arXiv subjects

Thomas Richter

Publications and source records attributed to Thomas Richter.

At least 55 records · Page 3Linked to original sources

arXiv2023

Temporal Scalability of Dynamic Volume Data Using Mesh Compensated Wavelet Lifting

Due to their high resolution, dynamic medical 2D+t and 3D+t volumes from computed tomography (CT) and magnetic resonance tomography (MR) reach a size which makes them very unhandy for teleradiologic applications. A lossless scalable representation offers the advantage of a down-scaled version which can be used for orientation or previewing, while the remaining information for reconstructing the full resolution is transmitted on demand. The wavelet transform offers the desired scalability. A very high quality of the lowpass sub-band is crucial in order to use it as a down-scaled representation. We propose an approach based on compensated wavelet lifting for obtaining a scalable representation of dynamic CT and MR volumes with very high quality. The mesh compensation is feasible to model the displacement in dynamic volumes which is mainly given by expansion and contraction of tissue over time. To achieve this, we propose an optimized estimation of the mesh compensation parameters to optimally fit for dynamic volumes. Within the lifting structure, the inversion of the motion compensation is crucial in the update step. We propose to take this inversion directly into account during the estimation step and can improve the quality of the lowpass sub-band by 0.63 and 0.43 dB on average for our tested dynamic CT and MR volumes at the cost of an increase of the rate by 2.4% and 1.2% on average.

eess.IV↗

arXiv2023

Analysis of mesh-based motion compensation in wavelet lifting of dynamical 3-D+t CT data

Factorized in the lifting structure, the wavelet transform can easily be extended by arbitrary compensation methods. Thereby, the transform can be adapted to displacements in the signal without losing the ability of perfect reconstruction. This leads to an improvement of scalability. In temporal direction of dynamic medical 3-D+t volumes from Computed Tomography, displacement is mainly given by expansion and compression of tissue. We show that these smooth movements can be well compensated with a mesh-based method. We compare the properties of triangle and quadrilateral meshes. We also show that with a mesh-based compensation approach coding results are comparable to the common slice wise coding with JPEG 2000 while a scalable representation in temporal direction can be achieved.

eess.IV↗

arXiv2023

Centroid adapted frequency selective extrapolation for reconstruction of lost image areas

Lost image areas with different size and arbitrary shape can occur in many scenarios such as error-prone communication, depth-based image rendering or motion compensated wavelet lifting. The goal of image reconstruction is to restore these lost image areas as close to the original as possible. Frequency selective extrapolation is a block-based method for efficiently reconstructing lost areas in images. So far, the actual shape of the lost area is not considered directly. We propose a centroid adaption to enhance the existing frequency selective extrapolation algorithm that takes the shape of lost areas into account. To enlarge the test set for evaluation we further propose a method to generate arbitrarily shaped lost areas. On our large test set, we obtain an average reconstruction gain of 1.29 dB.

eess.IV↗

arXiv2023

Efficient lossless coding of highpass bands from block-based motion compensated wavelet lifting using JPEG 2000

Lossless image coding is a crucial task especially in the medical area, e.g., for volumes from Computed Tomography or Magnetic Resonance Tomography. Besides lossless coding, compensated wavelet lifting offers a scalable representation of such huge volumes. While compensation methods increase the details in the lowpass band, they also vary the characteristics of the wavelet coefficients, so an adaption of the coefficient coder should be considered. We propose a simple invertible extension for JPEG 2000 that can reduce the filesize for lossless coding of the highpass band by 0.8% on average with peak rate saving of 1.1%.

