arXiv subjects

Michael Unser

Publications and source records attributed to Michael Unser.

At least 37 records · Page 2Linked to original sources

arXiv2025

SplineSplat: 3D Ray Tracing for Higher-Quality Tomography

We propose a method to efficiently compute tomographic projections of a 3D volume represented by a linear combination of shifted B-splines. To do so, we propose a ray-tracing algorithm that computes 3D line integrals with arbitrary projection geometries. One of the components of our algorithm is a neural network that computes the contribution of the basis functions efficiently. In our experiments, we consider well-posed cases where the data are sufficient for accurate reconstruction without the need for regularization. We achieve higher reconstruction quality than traditional voxel-based methods.

cs.CV↗

arXiv2025

Revisiting PSF models: unifying framework and high-performance implementation

Localization microscopy often relies on detailed models of point spread functions. For applications such as deconvolution or PSF engineering, accurate models for light propagation in imaging systems with high numerical aperture are required. Different models have been proposed based on 2D Fourier transforms or 1D Bessel integrals. The most precise ones combine a vectorial description of the electric field and precise aberration models. However, it may be unclear which model to choose, as there is no comprehensive comparison between the Fourier and Bessel approaches yet. Moreover, many existing libraries are written in Java (e.g. our previous PSF generator software) or MATLAB, which hinders the integration into deep learning algorithms. In this work, we start from the original Richards-Wolf integral and revisit both approaches in a systematic way. We present a unifying framework in which we prove the equivalence between the Fourier and Bessel strategies and detail a variety of correction factors applicable to both of them. Then, we provide a high-performance implementation of our theoretical framework in the form of an open-source library that is built on top of PyTorch, a popular library for deep learning. It enables us to benchmark the accuracy and computational speed of different models, thus allowing for an in-depth comparison of the existing models for the first time. We show that the Bessel strategy is optimal for axisymmetric beams while the Fourier approach can be applied to more general scenarios. Our work enables efficient PSF computation on CPU or GPU, which can then be included in simulation and optimization pipelines.

physics.optics↗

arXiv2025

A global inverse-problem approach to quantitative photo-switching optoacoustic mesoscopy

In this paper, we propose a global framework that includes a detailed model of the photo-switching and acoustic processes for photo-switching optoacoustic mesoscopy, based on the underlying physics. We efficiently implement two forward models as matrix-free linear operators and join them as one forward operator. Then, we reconstruct the concentration maps directly from the temporal series of acoustic signals through the resolution of one combined inverse problem. For robustness against noise and clean unmixing results, we adopt a hybrid regularization technique composed of the $l_1$ and total-variation regularizers applied to two different spaces. We use a proximal-gradient algorithm to solve the minimization problem. Our numerical results show that our regularized one-step approach is the most robust in terms of noise and experimental setup. It consistently achieves higher-quality images, as compared to two-step or unregularized methods.

physics.optics↗

arXiv2025

Mixed-Derivative Total Variation

The formulation of norms on continuous-domain Banach spaces with exact pixel-based discretization is advantageous for solving inverse problems (IPs). In this paper, we investigate a new regularization that is a convex combination of a TV term and the $\M(\R^2)$ norm of mixed derivatives. We show that the extreme points of the corresponding unit ball are indicator functions of polygons whose edges are aligned with either the $x_1$- or $x_2$-axis. We then apply this result to construct a new regularization for IPs, which can be discretized exactly by tensor products of first-order B-splines, or equivalently, pixels. Furthermore, we exactly discretize the loss of the denoising problem on its canonical pixel basis and prove that it admits a unique solution, which is also a solution to the underlying continuous-domain IP.

