Joint processing of visible (RGB) and near-infrared (NIR) images has recently found some appealing applications, which make joint capturing a pair of visible and NIR images an important problem. In this paper, we propose a new method to design color filter ar- rays (CFA) and demosaicing matrices for acquiring NIR and visible images using a single sensor. The proposed method modifies the optimum CFA algorithm proposed in [1] by taking advantage of the NIR/visible correlation in the design process. Simulation results show that by applying the proposed method, the quality of demosaiced NIR and visible images is increased by about 1 dB in peak signal-to-noise ratio over the results of the optimum CFA algorithm. It is also shown that better visual quality can be obtained by using the proposed algorithm.

Spectral factorization is a classical tool in signal processing and communications. It also plays a critical role in X-ray crystallography, in the context of phase retrieval. In this work, we study the problem of sparse spectral factorization, aiming to recover a one-dimensional sparse signal from its autocorrelation. We present a sufficient condi- tion for the recovery to be unique, and propose an iterative algorithm that can obtain the original signal (up to a sign change, time-shift and time-reversal). Numerical simulations verify the effectiveness of the proposed algorithm.

Consider a diffusion field induced by a finite number of localized and instantaneous sources. In this paper, we study the problem of estimating these sources (including their intensities, spatial locations, and activation time) from the spatiotemporal samples taken by a network of spatially distributed sensors. We propose two estimation algorithms, depending on whether the activation time of the sources is known. For the case of known activation
time, we present an annihilating filter based method to estimate the Euclidean distances between the sources and sensors, which can be subsequently used to localize the sources. For the case of a single source but with unknown activation time, we show that the diffusion field at any spatial location is a scaled and shifted version of a common prototype function, and that this function is the unique solution to a particular differential equation. This observation leads to an efficient algorithm that can estimate the unknown parameters of the source by solving
a system of linear equations. For both algorithms proposed in this work, the minimum number of sensors required is d + 1, where d is the spatial dimension of the field. This requirement is independent of the number of active sources.

We study the spatiotemporal sampling of a diffusion field generated by K point sources, aiming to fully reconstruct the unknown initial field distribution from the sample measurements. The sampling operator in our problem can be described by a matrix derived from the diffusion model. We analyze the important properties of the sampling matrices, leading to precise bounds on the spatial and temporal sampling densities under which perfect field reconstruction is feasible. Moreover, our analysis indicates that it is possible to compensate linearly for insufficient spatial sampling densities by oversampling in time. Numerical simulations on initial field reconstruction under different spatiotemporal sampling densities confirm our theoretical results.

We consider the problem of estimating room geometry from the acoustic room impulse response (RIR). Existing approaches addressing this problem exploit the knowledge of multiple RIRs. In contrast, we are interested in reconstructing the room geometry from a single RIR — a 1–D function of time. We discuss the uniqueness of the mapping between the geometry of a planar polygonal room and a single RIR. In addition to this theoretical analysis, we also propose an algorithm that performs the "blindfolded" room estimation. Furthermore, the derived results are used to construct an algorithm for localization in a known room using only a single RIR. Verification of the theoretical developments with numerical simulations is given before concluding the paper.

We propose a novel algorithm for sparse system identification in the frequency domain. Key to our result is the observation that the Fourier transform of the sparse impulse response is a simple sum of complex exponentials, whose parameters can be efficiently determined from only a narrow frequency band. From this perspective, we present a sub-Nyquist sampling scheme, and show that the original continuous-time system can be learned by considering an equivalent low-rate discrete system. The impulse response of that discrete system can then be adaptively obtained by a novel frequency-domain LMS filter, which exploits the parametric structure of the model. Numerical experiments confirm the effectiveness of the proposed scheme for sparse system identification tasks.

