arXiv subjects

Peter Maass

Publications and source records attributed to Peter Maass.

At least 19 recordsLinked to original sources

arXiv2026

A remark on an error analysis for classical and learned Tikhonov regularization schemes

This paper presents an error analysis of classical and learned Tikhonov regularization schemes for inverse problems. We first demonstrate, both theoretically and numerically, that using a fixed regularization parameter across varying noise levels-which is a common miss-specification in practice-has only a mild impact on the reconstruction error. As a special case, we then investigate scenarios where the true data resides in an unknown finite-dimensional subspace. Here, our results lead to an empirically supported strategy for estimating the unknown dimension based on numerical experiments. Finally, we examine the approach that motivated this study: a method where a sparsity-promoting term is learned from denoising tasks and subsequently applied to general inverse problems via a simple heuristic parameter selection. The corresponding error analysis is initially developed using classical concepts and subsequently refined through a more detailed investigation of the discretized setting.

math.NA↗

arXiv2025

Deep Learning Based Reconstruction Methods for Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is a powerful imaging modality widely used in medical diagnostics, industrial monitoring, and environmental studies. The EIT inverse problem is about inferring the internal conductivity distribution of the concerned object from the voltage measurements taken on its boundary. This problem is severely ill-posed, and requires advanced computational approaches for accurate and reliable image reconstruction. Recent innovations in both model-based reconstruction and deep learning have driven significant progress in the field. In this review, we explore learned reconstruction methods that employ deep neural networks for solving the EIT inverse problem. The discussion focuses on the complete electrode model, one popular mathematical model for real-world applications of EIT. We compare a wide variety of learned approaches, including fully-learned, post-processing and learned iterative methods, with several conventional model-based reconstruction techniques, e.g., sparsity regularization, regularized Gauss-Newton iteration and level set method. The evaluation is based on three datasets: a simulated dataset of ellipses, an out-of-distribution simulated dataset, and the KIT4 dataset, including real-world measurements. Our results demonstrate that learned methods outperform model-based methods for in-distribution data but face challenges in generalization, where hybrid methods exhibit a good balance of accuracy and adaptability.

eess.IV↗

arXiv2024

Visualize and Paint GAN Activations

We investigate how generated structures of GANs correlate with their activations in hidden layers, with the purpose of better understanding the inner workings of those models and being able to paint structures with unconditionally trained GANs. This gives us more control over the generated images, allowing to generate them from a semantic segmentation map while not requiring such a segmentation in the training data. To this end we introduce the concept of tileable features, allowing us to identify activations that work well for painting.

cs.CV↗

arXiv2024

Enhancing the analysis of murine neonatal ultrasonic vocalizations: Development, evaluation, and application of different mathematical models

Rodents employ a broad spectrum of ultrasonic vocalizations (USVs) for social communication. As these vocalizations offer valuable insights into affective states, social interactions, and developmental stages of animals, various deep learning approaches have aimed to automate both the quantitative (detection) and qualitative (classification) analysis of USVs. Here, we present the first systematic evaluation of different types of neural networks for USV classification. We assessed various feedforward networks, including a custom-built, fully-connected network and convolutional neural network, different residual neural networks (ResNets), an EfficientNet, and a Vision Transformer (ViT). Paired with a refined, entropy-based detection algorithm (achieving recall of 94.9% and precision of 99.3%), the best architecture (achieving 86.79% accuracy) was integrated into a fully automated pipeline capable of analyzing extensive USV datasets with high reliability. Additionally, users can specify an individual minimum accuracy threshold based on their research needs. In this semi-automated setup, the pipeline selectively classifies calls with high pseudo-probability, leaving the rest for manual inspection. Our study focuses exclusively on neonatal USVs. As part of an ongoing phenotyping study, our pipeline has proven to be a valuable tool for identifying key differences in USVs produced by mice with autism-like behaviors.

cs.SD↗

arXiv2024

Data-driven approaches for electrical impedance tomography image segmentation from partial boundary data

Electrical impedance tomography (EIT) plays a crucial role in non-invasive imaging, with both medical and industrial applications. In this paper, we present three data-driven reconstruction methods for EIT imaging. These three approaches were originally submitted to the Kuopio tomography challenge 2023 (KTC2023). First, we introduce a post-processing approach, which achieved first place at KTC2023. Further, we present a fully learned and a conditional diffusion approach. All three methods are based on a similar neural network as a backbone and were trained using a synthetically generated data set, providing with an opportunity for a fair comparison of these different data-driven reconstruction methods.

eess.IV↗

arXiv2023

Electrical Impedance Tomography: A Fair Comparative Study on Deep Learning and Analytic-based Approaches

