Supervised nonlinear spectral unmixing using a post-nonlinear mixing model for hyperspectral imagery

This paper presents a nonlinear mixing model for hyperspectral image unmixing. The proposed model assumes that the pixel reflectances are nonlinear functions of pure spectral components contaminated by an additive white Gaussian noise. These nonlinear functions are approximated using polynomial functions leading to a polynomial postnonlinear mixing model. A Bayesian algorithm and optimization methods are proposed to estimate the parameters involved in the model. The performance of the unmixing strategies is evaluated by simulations conducted on synthetic and real data

Group-Lasso on Splines for Spectrum Cartography

The unceasing demand for continuous situational awareness calls for innovative and large-scale signal processing algorithms, complemented by collaborative and adaptive sensing platforms to accomplish the objectives of layered sensing and control. Towards this goal, the present paper develops a spline-based approach to field estimation, which relies on a basis expansion model of the field of interest. The model entails known bases, weighted by generic functions estimated from the field's noisy samples. A novel field estimator is developed based on a regularized variational least-squares (LS) criterion that yields finitely-parameterized (function) estimates spanned by thin-plate splines. Robustness considerations motivate well the adoption of an overcomplete set of (possibly overlapping) basis functions, while a sparsifying regularizer augmenting the LS cost endows the estimator with the ability to select a few of these bases that ``better'' explain the data. This parsimonious field representation becomes possible, because the sparsity-aware spline-based method of this paper induces a group-Lasso estimator for the coefficients of the thin-plate spline expansions per basis. A distributed algorithm is also developed to obtain the group-Lasso estimator using a network of wireless sensors, or, using multiple processors to balance the load of a single computational unit. The novel spline-based approach is motivated by a spectrum cartography application, in which a set of sensing cognitive radios collaborate to estimate the distribution of RF power in space and frequency. Simulated tests corroborate that the estimated power spectrum density atlas yields the desired RF state awareness, since the maps reveal spatial locations where idle frequency bands can be reused for transmission, even when fading and shadowing effects are pronounced.Comment: Submitted to IEEE Transactions on Signal Processin

Dynamic Compressive Sensing of Time-Varying Signals via Approximate Message Passing

In this work the dynamic compressive sensing (CS) problem of recovering sparse, correlated, time-varying signals from sub-Nyquist, non-adaptive, linear measurements is explored from a Bayesian perspective. While there has been a handful of previously proposed Bayesian dynamic CS algorithms in the literature, the ability to perform inference on high-dimensional problems in a computationally efficient manner remains elusive. In response, we propose a probabilistic dynamic CS signal model that captures both amplitude and support correlation structure, and describe an approximate message passing algorithm that performs soft signal estimation and support detection with a computational complexity that is linear in all problem dimensions. The algorithm, DCS-AMP, can perform either causal filtering or non-causal smoothing, and is capable of learning model parameters adaptively from the data through an expectation-maximization learning procedure. We provide numerical evidence that DCS-AMP performs within 3 dB of oracle bounds on synthetic data under a variety of operating conditions. We further describe the result of applying DCS-AMP to two real dynamic CS datasets, as well as a frequency estimation task, to bolster our claim that DCS-AMP is capable of offering state-of-the-art performance and speed on real-world high-dimensional problems.Comment: 32 pages, 7 figure

Block-Sparse Recovery via Convex Optimization

Given a dictionary that consists of multiple blocks and a signal that lives in the range space of only a few blocks, we study the problem of finding a block-sparse representation of the signal, i.e., a representation that uses the minimum number of blocks. Motivated by signal/image processing and computer vision applications, such as face recognition, we consider the block-sparse recovery problem in the case where the number of atoms in each block is arbitrary, possibly much larger than the dimension of the underlying subspace. To find a block-sparse representation of a signal, we propose two classes of non-convex optimization programs, which aim to minimize the number of nonzero coefficient blocks and the number of nonzero reconstructed vectors from the blocks, respectively. Since both classes of problems are NP-hard, we propose convex relaxations and derive conditions under which each class of the convex programs is equivalent to the original non-convex formulation. Our conditions depend on the notions of mutual and cumulative subspace coherence of a dictionary, which are natural generalizations of existing notions of mutual and cumulative coherence. We evaluate the performance of the proposed convex programs through simulations as well as real experiments on face recognition. We show that treating the face recognition problem as a block-sparse recovery problem improves the state-of-the-art results by 10% with only 25% of the training data.Comment: IEEE Transactions on Signal Processin

Incremental Relaying for the Gaussian Interference Channel with a Degraded Broadcasting Relay

