arXiv subjects

Michael Unser

Publications and source records attributed to Michael Unser.

At least 19 recordsLinked to original sources

arXiv2026

The Generalized Twin Image: Understanding Regularized Phase Retrieval

Image priors can dramatically improve phase retrieval reconstruction, but their effectiveness strongly varies between imaging modalities. We introduce the generalized twin image, which extends the classical twin-image ambiguity from Fourier phase retrieval to any invertible sensing operator, to understand this effectiveness. The ambiguity extends to undersampled and oversampled measurements, remaining exact under undersampling and for oversampled coded Fourier measurements. We show that it provides a simple criterion to assess regularization effectiveness: the image prior should favor the ground truth over its generalized twin image. We propose the total-variation ratio as a simple measure of this preference, and derive closed-form estimates for structured random phase retrieval models. Experiments with a pretrained denoiser demonstrate that the ratio predicts regularized reconstruction performance across different sensing models.

physics.optics↗

arXiv2026

Guidance Without Labels: Flow Matching from Coupled Gaussian Mixtures

Flow matching models are typically trained to transport a Gaussian source to the data distribution, with source and target samples paired independently. We replace the source with a fitted Gaussian mixture and couple each data point to a mixture component through an entropic optimal transport problem. We show that the resulting transport reduces to the standard Gaussian-source problem up to a guidance-like correction, which enables the reuse of a pretrained velocity field. This correction is a closed-form function of the fitted mixture plus a term that we learn with a small correction network. We test our model on published checkpoints for CIFAR-10 and CelebA-HQ. It improves FID at every step count, and on CelebA-HQ it reaches the quality of the baseline with about half the steps. The closed-form term alone, with no training, already delivers most of this improvement; the small correction network accounts for the rest.

cs.LG↗

arXiv2026

Image Priors for Single-Shot Coded Diffraction Phase Retrieval

Phase retrieval recovers complex-valued objects from intensity-only measurements. In its classical Fourier form, reconstruction suffers from twin-image stagnation. Coded diffraction imaging, which adds a random phase mask before the Fourier transform, has so far required several masks. We show that an image prior makes single-shot coded diffraction imaging possible, with a single random mask followed by a Fourier transform. We validate this claim in simulation and in an optical experiment, in which a spatial light modulator displays the phase object together with the random mask. Our experimental validation covers five test signals and 530 images from the public Brno quantitative phase microscopy dataset of five adherent cell lines. With no image prior or with Fourier-only measurements, reconstruction of complex-valued objects from a single diffraction pattern consistently fails. An out-of-the-box pretrained image denoiser enables accurate reconstruction, in simulation even with fewer measurements than unknowns. Five regularization frameworks built on the same denoiser and one learned weakly convex regularizer all reconstruct the experimental images with peak signal-to-noise ratios within one decibel of each other. A single random phase mask and image priors thus turn coded diffraction imaging into a practical single-shot modality for quantitative phase imaging.

physics.optics↗

arXiv2026

Splines and Operators: Part I - Admissibility and Native Spaces

The variational theory of splines requires a native space: a Hilbert space over which the regularization energy $\|\mathrm{L} f\|_{L_2}^2$ is well defined and the minimum-energy interpolation problem admits a spline solution. In this paper, we characterize the operators $\mathrm{L}$ that enable the construction of such spaces. Our framework is based on the completion of a space $\mathcal{S}$ of test functions in a controlled Gelfand-triple configuration $\mathcal{S}\subset\mathcal{H}\subset\mathcal{S}'$, which we call a Hilbert-Gelfand space. This departs from the classical reproducing-kernel techniques. Our construction is substantially more general, as it supports arbitrary linear operators and singular behaviors, and it recovers the reproducing-kernel setting as a particular case. It is also complete and canonical: the admissibility conditions on $\mathrm{L}$ are necessary and sufficient, and $\mathrm{L}$ determines its native space as the unique Hilbert-Gelfand space on which it extends to a bounded surjection onto $L_2$. We first show that Hilbert-Gelfand spaces are in one-to-one correspondence with compatible Riesz pairs $(\mathrm{A},\mathrm{A}^{-1})$ of positive-definite operators $\mathcal{S}\to\mathcal{S}'$. We then extend this correspondence to conditional Riesz pairs, where $\mathrm{A}$ is only conditionally positive-definite, which is the scenario that is relevant to splines. The theory identifies the \emph{attainable null space} of the energy as the unique admissible conditioning, realizes the abstract completions as concrete spaces of distributions, and provides a verifiable admissibility condition in terms of a dual energy functional. The framework is illustrated with the fractional Laplacian, which covers polyharmonic and thin-plate splines, and with ordinary differential operators, which cover exponential splines.