Electrical Impedance Tomography (EIT) is a powerful imaging technique with diverse applications, e.g., medical diagnosis, industrial monitoring, and environmental studies. The EIT inverse problem is about inferring the internal conductivity distribution of an object from measurements taken on its boundary. It is severely ill-posed, necessitating advanced computational methods for accurate image reconstructions. Recent years have witnessed significant progress, driven by innovations in analytic-based approaches and deep learning. This review explores techniques for solving the EIT inverse problem, focusing on the interplay between contemporary deep learning-based strategies and classical analytic-based methods. Four state-of-the-art deep learning algorithms are rigorously examined, harnessing the representational capabilities of deep neural networks to reconstruct intricate conductivity distributions. In parallel, two analytic-based methods, rooted in mathematical formulations and regularisation techniques, are dissected for their strengths and limitations. These methodologies are evaluated through various numerical experiments, encompassing diverse scenarios that reflect real-world complexities. A suite of performance metrics is employed to assess the efficacy of these methods. These metrics collectively provide a nuanced understanding of the methods' ability to capture essential features and delineate complex conductivity patterns. One novel feature of the study is the incorporation of variable conductivity scenarios, introducing a level of heterogeneity that mimics textured inclusions. This departure from uniform conductivity assumptions mimics realistic scenarios where tissues or materials exhibit spatially varying electrical properties. Exploring how each method responds to such variable conductivity scenarios opens avenues for understanding their robustness and adaptability.

cs.LG↗

arXiv2023

Steerable Conditional Diffusion for Out-of-Distribution Adaptation in Medical Image Reconstruction

Denoising diffusion models have emerged as the go-to generative framework for solving inverse problems in imaging. A critical concern regarding these models is their performance on out-of-distribution tasks, which remains an under-explored challenge. Using a diffusion model on an out-of-distribution dataset, realistic reconstructions can be generated, but with hallucinating image features that are uniquely present in the training dataset. To address this discrepancy during train-test time and improve reconstruction accuracy, we introduce a novel sampling framework called Steerable Conditional Diffusion. Specifically, this framework adapts the diffusion model, concurrently with image reconstruction, based solely on the information provided by the available measurement. Utilising our proposed method, we achieve substantial enhancements in out-of-distribution performance across diverse imaging modalities, advancing the robust deployment of denoising diffusion models in real-world applications.

cs.CV↗

arXiv2023

Score-Based Generative Models for PET Image Reconstruction

Score-based generative models have demonstrated highly promising results for medical image reconstruction tasks in magnetic resonance imaging or computed tomography. However, their application to Positron Emission Tomography (PET) is still largely unexplored. PET image reconstruction involves a variety of challenges, including Poisson noise with high variance and a wide dynamic range. To address these challenges, we propose several PET-specific adaptations of score-based generative models. The proposed framework is developed for both 2D and 3D PET. In addition, we provide an extension to guided reconstruction using magnetic resonance images. We validate the approach through extensive 2D and 3D $\textit{in-silico}$ experiments with a model trained on patient-realistic data without lesions, and evaluate on data without lesions as well as out-of-distribution data with lesions. This demonstrates the proposed method's robustness and significant potential for improved PET reconstruction.

eess.IV↗

arXiv2023

Parameter Identification by Deep Learning of a Material Model for Granular Media

Classical physical modelling with associated numerical simulation (model-based), and prognostic methods based on the analysis of large amounts of data (data-driven) are the two most common methods used for the mapping of complex physical processes. In recent years, the efficient combination of these approaches has become increasingly important. Continuum mechanics in the core consists of conservation equations that -- in addition to the always necessary specification of the process conditions -- can be supplemented by phenomenological material models. The latter are an idealized image of the specific material behavior that can be determined experimentally, empirically, and based on a wealth of expert knowledge. The more complex the material, the more difficult the calibration is. This situation forms the starting point for this work's hybrid data-driven and model-based approach for mapping a complex physical process in continuum mechanics. Specifically, we use data generated from a classical physical model by the MESHFREE software to train a Principal Component Analysis-based neural network (PCA-NN) for the task of parameter identification of the material model parameters. The obtained results highlight the potential of deep-learning-based hybrid models for determining parameters, which are the key to characterizing materials occurring naturally, and their use in industrial applications (e.g. the interaction of vehicles with sand).

cs.CE↗

arXiv2023

Invertible residual networks in the context of regularization theory for linear inverse problems

Learned inverse problem solvers exhibit remarkable performance in applications like image reconstruction tasks. These data-driven reconstruction methods often follow a two-step scheme. First, one trains the often neural network-based reconstruction scheme via a dataset. Second, one applies the scheme to new measurements to obtain reconstructions. We follow these steps but parameterize the reconstruction scheme with invertible residual networks (iResNets). We demonstrate that the invertibility enables investigating the influence of the training and architecture choices on the resulting reconstruction scheme. For example, assuming local approximation properties of the network, we show that these schemes become convergent regularizations. In addition, the investigations reveal a formal link to the linear regularization theory of linear inverse problems and provide a nonlinear spectral regularization for particular architecture classes. On the numerical side, we investigate the local approximation property of selected trained architectures and present a series of experiments on the MNIST dataset that underpin and extend our theoretical findings.

