Ozan Öktem

Erik Jansson, Jonathan Krook, Klas Modin et al.

We address recovery of the three-dimensional backbone structure of single polypeptide proteins from single-particle cryo-electron microscopy (Cryo-SPA) data. Cryo-SPA produces noisy tomographic projections of electrostatic potentials of macromolecules. From these projections, we use methods from shape analysis to recover the three-dimensional backbone structure. Thus, we view the reconstruction problem as an indirect matching problem, where a point cloud representation of the protein backbone is deformed to match 2D tomography data. The deformations are obtained via the action of a matrix Lie group. By selecting a deformation energy, the optimality conditions are obtained, which lead to computational algorithms for optimal deformations. We showcase our approach on synthetic data, for which we recover the three-dimensional structure of the backbone.

Subhadip Mukherjee, Andreas Hauptmann, Ozan Öktem et al.

In recent years, deep learning has achieved remarkable empirical success for image reconstruction. This has catalyzed an ongoing quest for precise characterization of correctness and reliability of data-driven methods in critical use-cases, for instance in medical imaging. Notwithstanding the excellent performance and efficacy of deep learning-based methods, concerns have been raised regarding their stability, or lack thereof, with serious practical implications. Significant advances have been made in recent years to unravel the inner workings of data-driven image recovery methods, challenging their widely perceived black-box nature. In this article, we will specify relevant notions of convergence for data-driven image reconstruction, which will form the basis of a survey of learned methods with mathematically rigorous reconstruction guarantees. An example that is highlighted is the role of ICNN, offering the possibility to combine the power of deep learning with classical convex regularization theory for devising methods that are provably convergent. This survey article is aimed at both methodological researchers seeking to advance the frontiers of our understanding of data-driven image reconstruction methods as well as practitioners, by providing an accessible description of useful convergence concepts and by placing some of the existing empirical practices on a solid mathematical foundation.

Thomas Buddenkotte, Lorena Escudero Sanchez, Mireia Crispin-Ortuzar et al.

Uncertainty quantification in automated image analysis is highly desired in many applications. Typically, machine learning models in classification or segmentation are only developed to provide binary answers; however, quantifying the uncertainty of the models can play a critical role for example in active learning or machine human interaction. Uncertainty quantification is especially difficult when using deep learning-based models, which are the state-of-the-art in many imaging applications. The current uncertainty quantification approaches do not scale well in high-dimensional real-world problems. Scalable solutions often rely on classical techniques, such as dropout, during inference or training ensembles of identical models with different random seeds to obtain a posterior distribution. In this paper, we show that these approaches fail to approximate the classification probability. On the contrary, we propose a scalable and intuitive framework to calibrate ensembles of deep learning models to produce uncertainty quantification measurements that approximate the classification probability. On unseen test data, we demonstrate improved calibration, sensitivity (in two out of three cases) and precision when being compared with the standard approaches. We further motivate the usage of our method in active learning, creating pseudo-labels to learn from unlabeled images and human-machine collaboration.

Alma Eguizabal, Ozan Öktem, Mats U. Persson

Photon-counting CT (PCCT) offers improved diagnostic performance through better spatial and energy resolution, but developing high-quality image reconstruction methods that can deal with these large datasets is challenging. Model-based solutions incorporate models of the physical acquisition in order to reconstruct more accurate images, but are dependent on an accurate forward operator and present difficulties with finding good regularization. Another approach is deep-learning reconstruction, which has shown great promise in CT. However, fully data-driven solutions typically need large amounts of training data and lack interpretability. To combine the benefits of both methods, while minimizing their respective drawbacks, it is desirable to develop reconstruction algorithms that combine both model-based and data-driven approaches. In this work, we present a novel deep-learning solution for material decomposition in PCCT, based on an unrolled/unfolded iterative network. We evaluate two cases: a learned post-processing, which implicitly utilizes model knowledge, and a learned gradient-descent, which has explicit model-based components in the architecture. With our proposed techniques, we solve a challenging PCCT simulation case: three-material decomposition in abdomen imaging with low dose, iodine contrast, and a very small training sample support. In this scenario, our approach outperforms a maximum likelihood estimation, a variational method, as well as a fully-learned network.