math.NA↗

arXiv2025

A Statistical Benchmark for Diffusion Posterior Sampling Algorithms

We propose a statistical benchmark for diffusion posterior sampling (DPS) algorithms for Bayesian linear inverse problems. The benchmark synthesizes signals from sparse Lévy-process priors whose posteriors admit efficient Gibbs methods. These Gibbs methods can be used to obtain gold-standard posterior samples that can be compared to the samples obtained by the DPS algorithms. By using the Gibbs methods for the resolution of the denoising problems in the reverse diffusion, the framework also isolates the error that arises from the approximations to the likelihood score. We instantiate the benchmark with the minimum-mean-squared-error optimality gap and posterior coverage tests and provide numerical experiments for popular DPS algorithms on the inverse problems of denoising, deconvolution, imputation, and reconstruction from partial Fourier measurements. We release the benchmark code at https://github.com/zacmar/dps-benchmark. The repository exposes simple plug-in interfaces, reference scripts, and config-driven runs so that new algorithms can be added and evaluated with minimal effort. We invite researchers to contribute and report results.

eess.SP↗

arXiv2025

Generalized Ray Tracing with Basis functions for Tomographic Projections

This work aims at the precise and efficient computation of the x-ray projection of an image represented by a linear combination of general shifted basis functions that typically overlap. We achieve this with a suitable adaptation of ray tracing, which is one of the most efficient methods to compute line integrals. In our work, the cases in which the image is expressed as a spline are of particular relevance. The proposed implementation is applicable to any projection geometry as it computes the forward and backward operators over a collection of arbitrary lines. We validate our work with experiments in the context of inverse problems for image reconstruction and maximize the image quality for a given resolution of the reconstruction grid.

eess.IV↗

arXiv2025

Self-Calibrated Variance-Stabilizing Transformations for Real-World Image Denoising

Supervised deep learning has become the method of choice for image denoising. It involves the training of neural networks on large datasets composed of pairs of noisy and clean images. However, the necessity of training data that are specific to the targeted application constrains the widespread use of denoising networks. Recently, several approaches have been developed to overcome this difficulty by whether artificially generating realistic clean/noisy image pairs, or training exclusively on noisy images. In this paper, we show that, contrary to popular belief, denoising networks specialized in the removal of Gaussian noise can be efficiently leveraged in favor of real-world image denoising, even without additional training. For this to happen, an appropriate variance-stabilizing transform (VST) has to be applied beforehand. We propose an algorithm termed Noise2VST for the learning of such a model-free VST. Our approach requires only the input noisy image and an off-the-shelf Gaussian denoiser. We demonstrate through extensive experiments the efficiency and superiority of Noise2VST in comparison to existing methods trained in the absence of specific clean/noisy pairs.

cs.CV↗

arXiv2025

Sensitivity-Aware Density Estimation in Multiple Dimensions

We formulate an optimization problem to estimate probability densities in the context of multidimensional problems that are sampled with uneven probability. It considers detector sensitivity as an heterogeneous density and takes advantage of the computational speed and flexible boundary conditions offered by splines on a grid. We choose to regularize the Hessian of the spline via the nuclear norm to promote sparsity. As a result, the method is spatially adaptive and stable against the choice of the regularization parameter, which plays the role of the bandwidth. We test our computational pipeline on standard densities and provide software. We also present a new approach to PET rebinning as an application of our framework.

cs.LG↗

arXiv2025

Controlled Learning of Pointwise Nonlinearities in Neural-Network-Like Architectures

We present a general variational framework for the training of freeform nonlinearities in layered computational architectures subject to some slope constraints. The regularization that we add to the traditional training loss penalizes the second-order total variation of each trainable activation. The slope constraints allow us to impose properties such as 1-Lipschitz stability, firm non-expansiveness, and monotonicity/invertibility. These properties are crucial to ensure the proper functioning of certain classes of signal-processing algorithms (e.g., plug-and-play schemes, unrolled proximal gradient, invertible flows). We prove that the global optimum of the stated constrained-optimization problem is achieved with nonlinearities that are adaptive nonuniform linear splines. We then show how to solve the resulting function-optimization problem numerically by representing the nonlinearities in a suitable (nonuniform) B-spline basis. Finally, we illustrate the use of our framework with the data-driven design of (weakly) convex regularizers for the denoising of images and the resolution of inverse problems.