math.FA↗

arXiv2026

Splines and Operators: Part II - Resolution of Inverse Problems

We present a unified treatment of the data-fitting problems with quadratic (Hilbertian) regularization that arise in approximation theory, machine learning, engineering, and statistics. The generic problem is an optimization in the continuum: given $M$ linear measurements of an unknown function, one minimizes a convex data-fidelity term that is penalized by $\|\mathrm{L} f\|^2_{L_2}$, where $\mathrm{L} $ is a differential-like (spline-admissible) operator. The natural domain of the minimization is the native space $\mathcal{H}_\mathrm{L}$ of $\mathrm{L} $ - the largest Hilbert space over which the regularizer remains meaningful - for which we provide a concise and directly usable characterization. Central to our formulation is a measurement-adapted (conditioned) basis that decouples the null-space and spline components of the solution. This basis serves in three places. First, it tightens the formulation of our representer theorem, which confines the unique solution to an $M$-dimensional space without any orthogonality constraint on the coefficients. Second, it streamlines the implementation, as the exact discretization of the problem becomes an unconstrained convex program with closed-form solutions in the least-squares case, which covers ridge regression, Tikhonov regularization, and smoothing splines. Third, it makes the bridge with the minimum-mean-square-error estimation of generalized Gaussian processes, which is the topic of a companion paper.

math.FA↗

arXiv2026

Structured Random Model for Fast and Robust Phase Retrieval

Phase retrieval, a nonlinear problem prevalent in imaging applications, has been extensively studied using random models, some of which with i.i.d. sensing matrix components. While these models offer robust reconstruction guarantees, they are computationally expensive and impractical for real-world scenarios. In contrast, Fourier-based models, common in applications such as ptychography and coded diffraction imaging, are computationally more efficient but lack the theoretical guarantees of random models. Here, we introduce structured random models for phase retrieval that combine the efficiency of fast Fourier transforms with the versatility of random diagonal matrices. These models emulate i.i.d. random matrices at a fraction of the computational cost. Our approach demonstrates robust reconstructions comparable to fully random models using gradient descent and spectral methods. Furthermore, we establish that a minimum of two structured layers is necessary to achieve these structured-random properties. The proposed method is suitable for optical implementation and offers an efficient and robust alternative for phase retrieval in practical imaging applications.

physics.optics↗

arXiv2026

Structured Random Models for Phase Retrieval with Optical Diffusers

Phase retrieval is a nonlinear inverse problem that arises in a wide range of imaging modalities, from electron microscopy to Fourier ptychography. In particular, the reconstruction is facilitated when the sensing matrix is i.i.d. random, enabling strong theoretical guarantees and efficient reconstruction algorithms. However, its applicability is restricted by excessive computational costs. In this paper, we propose structured random models for phase retrieval, where we emulate a dense random matrix by a cascade of structured transforms and random diagonal matrices. We reduce the complexity from quadratic to log-linear at no cost in reconstruction performance. Through a spectral method initialization followed by gradient descent, robust reconstruction is obtained at an oversampling ratio as low as 2.5. Moreover, we observe that the reconstruction performance is solely determined by the singular-value distribution of the forward matrix. This class of models can directly be implemented with basic optical elements such as lenses and diffusers, paving the way for large-scale phase imaging with robust reconstruction guarantees.

physics.optics↗

arXiv2026

Generalized Splines and Gaussian Processes

For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes the form of a regularized least-squares data fit. In this chapter, we show that this equivalence extends to a much broader infinite-dimensional setting where generalized splines take the role of linear regressors and generalized Gaussian processes on a nuclear space $S$ are the counterpart of Gaussian random vectors. The scope of this extension is of the same nature as the switch from the classic notion of function to that of a distribution, also known as a "generalized function." Our formalism involves a whitening/regularization operator $L: S\to S'$ whose continuous extension induces a native Hilbert space $H\subset S'$ that plays a central role in our characterization. The presentation is self-contained for the most part and remarkably general and powerful. It allows for the recovery of all known instances of such equivalences; in particular, the methods involving innovations and reproducing-kernel Hilbert spaces developed by Kailath and his students, and the mathematical correspondence between fractional splines and Mandelbrot's fractional Brownian motion (fractals), with the former being the optimal estimators of the latter. It also covers general Bayesian methods for the resolution of infinite-dimensional inverse problems.

