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Julián Tachella

Publications and source records attributed to Julián Tachella.

At least 19 recordsLinked to original sources
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

Learning to Recorrupt: Noise Distribution Agnostic Self-Supervised Image Denoising

Self-supervised image denoising methods have traditionally relied on architectural constraints, pseudo-pair constructions, or specialized loss functions to avoid the trivial identity mapping. Among these, approaches such as Noisier2Noise or R2R create training pairs by adding synthetic noise to noisy images. While effective, these recorruption-based approaches require precise knowledge of the noise distribution, which is often unavailable. We present Learning to Recorrupt (L2R), a self-supervised framework that does not require exact knowledge of the noise distribution. Our method introduces a learnable recorruptor jointly optimized with the denoiser through a min--max saddle-point objective. The proposed method achieves state-of-the-art performance among methods without prior knowledge of the noise distribution across unconventional and heavy-tailed noise distributions, such as log-gamma and Laplace, as well as spatially correlated noise, while obtaining competitive results on real-world denoising benchmarks characterized by signal-dependent noise.

eess.IV↗
arXiv2026

UNet-AF: Alias-free UNet architectures

The simplicity and effectiveness of UNet architectures make them ubiquitous in image restoration, segmentation, and diffusion models. They are often assumed to be equivariant to translations, yet they traditionally consist of layers that are known to be prone to aliasing, which hinders their equivariance in practice. To overcome this limitation, we show how to build sub-pixel translation-equivariant UNet architectures by appropriately choosing their main components (convolution, pooling, downsampling, activation, and normalization layers) to be alias-free. We evaluate the proposed equivariant architectures against non-equivariant baselines on image restoration tasks and observe competitive performance with a significant increase in measured equivariance. Through extensive ablation studies, we also demonstrate the importance of every architectural choice to achieve high equivariance. Our implementation is available at https://github.com/jscanvic/UNet-AF

cs.CV↗
arXiv2026

Equivariant Splitting: Self-supervised learning from incomplete data

Self-supervised learning for inverse problems allows to train a reconstruction network from noise and/or incomplete data alone. These methods have the potential of enabling learning-based solutions when obtaining ground-truth references for training is expensive or even impossible. In this paper, we propose a new self-supervised learning strategy devised for the challenging setting where measurements are observed via a single incomplete observation model. We introduce a new definition of equivariance in the context of reconstruction networks, and show that the combination of self-supervised splitting losses and equivariant reconstruction networks results in unbiased estimates of the supervised loss. Through a series of experiments on image inpainting, accelerated magnetic resonance imaging, sparse-view computed tomography, and compressive sensing, we demonstrate that the proposed loss achieves state-of-the-art performance in settings with highly rank-deficient forward models. The code is available at https://github.com/vsechaud/Equivariant-Splitting

cs.CV↗
arXiv2026

Learning to reconstruct from saturated data: audio declipping and high-dynamic range imaging

Learning based methods are now ubiquitous for solving inverse problems, but their deployment in real-world applications is often hindered by the lack of ground truth references for training. Recent self-supervised learning strategies offer a promising alternative, avoiding the need for ground truth. However, most existing methods are limited to linear inverse problems. This work extends self-supervised learning to the non-linear problem of recovering audio and images from clipped measurements, by assuming that the signal distribution is approximately invariant to changes in amplitude. We provide sufficient conditions for learning to reconstruct from saturated signals alone and a self-supervised loss that can be used to train reconstruction networks. Experiments on both audio and image data show that the proposed approach is almost as effective as fully supervised approaches, despite relying solely on clipped measurements for training.

eess.IV↗
arXiv2026

Scale-Equivariant Imaging: Self-Supervised Learning for Image Super-Resolution and Deblurring

Self-supervised methods have recently proved to be nearly as effective as supervised ones in various imaging inverse problems, paving the way for learning-based approaches in scientific and medical imaging applications where ground truth data is hard or expensive to obtain. These methods critically rely on invariance to translations and/or rotations of the image distribution to learn from incomplete measurement data alone. However, existing approaches fail to obtain competitive performances in the problems of image super-resolution and deblurring, which play a key role in most imaging systems. In this work, we show that invariance to roto-translations is insufficient to learn from measurements that only contain low-frequency information. Instead, we propose scale-equivariant imaging, a new self-supervised approach that leverages the fact that many image distributions are approximately scale-invariant, enabling the recovery of high-frequency information lost in the measurement process. We demonstrate throughout a series of experiments on real datasets that the proposed method outperforms other self-supervised approaches, and obtains performances on par with fully supervised learning.