stat.ML↗

arXiv2025

Learning of Patch-Based Smooth-Plus-Sparse Models for Image Reconstruction

We aim at the solution of inverse problems in imaging, by combining a penalized sparse representation of image patches with an unconstrained smooth one. This allows for a straightforward interpretation of the reconstruction. We formulate the optimization as a bilevel problem. The inner problem deploys classical algorithms while the outer problem optimizes the dictionary and the regularizer parameters through supervised learning. The process is carried out via implicit differentiation and gradient-based optimization. We evaluate our method for denoising, super-resolution, and compressed-sensing magnetic-resonance imaging. We compare it to other classical models as well as deep-learning-based methods and show that it always outperforms the former and also the latter in some instances.

eess.IV↗

arXiv2025

Perturbative Fourier Ptychographic Microscopy for Fast Quantitative Phase Imaging

In computational phase imaging with a microscope equipped with an array of light emitting diodes as illumination unit, conventional Fourier ptychographic microscopy achieves high resolution and wide-field reconstructions but is constrained by a lengthy acquisition time. Conversely, differential phase contrast (DPC) offers fast imaging but is limited in resolution. Here, we introduce perturbative Fourier ptychographic microscopy (pFPM). pFPM is an extension of DPC that incorporates dark-field illumination to enable fast, high-resolution, wide-field quantitative phase imaging with few measurements. We interpret DPC as the initial iteration of a Gauss-Newton algorithm with quadratic regularization and generalize it to multiple iterations and more sophisticated regularizers. This broader framework is not restricted to bright-field measurements and allows us to overcome resolution limitations of DPC. We develop tailored annular dark-field illumination patterns that align with the perturbative interpretation and lead to an improvement in the quality of reconstruction with respect to other common illumination schemes. Consequently, our methodology combines an enhanced phase reconstruction algorithm with a specialized illumination strategy and offers significant advantages in both imaging speed and resolution.

physics.optics↗

arXiv2025

Random ReLU Neural Networks as Non-Gaussian Processes

We consider a large class of shallow neural networks with randomly initialized parameters and rectified linear unit activation functions. We prove that these random neural networks are well-defined non-Gaussian processes. As a by-product, we demonstrate that these networks are solutions to stochastic differential equations driven by impulsive white noise (combinations of random Dirac measures). These processes are parameterized by the law of the weights and biases as well as the density of activation thresholds in each bounded region of the input domain. We prove that these processes are isotropic and wide-sense self-similar with Hurst exponent 3/2. We also derive a remarkably simple closed-form expression for their autocovariance function. Our results are fundamentally different from prior work in that we consider a non-asymptotic viewpoint: The number of neurons in each bounded region of the input domain (i.e., the width) is itself a random variable with a Poisson law with mean proportional to the density parameter. Finally, we show that, under suitable hypotheses, as the expected width tends to infinity, these processes can converge in law not only to Gaussian processes, but also to non-Gaussian processes depending on the law of the weights. Our asymptotic results provide a new take on several classical results (wide networks converge to Gaussian processes) as well as some new ones (wide networks can converge to non-Gaussian processes).

stat.ML↗

arXiv2025

DEALing with Image Reconstruction: Deep Attentive Least Squares

State-of-the-art image reconstruction often relies on complex, highly parameterized deep architectures. We propose an alternative: a data-driven reconstruction method inspired by the classic Tikhonov regularization. Our approach iteratively refines intermediate reconstructions by solving a sequence of quadratic problems. These updates have two key components: (i) learned filters to extract salient image features, and (ii) an attention mechanism that locally adjusts the penalty of filter responses. Our method achieves performance on par with leading plug-and-play and learned regularizer approaches while offering interpretability, robustness, and convergent behavior. In effect, we bridge traditional regularization and deep learning with a principled reconstruction approach.

eess.IV↗

arXiv2025

Comparison of 2D Regular Lattices for the CPWL Approximation of Functions

We investigate the approximation error of functions with continuous and piecewise-linear (CPWL) representations. We focus on the CPWL search spaces generated by translates of box splines on two-dimensional regular lattices. We compute the approximation error in terms of the stepsize and angles that define the lattice. Our results show that hexagonal lattices are optimal, in the sense that they minimize the asymptotic approximation error.