We study a multichannel sampling scheme, where different channels observe scaled and shifted versions of a common bandlimited signal. The channel gains and offsets are unknown a priori, and each channel samples at sub-Nyquist rates. This setup appears in many practical signal processing applications, including time-interleaved ADC with timing skews, unsynchronized distributed sampling in sensor networks, and superresolution imaging. In this paper, we propose a new al- gorithm to efficiently estimate the unknown channel gains and offsets. Key to our algorithm is a novel linearization technique, which converts a system of trigonometric polynomial equations of the unknown parameters to an overparameterized linear system. The computation steps of the proposed algorithm boil down to forming a fixed data matrix from the discrete-time Fourier transforms of the observed channel samples and computing the singular value decomposition of that matrix. Numerical simulations verify the effectiveness, efficiency, and robustness of the proposed algorithm in the presence of noise. In the high SNR regime (40 dB and above), the proposed algorithm significantly outperforms a previous method in the literature in terms of estimation accuracy, at the same time being three orders of magnitudes faster.

We study the distributed sampling and centralized reconstruction of two correlated signals, modeled as the input and output of an unknown sparse filtering operation. This is akin to a Slepian-Wolf setup, but in the sampling rather than the lossless compression case. Two different scenarios are considered: In the case of universal reconstruction, we look for a sensing and recovery mechanism that works for all possible signals, whereas in what we call almost sure reconstruction, we allow to have a small set (with measure zero) of unrecoverable signals. We derive achievability bounds on the number of samples needed for both scenarios. Our results show that, only in the almost sure setup can we effectively exploit the signal correlations to achieve effective gains in sampling efficiency. In addition to the above theoretical analysis, we propose an efficient and robust distributed sampling and reconstruction algorithm based on annihilating filters. We evaluate the performance of our method in one synthetic scenario, and two practical applications, including the distributed audio sampling in binaural hearing aids and the efficient estimation of room impulse responses. The numerical results confirm the effectiveness and robustness of the proposed algorithm in both synthetic and practical setups.

Color image demosaicking is a key process in the digital imaging pipeline. In this paper, we study a well-known and influential demosaicking algorithm based on alternating projections (AP), proposed by Gunturk, Altunbasak and Mersereau in 2002. Since its publication, the AP algorithm has been widely cited and compared against in a series of more recent papers in the demosaicking literature. Despite good performances, a limitation of the AP algorithm is its high computational complexity. We provide three main contributions in this paper. First, we present a rigorous analysis of the convergence property of the AP demosaicking algorithm, showing that it is a contraction mapping, with a unique fixed point. Second, we show that this fixed point is in fact the solution to a constrained quadratic minimization problem, thus establishing the optimality of the AP algorithm. Finally, using the tool of polyphase representation, we show how to obtain the results of the AP algorithm in a single step, implemented as linear filtering in the polyphase domain. Replacing the original iterative procedure by the proposed one-step solution leads to substantial computational savings, by about an order of magnitude in our experiments.

We propose a Fourier analytical approach to the problems of alias-free sampling and critical sampling. Central to this approach are two Fourier conditions linking the above sam- pling criteria with the Fourier transform of the indicator function defined on the underlying frequency support. We present several examples to demonstrate the usefulness of the proposed Fourier conditions in the design of critically sampled multidimensional filter banks. In particular, we show that it is impossible to implement any cone-shaped fre- quency partitioning by a nonredundant filter bank, except for the 2-D case.

We consider the task of recovering correlated vectors at a central decoder based on fixed linear measurements ob- tained by distributed sensors. Two different scenarios are considered: In the case of universal reconstruction, we look for a sensing and recovery mechanism that works for all possible signals, whereas in the case of almost sure recon- struction, we allow to have a small set (with measure zero) of unrecoverable signals. We provide achievability bounds on the number of samples needed for both scenarios. The bounds show that only in the almost sure setup can we ef- fectively exploit the signal correlations to achieve effective gains in sampling efficiency. In addition, we propose an efficient and robust distributed sensing and reconstruction algorithm based on annihilating filters.

Digital camera sensors are inherently sensitive to the near- infrared (NIR) part of the light spectrum. In this paper, we propose a general design for color filter arrays that allow the joint capture of visible/NIR images using a single sensor. We pose the CFA design as a novel spatial domain optimization problem, and provide an efficient iterative procedure that finds (locally) optimal solutions. Numerical experiments confirm the effectiveness of the proposed CFA design, which can simultane- ously capture high quality visible and NIR image pairs.

