Lynton Ardizzone

Jana Fragemann, Lynton Ardizzone, Jan Egger et al.

Deep neural networks are commonly used for medical purposes such as image generation, segmentation, or classification. Besides this, they are often criticized as black boxes as their decision process is often not human interpretable. Encouraging the latent representation of a generative model to be disentangled offers new perspectives of control and interpretability. Understanding the data generation process could help to create artificial medical data sets without violating patient privacy, synthesizing different data modalities, or discovering data generating characteristics. These characteristics might unravel novel relationships that can be related to genetic traits or patient outcomes. In this paper, we give a comprehensive overview of popular generative models, like Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), and Flow-based Models. Furthermore, we summarize the different notions of disentanglement, review approaches to disentangle latent space representations and metrics to evaluate the degree of disentanglement. After introducing the theoretical frameworks, we give an overview of recent medical applications and discuss the impact and importance of disentanglement approaches for medical applications.

The-Gia Leo Nguyen, Lynton Ardizzone, Ullrich Köthe

Autoencoders are able to learn useful data representations in an unsupervised matter and have been widely used in various machine learning and computer vision tasks. In this work, we present methods to train Invertible Neural Networks (INNs) as (variational) autoencoders which we call INN (variational) autoencoders. Our experiments on MNIST, CIFAR and CelebA show that for low bottleneck sizes our INN autoencoder achieves results similar to the classical autoencoder. However, for large bottleneck sizes our INN autoencoder outperforms its classical counterpart. Based on the empirical results, we hypothesize that INN autoencoders might not have any intrinsic information loss and thereby are not bounded to a maximal number of layers (depth) after which only suboptimal results can be achieved.

Tim J. Adler, Jan-Hinrich Nölke, Annika Reinke et al.

Current deep learning-based solutions for image analysis tasks are commonly incapable of handling problems to which multiple different plausible solutions exist. In response, posterior-based methods such as conditional Diffusion Models and Invertible Neural Networks have emerged; however, their translation is hampered by a lack of research on adequate validation. In other words, the way progress is measured often does not reflect the needs of the driving practical application. Closing this gap in the literature, we present the first systematic framework for the application-driven validation of posterior-based methods in inverse problems. As a methodological novelty, it adopts key principles from the field of object detection validation, which has a long history of addressing the question of how to locate and match multiple object instances in an image. Treating modes as instances enables us to perform mode-centric validation, using well-interpretable metrics from the application perspective. We demonstrate the value of our framework through instantiations for a synthetic toy example and two medical vision use cases: pose estimation in surgery and imaging-based quantification of functional tissue parameters for diagnostics. Our framework offers key advantages over common approaches to posterior validation in all three examples and could thus revolutionize performance assessment in inverse problems.

Titus Leistner, Radek Mackowiak, Lynton Ardizzone et al.

Light field applications, especially light field rendering and depth estimation, developed rapidly in recent years. While state-of-the-art light field rendering methods handle semi-transparent and reflective objects well, depth estimation methods either ignore these cases altogether or only deliver a weak performance. We argue that this is due current methods only considering a single "true" depth, even when multiple objects at different depths contributed to the color of a single pixel. Based on the simple idea of outputting a posterior depth distribution instead of only a single estimate, we develop and explore several different deep-learning-based approaches to the problem. Additionally, we contribute the first "multimodal light field depth dataset" that contains the depths of all objects which contribute to the color of a pixel. This allows us to supervise the multimodal depth prediction and also validate all methods by measuring the KL divergence of the predicted posteriors. With our thorough analysis and novel dataset, we aim to start a new line of depth estimation research that overcomes some of the long-standing limitations of this field.

Jonas Haldemann, Victor Ksoll, Daniel Walter et al.