Jevgenija Rudzusika, Buda Bajić, Thomas Koehler et al.

Deep learning based computed tomography (CT) reconstruction has demonstrated outstanding performance on simulated 2D low-dose CT data. This applies in particular to domain adapted neural networks, which incorporate a handcrafted physics model for CT imaging. Empirical evidence shows that employing such architectures reduces the demand for training data and improves upon generalisation. However, their training requires large computational resources that quickly become prohibitive in 3D helical CT, which is the most common acquisition geometry used for medical imaging. Furthermore, clinical data also comes with other challenges not accounted for in simulations, like errors in flux measurement, resolution mismatch and, most importantly, the absence of the real ground truth. The necessity to have a computationally feasible training combined with the need to address these issues has made it difficult to evaluate deep learning based reconstruction on clinical 3D helical CT. This paper modifies a domain adapted neural network architecture, the Learned Primal-Dual (LPD), so that it can be trained and applied to reconstruction in this setting. We achieve this by splitting the helical trajectory into sections and applying the unrolled LPD iterations to those sections sequentially. To the best of our knowledge, this work is the first to apply an unrolled deep learning architecture for reconstruction on full-sized clinical data, like those in the Low dose CT image and projection data set (LDCT). Moreover, training and testing is done on a single GPU card with 24GB of memory.

Andreas Hauptmann, Ozan Öktem

Learned image reconstruction has become a pillar in computational imaging and inverse problems. Among the most successful approaches are learned iterative networks, which are formulated by unrolling classical iterative optimisation algorithms for solving variational problems. While the underlying algorithm is usually formulated in the functional analytic setting, learned approaches are often viewed as purely discrete. In this chapter we present a unified operator view for learned iterative networks. Specifically, we formulate a learned reconstruction operator, defining how to compute, and separately the learning problem, which defines what to compute. In this setting we present common approaches and show that many approaches are closely related in their core. We review linear as well as nonlinear inverse problems in this framework and present a short numerical study to conclude.

Daniel Burrows, Can Evren Yarman, Ozan Öktem

Recovering physical properties of objects in motion is a core task across scientific and industrial applications. When the relative motion between the object and the sensing apparatus provides sufficient angular coverage, Computerized Tomography offers a powerful means of reconstruction. For such scenarios, we propose a parametric spatiotemporal model applied to Gaussian Mixture Models (GMM), in which each constituent Gaussian is parameterized by its own angular velocity, projectile motion, and geometry. GMM are a suitable means of reconstruction because they (i) admit accurate approximations in object space and (ii) have a closed form expression under the ray transform; enabling efficient forward predictions and exact gradient computations in data space. By decoupling the reconstruction problem into two sub-inverse problems, we characterize solutions as minimizers of task-specific objective functions that are derived and solved by utilizing the properties of (ii). The resulting algorithm we provide is applicable to objects in Euclidean space of arbitrary dimension. We validate the method on a simulated 2D problem, achieving accurate reconstruction of a 5-Gaussian GMM with intersecting trajectories. This also provides a foundation for further experimentation in settings with noisy data, 3D objects, and non-rigid body dynamics.

Paul Häusner, Ozan Öktem, Jens Sjölund

The convergence of the conjugate gradient method for solving large-scale and sparse linear equation systems depends on the spectral properties of the system matrix, which can be improved by preconditioning. In this paper, we develop a computationally efficient data-driven approach to accelerate the generation of effective preconditioners. We, therefore, replace the typically hand-engineered preconditioners by the output of graph neural networks. Our method generates an incomplete factorization of the matrix and is, therefore, referred to as neural incomplete factorization (NeuralIF). Optimizing the condition number of the linear system directly is computationally infeasible. Instead, we utilize a stochastic approximation of the Frobenius loss which only requires matrix-vector multiplications for efficient training. At the core of our method is a novel message-passing block, inspired by sparse matrix theory, that aligns with the objective of finding a sparse factorization of the matrix. We evaluate our proposed method on both synthetic problem instances and on problems arising from the discretization of the Poisson equation on varying domains. Our experiments show that by using data-driven preconditioners within the conjugate gradient method we are able to speed up the convergence of the iterative procedure. The code is available at https://github.com/paulhausner/neural-incomplete-factorization.

