Michaël Unser

Related publications (706)

Delaunay-Triangulation-Based Learning With Hessian Total-Variation Regularization

Alexis Marie Frederic Goujon, Mehrsa Pourya, Michaël Unser

2023

Phase Retrieval: From Computational Imaging to Machine Learning: A tutorial

Jonathan Yuelin Dong, Thanh-An Michel Pham, Michaël Unser

Phase retrieval consists in the recovery of a complex-valued signal from intensity-only measurements. As it pervades a broad variety of applications, many researchers have striven to develop phase-retrieval algorithms. Classical approaches involve techniques as varied as generic gradient descent routines or specialized spectral methods, to name a few. However, the phase-recovery problem remains a challenge to this day. Recently, however, advances in machine learning have revitalized the study of phase retrieval in two ways: 1) significant theoretical advances have emerged from the analogy between phase retrieval and single-layer neural networks, and 2) practical breakthroughs have been obtained thanks to deep learning regularization. In this tutorial, we review phase retrieval under a unifying framework that encompasses classical and machine learning methods. We focus on three key elements: applications, an overview of recent reconstruction algorithms, and the latest theoretical results.

2023

Stability of Image-Reconstruction Algorithms

Pol del Aguila Pla, Michaël Unser

Robustness and stability of image-reconstruction algorithms have recently come under scrutiny. Their importance to medical imaging cannot be overstated. We review the known results for the topical variational regularization strategies ( ℓ2 and ℓ1 regularization) and present novel stability results for ℓp -regularized linear inverse problems for p∈(1,∞) . Our results guarantee Lipschitz continuity for small p and Hölder continuity for larger p . They generalize well to the Lp(Ω) function spaces.

2023

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Most modern imaging systems incorporate a computational pipeline to infer the image of interest from acquired measurements. The Bayesian approach to solve such ill-posed inverse problems involves the characterization of the posterior distribution of the image. It depends on the model of the imaging system and on prior knowledge on the image of interest. In this work, we present a Bayesian reconstruction framework for nonlinear imaging models where we specify the prior knowledge on the image through a deep generative model. We develop a tractable posterior-sampling scheme based on the Metropolis-adjusted Langevin algorithm for the class of nonlinear inverse problems where the forward model has a neural-network-like structure. This class includes most practical imaging modalities. We introduce the notion of augmented deep generative priors in order to suitably handle the recovery of quantitative images. We illustrate the advantages of our framework by applying it to two nonlinear imaging modalities—phase retrieval and optical diffraction tomography.

2022

Learning of Continuous and Piecewise-Linear Functions With Hessian Total-Variation Regularization

Shayan Aziznejad, Joaquim Gonçalves Garcia Barreto Campos, Michaël Unser

We develop a novel 2D functional learning framework that employs a sparsity-promoting regularization based on second-order derivatives. Motivated by the nature of the regularizer, we restrict the search space to the span of piecewise-linear box splines shifted on a 2D lattice. Our formulation of the infinite-dimensional problem on this search space allows us to recast it exactly as a finite-dimensional one that can be solved using standard methods in convex optimization. Since our search space is composed of continuous and piecewise-linear functions, our work presents itself as an alternative to training networks that deploy rectified linear units, which also construct models in this family. The advantages of our method are fourfold: the ability to enforce sparsity, favoring models with fewer piecewise-linear regions; the use of a rotation, scale and translation-invariant regularization; a single hyperparameter that controls the complexity of the model; and a clear model interpretability that provides a straightforward relation between the parameters and the overall learned function. We validate our framework in various experimental setups and compare it with neural networks.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2022

