Martin Benning

Martin Benning, Martin Burger

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from linear towards nonlinear regularization methods even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this development towards modern nonlinear regularization methods, including their analysis, applications, and issues for future research. In particular we will discuss variational methods and techniques derived from those, since they have attracted particular interest in the last years and link to other fields like image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions, and learning theory.

Martin Benning, Martin Burger

Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods. In the last decade nonlinear variational approaches such as $\ell^1$ or total variation regularizations became quite prominent regularization techniques with certain properties being superior to standard methods. In the analysis of those, singular values and vectors did not play any role so far, for the obvious reason that these problems are nonlinear, together with the issue of defining singular values and singular vectors. In this paper however we want to start a study of singular values and vectors for nonlinear variational regularization of linear inverse problems, with particular focus on singular one-homogeneous regularization functionals. A major role is played by the smallest singular value, which we define as the ground state of an appropriate functional combining the (semi-)norm introduced by the forward operator and the regularization functional. The optimality condition for the ground state further yields a natural generalization to higher singular values and vectors involving the subdifferential of the regularization functional. We carry over two main properties from the world of linear regularization. The first one is gaining information about scale, respectively the behavior of regularization techniques at different scales. This also leads to novel estimates at different scales, generalizing the estimates for the coefficients in the linear singular value expansion. The second one is to provide exact solutions for variational regularization methods. We will show that all singular vectors can be reconstructed up to a scalar factor by the standard Tikhonov-type regularization approach even in the presence of (small) noise. Moreover, we will show that they can even be reconstructed without any bias by the recently popularized inverse scale space method.

Martin Benning, Florian Knoll, Carola-Bibiane Schönlieb et al.

We are presenting a modification of the well-known Alternating Direction Method of Multipliers (ADMM) algorithm with additional preconditioning that aims at solving convex optimisation problems with nonlinear operator constraints. Connections to the recently developed Nonlinear Primal-Dual Hybrid Gradient Method (NL-PDHGM) are presented, and the algorithm is demonstrated to handle the nonlinear inverse problem of parallel Magnetic Resonance Imaging (MRI).

Veronica Corona, Martin Benning, Matthias J. Ehrhardt et al.

All imaging modalities such as computed tomography (CT), emission tomography and magnetic resonance imaging (MRI) require a reconstruction approach to produce an image. A common image processing task for applications that utilise those modalities is image segmentation, typically performed posterior to the reconstruction. We explore a new approach that combines reconstruction and segmentation in a unified framework. We derive a variational model that consists of a total variation regularised reconstruction from undersampled measurements and a Chan-Vese based segmentation. We extend the variational regularisation scheme to a Bregman iteration framework to improve the reconstruction and therefore the segmentation. We develop a novel alternating minimisation scheme that solves the non-convex optimisation problem with provable convergence guarantees. Our results for synthetic and real data show that both reconstruction and segmentation are improved compared to the classical sequential approach.

Marie Foged Schmidt, Martin Benning, Carola-Bibiane Schönlieb

We investigate the inverse scale space flow as a decomposition method for decomposing data into generalised singular vectors. We show that the inverse scale space flow, based on convex and absolutely one-homogeneous regularisation functionals, can decompose data represented by the application of a forward operator to a linear combination of generalised singular vectors into its individual singular vectors. We verify that for this decomposition to hold true, two additional conditions on the singular vectors are sufficient: orthogonality in the data space and inclusion of partial sums of the subgradients of the singular vectors in the subdifferential of the regularisation functional at zero. We also address the converse question of when the inverse scale space flow returns a generalised singular vector given that the initial data is arbitrary (and therefore not necessarily in the range of the forward operator). We prove that the inverse scale space flow is guaranteed to return a singular vector if the data satisfies a novel dual singular vector condition. We conclude the paper with numerical results that validate the theoretical results and that demonstrate the importance of the additional conditions required to guarantee the decomposition result.

Samira Kabri, Alexander Auras, Danilo Riccio et al.

The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computerized tomography (CT). As the (naive) solution does not depend on the measured data continuously, regularization is needed to re-establish a continuous dependence. In this work, we investigate simple, but yet still provably convergent approaches to learning linear regularization methods from data. More specifically, we analyze two approaches: One generic linear regularization that learns how to manipulate the singular values of the linear operator in an extension of our previous work, and one tailored approach in the Fourier domain that is specific to CT-reconstruction. We prove that such approaches become convergent regularization methods as well as the fact that the reconstructions they provide are typically much smoother than the training data they were trained on. Finally, we compare the spectral as well as the Fourier-based approaches for CT-reconstruction numerically, discuss their advantages and disadvantages and investigate the effect of discretization errors at different resolutions.