eess.IV↗
arXiv2026

Self-Supervised Learning from Noisy and Incomplete Data

Many important problems in science and engineering involve inferring a signal from noisy and/or incomplete observations, where the observation process is known. Historically, this problem has been tackled using hand-crafted regularization (e.g., sparsity, total-variation) to obtain meaningful estimates. Recent data-driven methods often offer better solutions by directly learning a solver from examples of ground-truth signals and associated observations. However, in many real-world applications, obtaining ground-truth references for training is expensive or impossible. Self-supervised learning methods offer a promising alternative by learning a solver from measurement data alone, bypassing the need for ground-truth references. This manuscript provides a comprehensive summary of different self-supervised methods for inverse problems, with a special emphasis on their theoretical underpinnings, and presents practical applications in imaging inverse problems.

stat.ML↗
arXiv2026

Efficient Unrolled Networks for Large-Scale 3D Inverse Problems

Deep learning-based methods have revolutionized the field of imaging inverse problems, yielding state-of-the-art performance across various imaging domains. The best performing networks incorporate the imaging operator within the network architecture, typically in the form of deep unrolling. However, in large-scale problems, such as 3D imaging, most existing methods fail to incorporate the operator in the architecture due to the prohibitive amount of memory required by global forward operators, which hinder typical patching strategies. In this work, we present a domain partitioning strategy and normal operator approximations that enable the training of end-to-end reconstruction models incorporating forward operators of arbitrarily large problems into their architecture. The proposed method achieves state-of-the-art performance on 3D X-ray cone-beam tomography and 3D multi-coil accelerated MRI, while requiring only a single GPU for both training and inference.

cs.CV↗
arXiv2025

Equivariant Deep Equilibrium Models for Imaging Inverse Problems

Equivariant imaging (EI) enables training signal reconstruction models without requiring ground truth data by leveraging signal symmetries. Deep equilibrium models (DEQs) are a powerful class of neural networks where the output is a fixed point of a learned operator. However, training DEQs with complex EI losses requires implicit differentiation through fixed-point computations, whose implementation can be challenging. We show that backpropagation can be implemented modularly, simplifying training. Experiments demonstrate that DEQs trained with implicit differentiation outperform those trained with Jacobian-free backpropagation and other baseline methods. Additionally, we find evidence that EI-trained DEQs approximate the proximal map of an invariant prior.

eess.IV↗
arXiv2025

Self-supervised learning for phase retrieval

In recent years, deep neural networks have emerged as a solution for inverse imaging problems. These networks are generally trained using pairs of images: one degraded and the other of high quality, the latter being called 'ground truth'. However, in medical and scientific imaging, the lack of fully sampled data limits supervised learning. Recent advances have made it possible to reconstruct images from measurement data alone, eliminating the need for references. However, these methods remain limited to linear problems, excluding non-linear problems such as phase retrieval. We propose a self-supervised method that overcomes this limitation in the case of phase retrieval by using the natural invariance of images to translations.

cs.IR↗
arXiv2025

Translation-Equivariance of Normalization Layers and Aliasing in Convolutional Neural Networks

The design of convolutional neural architectures that are exactly equivariant to continuous translations is an active field of research. It promises to benefit scientific computing, notably by making existing imaging systems more physically accurate. Most efforts focus on the design of downsampling/pooling layers, upsampling layers and activation functions, but little attention is dedicated to normalization layers. In this work, we present a novel theoretical framework for understanding the equivariance of normalization layers to discrete shifts and continuous translations. We also determine necessary and sufficient conditions for normalization layers to be equivariant in terms of the dimensions they operate on. Using real feature maps from ResNet-18 and ImageNet, we test those theoretical results empirically and find that they are consistent with our predictions.

cs.CV↗
arXiv2025

DeepInverse: A Python package for solving imaging inverse problems with deep learning

DeepInverse is an open-source PyTorch-based library for solving imaging inverse problems. The library covers all crucial steps in image reconstruction from the efficient implementation of forward operators (e.g., optics, MRI, tomography), to the definition and resolution of variational problems and the design and training of advanced neural network architectures. In this paper, we describe the main functionality of the library and discuss the main design choices.

eess.IV↗
arXiv2025

Generalized Recorrupted-to-Recorrupted: Self-Supervised Learning Beyond Gaussian Noise

