arXiv subjects

Peter Maass

Publications and source records attributed to Peter Maass.

30 records · Page 2Linked to original sources

arXiv2020

Deep image prior for 3D magnetic particle imaging: A quantitative comparison of regularization techniques on Open MPI dataset

Magnetic particle imaging (MPI) is an imaging modality exploiting the nonlinear magnetization behavior of (super-)paramagnetic nanoparticles to obtain a space- and often also time-dependent concentration of a tracer consisting of these nanoparticles. MPI has a continuously increasing number of potential medical applications. One prerequisite for successful performance in these applications is a proper solution to the image reconstruction problem. More classical methods from inverse problems theory, as well as novel approaches from the field of machine learning, have the potential to deliver high-quality reconstructions in MPI. We investigate a novel reconstruction approach based on a deep image prior, which builds on representing the solution by a deep neural network. Novel approaches, as well as variational and iterative regularization techniques, are compared quantitatively in terms of peak signal-to-noise ratios and structural similarity indices on the publicly available Open MPI dataset.

eess.IV↗

arXiv2020

Conditional Normalizing Flows for Low-Dose Computed Tomography Image Reconstruction

Image reconstruction from computed tomography (CT) measurement is a challenging statistical inverse problem since a high-dimensional conditional distribution needs to be estimated. Based on training data obtained from high-quality reconstructions, we aim to learn a conditional density of images from noisy low-dose CT measurements. To tackle this problem, we propose a hybrid conditional normalizing flow, which integrates the physical model by using the filtered back-projection as conditioner. We evaluate our approach on a low-dose CT benchmark and demonstrate superior performance in terms of structural similarity of our flow-based method compared to other deep learning based approaches.

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arXiv2020

Joint super-resolution image reconstruction and parameter identification in imaging operator: Analysis of bilinear operator equations, numerical solution, and application to magnetic particle imaging

One important property of imaging modalities and related applications is the resolution of image reconstructions which relies on various factors such as instrumentation or data processing. Restrictions in resolution can have manifold origins, e.g., limited resolution of available data, noise level in the data, and/or inexact model operators. In this work we investigate a novel data processing approach suited for inexact model operators. Here, two different information sources, high-dimensional model information and high-quality measurement on a lower resolution, are comprised in a hybrid approach. The joint reconstruction of a high resolution image and parameters of the imaging operator are obtained by minimizing a Tikhonov-type functional. The hybrid approach is analyzed for bilinear operator equations with respect to stability, convergence, and convergence rates. We further derive an algorithmic solution exploiting an algebraic reconstruction technique. The study is complemented by numerical results ranging from an academic test case to image reconstruction in magnetic particle imaging.

math.NA↗

arXiv2020

Error analysis for filtered back projection reconstructions in Besov spaces

Filtered back projection (FBP) methods are the most widely used reconstruction algorithms in computerized tomography (CT). The ill-posedness of this inverse problem allows only an approximate reconstruction for given noisy data. Studying the resulting reconstruction error has been a most active field of research in the 1990s and has recently been revived in terms of optimal filter design and estimating the FBP approximation errors in general Sobolev spaces. However, the choice of Sobolev spaces is suboptimal for characterizing typical CT reconstructions. A widely used model are sums of characteristic functions, which are better modelled in terms of Besov spaces $\mathrm{B}^{α,p}_q(\mathbb{R}^2)$. In particular $\mathrm{B}^{α,1}_1(\mathbb{R}^2)$ with $α\approx 1$ is a preferred model in image analysis for describing natural images. In case of noisy Radon data the total FBP reconstruction error $$\|f-f_L^δ\| \le \|f-f_L\|+ \|f_L - f_L^δ\|$$ splits into an approximation error and a data error, where $L$ serves as regularization parameter. In this paper, we study the approximation error of FBP reconstructions for target functions $f \in \mathrm{L}^1(\mathbb{R}^2) \cap \mathrm{B}^{α,p}_q(\mathbb{R}^2)$ with positive $α\not\in \mathbb{N}$ and $1 \leq p,q \leq \infty$. We prove that the $\mathrm{L}^p$-norm of the inherent FBP approximation error $f-f_L$ can be bounded above by \begin{equation*} \|f - f_L\|_{\mathrm{L}^p(\mathbb{R}^2)} \leq c_{α,q,W} \, L^{-α} \, |f|_{\mathrm{B}^{α,p}_q(\mathbb{R}^2)} \end{equation*} under suitable assumptions on the utilized low-pass filter's window function $W$. This then extends by classical methods to estimates for the total reconstruction error.