Jonathan Krook, Axel Janson, Joakim andén et al.

We present a geometry-aware method for heterogeneous single-particle cryogenic electron microscopy (cryo-EM) reconstruction that predicts atomic backbone conformations. To incorporate protein-structure priors, we represent the backbone as a graph and use a graph neural network (GNN) autodecoder that maps per-image latent variables to 3D displacements of a template conformation. The objective combines a data-discrepancy term based on a differentiable cryo-EM forward model with geometric regularization, and it supports unknown orientations via ellipsoidal support lifting (ESL) pose estimation. On synthetic datasets derived from molecular dynamics trajectories, the proposed GNN achieves higher accuracy compared to a multilayer perceptron (MLP) of comparable size, highlighting the benefits of a geometry-informed inductive bias.

Emilien Valat, Andreas Hauptmann, Ozan Öktem

Reconstructing images using Computed Tomography (CT) in an industrial context leads to specific challenges that differ from those encountered in other areas, such as clinical CT. Indeed, non-destructive testing with industrial CT will often involve scanning multiple similar objects while maintaining high throughput, requiring short scanning times, which is not a relevant concern in clinical CT. Under-sampling the tomographic data (sinograms) is a natural way to reduce the scanning time at the cost of image quality since the latter depends on the number of measurements. In such a scenario, post-processing techniques are required to compensate for the image artifacts induced by the sinogram sparsity. We introduce the Self-supervised Denoiser Framework (SDF), a self-supervised training method that leverages pre-training on highly sampled sinogram data to enhance the quality of images reconstructed from undersampled sinogram data. The main contribution of SDF is that it proposes to train an image denoiser in the sinogram space by setting the learning task as the prediction of one sinogram subset from another. As such, it does not require ground-truth image data, leverages the abundant data modality in CT, the sinogram, and can drastically enhance the quality of images reconstructed from a fraction of the measurements. We demonstrate that SDF produces better image quality, in terms of peak signal-to-noise ratio, than other analytical and self-supervised frameworks in both 2D fan-beam or 3D cone-beam CT settings. Moreover, we show that the enhancement provided by SDF carries over when fine-tuning the image denoiser on a few examples, making it a suitable pre-training technique in a context where there is little high-quality image data. Our results are established on experimental datasets, making SDF a strong candidate for being the building block of foundational image-enhancement models in CT.

Emilien Valat, Ozan Öktem

Ensuring proper generalization is a critical challenge in applying data-driven methods for solving inverse problems in imaging, as neural networks reconstructing an image must perform well across varied datasets and acquisition geometries. In X-ray Computed Tomography (CT), convolutional neural networks (CNNs) are widely used to filter the projection data but are ill-suited for this task as they apply grid-based convolutions to the sinogram, which inherently lies on a line manifold, not a regular grid. The CNNs, unaware of the geometry, are implicitly tied to it and require an excessive amount of parameters as they must infer the relations between measurements from the data rather than from prior information. The contribution of this paper is twofold. First, we introduce a graph data structure to represent CT acquisition geometries and tomographic data, providing a detailed explanation of the graph's structure for circular, cone-beam geometries. Second, we propose GLM, a hybrid neural network architecture that leverages both graph and grid convolutions to process tomographic data. We demonstrate that GLM outperforms CNNs when performance is quantified in terms of structural similarity and peak signal-to-noise ratio, despite the fact that GLM uses only a fraction of the trainable parameters. Compared to CNNs, GLM also requires significantly less training time and memory, and its memory requirements scale better. Crucially, GLM demonstrates robust generalization to unseen variations in the acquisition geometry, like when training only on fully sampled CT data and then testing on sparse-view CT data.

Aurora Poggi, Giuseppe Alessio D'Inverno, Hjalmar Brismar et al.

