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Deep neural networks provide state-of-the-art performance for image denoising, where the goal is to recover a near noise-free image from a noisy observation. The underlying principle is that neural networks trained on large data sets have empirically been shown to be able to generate natural images well from a low-dimensional latent representation of the image. Given such a generator network, a noisy image can be denoised by (i) finding the closest image in the range of the generator or by (ii) passing it through an encoder-generator architecture (known as an autoencoder). However, there is little theory to justify this success, let alone to predict the denoising performance as a function of the network parameters. In this paper, we consider the problem of denoising an image from additive Gaussian noise using the two generator-based approaches. In both cases, we assume the image is well described by a deep neural network with ReLU activations functions, mapping a $k$-dimensional code to an $n$-dimensional image. In the case of the autoencoder, we show that the feedforward network reduces noise energy by a factor of $O(k/n)$. In the case of optimizing over the range of a generative model, we state and analyze a simple gradient algorithm that minimizes a non-convex loss function and provably reduces noise energy by a factor of $O(k/n)$. We also demonstrate in numerical experiments that this denoising performance is, indeed, achieved by generative priors learned from data.

Because of its longevity and enormous information density, DNA is considered a promising data storage medium. In this work, we provide instructions for archiving digital information in the form of DNA and for subsequently retrieving it from the DNA. In principle, information can be represented in DNA by simply mapping the digital information to DNA and synthesizing it. However, imperfections in synthesis, sequencing, storage and handling of the DNA induce errors within the molecules, making error-free information storage challenging. The procedure discussed here enables error-free storage by protecting the information using error-correcting codes. Specifically, in this protocol, we provide the technical details and precise instructions for translating digital information to DNA sequences, physically handling the biomolecules, storing them and subsequently re-obtaining the information by sequencing the DNA. Along with the protocol, we provide computer code that automatically encodes digital information to DNA sequences and decodes the information back from DNA to a digital file. The required software is provided on a Github repository. The protocol relies on commercial DNA synthesis and DNA sequencing via Illumina dye sequencing, and requires 1–2 h of preparation time, 1/2 d for sequencing preparation and 2–4 h for data analysis. This protocol focuses on storage scales of ~100 kB to 15 MB, offering an ideal starting point for small experiments. It can be augmented to enable higher data volumes and random access to the data and also allows for future sequencing and synthesis technologies, by changing the parameters of the encoder/decoder to account for the corresponding error rates.

We demonstrate a compact, easy-to-build computational camera for single-shot three-dimensional (3D) imaging. Our lensless system consists solely of a diffuser placed in front of an image sensor. Every point within the volumetric field-of-view projects a unique pseudorandom pattern of caustics on the sensor. By using a physical approximation and simple calibration scheme, we solve the large-scale inverse problem in a computationally efficient way. The caustic patterns enable compressed sensing, which exploits sparsity in the sample to solve for more 3D voxels than pixels on the 2D sensor. Our 3D reconstruction grid is chosen to match the experimentally measured two-point optical resolution, resulting in 100 million voxels being reconstructed from a single 1.3 megapixel image. However, the effective resolution varies significantly with scene content. Because this effect is common to a wide range of computational cameras, we provide a new theory for analyzing resolution in such systems.

In this paper, we study the identification of a time-varying linear system from its response to a known input signal. More specifically, we consider systems whose response to the input signal is given by a weighted superposition of delayed and Doppler-shifted versions of the input. This problem arises in a multitude of applications such as wireless communications and radar imaging. Due to practical constraints, the input signal has finite bandwidth $B$, and the received signal is observed over a finite time interval of length $T$only. This gives rise to a delay and Doppler resolution of $1/B$and $1/T$. We show that this resolution limit can be overcome, i.e., we can exactly recover the continuous delay–Doppler pairs and the corresponding attenuation factors, by solving a convex optimization problem. This result holds provided that the distance between the delay–Doppler pairs is at least $2.37/B$in time or $2.37/T$in frequency. Furthermore, this result allows the total number of delay–Doppler pairs to be linear up to a log-factor in $BT$, the dimensionality of the response of the system, and thereby the limit for identifiability. Stated differently, we show that we can estimate the time–frequency components of a signal that is $S$-sparse in the continuousdictionary of time–frequency shifts of a random window function, from a number of measurements that is linear up to a log-factor in $S$.

The concentrations of nanoparticles present in colloidal dispersions are usually measured and given in mass concentration (e.g.mg/mL), and number concentrations can only be obtained by making assumptions about nanoparticle size and morphology. Additionally traditional nanoparticle concentration measures are not very sensitive, and only the presence/absence of millions/billions of particles occurring together can be obtained. Here, we describe a method, which not only intrinsically results in number concentrations, but is also sensitive enough to count individual nanoparticles, one by one. To make this possible, the sensitivity of the polymerase chain reaction (PCR) was combined with a binary (=0/1, yes/no) measurement arrangement, binomial statistics and DNA comprising monodisperse silica nanoparticles. With this method, individual tagged particles in the range of 60–250 nm could be detected and counted in drinking water in absolute number, utilizing a standard qPCR device within 1.5 h of measurement time. For comparison, the method was validated with single particle inductively coupled plasma mass spectrometry (sp-ICPMS).

With the total amount of worldwide data skyrocketing, the global data storage demand is predicted to grow to 1.75 × 1014GB by 2025. Traditional storage methods have difficulties keeping pace given that current storage media have a maximum density of 103GB/mm3. As such, data production will far exceed the capacity of currently available storage methods. The costs of maintaining and transferring data, as well as the limited lifespans and significant data losses associated with current technologies also demand advanced solutions for information storage. Nature offers a powerful alternative through the storage of information that defines living organisms in unique orders of four bases (A, T, C, G) located in molecules called deoxyribonucleic acid (DNA). DNA molecules as information carriers have many advantages over traditional storage media. Their high storage density, potentially low maintenance cost, ease of synthesis, and chemical modification make them an ideal alternative for information storage. To this end, rapid progress has been made over the past decade by exploiting user-defined DNA materials to encode information. In this review, we discuss the most recent advances of DNA-based data storage with a major focus on the challenges that remain in this promising field, including the current intrinsic low speed in data writing and reading and the high cost per byte stored. Alternatively, data storage relying on DNA nanostructures (as opposed to DNA sequence) as well as on other combinations of nanomaterials and biomolecules are proposed with promising technological and economic advantages. In summarizing the advances that have been made and underlining the challenges that remain, we provide a roadmap for the ongoing research in this rapidly growing field, which will enable the development of technological solutions to the global demand for superior storage methodologies.