math.NA↗

arXiv2020

Regularization by architecture: A deep prior approach for inverse problems

The present paper studies so-called deep image prior (DIP) techniques in the context of ill-posed inverse problems. DIP networks have been recently introduced for applications in image processing; also first experimental results for applying DIP to inverse problems have been reported. This paper aims at discussing different interpretations of DIP and to obtain analytic results for specific network designs and linear operators. The main contribution is to introduce the idea of viewing these approaches as the optimization of Tikhonov functionals rather than optimizing networks. Besides theoretical results, we present numerical verifications.

cs.LG↗

arXiv2019

Deep Relevance Regularization: Interpretable and Robust Tumor Typing of Imaging Mass Spectrometry Data

Neural networks have recently been established as a viable classification method for imaging mass spectrometry data for tumor typing. For multi-laboratory scenarios however, certain confounding factors may strongly impede their performance. In this work, we introduce Deep Relevance Regularization, a method of restricting what the neural network can focus on during classification, in order to improve the classification performance. We demonstrate how Deep Relevance Regularization robustifies neural networks against confounding factors on a challenging inter-lab dataset consisting of breast and ovarian carcinoma. We further show that this makes the relevance map -- a way of visualizing the discriminative parts of the mass spectrum -- sparser, thereby making the classifier easier to interpret

cs.LG↗

arXiv2019

Singular Values for ReLU Layers

Despite their prevalence in neural networks we still lack a thorough theoretical characterization of ReLU layers. This paper aims to further our understanding of ReLU layers by studying how the activation function ReLU interacts with the linear component of the layer and what role this interaction plays in the success of the neural network in achieving its intended task. To this end, we introduce two new tools: ReLU singular values of operators and the Gaussian mean width of operators. By presenting on the one hand theoretical justifications, results, and interpretations of these two concepts and on the other hand numerical experiments and results of the ReLU singular values and the Gaussian mean width being applied to trained neural networks, we hope to give a comprehensive, singular-value-centric view of ReLU layers. We find that ReLU singular values and the Gaussian mean width do not only enable theoretical insights, but also provide one with metrics which seem promising for practical applications. In particular, these measures can be used to distinguish correctly and incorrectly classified data as it traverses the network. We conclude by introducing two tools based on our findings: double-layers and harmonic pruning.

cs.LG↗

arXiv2019

A Projectional Ansatz to Reconstruction

Recently the field of inverse problems has seen a growing usage of mathematically only partially understood learned and non-learned priors. Based on first principles, we develop a projectional approach to inverse problems that addresses the incorporation of these priors, while still guaranteeing data consistency. We implement this projectional method (PM) on the one hand via very general Plug-and-Play priors and on the other hand, via an end-to-end training approach. To this end, we introduce a novel alternating neural architecture, allowing for the incorporation of highly customized priors from data in a principled manner. We also show how the recent success of Regularization by Denoising (RED) can, at least to some extent, be explained as an approximation of the PM. Furthermore, we demonstrate how the idea can be applied to stop the degradation of Deep Image Prior (DIP) reconstructions over time.

cs.LG↗

arXiv2019

On the Connection Between Adversarial Robustness and Saliency Map Interpretability

Recent studies on the adversarial vulnerability of neural networks have shown that models trained to be more robust to adversarial attacks exhibit more interpretable saliency maps than their non-robust counterparts. We aim to quantify this behavior by considering the alignment between input image and saliency map. We hypothesize that as the distance to the decision boundary grows,so does the alignment. This connection is strictly true in the case of linear models. We confirm these theoretical findings with experiments based on models trained with a local Lipschitz regularization and identify where the non-linear nature of neural networks weakens the relation.