Data-driven discovery of dynamics in biological systems allows for better observation and characterization of processes, such as calcium signaling in cell culture. Recent advancements in techniques allow the exploration of previously unattainable insights of dynamical systems, such as the Sparse Identification of Non-Linear Dynamics (SINDy), overcoming the limitations of more classic methodologies. The latter requires some prior knowledge of an effective library of candidate terms, which is not realistic for a real case study. Using inspiration from fields like traffic density estimation and control theory, we propose a methodology for characterization and performance analysis of calcium delivery in a family of cells. In this work, we compare the performance of the Constrained Regularized Least-Squares Method (CRLSM) and Physics-Informed Neural Networks (PINN) for system identification and parameter discovery for governing ordinary differential equations (ODEs). The CRLSM achieves a fairly good parameter estimate and a good data fit when using the learned parameters in the Consensus problem. On the other hand, despite the initial hypothesis, PINNs fail to match the CRLSM performance and, under the current configuration, do not provide fair parameter estimation. However, we have only studied a limited number of PINN architectures, and it is expected that additional hyperparameter tuning, as well as uncertainty quantification, could significantly improve the performance in future works.

Buda Bajić, Johannes A. J. Huber, Benedikt Neyses et al.

In the wood industry, logs are commonly quality screened by discrete X-ray scans on a moving conveyor belt from a few source positions. Typically, the measurements are obtained in a single two-dimensional (2D) plane (a "slice") by a sequential scanning geometry. The data from each slice alone does not carry sufficient information for a three-dimensional tomographic reconstruction in which biological features of interest in the log are well preserved. In the present work, we propose a learned iterative reconstruction method based on the Learned Primal-Dual neural network, suited for sequential scanning geometries. Our method accumulates information between neighbouring slices, instead of only accounting for single slices during reconstruction. Evaluations were performed by training U-Nets on segmentation of knots (branches), which are crucial features in wood processing. Our quantitative and qualitative evaluations show that with as few as five source positions our method yields reconstructions of logs that are sufficiently accurate to identify biological features like knots (branches), heartwood and sapwood.

Jevgenija Rudzusika, Thomas Koehler, Ozan Öktem

This work presents an approach for image reconstruction in clinical low-dose tomography that combines principles from sparse signal processing with ideas from deep learning. First, we describe sparse signal representation in terms of dictionaries from a statistical perspective and interpret dictionary learning as a process of aligning distribution that arises from a generative model with empirical distribution of true signals. As a result we can see that sparse coding with learned dictionaries resembles a specific variational autoencoder, where the decoder is a linear function and the encoder is a sparse coding algorithm. Next, we show that dictionary learning can also benefit from computational advancements introduced in the context of deep learning, such as parallelism and as stochastic optimization. Finally, we show that regularization by dictionaries achieves competitive performance in computed tomography (CT) reconstruction comparing to state-of-the-art model based and data driven approaches.

Héctor Andrade-Loarca, Gitta Kutyniok, Ozan Öktem et al.

We present a deep learning-based algorithm to jointly solve a reconstruction problem and a wavefront set extraction problem in tomographic imaging. The algorithm is based on a recently developed digital wavefront set extractor as well as the well-known microlocal canonical relation for the Radon transform. We use the wavefront set information about x-ray data to improve the reconstruction by requiring that the underlying neural networks simultaneously extract the correct ground truth wavefront set and ground truth image. As a necessary theoretical step, we identify the digital microlocal canonical relations for deep convolutional residual neural networks. We find strong numerical evidence for the effectiveness of this approach.

Subhadip Mukherjee, Marcello Carioni, Ozan Öktem et al.