Martin Benning, Carola-Bibiane Schönlieb, Tuomo Valkonen et al.

We explore anisotropic regularisation methods in the spirit of [Holler & Kunisch, 14]. Based on ground truth data, we propose a bilevel optimisation strategy to compute the optimal regularisation parameters of such a model for the application of video denoising. The optimisation poses a challenge in itself, as the dependency on one of the regularisation parameters is non-linear such that the standard existence and convergence theory does not apply. Moreover, we analyse numerical results of the proposed parameter learning strategy based on three exemplary video sequences and discuss the impact of these results on the actual modelling of dynamic inverse problems.

Xiaoyu Wang, Martin Benning

We introduce a novel mathematical formulation for the training of feed-forward neural networks with (potentially non-smooth) proximal maps as activation functions. This formulation is based on Bregman distances and a key advantage is that its partial derivatives with respect to the network's parameters do not require the computation of derivatives of the network's activation functions. Instead of estimating the parameters with a combination of first-order optimisation method and back-propagation (as is the state-of-the-art), we propose the use of non-smooth first-order optimisation methods that exploit the specific structure of the novel formulation. We present several numerical results that demonstrate that these training approaches can be equally well or even better suited for the training of neural network-based classifiers and (denoising) autoencoders with sparse coding compared to more conventional training frameworks.

Xiaoyu Wang, Martin Benning

We propose a novel framework for the regularised inversion of deep neural networks. The framework is based on the authors' recent work on training feed-forward neural networks without the differentiation of activation functions. The framework lifts the parameter space into a higher dimensional space by introducing auxiliary variables, and penalises these variables with tailored Bregman distances. We propose a family of variational regularisations based on these Bregman distances, present theoretical results and support their practical application with numerical examples. In particular, we present the first convergence result (to the best of our knowledge) for the regularised inversion of a single-layer perceptron that only assumes that the solution of the inverse problem is in the range of the regularisation operator, and that shows that the regularised inverse provably converges to the true inverse if measurement errors converge to zero.

Tatiana Gaintseva, Andrew Stepanov, Ziquan Liu et al.

Steering intermediate representations has emerged as a powerful strategy for controlling generative models, particularly in post-deployment alignment and safety settings. However, despite its empirical success, it currently lacks a comprehensive theoretical framework. In this paper, we bridge this gap by formalizing the theory of concept steering. First, we establish a link between steering and affine concept erasure, proving that the standard approach for removing unwanted behaviors is a special case of LEACE (a closed-form method for affine erasure). Next, we formulate a principled theoretical framework for concept switching, LEACE-Switch, and characterize the assumptions under which it provides an optimal affine solution. Building on this analysis, we then introduce MidSteer (Minimal Disturbance concept Steering), a more general affine framework for concept manipulation that relaxes these assumptions and enables directed, minimal-disturbance transformations. We demonstrate that MidSteer performs favorably across a range of tasks, modalities, and architectures, including vision diffusion models and large language models.

Abbas Khan, Muhammad Asad, Martin Benning et al.

Cardiac image segmentation is essential for automated cardiac function assessment and monitoring of changes in cardiac structures over time. Inspired by coarse-to-fine approaches in image analysis, we propose a novel multitask compositional segmentation approach that can simultaneously localize the heart in a cardiac image and perform part-based segmentation of different regions of interest. We demonstrate that this compositional approach achieves better results than direct segmentation of the anatomies. Further, we propose a novel Cross-Modal Feature Integration (CMFI) module to leverage the metadata related to cardiac imaging collected during image acquisition. We perform experiments on two different modalities, MRI and ultrasound, using public datasets, Multi-disease, Multi-View, and Multi-Centre (M&Ms-2) and Multi-structure Ultrasound Segmentation (CAMUS) data, to showcase the efficiency of the proposed compositional segmentation method and Cross-Modal Feature Integration module incorporating metadata within the proposed compositional segmentation network. The source code is available: https://github.com/kabbas570/CompSeg-MetaData.

Abbas Khan, Muhammad Asad, Martin Benning et al.

