MUSIC for multidimensional spectral estimation: stability and super-resolution

This paper presents a performance analysis of the MUltiple SIgnal Classification (MUSIC) algorithm applied on $D$ dimensional single-snapshot spectral estimation while $s$ true frequencies are located on the continuum of a bounded domain. Inspired by the matrix pencil form, we construct a D-fold Hankel matrix from the measurements and exploit its Vandermonde decomposition in the noiseless case. MUSIC amounts to identifying a noise subspace, evaluating a noise-space correlation function, and localizing frequencies by searching the $s$ smallest local minima of the noise-space correlation function. In the noiseless case, $(2s)^D$ measurements guarantee an exact reconstruction by MUSIC as the noise-space correlation function vanishes exactly at true frequencies. When noise exists, we provide an explicit estimate on the perturbation of the noise-space correlation function in terms of noise level, dimension $D$, the minimum separation among frequencies, the maximum and minimum amplitudes while frequencies are separated by two Rayleigh Length (RL) at each direction. As a by-product the maximum and minimum non-zero singular values of the multidimensional Vandermonde matrix whose nodes are on the unit sphere are estimated under a gap condition of the nodes. Under the 2-RL separation condition, if noise is i.i.d. gaussian, we show that perturbation of the noise-space correlation function decays like $\sqrt{\log(\#(\mathbf{N}))/\#(\mathbf{N})}$ as the sample size $\#(\mathbf{N})$ increases. When the separation among frequencies drops below 2 RL, our numerical experiments show that the noise tolerance of MUSIC obeys a power law with the minimum separation of frequencies.Comment: To appear in IEEE Transactions on Signal Processin

Fourier Phase Retrieval: Uniqueness and Algorithms

The problem of recovering a signal from its phaseless Fourier transform measurements, called Fourier phase retrieval, arises in many applications in engineering and science. Fourier phase retrieval poses fundamental theoretical and algorithmic challenges. In general, there is no unique mapping between a one-dimensional signal and its Fourier magnitude and therefore the problem is ill-posed. Additionally, while almost all multidimensional signals are uniquely mapped to their Fourier magnitude, the performance of existing algorithms is generally not well-understood. In this chapter we survey methods to guarantee uniqueness in Fourier phase retrieval. We then present different algorithmic approaches to retrieve the signal in practice. We conclude by outlining some of the main open questions in this field

Face Recognition in Low Quality Images: A Survey

Low-resolution face recognition (LRFR) has received increasing attention over the past few years. Its applications lie widely in the real-world environment when high-resolution or high-quality images are hard to capture. One of the biggest demands for LRFR technologies is video surveillance. As the the number of surveillance cameras in the city increases, the videos that captured will need to be processed automatically. However, those videos or images are usually captured with large standoffs, arbitrary illumination condition, and diverse angles of view. Faces in these images are generally small in size. Several studies addressed this problem employed techniques like super resolution, deblurring, or learning a relationship between different resolution domains. In this paper, we provide a comprehensive review of approaches to low-resolution face recognition in the past five years. First, a general problem definition is given. Later, systematically analysis of the works on this topic is presented by catogory. In addition to describing the methods, we also focus on datasets and experiment settings. We further address the related works on unconstrained low-resolution face recognition and compare them with the result that use synthetic low-resolution data. Finally, we summarized the general limitations and speculate a priorities for the future effort.Comment: There are some mistakes addressing in this paper which will be misleading to the reader and we wont have a new version in short time. We will resubmit once it is being corecte

Compressive Multidimensional Harmonic Retrieval with Prior Knowledge

This paper concerns the problem of estimating multidimensional (MD) frequencies using prior knowledge of the signal spectral sparsity from partial time samples. In many applications, such as radar, wireless communications, and super-resolution imaging, some structural information about the signal spectrum might be known beforehand. Suppose that the frequencies lie in given intervals, the goal is to improve the frequency estimation performance by using the prior information. We study the MD Vandermonde decomposition of block Toeplitz matrices in which the frequencies are restricted to given intervals. We then propose to solve the frequency-selective atomic norm minimization by converting them into semidefinite program based on the MD Vandermonde decomposition. Numerical simulation results are presented to illustrate the good performance of the proposed method

Achieving Super-Resolution in Multi-Rate Sampling Systems via Efficient Semidefinite Programming

Super-resolution theory aims to estimate the discrete components lying in a continuous space that constitute a sparse signal with optimal precision. This work investigates the potential of recent super-resolution techniques for spectral estimation in multi-rate sampling systems. It shows that, under the existence of a common supporting grid, and under a minimal separation constraint, the frequencies of a spectrally sparse signal can be exactly jointly recovered from the output of a semidefinite program (SDP). The algorithmic complexity of this approach is discussed, and an equivalent SDP of minimal dimension is derived by extending the Gram parametrization properties of sparse trigonometric polynomials