We propose an unsupervised approach for learning end-to-end reconstruction operators for ill-posed inverse problems. The proposed method combines the classical variational framework with iterative unrolling, which essentially seeks to minimize a weighted combination of the expected distortion in the measurement space and the Wasserstein-1 distance between the distributions of the reconstruction and ground-truth. More specifically, the regularizer in the variational setting is parametrized by a deep neural network and learned simultaneously with the unrolled reconstruction operator. The variational problem is then initialized with the reconstruction of the unrolled operator and solved iteratively till convergence. Notably, it takes significantly fewer iterations to converge, thanks to the excellent initialization obtained via the unrolled operator. The resulting approach combines the computational efficiency of end-to-end unrolled reconstruction with the well-posedness and noise-stability guarantees of the variational setting. Moreover, we demonstrate with the example of X-ray computed tomography (CT) that our approach outperforms state-of-the-art unsupervised methods, and that it outperforms or is on par with state-of-the-art supervised learned reconstruction approaches.

Subhadip Mukherjee, Ozan Öktem, Carola-Bibiane Schönlieb

In numerous practical applications, especially in medical image reconstruction, it is often infeasible to obtain a large ensemble of ground-truth/measurement pairs for supervised learning. Therefore, it is imperative to develop unsupervised learning protocols that are competitive with supervised approaches in performance. Motivated by the maximum-likelihood principle, we propose an unsupervised learning framework for solving ill-posed inverse problems. Instead of seeking pixel-wise proximity between the reconstructed and the ground-truth images, the proposed approach learns an iterative reconstruction network whose output matches the ground-truth in distribution. Considering tomographic reconstruction as an application, we demonstrate that the proposed unsupervised approach not only performs on par with its supervised variant in terms of objective quality measures but also successfully circumvents the issue of over-smoothing that supervised approaches tend to suffer from. The improvement in reconstruction quality comes at the expense of higher training complexity, but, once trained, the reconstruction time remains the same as its supervised counterpart.

Subhadip Mukherjee, Sören Dittmer, Zakhar Shumaylov et al.

We consider the variational reconstruction framework for inverse problems and propose to learn a data-adaptive input-convex neural network (ICNN) as the regularization functional. The ICNN-based convex regularizer is trained adversarially to discern ground-truth images from unregularized reconstructions. Convexity of the regularizer is desirable since (i) one can establish analytical convergence guarantees for the corresponding variational reconstruction problem and (ii) devise efficient and provable algorithms for reconstruction. In particular, we show that the optimal solution to the variational problem converges to the ground-truth if the penalty parameter decays sub-linearly with respect to the norm of the noise. Further, we prove the existence of a sub-gradient-based algorithm that leads to a monotonically decreasing error in the parameter space with iterations. To demonstrate the performance of our approach for solving inverse problems, we consider the tasks of deblurring natural images and reconstructing images in computed tomography (CT), and show that the proposed convex regularizer is at least competitive with and sometimes superior to state-of-the-art data-driven techniques for inverse problems.

Héctor Andrade-Loarca, Gitta Kutyniok, Ozan Öktem

Semantic edge detection has recently gained a lot of attention as an image processing task, mainly due to its wide range of real-world applications. This is based on the fact that edges in images contain most of the semantic information. Semantic edge detection involves two tasks, namely pure edge detecion and edge classification. Those are in fact fundamentally distinct in terms of the level of abstraction that each task requires, which is known as the distracted supervision paradox that limits the possible performance of a supervised model in semantic edge detection. In this work, we will present a novel hybrid method to avoid the distracted supervision paradox and achieve high-performance in semantic edge detection. Our approach is based on a combination of the model-based concept of shearlets, which provides probably optimally sparse approximations of a model-class of images, and the data-driven method of a suitably designed convolutional neural netwok. Finally, we present several applications such as tomographic reconstruction and show that our approach signifiantly outperforms former methods, thereby indicating the value of such hybrid methods for the area in biomedical imaging.

Ozan Öktem, Camille Pouchol, Olivier Verdier

Patient movement in emission tomography deteriorates reconstruction quality because of motion blur. Gating the data improves the situation somewhat: each gate contains a movement phase which is approximately stationary. A standard method is to use only the data from a few gates, with little movement between them. However, the corresponding loss of data entails an increase of noise. Motion correction algorithms have been implemented to take into account all the gated data, but they do not scale well, especially not in 3D. We propose a novel motion correction algorithm which addresses the scalability issue. Our approach is to combine an enhanced ML-EM algorithm with deep learning based movement registration. The training is unsupervised, and with artificial data. We expect this approach to scale very well to higher resolutions and to 3D, as the overall cost of our algorithm is only marginally greater than that of a standard ML-EM algorithm. We show that we can significantly decrease the noise corresponding to a limited number of gates.

