A. Agaskar and Y. M. Lu, “A Spectral Graph Uncertainty Principle,” IEEE Transactions on Information Theory, vol. 59, no. 7, pp. 4338-4356, 2013.Abstract

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral ``spreads'' are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within $\varepsilon$ by a fast approximation algorithm requiring $\mathcal{O}(\varepsilon^{-1/2})$ typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erdos-Renyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.

C. Hu and Y. M. Lu, “Adaptive time-sequential binary sensing for high dynamic range imaging,” in Proc. SPIE Conference on Advanced Photon Counting Techniques VI, Baltimore, MD, 2012.Abstract

We present a novel image sensor for high dynamic range imaging. The sensor performs an adaptive one-bit quantization at each pixel, with the pixel output switched from 0 to 1 only if the number of photons reaching that pixel is greater than or equal to a quantization threshold. With an oracle knowledge of the incident light intensity, one can pick an optimal threshold (for that light intensity) and the corresponding Fisher information contained in the output sequence follows closely that of an ideal unquantized sensor over a wide range of intensity values. This observation suggests the potential gains one may achieve by adaptively updating the quantization thresholds. As the main contribution of this work, we propose a time-sequential threshold-updating rule that asymptotically approaches the performance of the oracle scheme. With every threshold mapped to a number of ordered states, the dynamics of the proposed scheme can be modeled as a parametric Markov chain. We show that the frequencies of different thresholds converge to a steady-state distribution that is concentrated around
the optimal choice. Moreover, numerical experiments show that the theoretical performance measures (Fisher information and Cramer-Rao bounds) can be achieved by a maximum likelihood estimator, which is guaranteed to find globally optimal solution due to the concavity of the log-likelihood functions. Compared with conventional image sensors and the strategy that utilizes a constant single-photon threshold considered in previous work, the
proposed scheme attains orders of magnitude improvement in terms of sensor dynamic ranges.

C. Hu, L. Cheng, and Y. M. Lu, “Graph-Based regularization for color image demosaicking,” in Proc. IEEE International Conference on Image Processing, Orlando, FL, 2012.Abstract

We present a novel regularization framework for demosaicking by viewing images as smooth signals defined on weighted graphs. The restoration problem is formulated as a minimization of variation of these graph-domain signals. As an initial step, we build a weight matrix which measures the similarity between every pair of pixels, from an estimate of the full color image. Subsequently, a two-stage optimization is carried out: first, we assume that the graph Laplacian is signal dependent and solve a non-quadratic problem by gradient descent; then, we pose a variational problem on graphs with a fixed Laplacian, subject to the constraint of consistency given by available samples in each color channel. Performance evaluation shows that our approach can improve existing demosaicking methods both quantitively and visually, by reducing color artifacts.

Z. Sadeghipoor, Y. M. Lu, and S. Süsstrunk, “Optimal spectral sensitivity functions for single sensor color imaging,” in Proc. SPIE Conference on Digital Photography VIII, Burlingame, 2012.Abstract

A cost-effective and convenient approach for color imaging is to use a single sensor and mount a color filter array
(CFA) in front of it, such that at each spatial position the scene information in one color channel is captured. To
estimate the missing colors at each pixel, a demosaicing algorithm is applied to the CFA samples. Besides the
filter arrangement and the demosaicing method, the spectral sensitivity functions of the CFA filters considerably
affect the quality of the demosaiced image. In this paper, we extend the algorithm presented by Lu and Vetterli,
originally proposed for designing the optimum CFA, to compute the optimum spectral sensitivities. The proposed
algorithm solves a constrained optimization problem to find optimum spectral sensitivities and the corresponding
linear demosaicing method. An important constraint for this problem is the smoothness of spectral sensitivities,
which is imposed by modeling these functions as a linear combination of several smooth kernels. Simulation
results verify the effectiveness of the proposed algorithm in finding optimal spectral sensitivity functions, which
outperform measured camera sensitivity functions.

A. Agaskar and Y. M. Lu, “Uncertainty principles for signals defined on graphs: Bounds and characterizations,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, Kyoto, 2012.Abstract

The classical uncertainty principle provides a fundamental tradeoff in the localization of a signal in the time and frequency domains. In this paper we describe a similar tradeoff for signals defined on graphs. We describe the notions of "spread" in the graph and spectral domains, using the eigenvectors of the graph Laplacian as a surrogate Fourier basis. We then describe how to find signals that, among all signals with the same spectral spread, have the smallest graph spread about a given vertex. For every possible spectral spread, the desired signal is the solution to an eigenvalue problem. Since localization in graph and spectral domains is a desirable property of the elements of wavelet frames on graphs, we compare the performance of some existing wavelet transforms to the obtained bound.

See, also, the extended journal version: A. Agaskar and Y. M. Lu, A Spectral Graph Uncertainty Principle, IEEE Transactions on Information Theory, to appear.
Y. Xiong and Y. M. Lu, “Blind estimation and low-rate sampling of sparse MIMO systems with common support,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, Kyoto, 2012.Abstract

We present a blind estimation algorithm for multi-input and multi-output (MIMO) systems with sparse common support. Key to the proposed algorithm is a matrix generalization of the classical annihilating filter technique, which allows us to estimate the nonlinear parameters of the channels through an efficient and noniterative procedure. An attractive property of the proposed algorithm is that it only needs the sensor measurements at a narrow frequency band. By exploiting this feature, we can derive efficient sub-Nyquist sampling schemes which significantly reduce the number of samples that need to be retained at each sensor. Numerical simulations verify the accuracy of the proposed estimation algorithm and its robustness in the presence of noise.