Synthesis-based Robust Low Resolution Face Recognition

Recognition of low resolution face images is a challenging problem in many practical face recognition systems. Methods have been proposed in the face recognition literature for the problem which assume that the probe is low resolution, but a high resolution gallery is available for recognition. These attempts have been aimed at modifying the probe image such that the resultant image provides better discrimination. We formulate the problem differently by leveraging the information available in the high resolution gallery image and propose a dictionary learning approach for classifying the low-resolution probe image. An important feature of our algorithm is that it can handle resolution change along with illumination variations. Furthermore, we also kernelize the algorithm to handle non-linearity in data and present a joint dictionary learning technique for robust recognition at low resolutions. The effectiveness of the proposed method is demonstrated using standard datasets and a challenging outdoor face dataset. It is shown that our method is efficient and can perform significantly better than many competitive low resolution face recognition algorithms

Compressive Fourier Transform Spectroscopy

We describe an approach based on compressive-sampling which allows for a considerable reduction in the acquisition time in Fourier-transform spectroscopy. In this approach, an N-point Fourier spectrum is resolved from much less than N time-domain measurements using a compressive-sensing reconstruction algorithm. We demonstrate the technique by resolving sparse vibrational spectra using <25% of the Nyquist rate samples in single-pulse CARS experiments. The method requires no modifications to the experimental setup and can be directly applied to any Fourier-transform spectroscopy measurement, in particular multidimensional spectroscopy

Efficient Two-Dimensional Line Spectrum Estimation Based on Decoupled Atomic Norm Minimization

This paper presents an efficient optimization technique for gridless {2-D} line spectrum estimation, named decoupled atomic norm minimization (D-ANM). The framework of atomic norm minimization (ANM) is considered, which has been successfully applied in 1-D problems to allow super-resolution frequency estimation for correlated sources even when the number of snapshots is highly limited. The state-of-the-art 2-D ANM approach vectorizes the 2-D measurements to their 1-D equivalence, which incurs huge computational cost and may become too costly for practical applications. We develop a novel decoupled approach of 2-D ANM via semi-definite programming (SDP), which introduces a new matrix-form atom set to naturally decouple the joint observations in both dimensions without loss of optimality. Accordingly, the original large-scale 2-D problem is equivalently reformulated via two decoupled one-level Toeplitz matrices, which can be solved by simple 1-D frequency estimation with pairing. Compared with the conventional vectorized approach, the proposed D-ANM technique reduces the computational complexity by several orders of magnitude with respect to the problem size. It also retains the benefits of ANM in terms of precise signal recovery, small number of required measurements, and robustness to source correlation. The complexity benefits are particularly attractive for large-scale antenna systems such as massive MIMO, radar signal processing and radio astronomy

Nonlinear Prediction of Multidimensional Signals via Deep Regression with Applications to Image Coding

Deep convolutional neural networks (DCNN) have enjoyed great successes in many signal processing applications because they can learn complex, non-linear causal relationships from input to output. In this light, DCNNs are well suited for the task of sequential prediction of multidimensional signals, such as images, and have the potential of improving the performance of traditional linear predictors. In this research we investigate how far DCNNs can push the envelop in terms of prediction precision. We propose, in a case study, a two-stage deep regression DCNN framework for nonlinear prediction of two-dimensional image signals. In the first-stage regression, the proposed deep prediction network (PredNet) takes the causal context as input and emits a prediction of the present pixel. Three PredNets are trained with the regression objectives of minimizing $\ell_1$, $\ell_2$ and $\ell_\infty$ norms of prediction residuals, respectively. The second-stage regression combines the outputs of the three PredNets to generate an even more precise and robust prediction. The proposed deep regression model is applied to lossless predictive image coding, and it outperforms the state-of-the-art linear predictors by appreciable margin

Super-Resolution using Convolutional Neural Networks without Any Checkerboard Artifacts

It is well-known that a number of excellent super-resolution (SR) methods using convolutional neural networks (CNNs) generate checkerboard artifacts. A condition to avoid the checkerboard artifacts is proposed in this paper. So far, checkerboard artifacts have been mainly studied for linear multirate systems, but the condition to avoid checkerboard artifacts can not be applied to CNNs due to the non-linearity of CNNs. We extend the avoiding condition for CNNs, and apply the proposed structure to some typical SR methods to confirm the effectiveness of the new scheme. Experiment results demonstrate that the proposed structure can perfectly avoid to generate checkerboard artifacts under two loss conditions: mean square error and perceptual loss, while keeping excellent properties that the SR methods have.Comment: To appear in Proc. ICIP2018 October 07-10, 2018, Athens, Greec