We provide a short, self-contained introduction to deep neural networks that is aimed at mathematically inclined readers. We promote the use of a vect--matrix formalism that is well suited to the compositional structure of these networks and that facilitates the derivation/description of the backpropagation algorithm. We present a detailed analysis of supervised learning for the two most common scenarios, (i) multivariate regression and (ii) classification, which rely on the minimization of least squares and cross-entropy criteria, respectively.

In this chapter, we provide an overview of a recent image-reconstruction method that uses a deep generative algorithm for dynamic magnetic resonance-imaging (dMRI). We begin by briefly introducing the imaging modality of dMRI, the associated image-reconstruction problem, and existing reconstruction approaches. Next, we introduce the time-dependent deep image prior (TD-DIP), which exploits the structure of convolutional neural networks (CNNs) as a regularizing prior. We show some representative results and discuss the pros and cons of this regularizing paradigm. Finally, we discuss a few potential remaining limitations.

CryoGAN uses ideas from deep generative adversarial learning to perform image reconstruction in single-particle cryo-electron microscopy (cryo-EM). In this chapter, we begin by introducing single-particle cryo-EM. We then formulate the associated image-reconstruction problem and discuss the main solutions found in the literature. Next, we describe the CryoGAN algorithm and show some representative results. Finally, we discuss what our experiences with Cryo-GAN suggest about the advantages and disadvantages of such deep generative adversarial methods in single-particle cryo-EM and beyond.

Quantitative phase imaging (QPI) refers to label-free techniques that produce images containing morphological information. In this chapter, we focus on 2D phase imaging with a holographic setup. In such a setting, the complex-valued measurements contain both intensity and phase information. The phase is related to the distribution of the refractive index of the underlying specimen. In practice, the collected phase happens to be wrapped (i.e., modulo 2π of the original phase) and one gains quantitative information on the sample only once the measurements are unwrapped. The process of phase unwrapping relies on the solution of an inverse problem, for which numerous methods exist. However, it is challenging to unwrap the phases of particularly complex or thick specimens such as organoids. Under such extreme conditions, classical methods often exhibit unwrapping errors. In this chapter we first formulate the problem of phase unwrapping and review the existing methods to solve it. Then, we present an application of a regularizing neural network to phase unwrapping, which allows us to outline the advantages of a training-free approach, i.e., a deep image prior, over classical methods or supervised learning.

The stochastic processes that we wish to characterize are those generated by linear transformation of non-Gaussian white noise. If we were operating in the discrete domain and restricting ourselves to a finite number of dimensions, we would be able to use any sequence of i.i.d. random variables wn as system input and rely on conventional multivariate statistics to characterize the output. This strongly suggests that the specification of the mixing matrix (L−1) and the probability density function (pdf) of the innovation is sufficient to obtain a complete description of a linear stochastic process, at least in the discrete setting.

But our goal is more ambitious since we place ourselves in the context of continuously defined processes. The situation is then not quite as straightforward because: (1) we are dealing with infinite-dimensional objects, (2) it is much harder to properly define the notion of continuous-domain white noise, and (3) there are theoretical restrictions on the class of admissible innovations. While this calls for an advanced mathematical machinery, the payoff is that the continuous-domain formalism lends itself better to analytical computations, by virtue of the powerful tools of functional and harmonic analysis. Another benefit is that the non-Gaussian members of the family are necessarily sparse as a consequence of the theory which rests upon the powerful characterization and existence theorems by Lévy–Khintchine, Minlos, Bochner, and Gelfand–Vilenkin.

As in the subsequent chapters, we start by providing some intuition in the first section and then proceed with a more formal characterization.

In order to obtain an uncoupled representation of a sparse process s = L−1w of the type described in the previous chapters, it is essential that we somehow invert the integral operator L−1. The ideal scenario would be to apply the differential operator L = (L−1)−1 to uncover the innovation w that is independent at every point. Unfortunately, this is not feasible in practice because we do not have access to the signal s(r) over the entire domain r ∈ ℝd, but only to its sampled values on a lattice or, more generally, to a series of coefficients in some appropriate basis. Our analysis options, as already alluded to in Chapter 2, are essentially twofold: the application of a discrete version of the operator, or an operator-like wavelet analysis.

This chapter is devoted to the in-depth investigation of these two modes of representation. As with the other chapters, we start with a concrete example (Lévy process) to lay down the key ideas in Section 8.1. Our primary tool for deriving the transform-domain pdfs is the characteristic functional, reviewed in Section 8.2 and properly extended so that it can handle arbitrary analysis functions φ ∈ Lp(ℝd). In Section 8.3, we investigate the decoupling ability of finite-difference-type operators and determine the statistical distribution of the resulting generalized increments. In Section 8.4, we show how a sparse process can be expanded in a matched-wavelet basis and provide the complete multivariate description of the transform-domain statistics, including general formulas for the wavelet cumulants.

Sparsity: Occam's razor of modern signal processing?

The hypotheses of Gaussianity and stationarity play a central role in the standard statistical formulation of signal processing. They fully justify the use of the Fourier transform as the optimal signal representation and naturally lead to the derivation of optimal linear filtering algorithms for a large variety of statistical estimation tasks. This classical view of signal processing is elegant and reassuring, but it is not at the forefront of research anymore.

Starting with the discovery of the wavelet transform in the late 1980s [Dau88, Mal89], researchers in signal processing have progressively moved away from the Fourier transform and have uncovered powerful alternatives. Consequently, they have ceased modeling signals as Gaussian stationary processes and have adopted a more deterministic, approximation-theoretic point of view. The key developments that are presently reshaping the field, and which are central to the theory presented in this book, are summarized below.

• Novel transforms and dictionaries for the representation of signals. New redundant and non-redundant representations of signals (wavelets, local cosine, curvelets) have emerged since the mid 1990s and have led to better algorithms for data compression, data processing, and feature extraction. The most prominent example is the wavelet based JPEG-2000 standard for image compression [CSE00], which outperforms the widely-used JPEG method based on the DCT (discrete cosine transform). Another illustration is wavelet-domain image denoising, which provides a good alternative to more traditional linear filtering [Don95]. The various dictionaries of basis functions that have been proposed so far are tailored to specific types of signals; there does not appear to be one that fits all.