The characterization of an exoplanet's interior is an inverse problem, which requires statistical methods such as Bayesian inference in order to be solved. Current methods employ Markov Chain Monte Carlo (MCMC) sampling to infer the posterior probability of planetary structure parameters for a given exoplanet. These methods are time consuming since they require the calculation of a large number of planetary structure models. To speed up the inference process when characterizing an exoplanet, we propose to use conditional invertible neural networks (cINNs) to calculate the posterior probability of the internal structure parameters. cINNs are a special type of neural network which excel in solving inverse problems. We constructed a cINN using FrEIA, which was then trained on a database of $5.6\cdot 10^6$ internal structure models to recover the inverse mapping between internal structure parameters and observable features (i.e., planetary mass, planetary radius and composition of the host star). The cINN method was compared to a Metropolis-Hastings MCMC. For that we repeated the characterization of the exoplanet K2-111 b, using both the MCMC method and the trained cINN. We show that the inferred posterior probability of the internal structure parameters from both methods are very similar, with the biggest differences seen in the exoplanet's water content. Thus cINNs are a possible alternative to the standard time-consuming sampling methods. Indeed, using cINNs allows for orders of magnitude faster inference of an exoplanet's composition than what is possible using an MCMC method, however, it still requires the computation of a large database of internal structures to train the cINN. Since this database is only computed once, we found that using a cINN is more efficient than an MCMC, when more than 10 exoplanets are characterized using the same cINN.

Lynton Ardizzone, Jakob Kruse, Carsten Lüth et al.

We introduce a new architecture called a conditional invertible neural network (cINN), and use it to address the task of diverse image-to-image translation for natural images. This is not easily possible with existing INN models due to some fundamental limitations. The cINN combines the purely generative INN model with an unconstrained feed-forward network, which efficiently preprocesses the conditioning image into maximally informative features. All parameters of a cINN are jointly optimized with a stable, maximum likelihood-based training procedure. Even though INN-based models have received far less attention in the literature than GANs, they have been shown to have some remarkable properties absent in GANs, e.g. apparent immunity to mode collapse. We find that our cINNs leverage these properties for image-to-image translation, demonstrated on day to night translation and image colorization. Furthermore, we take advantage of our bidirectional cINN architecture to explore and manipulate emergent properties of the latent space, such as changing the image style in an intuitive way.

Jakob Kruse, Lynton Ardizzone, Carsten Rother et al.

Recent work demonstrated that flow-based invertible neural networks are promising tools for solving ambiguous inverse problems. Following up on this, we investigate how ten invertible architectures and related models fare on two intuitive, low-dimensional benchmark problems, obtaining the best results with coupling layers and simple autoencoders. We hope that our initial efforts inspire other researchers to evaluate their invertible architectures in the same setting and put forth additional benchmarks, so our evaluation may eventually grow into an official community challenge.

Darya Trofimova, Tim Adler, Lisa Kausch et al.

Image registration is the basis for many applications in the fields of medical image computing and computer assisted interventions. One example is the registration of 2D X-ray images with preoperative three-dimensional computed tomography (CT) images in intraoperative surgical guidance systems. Due to the high safety requirements in medical applications, estimating registration uncertainty is of a crucial importance in such a scenario. However, previously proposed methods, including classical iterative registration methods and deep learning-based methods have one characteristic in common: They lack the capacity to represent the fact that a registration problem may be inherently ambiguous, meaning that multiple (substantially different) plausible solutions exist. To tackle this limitation, we explore the application of invertible neural networks (INN) as core component of a registration methodology. In the proposed framework, INNs enable going beyond point estimates as network output by representing the possible solutions to a registration problem by a probability distribution that encodes different plausible solutions via multiple modes. In a first feasibility study, we test the approach for a 2D 3D registration setting by registering spinal CT volumes to X-ray images. To this end, we simulate the X-ray images taken by a C-Arm with multiple orientations using the principle of digitially reconstructed radiographs (DRRs). Due to the symmetry of human spine, there are potentially multiple substantially different poses of the C-Arm that can lead to similar projections. The hypothesis of this work is that the proposed approach is able to identify multiple solutions in such ambiguous registration problems.

Jan-Hinrich Nölke, Tim Adler, Janek Gröhl et al.