Andreas Hauptmann, Jonas Adler, Simon Arridge et al.

Model-based learned iterative reconstruction methods have recently been shown to outperform classical reconstruction algorithms. Applicability of these methods to large scale inverse problems is however limited by the available memory for training and extensive training times, the latter due to computationally expensive forward models. As a possible solution to these restrictions we propose a multi-scale learned iterative reconstruction scheme that computes iterates on discretisations of increasing resolution. This procedure does not only reduce memory requirements, it also considerably speeds up reconstruction and training times, but most importantly is scalable to large scale inverse problems with non-trivial forward operators, such as those that arise in many 3D tomographic applications. In particular, we propose a hybrid network that combines the multi-scale iterative approach with a particularly expressive network architecture which in combination exhibits excellent scalability in 3D. Applicability of the algorithm is demonstrated for 3D cone beam computed tomography from real measurement data of an organic phantom. Additionally, we examine scalability and reconstruction quality in comparison to established learned reconstruction methods in two dimensions for low dose computed tomography on human phantoms.

Héctor Andrade-Loarca, Gitta Kutyniok, Ozan Öktem et al.

Microlocal analysis provides deep insight into singularity structures and is often crucial for solving inverse problems, predominately, in imaging sciences. Of particular importance is the analysis of wavefront sets and the correct extraction of those. In this paper, we introduce the first algorithmic approach to extract the wavefront set of images, which combines data-based and model-based methods. Based on a celebrated property of the shearlet transform to unravel information on the wavefront set, we extract the wavefront set of an image by first applying a discrete shearlet transform and then feeding local patches of this transform to a deep convolutional neural network trained on labeled data. The resulting algorithm outperforms all competing algorithms in edge-orientation and ramp-orientation detection.

Chong Chen, Barbara Gris, Ozan Öktem

We propose a new variational model for joint image reconstruction and motion estimation in spatiotemporal imaging, which is investigated along a general framework that we present with shape theory. This model consists of two components, one for conducting modified static image reconstruction, and the other performs sequentially indirect image registration. For the latter, we generalize the large deformation diffeomorphic metric mapping framework into the sequentially indirect registration setting. The proposed model is compared theoretically against alternative approaches (optical flow based model and diffeomorphic motion models), and we demonstrate that the proposed model has desirable properties in terms of the optimal solution. The theoretical derivations and efficient algorithms are also presented for a time-discretized scenario of the proposed model, which show that the optimal solution of the time-discretized version is consistent with that of the time-continuous one, and most of the computational components is the easy-implemented linearized deformation. The complexity of the algorithm is analyzed as well. This work is concluded by some numerical examples in 2D space + time tomography with very sparse and/or highly noisy data.

Jonas Adler, Ozan Öktem

Characterizing statistical properties of solutions of inverse problems is essential for decision making. Bayesian inversion offers a tractable framework for this purpose, but current approaches are computationally unfeasible for most realistic imaging applications in the clinic. We introduce two novel deep learning based methods for solving large-scale inverse problems using Bayesian inversion: a sampling based method using a WGAN with a novel mini-discriminator and a direct approach that trains a neural network using a novel loss function. The performance of both methods is demonstrated on image reconstruction in ultra low dose 3D helical CT. We compute the posterior mean and standard deviation of the 3D images followed by a hypothesis test to assess whether a "dark spot" in the liver of a cancer stricken patient is present. Both methods are computationally efficient and our evaluation shows very promising performance that clearly supports the claim that Bayesian inversion is usable for 3D imaging in time critical applications.

Jonas Adler, Sebastian Lunz, Olivier Verdier et al.