math.ST↗

arXiv2026

Geometry Calibration in Tomography with a Differentiable Ray-Based Model

Geometric misalignments between the nominal and true acquisition parameters in tomography degrade reconstructions. We propose a framework that jointly reconstructs the volume and calibrates the acquisition geometry for arbitrary source--detector configurations. The core of our framework is an x-ray transform operator whose gradients with respect to the acquisition geometry can be efficiently computed with a ray-tracing method of structure and computational complexity similar to those of the forward operator. We represent the volume in a B-spline basis to provide a continuously differentiable model. This results in a better-behaved optimization landscape compared to voxel-based representations. We validate our framework with CT, micro-CT, nano-CT, and positron emission tomography data under a variety of geometric misalignments.

eess.IV↗

arXiv2026

Breaking the Weak Recovery Limit in Random Phase Retrieval with Learned Regularizers

We seek to recover an unknown signal from nonlinear amplitude-only measurements, a challenging inverse problem. Strong theoretical guarantees have been established for idealized random measurements, defining the sampling ratio required for signal recovery. However, these results neglect signal priors, which can fundamentally shift these limits, potentially enabling reconstruction with far fewer measurements and simpler models. We evaluate a variety of image priors in the context of severe undersampling with physically-grounded random measurement models. Our results show that these priors enable accurate recovery well below the weak recovery limit, the theoretical threshold required for recovery better than a random guess.

eess.IV↗

arXiv2026

The Gaussian Latent Machine: Efficient Prior and Posterior Sampling for Inverse Problems

We consider the problem of sampling from a product-of-experts-type model that encompasses many standard prior and posterior distributions commonly found in Bayesian imaging. We show that this model can be easily lifted into a novel latent variable model, which we refer to as a Gaussian latent machine. This leads to a general sampling approach that unifies and generalizes many existing sampling algorithms in the literature. Most notably, it yields a highly efficient and effective two-block Gibbs sampling approach in the general case, while also specializing to direct sampling algorithms in particular cases. Finally, we present detailed numerical experiments that demonstrate the efficiency and effectiveness of our proposed sampling approach across a wide range of prior and posterior sampling problems from Bayesian imaging.

eess.IV↗

arXiv2026

Multivariate Fields of Experts for Convergent Image Reconstruction

We introduce the multivariate fields of experts, a new framework for the learning of image priors. Our model generalizes existing fields of experts methods by incorporating multivariate potential functions constructed via Moreau envelopes of the $\ell_\infty$-norm. We demonstrate the effectiveness of our proposal across a range of inverse problems that include image denoising, deblurring, compressed-sensing magnetic-resonance imaging, and computed tomography. The proposed approach outperforms comparable univariate models and achieves performance close to that of deep-learning-based regularizers while being significantly faster, requiring fewer parameters, and being trained on substantially fewer data. In addition, our model retains a high level of interpretability due to its structured design. It is supported by theoretical convergence guarantees which ensure reliability in sensitive reconstruction tasks.

eess.IV↗

arXiv2026

Flower: A Flow-Matching Solver for Inverse Problems

We introduce Flower, a solver for linear inverse problems. It leverages a pre-trained flow model to produce reconstructions that are consistent with the observed measurements. Flower operates through an iterative procedure over three steps: (i) a flow-consistent destination estimation, where the velocity network predicts a denoised target; (ii) a refinement step that projects the estimated destination onto a feasible set defined by the forward operator; and (iii) a time-progression step that re-projects the refined destination along the flow trajectory. We provide a theoretical analysis that demonstrates how Flower approximates Bayesian posterior sampling, thereby unifying perspectives from plug-and-play methods and generative inverse solvers. On the practical side, Flower achieves state-of-the-art reconstruction quality while using nearly identical hyperparameters across various linear inverse problems. Our code is available at https://github.com/mehrsapo/Flower.