Multispectral photoacoustic imaging (PAI) is an emerging imaging modality which enables the recovery of functional tissue parameters such as blood oxygenation. However, the underlying inverse problems are potentially ill-posed, meaning that radically different tissue properties may - in theory - yield comparable measurements. In this work, we present a new approach for handling this specific type of uncertainty by leveraging the concept of conditional invertible neural networks (cINNs). Specifically, we propose going beyond commonly used point estimates for tissue oxygenation and converting single-pixel initial pressure spectra to the full posterior probability density. This way, the inherent ambiguity of a problem can be encoded with multiple modes in the output. Based on the presented architecture, we demonstrate two use cases which leverage this information to not only detect and quantify but also to compensate for uncertainties: (1) photoacoustic device design and (2) optimization of photoacoustic image acquisition. Our in silico studies demonstrate the potential of the proposed methodology to become an important building block for uncertainty-aware reconstruction of physiological parameters with PAI.

Jens Müller, Robert Schmier, Lynton Ardizzone et al.

Standard supervised learning breaks down under data distribution shift. However, the principle of independent causal mechanisms (ICM, Peters et al. (2017)) can turn this weakness into an opportunity: one can take advantage of distribution shift between different environments during training in order to obtain more robust models. We propose a new gradient-based learning framework whose objective function is derived from the ICM principle. We show theoretically and experimentally that neural networks trained in this framework focus on relations remaining invariant across environments and ignore unstable ones. Moreover, we prove that the recovered stable relations correspond to the true causal mechanisms under certain conditions. In both regression and classification, the resulting models generalize well to unseen scenarios where traditionally trained models fail.

Radek Mackowiak, Lynton Ardizzone, Ullrich Köthe et al.

With the maturing of deep learning systems, trustworthiness is becoming increasingly important for model assessment. We understand trustworthiness as the combination of explainability and robustness. Generative classifiers (GCs) are a promising class of models that are said to naturally accomplish these qualities. However, this has mostly been demonstrated on simple datasets such as MNIST and CIFAR in the past. In this work, we firstly develop an architecture and training scheme that allows GCs to operate on a more relevant level of complexity for practical computer vision, namely the ImageNet challenge. Secondly, we demonstrate the immense potential of GCs for trustworthy image classification. Explainability and some aspects of robustness are vastly improved compared to feed-forward models, even when the GCs are just applied naively. While not all trustworthiness problems are solved completely, we observe that GCs are a highly promising basis for further algorithms and modifications. We release our trained model for download in the hope that it serves as a starting point for other generative classification tasks, in much the same way as pretrained ResNet architectures do for discriminative classification.

Stefan T. Radev, Ulf K. Mertens, Andreas Voss et al.

Estimating the parameters of mathematical models is a common problem in almost all branches of science. However, this problem can prove notably difficult when processes and model descriptions become increasingly complex and an explicit likelihood function is not available. With this work, we propose a novel method for globally amortized Bayesian inference based on invertible neural networks which we call BayesFlow. The method uses simulation to learn a global estimator for the probabilistic mapping from observed data to underlying model parameters. A neural network pre-trained in this way can then, without additional training or optimization, infer full posteriors on arbitrary many real datasets involving the same model family. In addition, our method incorporates a summary network trained to embed the observed data into maximally informative summary statistics. Learning summary statistics from data makes the method applicable to modeling scenarios where standard inference techniques with hand-crafted summary statistics fail. We demonstrate the utility of BayesFlow on challenging intractable models from population dynamics, epidemiology, cognitive science and ecology. We argue that BayesFlow provides a general framework for building amortized Bayesian parameter estimation machines for any forward model from which data can be simulated.

Lynton Ardizzone, Radek Mackowiak, Carsten Rother et al.

The Information Bottleneck (IB) objective uses information theory to formulate a task-performance versus robustness trade-off. It has been successfully applied in the standard discriminative classification setting. We pose the question whether the IB can also be used to train generative likelihood models such as normalizing flows. Since normalizing flows use invertible network architectures (INNs), they are information-preserving by construction. This seems contradictory to the idea of a bottleneck. In this work, firstly, we develop the theory and methodology of IB-INNs, a class of conditional normalizing flows where INNs are trained using the IB objective: Introducing a small amount of {\em controlled} information loss allows for an asymptotically exact formulation of the IB, while keeping the INN's generative capabilities intact. Secondly, we investigate the properties of these models experimentally, specifically used as generative classifiers. This model class offers advantages such as improved uncertainty quantification and out-of-distribution detection, but traditional generative classifier solutions suffer considerably in classification accuracy. We find the trade-off parameter in the IB controls a mix of generative capabilities and accuracy close to standard classifiers. Empirically, our uncertainty estimates in this mixed regime compare favourably to conventional generative and discriminative classifiers.

