Consistent Basis Pursuit for Signal and Matrix Estimates in Quantized Compressed Sensing

This paper focuses on the estimation of low-complexity signals when they are observed through $M$ uniformly quantized compressive observations. Among such signals, we consider 1-D sparse vectors, low-rank matrices, or compressible signals that are well approximated by one of these two models. In this context, we prove the estimation efficiency of a variant of Basis Pursuit Denoise, called Consistent Basis Pursuit (CoBP), enforcing consistency between the observations and the re-observed estimate, while promoting its low-complexity nature. We show that the reconstruction error of CoBP decays like $M^{-1/4}$ when all parameters but $M$ are fixed. Our proof is connected to recent bounds on the proximity of vectors or matrices when (i) those belong to a set of small intrinsic "dimension", as measured by the Gaussian mean width, and (ii) they share the same quantized (dithered) random projections. By solving CoBP with a proximal algorithm, we provide some extensive numerical observations that confirm the theoretical bound as $M$ is increased, displaying even faster error decay than predicted. The same phenomenon is observed in the special, yet important case of 1-bit CS.Comment: Keywords: Quantized compressed sensing, quantization, consistency, error decay, low-rank, sparsity. 10 pages, 3 figures. Note abbout this version: title change, typo corrections, clarification of the context, adding a comparison with BPD

A Geometrical Study of Matching Pursuit Parametrization

This paper studies the effect of discretizing the parametrization of a dictionary used for Matching Pursuit decompositions of signals. Our approach relies on viewing the continuously parametrized dictionary as an embedded manifold in the signal space on which the tools of differential (Riemannian) geometry can be applied. The main contribution of this paper is twofold. First, we prove that if a discrete dictionary reaches a minimal density criterion, then the corresponding discrete MP (dMP) is equivalent in terms of convergence to a weakened hypothetical continuous MP. Interestingly, the corresponding weakness factor depends on a density measure of the discrete dictionary. Second, we show that the insertion of a simple geometric gradient ascent optimization on the atom dMP selection maintains the previous comparison but with a weakness factor at least two times closer to unity than without optimization. Finally, we present numerical experiments confirming our theoretical predictions for decomposition of signals and images on regular discretizations of dictionary parametrizations.Comment: 26 pages, 8 figure

Natural Image Noise Dataset

Convolutional neural networks have been the focus of research aiming to solve image denoising problems, but their performance remains unsatisfactory for most applications. These networks are trained with synthetic noise distributions that do not accurately reflect the noise captured by image sensors. Some datasets of clean-noisy image pairs have been introduced but they are usually meant for benchmarking or specific applications. We introduce the Natural Image Noise Dataset (NIND), a dataset of DSLR-like images with varying levels of ISO noise which is large enough to train models for blind denoising over a wide range of noise. We demonstrate a denoising model trained with the NIND and show that it significantly outperforms BM3D on ISO noise from unseen images, even when generalizing to images from a different type of camera. The Natural Image Noise Dataset is published on Wikimedia Commons such that it remains open for curation and contributions. We expect that this dataset will prove useful for future image denoising applications.Comment: NTIRE at CVPR 201

Reversible watermarking scheme with image-independent embedding capacity

Permanent distortion is one of the main drawbacks of all the irreversible watermarking schemes. Attempts to recover the original signal after the signal passing the authentication process are being made starting just a few years ago. Some common problems, such as salt-and-pepper artefacts owing to intensity wraparound and low embedding capacity, can now be resolved. However, some significant problems remain unsolved. First, the embedding capacity is signal-dependent, i.e., capacity varies significantly depending on the nature of the host signal. The direct impact of this is compromised security for signals with low capacity. Some signals may be even non-embeddable. Secondly, while seriously tackled in irreversible watermarking schemes, the well-known problem of block-wise dependence, which opens a security gap for the vector quantisation attack and transplantation attack, are not addressed by researchers of the reversible schemes. This work proposes a reversible watermarking scheme with near-constant signal-independent embedding capacity and immunity to the vector quantisation attack and transplantation attack