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This paper considers the problem of estimating the delays of a weighted superposition of pulses, called stream of pulses, in a noisy environment. We show that the delays can be estimated using a tractable convex optimization problem with a localization error proportional to the square root of the noise level. Furthermore, all false detections produced by the algorithm have small amplitudes. Numerical and in-vitro ultrasound experiments corroborate the theoretical results and demonstrate their applicability for the ultrasound imaging signal processing.

We model the inverse physical problem of identifying an underwater obstacle by using the acoustic wave equation. Measurements are collected in a set of sensors placed in the medium. We use this partial information to approximate a segmentation of the obstacle in its correct location. This is an ill-posed problem. We propose a novel deep learning architecture that takes as input the sensor data and computes an approximate segmentation map of the obstacle.

In recent years, the theory of compressed sensing has emerged as an alternative for the Shannon sampling theorem, suggesting that compressible signals can be reconstructed from far fewer samples than required by the Shannon sampling theorem. In fact the theory advocates that nonadaptive, “random” functionals are in some sense optimal for this task. However, in practice, compressed sensing is very difficult to implement for large data sets, particularly because the recovery algorithms require significant computational resources. In this work, we present a new alternative method for simultaneous image acquisition and compression called adaptive compressed sampling. We exploit wavelet tree structures found in natural images to replace the “universal” acquisition of incoherent measurements with a direct and fast method for adaptive wavelet tree acquisition. The main advantages of this direct approach are that no complex recovery algorithm is in fact needed and that it allows more control over the compressed image quality, in particular, the sharpness of edges. Our experimental results show, by way of software simulations, that our adaptive algorithms perform better than existing nonadaptive methods in terms of image quality and speed.

We present a new image coding algorithm, the geometric piecewise polynomials (GPP) method, that draws on recent developments in the theory of adaptive multivariate piecewise polynomials approximation. The algorithm relies on a segmentation stage whose goal is to minimize a functional that is conceptually similar to the Mumford-Shah functional except that it measures the smoothness of the segmentation instead of the length. The initial segmentation is ldquoprunedrdquo and the remaining curve portions are lossy encoded. The image is then further partitioned and approximated by low order polynomials on the subdomains. We show examples where our algorithm outperforms state-of-the-art wavelet coding in the low bit-rate range. The GPP algorithm significantly outperforms wavelet based coding methods on graphic and cartoon images. Also, at the bit rate 0.05 bits per pixel, the GPP algorithm achieves on the test image Cameraman, which has a geometric structure, a PSNR of 21.5 dB, while the JPEG2000 Kakadu software obtains PSNR of 20 dB. For the test image Lena, the GPP algorithm obtains the same PSNR as JPEG2000, but with better visual quality at 0.03 bpp.

This paper describes a new and efficient method for low bit-rate image coding which is based on recent development in the theory of multivariate nonlinear piecewise polynomial approximation. It combines a binary space partition scheme with geometric wavelet (GW) tree approximation so as to efficiently capture curve singularities and provide a sparse representation of the image. The GW method successfully competes with state-of-the-art wavelet methods such as the EZW, SPIHT, and EBCOT algorithms. We report a gain of about 0.4 dB over the SPIHT and EBCOT algorithms at the bit-rate 0.0625 bits-per-pixels (bpp). It also outperforms other recent methods that are based on "sparse geometric representation." For example, we report a gain of 0.27 dB over the Bandelets algorithm at 0.1 bpp. Although the algorithm is computationally intensive, its time complexity can be significantely reduced by collecting a "global" GW n-term approximation to the image from a collection of GW trees, each constructed separately over tiles of the image.

The binary space partition (BSP) technique is a simple and efficient method to adaptively partition an initial given domain to match the geometry of a given input function. As such, the BSP technique has been widely used by practitioners, but up until now no rigorous mathematical justification for it has been offered. Here we attempt to put the technique on sound mathematical foundations, and we offer an enhancement of the BSP algorithm in the spirit of what we are going to call geometric wavelets. This new approach to sparse geometric representation is based on recent developments in the theory of multivariate nonlinear piecewise polynomial approximation. We provide numerical examples of n-term geometric wavelet approximations of known test images and compare them with dyadic wavelet approximation. We also discuss applications to image denoising and compression.

The problem of resolving the fine details of a signal from its coarse scale measurements or, as it is commonly referred to in the literature, the super-resolution problem arises naturally in engineering and physics in a variety of settings. We suggest a unified convex optimization approach for super-resolution. The key is the construction of an interpolating polynomial in the measurements space based on localized kernels. We also show that the localized kernels act as the connecting thread to another wide-spread problem of stream of pulses.

This paper considers the problem of recovering the delays and amplitudes of a weighted superposition of pulses. This problem is motivated by a variety of applications such as ultrasound and radar. We show that for univariate and bivariate stream of pulses, one can recover the delays and weights to any desired accuracy by solving a tractable convex optimization problem, provided that a pulse-dependent separation condition is satisfied. The main result of this paper states that the recovery is robust to additive noise or model mismatch., Small modifications

This paper considers the problem of recovering an ensemble of Diracs on a sphere from its low resolution measurements. The Diracs can be located at any location on the sphere, not necessarily on a grid. We show that under a separation condition, one can recover the ensemble with high precision by a three-stage algorithm, which consists of solving a semi-definite program, root finding and least-square fitting. The algorithm's computation time depends solely on the number of measurements, and not on the required solution accuracy. We also show that in the special case of non-negative ensembles, a sparsity condition is sufficient for recovery. Furthermore, in the discrete setting, we estimate the recovery error in the presence of noise as a function of the noise level and the super-resolution factor.