Convolutional Neural Networks (CNNs) and Transformer-based self-attention models have become the standard for medical image segmentation. This paper demonstrates that convolution and self-attention, while widely used, are not the only effective methods for segmentation. Breaking with convention, we present a Convolution and self-Attention-free Mamba-based semantic Segmentation Network named CAMS-Net. Specifically, we design Mamba-based Channel Aggregator and Spatial Aggregator, which are applied independently in each encoder-decoder stage. The Channel Aggregator extracts information across different channels, and the Spatial Aggregator learns features across different spatial locations. We also propose a Linearly Interconnected Factorized Mamba (LIFM) block to reduce the computational complexity of a Mamba block and to enhance its decision function by introducing a non-linearity between two factorized Mamba blocks. Our model outperforms the existing state-of-the-art CNN, self-attention, and Mamba-based methods on CMR and M&Ms-2 Cardiac segmentation datasets, showing how this innovative, convolution, and self-attention-free method can inspire further research beyond CNN and Transformer paradigms, achieving linear complexity and reducing the number of parameters. Source code and pre-trained models are available at: https://github.com/kabbas570/CAMS-Net.

Xiaoyu Wang, Martin Benning, Audrey Repetti

Unfolded proximal neural networks (PNNs) form a family of methods that combines deep learning and proximal optimization approaches. They consist in designing a neural network for a specific task by unrolling a proximal algorithm for a fixed number of iterations, where linearities can be learned from prior training procedure. PNNs have shown to be more robust than traditional deep learning approaches while reaching at least as good performances, in particular in computational imaging. However, training PNNs still depends on the efficiency of available training algorithms. In this work, we propose a lifted training formulation based on Bregman distances for unfolded PNNs. Leveraging the deterministic mini-batch block-coordinate forward-backward method, we design a bespoke computational strategy beyond traditional back-propagation methods for solving the resulting learning problem efficiently. We assess the behaviour of the proposed training approach for PNNs through numerical simulations on image denoising, considering a denoising PNN whose structure is based on dual proximal-gradient iterations.

Tatiana Gaintseva, Martin Benning, Gregory Slabaugh

In this paper we propose a novel modification of Contrastive Language-Image Pre-Training (CLIP) guidance for the task of unsupervised backlit image enhancement. Our work builds on the state-of-the-art CLIP-LIT approach, which learns a prompt pair by constraining the text-image similarity between a prompt (negative/positive sample) and a corresponding image (backlit image/well-lit image) in the CLIP embedding space. Learned prompts then guide an image enhancement network. Based on the CLIP-LIT framework, we propose two novel methods for CLIP guidance. First, we show that instead of tuning prompts in the space of text embeddings, it is possible to directly tune their embeddings in the latent space without any loss in quality. This accelerates training and potentially enables the use of additional encoders that do not have a text encoder. Second, we propose a novel approach that does not require any prompt tuning. Instead, based on CLIP embeddings of backlit and well-lit images from training data, we compute the residual vector in the embedding space as a simple difference between the mean embeddings of the well-lit and backlit images. This vector then guides the enhancement network during training, pushing a backlit image towards the space of well-lit images. This approach further dramatically reduces training time, stabilizes training and produces high quality enhanced images without artifacts, both in supervised and unsupervised training regimes. Additionally, we show that residual vectors can be interpreted, revealing biases in training data, and thereby enabling potential bias correction.

Abbas Khan, Muhammad Asad, Martin Benning et al.

Diagnosis of cardiovascular disease using automated methods often relies on the critical task of cardiac image segmentation. We propose a novel strategy that performs segmentation using specialist networks that focus on a single anatomy (left ventricle, right ventricle, or myocardium). Given an input long-axis cardiac MR image, our method performs a ternary segmentation in the first stage to identify these anatomical regions, followed by cropping the original image to focus subsequent processing on the anatomical regions. The specialist networks are coupled through an attention mechanism that performs cross-attention to interlink features from different anatomies, serving as a soft relative shape prior. Central to our approach is an additive attention block (E-2A block), which is used throughout our architecture thanks to its efficiency.

Yuan Jin, Antonio Pepe, Gian Marco Melito et al.

The automated analysis of the aortic vessel tree (AVT) from computed tomography angiography (CTA) holds immense clinical potential, but its development has been impeded by a lack of shared, high-quality data. We launched the SEG.A. challenge to catalyze progress in this field by introducing a large, publicly available, multi-institutional dataset for AVT segmentation. The challenge benchmarked automated algorithms on a hidden test set, with subsequent optional tasks in surface meshing for computational simulations. Our findings reveal a clear convergence on deep learning methodologies, with 3D U-Net architectures dominating the top submissions. A key result was that an ensemble of the highest-ranking algorithms significantly outperformed individual models, highlighting the benefits of model fusion. Performance was strongly linked to algorithmic design, particularly the use of customized post-processing steps, and the characteristics of the training data. This initiative not only establishes a new performance benchmark but also provides a lasting resource to drive future innovation toward robust, clinically translatable tools.