cs.CV↗

arXiv2026

A Mini-Batch Quasi-Newton Proximal Method for Constrained Total-Variation Nonlinear Image Reconstruction

Over the years, computational imaging with accurate nonlinear physical models has garnered considerable interest due to its ability to achieve high-quality reconstructions. However, using such nonlinear models for reconstruction is computationally demanding. A popular choice for solving the corresponding inverse problems is the accelerated stochastic proximal method (ASPM), with the caveat that each iteration is still expensive. To overcome this issue, we propose a mini-batch quasi-Newton proximal method (BQNPM) tailored to image reconstruction problems with constrained total variation regularization. Compared to ASPM, BQNPM requires fewer iterations to converge. Moreover, we propose an efficient approach to compute a weighted proximal mapping at a cost similar to that of the proximal mapping in ASPM. We also analyze the convergence of BQNPM in the nonconvex setting. We assess the performance of BQNPM on three-dimensional inverse-scattering problems with linear and nonlinear physical models. Our results on simulated and real data demonstrate the effectiveness and efficiency of BQNPM, while also validating our theoretical analysis.

math.OC↗

arXiv2026

Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizers

This paper addresses the task of learning convex regularizers to guide the reconstruction of images from limited data. By imposing that the reconstruction be amplitude-equivariant, we narrow down the class of admissible functionals to those that can be expressed as a power of a seminorm. We then show that such functionals can be approximated to arbitrary precision with the help of polyhedral norms. In particular, we identify two dual parameterizations of such systems: (i) a synthesis form with an $\ell_1$-penalty that involves some learnable dictionary; and (ii) an analysis form with an $\ell_\infty$-penalty that involves a trainable regularization operator. After having provided geometric insights and proved that the two forms are universal, we propose an implementation that relies on a specific architecture (tight frame with a weighted $\ell_1$ penalty) that is easy to train. We illustrate its use for denoising and the reconstruction of biomedical images. We find that the proposed framework outperforms the sparsity-based methods of compressed sensing, while it offers essentially the same convergence and robustness guarantees.

stat.ML↗

arXiv2026

Revisiting Deep Information Propagation: Fractal Frontier and Finite-size Effects

Information propagation characterizes how input correlations evolve across layers in deep neural networks. This framework has been well studied using mean-field theory, which assumes infinitely wide networks. However, these assumptions break down for practical, finite-size networks. In this work, we study information propagation in randomly initialized neural networks with finite width and reveal that the boundary between ordered and chaotic regimes exhibits a fractal structure. This shows the fundamental complexity of neural network dynamics, in a setting that is independent of input data and optimization. To extend this analysis beyond multilayer perceptrons, we leverage recently introduced Fourier-based structured transforms, and show that information propagation in convolutional neural networks also follow the same behavior. In practice, our investigation highlights the importance of finite network depth with respect to the tradeoff between separation and robustness.

cs.LG↗

arXiv2025

Variational Tensor-Product Splines

Multidimensional continuous-domain inverse problems are often solved by the minimization of a loss functional, formed as the sum of a data fidelity and a regularization. In this work, we present a new construction where the regularization is itself built as the sum of two terms: i) the M norm of the regularizing operator L1 b L2, with L1 and L2 being two one-dimensional differential operators; ii) a bounded-variation norm that regularizes on the infinite-dimensional nullspace of L1 b L2. In this construction, we show that the extreme points of the solution set are the tensor product of one-dimensional splines, with a number of atoms upper-bounded in term of the number of data points. Further, when the data of the inverse problem is localized, we reveal that the term ii) must take the form of a sum of bounded-variation norms, precomposed with partial derivative of different orders.

math.NA↗

arXiv2025

Sampling in BV-Type Spaces

The sampling of functions of bounded variation (BV) is a long-standing problem in op- timization. The ability to sample such functions has relevance in the field of variational inverse problems, where the standard theory fails to guarantee the mere existence of solutions when the loss functional involves samples of BV functions. In this paper, we prove the continuity of sampling functionals and show that the differential operator D admits a unique local inverse. This canonical inversion enables us to formulate an existence theorem for a class of regularized optimization problems that incorporate samples of BV functions. Finally, we characterize the solution set in terms of its extreme points.

math.OC↗