The paper considers the problem of performing a task defined on a model parameter that is only observed indirectly through noisy data in an ill-posed inverse problem. A key aspect is to formalize the steps of reconstruction and task as appropriate estimators (non-randomized decision rules) in statistical estimation problems. The implementation makes use of (deep) neural networks to provide a differentiable parametrization of the family of estimators for both steps. These networks are combined and jointly trained against suitable supervised training data in order to minimize a joint differentiable loss function, resulting in an end-to-end task adapted reconstruction method. The suggested framework is generic, yet adaptable, with a plug-and-play structure for adjusting both the inverse problem and the task at hand. More precisely, the data model (forward operator and statistical model of the noise) associated with the inverse problem is exchangeable, e.g., by using neural network architecture given by a learned iterative method. Furthermore, any task that is encodable as a trainable neural network can be used. The approach is demonstrated on joint tomographic image reconstruction, classification and joint tomographic image reconstruction segmentation.

Sebastian Lunz, Ozan Öktem, Carola-Bibiane Schönlieb

Inverse Problems in medical imaging and computer vision are traditionally solved using purely model-based methods. Among those variational regularization models are one of the most popular approaches. We propose a new framework for applying data-driven approaches to inverse problems, using a neural network as a regularization functional. The network learns to discriminate between the distribution of ground truth images and the distribution of unregularized reconstructions. Once trained, the network is applied to the inverse problem by solving the corresponding variational problem. Unlike other data-based approaches for inverse problems, the algorithm can be applied even if only unsupervised training data is available. Experiments demonstrate the potential of the framework for denoising on the BSDS dataset and for computed tomography reconstruction on the LIDC dataset.

Jonas Adler, Axel Ringh, Ozan Öktem et al.

We propose using the Wasserstein loss for training in inverse problems. In particular, we consider a learned primal-dual reconstruction scheme for ill-posed inverse problems using the Wasserstein distance as loss function in the learning. This is motivated by miss-alignments in training data, which when using standard mean squared error loss could severely degrade reconstruction quality. We prove that training with the Wasserstein loss gives a reconstruction operator that correctly compensates for miss-alignments in certain cases, whereas training with the mean squared error gives a smeared reconstruction. Moreover, we demonstrate these effects by training a reconstruction algorithm using both mean squared error and optimal transport loss for a problem in computerized tomography.

Jonas Adler, Ozan Öktem

We propose the Learned Primal-Dual algorithm for tomographic reconstruction. The algorithm accounts for a (possibly non-linear) forward operator in a deep neural network by unrolling a proximal primal-dual optimization method, but where the proximal operators have been replaced with convolutional neural networks. The algorithm is trained end-to-end, working directly from raw measured data and it does not depend on any initial reconstruction such as FBP. We compare performance of the proposed method on low dose CT reconstruction against FBP, TV, and deep learning based post-processing of FBP. For the Shepp-Logan phantom we obtain >6dB PSNR improvement against all compared methods. For human phantoms the corresponding improvement is 6.6dB over TV and 2.2dB over learned post-processing along with a substantial improvement in the SSIM. Finally, our algorithm involves only ten forward-back-projection computations, making the method feasible for time critical clinical applications.

Chong Chen, Ozan Öktem

The paper adapts the large deformation diffeomorphic metric mapping framework for image registration to the indirect setting where a template is registered against a target that is given through indirect noisy observations. The registration uses diffeomorphisms that transform the template through a (group) action. These diffeomorphisms are generated by solving a flow equation that is defined by a velocity field with certain regularity. The theoretical analysis includes a proof that indirect image registration has solutions (existence) that are stable and that converge as the data error tends so zero, so it becomes a well-defined regularization method. The paper concludes with examples of indirect image registration in 2D tomography with very sparse and/or highly noisy data.

Jonas Adler, Ozan Öktem

We propose a partially learned approach for the solution of ill posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularization theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularizing functional. The method results in a gradient-like iterative scheme, where the "gradient" component is learned using a convolutional network that includes the gradients of the data discrepancy and regularizer as input in each iteration. We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against FBP and TV reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the TV reconstruction while being significantly faster, giving reconstructions of 512 x 512 volumes in about 0.4 seconds using a single GPU.