is a generalization of the classical sampling of signals with Finite Rate of Innovation (FRI) and thus called SCS-FRI. It is applicable to OFDM and Walsh- Hadamard coded (CDMA downlink) communications since SCS-FRI is shown to work not only on contiguous DFT pilots but also uniformly scattered ones. The support estimation performances compare favorably to theoretical lower-bounds, and importantly this translates into a sub- stantial equalization gain at the receiver compared to the widely used spectrum lowpass interpolation method

Time of flight estimation method using beamforming for acoustic tomography

It is disclosed an acoustic tomography method to improve the time of flight estimation, said method comprising the steps of: sequentially triggering a set of N transmitters so as to generate a sequence of N acoustic waves through a volume being scanned; receiving each of said acoustic waves after transmission through said volume with a set of M receivers, which are called received signals; delaying by varying delays the N different said received signals that each receiver receives from the N different transmitters, and adding them together to form a new received signal, which is called transmit-beamformed signal for that receiver; delaying by varying delays the M different said transmit-beamformed signals for each receiver and adding them together at each receiver to form a new signal, which we call transmit-receive-beamformed signal