Recorrupted-to-Recorrupted (R2R) has emerged as a methodology for training deep networks for image restoration in a self-supervised manner from noisy measurement data alone, demonstrating equivalence in expectation to the supervised squared loss in the case of Gaussian noise. However, its effectiveness with non-Gaussian noise remains unexplored. In this paper, we propose Generalized R2R (GR2R), extending the R2R framework to handle a broader class of noise distribution as additive noise like log-Rayleigh and address the natural exponential family including Poisson and Gamma noise distributions, which play a key role in many applications including low-photon imaging and synthetic aperture radar. We show that the GR2R loss is an unbiased estimator of the supervised loss and that the popular Stein's unbiased risk estimator can be seen as a special case. A series of experiments with Gaussian, Poisson, and Gamma noise validate GR2R's performance, showing its effectiveness compared to other self-supervised methods.

eess.IV↗
arXiv2025

UNSURE: self-supervised learning with Unknown Noise level and Stein's Unbiased Risk Estimate

Recently, many self-supervised learning methods for image reconstruction have been proposed that can learn from noisy data alone, bypassing the need for ground-truth references. Most existing methods cluster around two classes: i) Stein's Unbiased Risk Estimate (SURE) and similar approaches that assume full knowledge of the noise distribution, and ii) Noise2Self and similar cross-validation methods that require very mild knowledge about the noise distribution. The first class of methods tends to be impractical, as the noise level is often unknown in real-world applications, and the second class is often suboptimal compared to supervised learning. In this paper, we provide a theoretical framework that characterizes this expressivity-robustness trade-off and propose a new approach based on SURE, but unlike the standard SURE, does not require knowledge about the noise level. Throughout a series of experiments, we show that the proposed estimator outperforms other existing self-supervised methods on various imaging inverse problems.

stat.ML↗
arXiv2024

Equivariance-based self-supervised learning for audio signal recovery from clipped measurements

In numerous inverse problems, state-of-the-art solving strategies involve training neural networks from ground truth and associated measurement datasets that, however, may be expensive or impossible to collect. Recently, self-supervised learning techniques have emerged, with the major advantage of no longer requiring ground truth data. Most theoretical and experimental results on self-supervised learning focus on linear inverse problems. The present work aims to study self-supervised learning for the non-linear inverse problem of recovering audio signals from clipped measurements. An equivariance-based selfsupervised loss is proposed and studied. Performance is assessed on simulated clipped measurements with controlled and varied levels of clipping, and further reported on standard real music signals. We show that the performance of the proposed equivariance-based self-supervised declipping strategy compares favorably to fully supervised learning while only requiring clipped measurements alone for training.

eess.AS↗
arXiv2023

Learning to Reconstruct Signals From Binary Measurements

Recent advances in unsupervised learning have highlighted the possibility of learning to reconstruct signals from noisy and incomplete linear measurements alone. These methods play a key role in medical and scientific imaging and sensing, where ground truth data is often scarce or difficult to obtain. However, in practice, measurements are not only noisy and incomplete but also quantized. Here we explore the extreme case of learning from binary observations and provide necessary and sufficient conditions on the number of measurements required for identifying a set of signals from incomplete binary data. Our results are complementary to existing bounds on signal recovery from binary measurements. Furthermore, we introduce a novel self-supervised learning approach, which we name SSBM, that only requires binary data for training. We demonstrate in a series of experiments with real datasets that SSBM performs on par with supervised learning and outperforms sparse reconstruction methods with a fixed wavelet basis by a large margin.

eess.SP↗
arXiv2022

Sensing Theorems for Unsupervised Learning in Linear Inverse Problems

Solving an ill-posed linear inverse problem requires knowledge about the underlying signal model. In many applications, this model is a priori unknown and has to be learned from data. However, it is impossible to learn the model using observations obtained via a single incomplete measurement operator, as there is no information about the signal model in the nullspace of the operator, resulting in a chicken-and-egg problem: to learn the model we need reconstructed signals, but to reconstruct the signals we need to know the model. Two ways to overcome this limitation are using multiple measurement operators or assuming that the signal model is invariant to a certain group action. In this paper, we present necessary and sufficient sensing conditions for learning the signal model from measurement data alone which only depend on the dimension of the model and the number of operators or properties of the group action that the model is invariant to. As our results are agnostic of the learning algorithm, they shed light into the fundamental limitations of learning from incomplete data and have implications in a wide range set of practical algorithms, such as dictionary learning, matrix completion and deep neural networks.

stat.ML↗