M. N. Do and Y. M. Lu, “Multidimensional Filter Banks and Multiscale Geometric Representations,” Foundation and Trends in Signal Processing, vol. 5, no. 3, pp. 157-264, 2012.Abstract

Thanks to the explosive growth of sensing devices and capabilities, multidimensional (MD) signals — such as images, videos, multispectral images, light fields, and biomedical data volumes — have become ubiquitous.
Multidimensional filter banks and the associated constructions provide a unified framework and an efficient computational tool in the formation, representation, and processing of these multidimensional data sets. In this survey we aim to provide a systematic development of the theory and constructions of multidimensional filter banks. We thoroughly review several tools that have been shown to be particularly effective in the design and analysis of multidimensional filter banks, including sampling lattices, multidimensional bases and frames, polyphase representations, Gröbner bases, mapping methods, frequency domain constructions, ladder structures and lifting schemes. We then focus on the construction of filter banks and signal representations that can capture directional and geometric features, which are unique and key properties of many multidimensional signals. Next, we study the connection between iterated multidimensional filter banks in the discrete domain and the associated multiscale signal representations in the continuous domain through a directional multiresolution analysis framework. Finally, we show several examples to demonstrate the power of multidimensional filter banks and geometric signal representations in applications.

F. Yang, Y. M. Lu, L. Sbaiz, and M. Vetterli, “Bits from Photons: Oversampled Image Acquisition Using Binary Poisson Statistics,” IEEE Transactions on Image Processing, vol. 21, no. 4, pp. 1421-1436, 2012. Extended Version with Complete ProofAbstract

We study a new image sensor that is reminiscent of traditional photographic film. Each pixel in the sensor has a binary response, giving only a one-bit quantized measurement of the local light intensity. To analyze its performance, we formulate the oversampled binary sensing scheme as a parameter estimation problem based on quantized Poisson statistics. We show that, with a single-photon quantization threshold and large oversampling factors, the Cramér-Rao lower bound (CRLB) of the estimation variance approaches that of an ideal unquantized sensor, that is, as if there were no quantization in the sensor measurements. Furthermore, the CRLB is shown to be asymptotically achievable by the maximum likelihood estimator (MLE). By showing that the log-likelihood function of our problem is concave, we guarantee the global optimality of iterative algorithms in finding the MLE. Numerical results on both synthetic data and images taken by a prototype sensor verify our theoretical analysis and demonstrate the effectiveness of our image reconstruction algorithm. They also suggest the potential application of the oversampled binary sensing scheme in high dynamic range photography.

A. Agaskar and Y. M. Lu, “An uncertainty principle for functions defined on graphs,” in Proc. SPIE Conf. on Wavelets and Sparsity, San Diego, 2011.Abstract

The classical uncertainty principle provides a fundamental tradeoff in the localization of a function in the time
and frequency domains. In this paper we extend this classical result to functions defined on graphs. We justify
the use of the graph Laplacian’s eigenbasis as a surrogate for the Fourier basis for graphs, and define the notions of "spread" in the graph and spectral domains. We then establish an analogous uncertainty principle relating the two quantities, showing the degree to which a function can be simultaneously localized in the graph and spectral domains.

See, also, the extended journal version: A. Agaskar and Y. M. Lu, A Spectral Graph Uncertainty Principle, IEEE Transactions on Information Theory, to appear.
Z. Sadeghipoor, Y. M. Lu, and S. Susstrunk, “Correlation-Based joint acquisition and demosaicing of visible and near-infrared images,” in Proc. IEEE International Conference on Image Processing, Brussels, 2011.Abstract

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.

Y. M. Lu and M. Vetterli, “Sparse spectral factorization: Unicity and reconstruction algorithms,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, Prague, 2011.Abstract

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.

Y. M. Lu, P. L. Dragotti, and M. Vetterli, “Localizing point sources in diffusion fields from spatiotemporal measurements,” in Proc. Int. Conf. Sampling Theory and applications (SampTA), Singapore, 2011.Abstract

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.

J. Ranieri, A. Chebira, Y. M. Lu, and M. Vetterli, “Sampling and reconstructing diffusion fields with localized sources,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, Prague, 2011.Abstract

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.

I. Dokmanić, Y. M. Lu, and M. Vetterli, “Can one hear the shape of a room: The 2-D polygonal case,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, Prague, 2011.Abstract

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.


(This paper received the Best Student Paper Award of ICASSP.)

M. McCormick, Y. M. Lu, and M. Vetterli, “Learning sparse systems at sub-Nyquist rates: A frequency-domain approach,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, Dallas, 2010.Abstract

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.

Y. M. Lu and M. Vetterli, “Multichannel sampling with unknown gains and offsets: A fast reconstruction algorithm,” in Proc. Allerton Conference on Communication, Control and Computing. Monticello, IL, 2010.Abstract

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.

F. Yang, Y. M. Lu, L. Sbaiz, and M. Vetterli, “An optimal algorithm for reconstructing images from binary measurements,” in Proc. SPIE Conf. Computational Imaging VIII, San Jose, CA, 2010.
A. Hormati, O. Roy, Y. M. Lu, and M. Vetterli, “Distributed Sampling of Correlated Signals Linked by Sparse Filtering: Theory and Applications,” IEEE Transactions on Signal Processing, vol. 58, no. 3, pp. 1095-1109, 2010.Abstract

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.

Y. M. Lu, M. Karzand, and M. Vetterli, “Demosaicking by Alternating Projections: Theory and Fast One-Step Implementation,” IEEE Transactions on Image Processing, vol. 19, no. 8, pp. 2085-2098, 2010. MATLAB code and imagesAbstract

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.

Y. M. Lu, M. N. Do, and R. Laugesen, “Computable Fourier conditions for alias-free sampling and critical sampling.” in Proc. Int. Conf. Sampling Theory and applications (SampTA), Marseille, France, 2009.Abstract

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.