This paper studies incremental relay strategies for a two-user Gaussian relay-interference channel with an in-band-reception and out-of-band-transmission relay, where the link between the relay and the two receivers is modelled as a degraded broadcast channel. It is shown that generalized hash-and-forward (GHF) can achieve the capacity region of this channel to within a constant number of bits in a certain weak relay regime, where the transmitter-to-relay link gains are not unboundedly stronger than the interference links between the transmitters and the receivers. The GHF relaying strategy is ideally suited for the broadcasting relay because it can be implemented in an incremental fashion, i.e., the relay message to one receiver is a degraded version of the message to the other receiver. A generalized-degree-of-freedom (GDoF) analysis in the high signal-to-noise ratio (SNR) regime reveals that in the symmetric channel setting, each common relay bit can improve the sum rate roughly by either one bit or two bits asymptotically depending on the operating regime, and the rate gain can be interpreted as coming solely from the improvement of the common message rates, or alternatively in the very weak interference regime as solely coming from the rate improvement of the private messages. Further, this paper studies an asymmetric case in which the relay has only a single single link to one of the destinations. It is shown that with only one relay-destination link, the approximate capacity region can be established for a larger regime of channel parameters. Further, from a GDoF point of view, the sum-capacity gain due to the relay can now be thought as coming from either signal relaying only, or interference forwarding only.Comment: To appear in IEEE Trans. on Inf. Theor

Zigzag Codes: MDS Array Codes with Optimal Rebuilding

MDS array codes are widely used in storage systems to protect data against erasures. We address the \emph{rebuilding ratio} problem, namely, in the case of erasures, what is the fraction of the remaining information that needs to be accessed in order to rebuild \emph{exactly} the lost information? It is clear that when the number of erasures equals the maximum number of erasures that an MDS code can correct then the rebuilding ratio is 1 (access all the remaining information). However, the interesting and more practical case is when the number of erasures is smaller than the erasure correcting capability of the code. For example, consider an MDS code that can correct two erasures: What is the smallest amount of information that one needs to access in order to correct a single erasure? Previous work showed that the rebuilding ratio is bounded between 1/2 and 3/4, however, the exact value was left as an open problem. In this paper, we solve this open problem and prove that for the case of a single erasure with a 2-erasure correcting code, the rebuilding ratio is 1/2. In general, we construct a new family of $r$-erasure correcting MDS array codes that has optimal rebuilding ratio of $\frac{e}{r}$ in the case of $e$ erasures, $1 \le e \le r$. Our array codes have efficient encoding and decoding algorithms (for the case $r=2$ they use a finite field of size 3) and an optimal update property.Comment: 23 pages, 5 figures, submitted to IEEE transactions on information theor

Geographic Gossip: Efficient Averaging for Sensor Networks

Gossip algorithms for distributed computation are attractive due to their simplicity, distributed nature, and robustness in noisy and uncertain environments. However, using standard gossip algorithms can lead to a significant waste in energy by repeatedly recirculating redundant information. For realistic sensor network model topologies like grids and random geometric graphs, the inefficiency of gossip schemes is related to the slow mixing times of random walks on the communication graph. We propose and analyze an alternative gossiping scheme that exploits geographic information. By utilizing geographic routing combined with a simple resampling method, we demonstrate substantial gains over previously proposed gossip protocols. For regular graphs such as the ring or grid, our algorithm improves standard gossip by factors of $n$ and $\sqrt{n}$ respectively. For the more challenging case of random geometric graphs, our algorithm computes the true average to accuracy $\epsilon$ using $O(\frac{n^{1.5}}{\sqrt{\log n}} \log \epsilon^{-1})$ radio transmissions, which yields a $\sqrt{\frac{n}{\log n}}$ factor improvement over standard gossip algorithms. We illustrate these theoretical results with experimental comparisons between our algorithm and standard methods as applied to various classes of random fields.Comment: To appear, IEEE Transactions on Signal Processin

Power Allocation Based on SEP Minimization in Two-Hop Decode-and-Forward Relay Networks

The problem of optimal power allocation among the relays in a two-hop decode-and-forward cooperative relay network with independent Rayleigh fading channels is considered. It is assumed that only the relays that decode the source message correctly contribute in data transmission. Moreover, only the knowledge of statistical channel state information is available. A new simple closed-form expression for the average symbol error probability is derived. Based on this expression, a new power allocation method that minimizes the average symbol error probability and takes into account the constraints on the total average power of all the relay nodes and maximum instant power of each relay node is developed. The corresponding optimization problem is shown to be a convex problem that can be solved using interior point methods. However, an approximate closed-form solution is obtained and shown to be practically more appealing due to significant complexity reduction. The accuracy of the approximation is discussed. Moreover, the so obtained closed-form solution gives additional insights into the optimal power allocation problem. Simulation results confirm the improved performance of the proposed power allocation scheme as compared to other schemes.Comment: 27 pages, 5 figures, submitted to the IEEE Trans. Signal Processing in Feb. 201

Compact focusing grating couplers for silicon-on-insulator integrated circuits

Abstract—We report experimental results on compact and broadband focusing grating couplers, both in silicon-on-insulator (SOI) and gold on SOI. An eight-fold length reduction of the coupling structure from fiber to photonic wire in SOI, as com-pared to a linear grating and adiabatic taper, is obtained, without performance penalty. A proof of principle is given for a focusing grating coupler in gold on SOI, with 20 % fiber-to-focus efficiency. Index Terms—Focusing grating coupler, gold, integrated optics, metal, nanophotonic waveguides, silicon-on-insulator (SOI). I