Tim J. Adler, Leonardo Ayala, Lynton Ardizzone et al.

Multispectral optical imaging is becoming a key tool in the operating room. Recent research has shown that machine learning algorithms can be used to convert pixel-wise reflectance measurements to tissue parameters, such as oxygenation. However, the accuracy of these algorithms can only be guaranteed if the spectra acquired during surgery match the ones seen during training. It is therefore of great interest to detect so-called out of distribution (OoD) spectra to prevent the algorithm from presenting spurious results. In this paper we present an information theory based approach to OoD detection based on the widely applicable information criterion (WAIC). Our work builds upon recent methodology related to invertible neural networks (INN). Specifically, we make use of an ensemble of INNs as we need their tractable Jacobians in order to compute the WAIC. Comprehensive experiments with in silico, and in vivo multispectral imaging data indicate that our approach is well-suited for OoD detection. Our method could thus be an important step towards reliable functional imaging in the operating room.

Lynton Ardizzone, Carsten Lüth, Jakob Kruse et al.

In this work, we address the task of natural image generation guided by a conditioning input. We introduce a new architecture called conditional invertible neural network (cINN). The cINN combines the purely generative INN model with an unconstrained feed-forward network, which efficiently preprocesses the conditioning input into useful features. All parameters of the cINN are jointly optimized with a stable, maximum likelihood-based training procedure. By construction, the cINN does not experience mode collapse and generates diverse samples, in contrast to e.g. cGANs. At the same time our model produces sharp images since no reconstruction loss is required, in contrast to e.g. VAEs. We demonstrate these properties for the tasks of MNIST digit generation and image colorization. Furthermore, we take advantage of our bi-directional cINN architecture to explore and manipulate emergent properties of the latent space, such as changing the image style in an intuitive way.

Tim J. Adler, Lynton Ardizzone, Anant Vemuri et al.

Purpose: Optical imaging is evolving as a key technique for advanced sensing in the operating room. Recent research has shown that machine learning algorithms can be used to address the inverse problem of converting pixel-wise multispectral reflectance measurements to underlying tissue parameters, such as oxygenation. Assessment of the specific hardware used in conjunction with such algorithms, however, has not properly addressed the possibility that the problem may be ill-posed. Methods: We present a novel approach to the assessment of optical imaging modalities, which is sensitive to the different types of uncertainties that may occur when inferring tissue parameters. Based on the concept of invertible neural networks, our framework goes beyond point estimates and maps each multispectral measurement to a full posterior probability distribution which is capable of representing ambiguity in the solution via multiple modes. Performance metrics for a hardware setup can then be computed from the characteristics of the posteriors. Results: Application of the assessment framework to the specific use case of camera selection for physiological parameter estimation yields the following insights: (1) Estimation of tissue oxygenation from multispectral images is a well-posed problem, while (2) blood volume fraction may not be recovered without ambiguity. (3) In general, ambiguity may be reduced by increasing the number of spectral bands in the camera. Conclusion: Our method could help to optimize optical camera design in an application-specific manner.

Lynton Ardizzone, Jakob Kruse, Sebastian Wirkert et al.

In many tasks, in particular in natural science, the goal is to determine hidden system parameters from a set of measurements. Often, the forward process from parameter- to measurement-space is a well-defined function, whereas the inverse problem is ambiguous: one measurement may map to multiple different sets of parameters. In this setting, the posterior parameter distribution, conditioned on an input measurement, has to be determined. We argue that a particular class of neural networks is well suited for this task -- so-called Invertible Neural Networks (INNs). Although INNs are not new, they have, so far, received little attention in literature. While classical neural networks attempt to solve the ambiguous inverse problem directly, INNs are able to learn it jointly with the well-defined forward process, using additional latent output variables to capture the information otherwise lost. Given a specific measurement and sampled latent variables, the inverse pass of the INN provides a full distribution over parameter space. We verify experimentally, on artificial data and real-world problems from astrophysics and medicine, that INNs are a powerful analysis tool to find multi-modalities in parameter space, to uncover parameter correlations, and to identify unrecoverable parameters.