math.NA↗

arXiv2025

Model-based temporal unmixing towards quantitative photo-switching optoacoustic tomography

Optoacoustic (OA) imaging combined with reversibly photoswitchable proteins has emerged as a promising technology for the high-sensitivity and multiplexed imaging of cells in live tissues in preclinical research. Through carefully-designed illumination schedules of ON and OFF laser pulses, the resulting OA signal is a multiplex of different reporter species and the background. We propose a model-based variational framework to computationally unmix and image different species of photo-switching reporters using optoacoustic tomography. It is based on a detailed mathematical description of the photo-switching mechanism, which models how relevant physical parameters such as the kinetic constants and light fluence impact the switching signal. We introduce an algorithm that operates on images, as opposed to traditional pixelwise approaches. It takes the form of an iterative inversion combined with tailored $\ell_1$ and total-variation regularization to increase the robustness to noise and to improve the unmixing quality. We show that our method is able to disentangle multiple spatially overlapping labels and to recover continuous maps of quantities of interest on controlled phantoms and mice experiments.

physics.optics↗

arXiv2025

Function-Space Optimality of Neural Architectures with Multivariate Nonlinearities

We investigate the function-space optimality (specifically, the Banach-space optimality) of a large class of shallow neural architectures with multivariate nonlinearities/activation functions. To that end, we construct a new family of Banach spaces defined via a regularization operator, the $k$-plane transform, and a sparsity-promoting norm. We prove a representer theorem that states that the solution sets to learning problems posed over these Banach spaces are completely characterized by neural architectures with multivariate nonlinearities. These optimal architectures have skip connections and are tightly connected to orthogonal weight normalization and multi-index models, both of which have received recent interest in the neural network community. Our framework is compatible with a number of classical nonlinearities including the rectified linear unit (ReLU) activation function, the norm activation function, and the radial basis functions found in the theory of thin-plate/polyharmonic splines. We also show that the underlying spaces are special instances of reproducing kernel Banach spaces and variation spaces. Our results shed light on the regularity of functions learned by neural networks trained on data, particularly with multivariate nonlinearities, and provide new theoretical motivation for several architectural choices found in practice.

stat.ML↗

arXiv2024

Point-Spread-Function Engineering in MINFLUX: Optimality of Donut and Half-Moon Excitation Patterns

Localization microscopy enables imaging with resolutions that surpass the conventional optical diffraction limit. Notably, the MINFLUX method achieves super-resolution by shaping the excitation point-spread function (PSF) to minimize the required photon flux for a given precision. Various beam shapes have recently been proposed to improve localization efficiency, yet their optimality remains an open question. In this work, we deploy a numerical and theoretical framework to determine optimal excitation patterns for MINFLUX. Such a computational approach allows us to search for new beam patterns in a fast and low-cost fashion, and to avoid time-consuming and expensive experimental explorations. We show that the conventional donut beam is a robust optimum when the excitation beams are all constrained to the same shape. Further, our PSF engineering framework yields two pairs of half-moon beams (orthogonal to each other) which can improve the theoretical localization precision by a factor of about two.

physics.optics↗

arXiv2024

Parseval Convolution Operators and Neural Networks

We first establish a kernel theorem that characterizes all linear shift-invariant (LSI) operators acting on discrete multicomponent signals. This result naturally leads to the identification of the Parseval convolution operators as the class of energy-preserving filterbanks. We then present a constructive approach for the design/specification of such filterbanks via the chaining of elementary Parseval modules, each of which being parameterized by an orthogonal matrix or a 1-tight frame. Our analysis is complemented with explicit formulas for the Lipschitz constant of all the components of a convolutional neural network (CNN), which gives us a handle on their stability. Finally, we demonstrate the usage of those tools with the design of a CNN-based algorithm for the iterative reconstruction of biomedical images. Our algorithm falls within the plug-and-play framework for the resolution of inverse problems. It yields better-quality results than the sparsity-based methods used in compressed sensing, while offering essentially the same convergence and robustness guarantees.

eess.SP↗