Most digital cameras employ a spatial subsampling process, implemented as a color filter array (CFA), to capture color images. The choice of CFA patterns has a great impact on the performance of subsequent reconstruction (demosaicking) algorithms. In this work, we propose a quantitative theory for optimal CFA design. We view the CFA sampling process as an encoding (low-dimensional approximation) operation and, correspondingly, demosaicking as the best decoding (reconstruction) operation. Finding the optimal CFA is thus equivalent to finding the optimal approximation scheme for the original signals with minimum information loss. We present several quantitative conditions for optimal CFA design, and propose an efficient computational procedure to search for the best CFAs that satisfy these conditions. Numerical experiments show that the optimal CFA patterns designed from the proposed procedure can effectively retain the information of the original full-color images. In particular, with the designed CFA patterns, high quality demosaicking can be achieved by using simple and efficient linear filtering operations in the polyphase domain. The visual qualities of the reconstructed images are competitive to those obtained by the state-of-the-art adaptive demosaicking algorithms based on the Bayer pattern.

We propose a Fourier analytical condition linking alias-free sampling with the Fourier transform of the indicator function defined on the given frequency support. Our discussions center around how to develop practical computation algorithms based on the proposed analytical condition. We address several issues along this line, including the derivation of simple closed-form expressions for the Fourier transforms of the indicator functions defined on arbitrary polygonal and polyhedral domains; a complete and nonredundant enumeration of all quantized sampling lattices via the Hermite normal forms of integer matrices; and a quantitative analysis of the approximation of the original infinite Fourier condition by using finite computations. Combining these results, we propose a computational testing procedure that can efficiently search for the optimal alias-free sampling lattices for a given polygonal or polyhedral shaped frequency domain. Several examples are presented to show the potential of the proposed algorithm in multidimensional filter bank design, as well as in applications involving the design of efficient sampling patterns for multidimensional band-limited signals.

We study the spatio-temporal sampling of a diffusion field driven by K unknown instantaneous source distributions. Exploiting the spatio-temporal correlation offered by the diffusion model, we show that it is possible to compensate for insufficient spatial sampling densities (i.e. sub-Nyquist sampling) by increasing the temporal sampling rate, as long as their product remains roughly a constant. Combining a distributed sparse sampling scheme and an adaptive feedback mechanism, the proposed sampling algorithm can accurately and efficiently estimate the unknown sources and reconstruct the field. The total number of samples to be transmitted through the network is roughly equal to the number of degrees of freedom of the field, plus some additional costs for in-network averaging.

We study the spatial-temporal sampling of a linear diffusion field, and show that it is possible to compensate for insufficient spatial sampling densities by oversampling in time. Our work is motivated by the following issue often encountered in sensor network sampling, namely increasing the temporal sampling density is often easier and less expensive than increasing the spatial sampling density of the network. For the case of sampling a diffusion field, we show that, to achieve trade-off between spatial and temporal sampling, the spatial arrangement of the sensors must satisfy certain conditions. We provide in this paper the precise relationships between the achievable reduction of spatial sampling density, the required temporal oversampling rate, the spatial arrangement of the sensors, and the bound for the condition numbers of the resulting sampling and reconstruction procedures.

We consider the task of recovering correlated vectors at a central decoder based on fixed linear measurements obtained by distributed sensors. A general formulation of the problem is proposed, under both a universal and an almost sure reconstruction requirement. We then study a specific correlation model which involves a filter that is sparse in the time domain. While this sparsity assumption does not allow reducing the description cost in the universal case, we show that large gains can be achieved in the almost sure scenario by means of a novel distributed scheme based on annihilating filters. The robustness of the proposed method is also investigated.

Color image demosaicking is a key process in the digital imaging pipeline. In this paper, we present a rigorous treatment of a classical demosaicking algorithm based on alternating projections (AP). Since its publication, the AP algorithm has been widely cited and served as a benchmark in a flurry of papers in the demosaicking literature. Despite its impressive performances, a relative weakness of the AP algorithm is its high computational complexity. In our work, we provide a rigorous analysis of the convergence of the AP algorithm based on the concept of contraction mapping. Furthermore, we propose an efficient noniterative implementation of the AP algorithm in the polyphase domain. Numerical experiments show that the proposed noniterative implementation achieves the same results obtained by the original AP algorithm at convergence, but is about an order of magnitude faster than the latter.