Hanjie Pan

Sampling Curves With Finite Rate of Innovation

In this paper, we extend the theory of sampling signals with finite rate of innovation (FRI) to a specific class of two-dimensional curves, which are defined implicitly as the zeros of a mask function. Here the mask function has a parametric representation as a weighted summation of a finite number of complex exponentials, and therefore, has finite rate of innovation . An associated edge image, which is discontinuous on the predefined parametric curve, is proved to satisfy a set of linear annihilation equations. We show that it is possible to reconstruct the parameters of the curve (i.e., to detect the exact edge positions in the continuous domain) based on the annihilation equations. Robust reconstruction algorithms are also developed to cope with scenarios with model mismatch. Moreover, the annihilation equations that characterize the curve are linear constraints that can be easily exploited in optimization problems for further image processing (e.g., image up-sampling). We demonstrate one potential application of the annihilation algorithm with examples in edge-preserving interpolation. Experimental results with both synthetic curves as well as edges of natural images clearly show the effectiveness of the annihilation constraint in preserving sharp edges, and improving SNRs.

An Iterative Linear Expansion of Thresholds for

This paper proposes a novel algorithmic framework to solve image restoration problems under sparsity assumptions. As usual, the reconstructed image is the minimum of an objective functional that consists of a data fidelity term and an ℓ1 regularization. However, instead of estimating the reconstructed image that minimizes the objective functional directly, we focus on the restoration process that maps the degraded measurements to the reconstruction. Our idea amounts to parameterize the process as a linear combination of few elementary thresholding functions (LET) and to solve the linear weighting coefficients by minimizing the objective functional. It is then possible to update the thresholding functions and to iterate this process ( i-LET). The key advantage of such a linear parametrization is that the problem size reduces dramatically-each time we only need to solve an optimization problem over the dimension of the linear coefficients (typically less than 10) instead of the whole image dimension. With the elementary thresholding functions satisfying certain constraints, a global convergence of the iterated LET algorithm is guaranteed. Experiments on several test images over a wide range of noise levels and different types of convolution kernels clearly indicate that the proposed framework usually outperforms state-of-the-art algorithms in terms of both the CPU time and the number of iterations.

\ell_{1}

It is a classic problem to estimate continuous-time sparse signals, like point sources in a direction-of-arrival problem, or pulses in a time-of-flight measurement. The earliest occurrence is the estimation of sinusoids in time series using Pronys method. This is at the root of a substantial line of work on high resolution spectral estimation. The estimation of continuous-time sparse signals from discrete-time samples is the goal of the sampling theory for finite rate of innovation (FRI) signals. Both spectral estimation and FRI sampling usually assume uniform sampling. But not all measurements are obtained uniformly, as exemplified by a concrete radioastronomy problem we set out to solve. Thus, we develop the theory and algorithm to reconstruct sparse signals, typically sum of sinusoids, from nonuniform samples. We achieve this by identifying a linear transformation that relates the unknown uniform samples of sinusoids to the given measurements. These uniform samples are known to satisfy the annihilation equations. A valid solution is then obtained by solving a constrained minimization such that the reconstructed signal is consistent with the given measurements and satisfies the annihilation constraint. Thanks to this new approach, we unify a variety of FRI-based methods. We demonstrate the versatility and robustness of the proposed approach with five FRI reconstruction problems, namely Dirac reconstructions with irregular time or Fourier domain samples, FRI curve reconstructions, Dirac reconstructions on the sphere, and point source reconstructions in radioastronomy. The proposed algorithm improves substantially over state-of-the-art methods and is able to reconstruct point sources accurately from irregularly sampled Fourier measurements under severe noise conditions.

-Based Image Restoration

We focus on image restoration that consists in regularizing a quadratic data-fidelity term with the standard ℓ1 sparse-enforcing norm. We propose a novel algorithmic approach to solve this optimization problem. Our idea amounts to approximating the result of the restoration as a linear sum of basic thresholds (e.g. soft-thresholds) weighted by unknown coefficients. The few coefficients of this expansion are obtained by minimizing the equivalent low-dimensional ℓ1-norm regularized objective function, which can be solved efficiently with standard convex optimization techniques, e.g. iterative reweighted least square (IRLS). By iterating this process, we claim that we reach the global minimum of the objective function. Experimentally we discover that very few iterations are required before we reach the convergence.

Towards Generalized FRI Sampling With an Application to Source Resolution in Radioastronomy

In this paper we present FRIDA—an algorithm for estimating directions of arrival of multiple wideband sound sources. FRIDA combines multi-band information coherently and achieves state-of-the-art resolution at extremely low signal-to-noise ratios. It works for arbitrary array layouts, but unlike the various steered response power and subspace methods, it does not require a grid search. FRIDA leverages recent advances in sampling signals with a finite rate of innovation. It is based on the insight that for any array layout, the entries of the spatial covariance matrix can be linearly transformed into a uniformly sampled sum of sinusoids.