Xiaoyu Wang, Alexandra Valavanis, Azhir Mahmood et al.

The training of deep neural networks predominantly relies on a combination of gradient-based optimisation and back-propagation for the computation of the gradient. While incredibly successful, this approach faces challenges such as vanishing or exploding gradients, difficulties with non-smooth activations, and an inherently sequential structure that limits parallelisation. Lifted training methods offer an alternative by reformulating the nested optimisation problem into a higher-dimensional, constrained optimisation problem where the constraints are no longer enforced directly but penalised with penalty terms. This chapter introduces a unified framework that encapsulates various lifted training strategies, including the Method of Auxiliary Coordinates, Fenchel Lifted Networks, and Lifted Bregman Training, and demonstrates how diverse architectures, such as Multi-Layer Perceptrons, Residual Neural Networks, and Proximal Neural Networks fit within this structure. By leveraging tools from convex optimisation, particularly Bregman distances, the framework facilitates distributed optimisation, accommodates non-differentiable proximal activations, and can improve the conditioning of the training landscape. We discuss the implementation of these methods using block-coordinate descent strategies, including deterministic implementations enhanced by accelerated and adaptive optimisation techniques, as well as implicit stochastic gradient methods. Furthermore, we explore the application of this framework to inverse problems, detailing methodologies for both the training of specialised networks (e.g., unrolled architectures) and the stable inversion of pre-trained networks. Numerical results on standard imaging tasks validate the effectiveness and stability of the lifted Bregman approach compared to conventional training, particularly for architectures employing proximal activations.

Dao Lin, Jian Zhang, Martin Benning

Rotating synthetic aperture (RSA) imaging system captures images of the target scene at different rotation angles by rotating a rectangular aperture. Deblurring acquired RSA images plays a critical role in reconstructing a latent sharp image underlying the scene. In the past decade, the emergence of blind convolution technology has revolutionised this field by its ability to model complex features from acquired images. Most of the existing methods attempt to solve the above ill-posed inverse problem through maximising a posterior. Despite this progress, researchers have paid limited attention to exploring low-dimensional manifold structures of the latent image within a high-dimensional ambient-space. Here, we propose a novel method to process RSA images using manifold fitting and penalisation in the content of multi-frame blind convolution. We develop fast algorithms for implementing the proposed procedure. Simulation studies demonstrate that manifold-based deconvolution can outperform conventional deconvolution algorithms in the sense that it can generate a sharper estimate of the latent image in terms of estimating pixel intensities and preserving structural details.

Tatiana Gaintseva, Laida Kushnareva, German Magai et al.

With growing abilities of generative models, artificial content detection becomes an increasingly important and difficult task. However, all popular approaches to this problem suffer from poor generalization across domains and generative models. In this work, we focus on the robustness of AI-generated image (AIGI) detectors. We analyze existing state-of-the-art AIGI detection methods based on frozen CLIP embeddings and show how to interpret them, shedding light on how images produced by various AI generators differ from real ones. Next we propose two ways to improve robustness: based on removing harmful components of the embedding vector and based on selecting the best performing attention heads in the image encoder model. Our methods increase the mean out-of-distribution (OOD) classification score by up to 6% for cross-model transfer. We also propose a new dataset for AIGI detection and use it in our evaluation; we believe this dataset will help boost further research. The dataset and code are provided as a supplement.

Abbas Khan, Muhammad Asad, Martin Benning et al.

We propose a novel multi-stage trans-dimensional architecture for multi-view cardiac image segmentation. Our method exploits the relationship between long-axis (2D) and short-axis (3D) magnetic resonance (MR) images to perform a sequential 3D-to-2D-to-3D segmentation, segmenting the long-axis and short-axis images. In the first stage, 3D segmentation is performed using the short-axis image, and the prediction is transformed to the long-axis view and used as a segmentation prior in the next stage. In the second step, the heart region is localized and cropped around the segmentation prior using a Heart Localization and Cropping (HLC) module, focusing the subsequent model on the heart region of the image, where a 2D segmentation is performed. Similarly, we transform the long-axis prediction to the short-axis view, localize and crop the heart region and again perform a 3D segmentation to refine the initial short-axis segmentation. We evaluate our proposed method on the Multi-Disease, Multi-View & Multi-Center Right Ventricular Segmentation in Cardiac MRI (M&Ms-2) dataset, where our method outperforms state-of-the-art methods in segmenting cardiac regions of interest in both short-axis and long-axis images. The pre-trained models, source code, and implementation details will be publicly available.