math.NA↗

arXiv2023

SVD-DIP: Overcoming the Overfitting Problem in DIP-based CT Reconstruction

The deep image prior (DIP) is a well-established unsupervised deep learning method for image reconstruction; yet it is far from being flawless. The DIP overfits to noise if not early stopped, or optimized via a regularized objective. We build on the regularized fine-tuning of a pretrained DIP, by adopting a novel strategy that restricts the learning to the adaptation of singular values. The proposed SVD-DIP uses ad hoc convolutional layers whose pretrained parameters are decomposed via the singular value decomposition. Optimizing the DIP then solely consists in the fine-tuning of the singular values, while keeping the left and right singular vectors fixed. We thoroughly validate the proposed method on real-measured $\mu$CT data of a lotus root as well as two medical datasets (LoDoPaB and Mayo). We report significantly improved stability of the DIP optimization, by overcoming the overfitting to noise.

eess.IV↗

arXiv2023

Model Stitching and Visualization How GAN Generators can Invert Networks in Real-Time

In this work, we propose a fast and accurate method to reconstruct activations of classification and semantic segmentation networks by stitching them with a GAN generator utilizing a 1x1 convolution. We test our approach on images of animals from the AFHQ wild dataset, ImageNet1K, and real-world digital pathology scans of stained tissue samples. Our results show comparable performance to established gradient descent methods but with a processing time that is two orders of magnitude faster, making this approach promising for practical applications.

cs.CV↗

arXiv2022

Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward neural networks, recurrent neural networks, or convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications.

cs.LG↗

arXiv2022

SELTO: Sample-Efficient Learned Topology Optimization

Recent developments in Deep Learning (DL) suggest a vast potential for Topology Optimization (TO). However, while there are some promising attempts, the subfield still lacks a firm footing regarding basic methods and datasets. We aim to address both points. First, we explore physics-based preprocessing and equivariant networks to create sample-efficient components for TO DL pipelines. We evaluate them in a large-scale ablation study using end-to-end supervised training. The results demonstrate a drastic improvement in sample efficiency and the predictions' physical correctness. Second, to improve comparability and future progress, we publish the two first TO datasets containing problems and corresponding ground truth solutions.

cs.LG↗

arXiv2022

PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularization

Learning neural networks using only few available information is an important ongoing research topic with tremendous potential for applications. In this paper, we introduce a powerful regularizer for the variational modeling of inverse problems in imaging. Our regularizer, called patch normalizing flow regularizer (patchNR), involves a normalizing flow learned on small patches of very few images. In particular, the training is independent of the considered inverse problem such that the same regularizer can be applied for different forward operators acting on the same class of images. By investigating the distribution of patches versus those of the whole image class, we prove that our model is indeed a MAP approach. Numerical examples for low-dose and limited-angle computed tomography (CT) as well as superresolution of material images demonstrate that our method provides very high quality results. The training set consists of just six images for CT and one image for superresolution. Finally, we combine our patchNR with ideas from internal learning for performing superresolution of natural images directly from the low-resolution observation without knowledge of any high-resolution image.

cs.LG↗

arXiv2021

Regularized Orthogonal Nonnegative Matrix Factorization and $K$-means Clustering

In this work, we focus on connections between $K$-means clustering approaches and Orthogonal Nonnegative Matrix Factorization (ONMF) methods. We present a novel framework to extract the distance measure and the centroids of the $K$-means method based on first order conditions of the considered ONMF objective function, which exploits the classical alternating minimization schemes of Nonnegative Matrix Factorization (NMF) algorithms. While this technique is characterized by a simple derivation procedure, it can also be applied to non-standard regularized ONMF models. Using this framework, we consider in this work ONMF models with $\ell_1$ and standard $\ell_2$ discrepancy terms with an additional elastic net regularization on both factorization matrices and derive the corresponding distance measures and centroids of the generalized $K$-means clustering model. Furthermore, we give an intuitive view of the obtained results, examine special cases and compare them to the findings described in the literature.

math.NA↗

arXiv2021

Conditional Invertible Neural Networks for Medical Imaging

Over the last years, deep learning methods have become an increasingly popular choice to solve tasks from the field of inverse problems. Many of these new data-driven methods have produced impressive results, although most only give point estimates for the reconstruction. However, especially in the analysis of ill-posed inverse problems, the study of uncertainties is essential. In our work, we apply generative flow-based models based on invertible neural networks to two challenging medical imaging tasks, i.e. low-dose computed tomography and accelerated medical resonance imaging. We test different architectures of invertible neural networks and provide extensive ablation studies. In most applications, a standard Gaussian is used as the base distribution for a flow-based model. Our results show that the choice of a radial distribution can improve the quality of reconstructions.

eess.IV↗

arXiv2020

Ground Truth Free Denoising by Optimal Transport

We present a learned unsupervised denoising method for arbitrary types of data, which we explore on images and one-dimensional signals. The training is solely based on samples of noisy data and examples of noise, which -- critically -- do not need to come in pairs. We only need the assumption that the noise is independent and additive (although we describe how this can be extended). The method rests on a Wasserstein Generative Adversarial Network setting, which utilizes two critics and one generator.

cs.CV↗