stat.ML↗

arXiv2018

Joint bi-modal image reconstruction of DOT and XCT with an extended Mumford-Shah functional

Feature similarity measures are indispensable for joint image reconstruction in multi-modality medical imaging, which enable joint multi-modal image reconstruction (JmmIR) by communication of feature information from one modality to another, and vice versa. In this work, we establish an image similarity measure in terms of image edges from Tversky's theory of feature similarity in psychology. For joint bi-modal image reconstruction (JbmIR), it is found that this image similarity measure is an extended Mumford-Shah functional with a-priori edge information proposed previously from the perspective of regularization approach. This image similarity measure consists of Hausdorff measures of the common and different parts of image edges from both modalities. By construction, it posits that two images are more similar if they have more common edges and fewer unique/distinctive features, and will not force the nonexistent structures to be reconstructed when applied to JbmIR. With the Gamma-approximation of the JbmIR functional, an alternating minimization method is proposed for the JbmIR of diffuse optical tomography and x-ray computed tomography. The performance of the proposed method is evaluated by three numerical phantoms. It is found that the proposed method improves the reconstructed image quality by more than 10% compared to single modality image reconstruction (SmIR) in terms of the structural similarity index measure (SSIM)

eess.IV↗

arXiv2018

A Survey on Surrogate Approaches to Non-negative Matrix Factorization

Motivated by applications in hyperspectral imaging we investigate methods for approximating a high-dimensional non-negative matrix $\mathbf{\mathit{Y}}$ by a product of two lower-dimensional, non-negative matrices $\mathbf{\mathit{K}}$ and $\mathbf{\mathit{X}}.$ This so-called non-negative matrix factorization is based on defining suitable Tikhonov functionals, which combine a discrepancy measure for $\mathbf{\mathit{Y}}\approx\mathbf{\mathit{KX}}$ with penalty terms for enforcing additional properties of $\mathbf{\mathit{K}}$ and $\mathbf{\mathit{X}}$. The minimization is based on alternating minimization with respect to $\mathbf{\mathit{K}}$ or $\mathbf{\mathit{X}}$, where in each iteration step one replaces the original Tikhonov functional by a locally defined surrogate functional. The choice of surrogate functionals is crucial: It should allow a comparatively simple minimization and simultaneously its first order optimality condition should lead to multiplicative update rules, which automatically preserve non-negativity of the iterates. We review the most standard construction principles for surrogate functionals for Frobenius-norm and Kullback-Leibler discrepancy measures. We extend the known surrogate constructions by a general framework, which allows to add a large variety of penalty terms. The paper finishes by deriving the corresponding alternating minimization schemes explicitely and by applying these methods to MALDI imaging data.

cs.LG↗

arXiv2017

Deep Learning for Tumor Classification in Imaging Mass Spectrometry

Motivation: Tumor classification using Imaging Mass Spectrometry (IMS) data has a high potential for future applications in pathology. Due to the complexity and size of the data, automated feature extraction and classification steps are required to fully process the data. Deep learning offers an approach to learn feature extraction and classification combined in a single model. Commonly these steps are handled separately in IMS data analysis, hence deep learning offers an alternative strategy worthwhile to explore. Results: Methodologically, we propose an adapted architecture based on deep convolutional networks to handle the characteristics of mass spectrometry data, as well as a strategy to interpret the learned model in the spectral domain based on a sensitivity analysis. The proposed methods are evaluated on two challenging tumor classification tasks and compared to a baseline approach. Competitiveness of the proposed methods are shown on both tasks by studying the performance via cross-validation. Moreover, the learned models are analyzed by the proposed sensitivity analysis revealing biologically plausible effects as well as confounding factors of the considered task. Thus, this study may serve as a starting point for further development of deep learning approaches in IMS classification tasks.

stat.ML↗