Sparse image restoration using iterated linear expansion of thresholds

Recently, the type of compound regularizers has become a popular choice for signal reconstruction. The estimation quality is generally sensitive to the values of multiple regularization parameters. In this work, based on BDF algorithm, we develop a data-driven optimization scheme based on minimization of Steins unbiased risk estimate (SURE)— statistically equivalent to mean squared error (MSE). We propose a recursive evaluation of SURE to monitor the MSE during BDF iteration; the optimal values of the multiple parameters are then identified by the minimum SURE. Monte-Carlo simulation is applied to compute SURE for large-scale data. We exemplify the proposed method with image deconvolution. Numerical experiments show that the proposed method leads to highly accurate estimates of regularization parameters and nearly optimal restoration performance.

FRIDA: FRI-based DOA estimation for arbitrary array layouts

In our demo, we present two hardware platforms for prototyping audio array signal processing. Pyramic is a 48-channel microphone array fitted on an FPGA and Compact Six is a portable microphone array with six microphones, closer to the technical constraints of consumer electronics. A browser based interface was developed that allows the user to interact with the audio stream from the arrays in real time. The software component of this demo is a Python module with implementations of basic audio signal processing blocks and popular techniques like STFT, beamforming, and DoA. Both the hardware design files and the software are open source and freely shared. As part of a collaboration with IBM Research, their beamforming and imaging technologies will also be portrayed. The hardware will be demonstrated through an installation processing the microphone signals into light patterns on a circular LED array. The demo will be interactive and let visitors play with different algorithms for DoA (SRP, FRIDA [1], Bluebild) and beamforming (MVDR, Flexibeam [2]). The availability of an open platform with reference implementations encourages reproducible research and minimizes setup-time when testing and benchmarking new audio array signal processing algorithms. It can also serve as a useful educational tool, providing a means to work with real-life signals.

Optimization of compound regularization parameters based on Stein's unbiased risk estimate

Context. Two main classes of imaging algorithms have emerged in radio interferometry: the CLEAN algorithm and its multiple variants, and compressed-sensing inspired methods. They are both discrete in nature, and estimate source locations and intensities on a regular grid. For the traditional CLEAN-based imaging pipeline, the resolution power of the tool is limited by the width of the synthesized beam, which is inversely proportional to the largest baseline. The finite rate of innovation (FRI) framework is a robust method to find the locations of point-sources in a continuum without grid imposition. The continuous formulation makes the FRI recovery performance only dependent on the number of measurements and the number of sources in the sky. FRI can theoretically find sources below the perceived tool resolution. To date, FRI had never been tested in the extreme conditions inherent to radio astronomy: weak signal / high noise, huge data sets, large numbers of sources. Aims. The aims were (i) to adapt FRI to radio astronomy, (ii) verify it can recover sources in radio astronomy conditions with more accurate positioning than CLEAN, and possibly resolve some sources that would otherwise be missed, (iii) show that sources can be found using less data than would otherwise be required to find them, and (v) show that FRI does not lead to an augmented rate of false positives. Methods. We implemented a continuous domain sparse reconstruction algorithm in Python. The angular resolution performance of the new algorithm was assessed under simulation, and with visibility measurements from the LOFAR telescope. Existing catalogs were used to confirm the existence of sources. Results. We adapted the FRI framework to radio interferometry, and showed that it is possible to determine accurate off-grid point source locations and their corresponding intensities. In addition, FRI-based sparse reconstruction required less integration time and smaller baselines to reach a comparable reconstruction quality compared to a conventional method. The achieved angular resolution is higher than the perceived instrument resolution, and very close sources can be reliably distinguished. The proposed approach has cubic complexity in the total number (typically around a few thousand) of uniform Fourier data of the sky image estimated from the reconstruction. It is also demonstrated that the method is robust to the presence of extended-sources, and that false-positives can be addressed by choosing an adequate model order to match the noise level.

Hardware and software for reproducible research in audio array signal processing

We propose a novel edge detection algorithm with sub-pixel accuracy based on annihilation of signals with finite rate of innovation [1, 2]. We show that the Fourier domain annihilation equations can be interpreted as spatial domain multiplications. From this new perspective, we obtain an accurate estimation of the edge model by assuming a simple parametric form within each localised block. Further, we build a locally adaptive global mask function (i.e, our edge model) for the whole image. The mask function is then used as an edge-preserving constraint in further processing. Numerical experiments on both edge localisations and image up-sampling show the effectiveness of the proposed approach, which out-performs state-of-the-art method.