William Mantzel

Random channel coding and blind deconvolution

Blind deconvolution arises naturally when dealing with finite multipath interference on a signal. In this paper we present a new method to protect the signals from the effects of sparse multipath channels — we modulate/encode the signal using random waveforms before transmission and estimate the channel and signal from the observations, without any prior knowledge of the channel other than that it is sparse. The problem can be articulated as follows. The original message x is encoded with an overdetermined m × n (m > n) matrix A whose entries are randomly chosen; the encoded message is given by Ax. The received signal is the convolution of the encoded message with h, the S-sparse impulse response of the channel. We explore three different schemes to recover the message x and the channel h simultaneously. The first scheme recasts the problem as a block l1 optimization program. The second scheme imposes a rank-1 structure on the estimated signal. The third scheme uses nuclear norm as a proxy for rank, to recover the x and h. The simulation results are presented to demonstrate the efficiency of the random coding and proposed recovery schemes.

Channel protection: Random coding meets sparse channels

Multipath interference is an ubiquitous phenomenon in modern communication systems. The conventional way to compensate for this effect is to equalize the channel by estimating its impulse response by transmitting a set of training symbols. The primary drawback to this type of approach is that it can be unreliable if the channel is changing rapidly. In this paper, we show that randomly encoding the signal can protect it against channel uncertainty when the channel is sparse. Before transmission, the signal is mapped into a slightly longer codeword using a random matrix. From the received signal, we are able to simultaneously estimate the channel and recover the transmitted signal. We discuss two schemes for the recovery. Both of them exploit the sparsity of the underlying channel. We show that if the channel impulse response is sufficiently sparse, the transmitted signal can be recovered reliably.

Round-robin multiple-source localization

This paper introduces a round-robin approach for multi-source localization based on matched-field processing. Each new source location is estimated from the ambiguity function after nulling from the data vector the current source location estimates using a robust projection matrix. This projection matrix effectively minimizes mean-square energy near current source location estimates subject to a rank constraint that prevents excessive interference with sources outside of these neighborhoods. Numerical simulations are presented for multiple sources transmitting through a fixed (and presumed known) generic Pekeris ocean waveguide in the single-frequency and broadband-coherent cases that illustrate the performance of the proposed approach which compares favorably against other previously published approaches. Furthermore, the efficacy with which randomized back-propagations may also be incorporated for computational advantage is also presented.

Randomized group testing for acoustic source localization

Undersea localization requires a computationally expensive partial differential equation simulation to test each candidate hypothesis location via matched filter. We propose a method of batch testing that effectively yields a test sequence output of random combinations of location-specific matched filter correlations, such that the computational run time varies with the number of tests instead of the number of locations. We show that by finding the most likely location that could have accounted for these batch test outputs, we are able to perform almost as well as if we had computed each locations matched filter. In particular, we show that we can reliably resolve the targets location up to the resolution of incoherence using only logarithmically many measurements when the number of candidate locations is less than the dimension of the matched filter. In this way, our random mask pattern not only performs substantially the same as cleverly designed deterministic masks in classical batch testing scenarios, but also naturally extends to other scenarios when the design of such deterministic masks may be less obvious.

Poutine: A correlation estimator for ergodic stationary signals

In this work, we present POUTINE, a novel estimator of the auto-correlation function (or more generally, the cross-correlation function) of ergodic stationary signals, an important task in a variety of applications. This estimator sparsely and non-adaptively samples the process via Bernoulli selection, generalizing the classical estimator in a natural way, and offering significant sampling reductions while sacrificing a modest degree of accuracy. Both the mean and variance of our estimator are explicitly analyzed, and in particular, we show that POUTINE gives an unbiased estimate of the classical estimator, which in turn gives an unbiased estimate of the underlying second-order statistics of interest. Furthermore, we show that POUTINE is a consistent estimator with variance approaching zero asymptotically. We demonstrate favorable performance of this approach for a simple stochastic process.

Compressing the computational burden of matched‐field processing for sound source localization.

Sound source localization in shallow water environment is done using matched field processing (MFP), also referred to as time‐reversal imaging or backprojection method. Standard MFP is usually implemented by matching (using a cross‐correlation operation) the received acoustic data from the sought source with modeled data (or replicas) for point source located at multiple test locations over the a priori search area. Consequently, a direct implementation of MFP (i.e., brute force search) over a large search area can be highly computationally demanding especially when attempting to locate repeatedly several stationary or moving unknown sources in a complex environment. We formulated instead a compressive MFP approach allowing for significant computation time savings with a computational cost that scales with the logarithm of the number of independent‐or uncorrelated‐ test locations. This approach leverages the key concepts behind group testing and compressed sensing by computing instead the expected value o...

Functional vanishing point estimation via a filtered-Radon operator

When available, vanishing points in a scene are a key factor in effectively recovering absolute camera orientation, thus simplifying the structure-from-motion problem. We present a novel method for estimating vanishing points without explicitly detecting line features. This approach first maps images into line-space with a filtered- Radon operator, allowing subtle line textures to contribute, and improving the angular resolution of broken or occluded segments of the same line. Then, we use a robust coarse-to-fine method to jointly estimate the three vanishing points. We evaluate our method on video sequences, demonstrating robustness to clutter lines as well as the ability to effectively utilize subtle edge-texture information.

Random encoding for robust recovery and channel identification.

By encoding a signal in a randomly chosen subspace before transmission, this transmitted signal proves robust to sparse errors in the channel. In this talk, we will further demonstrate the robustness of such signals to unknown convolutive channels. In particular, we show that if the channel is not especially resonant at any given frequency (e.g., the channel is sparse or random), we are able to simultaneously recover both the transmitted signal and the channel characteristics by solving a rank minimization in X, the rank‐1 outer product of the two unknown vectors, subject to the receiver observations. We demonstrate favorable simulated performance for sparse and random channels. Using a parallel approach to classical compressed sensing theory, we prove a restricted isometry property about our linear operator that guarantees a high probability of recovery when the number of observations is at least linear with the dimension of the message and the length of the channel.