Russell Sammut Bonnici, Charalampos Saitis, Martin Benning

This research project investigates the application of deep learning to timbre transfer, where the timbre of a source audio can be converted to the timbre of a target audio with minimal loss in quality. The adopted approach combines Variational Autoencoders with Generative Adversarial Networks to construct meaningful representations of the source audio and produce realistic generations of the target audio and is applied to the Flickr 8k Audio dataset for transferring the vocal timbre between speakers and the URMP dataset for transferring the musical timbre between instruments. Furthermore, variations of the adopted approach are trained, and generalised performance is compared using the metrics SSIM (Structural Similarity Index) and FAD (Frechét Audio Distance). It was found that a many-to-many approach supersedes a one-to-one approach in terms of reconstructive capabilities, and that the adoption of a basic over a bottleneck residual block design is more suitable for enriching content information about a latent space. It was also found that the decision on whether cyclic loss takes on a variational autoencoder or vanilla autoencoder approach does not have a significant impact on reconstructive and adversarial translation aspects of the model.

Xiaoyu Wang, Martin Benning

We present a generalisation of Rosenblatt's traditional perceptron learning algorithm to the class of proximal activation functions and demonstrate how this generalisation can be interpreted as an incremental gradient method applied to a novel energy function. This novel energy function is based on a generalised Bregman distance, for which the gradient with respect to the weights and biases does not require the differentiation of the activation function. The interpretation as an energy minimisation algorithm paves the way for many new algorithms, of which we explore a novel variant of the iterative soft-thresholding algorithm for the learning of sparse perceptrons.

Ferdia Sherry, Martin Benning, Juan Carlos De los Reyes et al.

The discovery of the theory of compressed sensing brought the realisation that many inverse problems can be solved even when measurements are "incomplete". This is particularly interesting in magnetic resonance imaging (MRI), where long acquisition times can limit its use. In this work, we consider the problem of learning a sparse sampling pattern that can be used to optimally balance acquisition time versus quality of the reconstructed image. We use a supervised learning approach, making the assumption that our training data is representative enough of new data acquisitions. We demonstrate that this is indeed the case, even if the training data consists of just 7 training pairs of measurements and ground-truth images; with a training set of brain images of size 192 by 192, for instance, one of the learned patterns samples only 35% of k-space, however results in reconstructions with mean SSIM 0.914 on a test set of similar images. The proposed framework is general enough to learn arbitrary sampling patterns, including common patterns such as Cartesian, spiral and radial sampling.

Martin Benning, Elena Celledoni, Matthias J. Ehrhardt et al.

We consider recent work of Haber and Ruthotto 2017 and Chang et al. 2018, where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. We review the first order conditions for optimality, and the conditions ensuring optimality after discretisation. This leads to a class of algorithms for solving the discrete optimal control problem which guarantee that the corresponding discrete necessary conditions for optimality are fulfilled. The differential equation setting lends itself to learning additional parameters such as the time discretisation. We explore this extension alongside natural constraints (e.g. time steps lie in a simplex). We compare these deep learning algorithms numerically in terms of induced flow and generalisation ability.

Martin Benning, Michael Möller, Raz Z. Nossek et al.

In this paper we demonstrate that the framework of nonlinear spectral decompositions based on total variation (TV) regularization is very well suited for image fusion as well as more general image manipulation tasks. The well-localized and edge-preserving spectral TV decomposition allows to select frequencies of a certain image to transfer particular features, such as wrinkles in a face, from one image to another. We illustrate the effectiveness of the proposed approach in several numerical experiments, including a comparison to the competing techniques of Poisson image editing, linear osmosis, wavelet fusion and Laplacian pyramid fusion. We conclude that the proposed spectral TV image decomposition framework is a valuable tool for semi- and fully-automatic image editing and fusion.

Michael Moeller, Martin Benning, Carola Schönlieb et al.

This paper deals with the problem of reconstructing a depth map from a sequence of differently focused images, also known as depth from focus or shape from focus. We propose to state the depth from focus problem as a variational problem including a smooth but nonconvex data fidelity term, and a convex nonsmooth regularization, which makes the method robust to noise and leads to more realistic depth maps. Additionally, we propose to solve the nonconvex minimization problem with a linearized alternating directions method of multipliers (ADMM), allowing to minimize the energy very efficiently. A numerical comparison to classical methods on simulated as well as on real data is presented.