We present a new family of active surfaces for the semiautomatic segmentation of volumetric objects in 3D biomedical images. We represent our deformable model by a subdivision surface encoded by a small set of control points and generated through a geometric refinement process. The subdivision operator confers important properties to the surface such as smoothness, reproduction of desirable shapes and interpolation of the control points. We deform the subdivision surface through the minimization of suitable gradient-based and region-based energy terms that we have designed for that purpose. In addition, we provide an easy way to combine these energies with convolutional neural networks. Our active subdivision surface satisfies the property of multiresolution, which allows us to adopt a coarse-to-fine optimization strategy. This speeds up the computations and decreases its dependence on initialization compared to singleresolution active surfaces. Performance evaluations on both synthetic and real biomedical data show that our active subdivision surface is robust in the presence of noise and outperforms current state-of-the-art methods. In addition, we provide a software that gives full control over the active subdivision surface via an intuitive manipulation of the control points.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2021

Continuous-Domain Formulation of Inverse Problems for Composite Sparse-Plus-Smooth Signals

Shayan Aziznejad, Thomas Jean Debarre, Michaël Unser

We present a novel framework for the reconstruction of 1D composite signals assumed to be a mixture of two additive components, one sparse and the other smooth, given a finite number of linear measurements. We formulate the reconstruction problem as a continuous-domain regularized inverse problem with multiple penalties. We prove that these penalties induce reconstructed signals that indeed take the desired form of the sum of a sparse and a smooth component. We then discretize this problem using Riesz bases, which yields a discrete problem that can be solved by standard algorithms. Our discretization is exact in the sense that we are solving the continuous-domain problem over the search space specified by our bases without any discretization error. We propose a complete algorithmic pipeline and demonstrate its feasibility on simulated data.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2021

We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a finite sum of components, where each component belongs to its own prescribed Banach space; moreover, the problem is regularized by penalizing some composite norm of the solution. We establish general conditions for existence and derive the generic parametric representation of the solution components. These representations fall into three categories depending on the underlying regularization norm: (i) a linear expansion in terms of predefined “kernels” when the component space is a reproducing kernel Hilbert space (RKHS), (ii) a non-linear (duality) mapping of a linear combination of measurement functionals when the component Banach space is strictly convex, and, (iii) an adaptive expansion in terms of a small number of atoms within a larger dictionary when the component Banach space is not strictly convex. Our approach generalizes and unifies a number of multi-kernel (RKHS) and sparse-dictionary learning techniques for compressed sensing available in the literature. It also yields the natural extension of the classical spline-fitting techniques in (semi-)RKHS to the abstract Banach-space setting.

2021

CryoGAN: A New Reconstruction Paradigm for Single-Particle Cryo-EM Via Deep Adversarial Learning

Laurène Donati, Harshit Gupta, Michael Thompson McCann, Michaël Unser

We present CryoGAN, a new paradigm for single-particle cryo-electron microscopy (cryo-EM) reconstruction based on unsupervised deep adversarial learning. In single-particle cryo-EM, the structure of a biomolecule needs to be reconstructed from a large set of noisy tomographic projections with unknown orientations. Current reconstruction techniques are based on a marginalized maximum-likelihood formulation that requires calculations over the set of all possible poses for each projection image, a computationally demanding procedure. Our approach is to seek a 3D structure that has simulated projections that match the real data in a distributional sense, thereby sidestepping pose estimation or marginalization. We prove that, in an idealized mathematical model of cryo-EM, this approach results in recovery of the correct structure. Motivated by distribution matching, we propose CryoGAN, a specialized GAN that consists of a 3D structure, a cryo-EM physics simulator, and a discriminator neural network. During reconstruction, the 3D structure is optimized so that its projections obtained through the simulator resemble real data (to the discriminator). Simultaneously, the discriminator is trained to distinguish real projections from simulated projections. CryoGAN takes as input only real projection images and the distribution of the cryo-EM imaging parameters. It involves neither prior training nor an initial estimation of the 3D structure. CryoGAN currently achieves a 10.8 angstrom resolution on a realistic synthetic dataset. Preliminary results on experimental beta-galactosidase and 80S ribosome data demonstrate the ability of CryoGAN to exploit data statistics under standard experimental imaging conditions. We believe that this paradigm opens the door to a family of novel likelihood-free algorithms for cryo-EM reconstruction.

IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC2021