eess.IV↗

arXiv2022

3-D mesh compensated wavelet lifting for 3-D+t medical CT data

For scalable coding, a high quality of the lowpass band of a wavelet transform is crucial when it is used as a downscaled version of the original signal. However, blur and motion can lead to disturbing artifacts. By incorporating feasible compensation methods directly into the wavelet transform, the quality of the lowpass band can be improved. The displacement in dynamic medical 3-D+t volumes from Computed Tomography is mainly given by expansion and compression of tissue over time and can be modeled well by mesh-based methods. We extend a 2-D mesh-based compensation method to three dimensions to obtain a volume compensation method that can additionally compensate deforming displacements in the third dimension. We show that a 3-D mesh can obtain a higher quality of the lowpass band by 0.28 dB with less than 40% of the model parameters of a comparable 2-D mesh. Results from lossless coding with JPEG 2000 3D and SPECK3D show that the compensated subbands using a 3-D mesh need about 6% less data compared to using a 2-D mesh.

eess.IV↗

arXiv2022

A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem

We analyze neural network solutions to partial differential equations obtained with Physics Informed Neural Networks. In particular, we apply tools of classical finite element error analysis to obtain conclusions about the error of the Deep Ritz method applied to the Laplace and the Stokes equations. Further, we develop an a posteriori error estimator for neural network approximations of partial differential equations. The proposed approach is based on the dual weighted residual estimator. It is destined to serve as a stopping criterion that guarantees the accuracy of the solution independently of the design of the neural network training. The result is equipped with computational examples for Laplace and Stokes problems.

math.NA↗

arXiv2022

Error analysis for a parabolic PDE model problem on a coupled moving domain in a fully Eulerian framework

We introduce an unfitted finite element method with Lagrange-multipliers to study an Eulerian time stepping scheme for moving domain problems applied to a model problem where the domain motion is implicit to the problem. We consider a parabolic partial differential equation (PDE) in the bulk domain, and the domain motion is described by an ordinary differential equation (ODE), coupled to the bulk partial differential equation through the transfer of forces at the moving interface. The discretisation is based on an unfitted finite element discretisation on a time-independent mesh. The method-of-lines time discretisation is enabled by an implicit extension of the bulk solution through additional stabilisation, as introduced by Lehrenfeld & Olshanskii (ESAIM: M2AN, 53:585-614, 2019). The analysis of the coupled problem relies on the Lagrange-multiplier formulation, the fact that the Lagrange-multiplier solution is equal to the normal stress at the interface and that the motion of the interface is given through rigid body motion. This paper covers the complete stability analysis of the method and an error estimate in the energy norm, under an assumption on the discrete interface velocity. This includes the dynamic error in the domain motion resulting from the discretised ODE and the forces from the discretised PDE. To the best of our knowledge this is the first error analysis of this type of coupled moving domain problem in a fully Eulerian framework. Numerical examples illustrate the theoretical results.

math.NA↗

arXiv2022

Spatio-temporal prediction in video coding by non-local means refined motion compensation

The prediction step is a very important part of hybrid video codecs. In this contribution, a novel spatio-temporal prediction algorithm is introduced. For this, the prediction is carried out in two steps. Firstly, a preliminary temporal prediction is conducted by motion compensation. Afterwards, spatial refinement is carried out for incorporating spatial redundancies from already decoded neighboring blocks. Thereby, the spatial refinement is achieved by applying Non-Local Means de-noising to the union of the motion compensated block and the already decoded blocks. Including the spatial refinement into H.264/AVC, a rate reduction of up to 14 % or respectively a gain of up to 0.7 dB PSNR compared to unrefined motion compensated prediction can be achieved.

eess.IV↗

arXiv2022

Reconstruction of images taken by a pair of non-regular sampling sensors using correlation based matching

Multi-view image acquisition systems with two or more cameras can be rather costly due to the number of high resolution image sensors that are required. Recently, it has been shown that by covering a low resolution sensor with a non-regular sampling mask and by using an efficient algorithm for image reconstruction, a high resolution image can be obtained. In this paper, a stereo image reconstruction setup for multi-view scenarios is proposed. A scene is captured by a pair of non-regular sampling sensors and by incorporating information from the adjacent view, the reconstruction quality can be increased. Compared to a state-of-the-art single-view reconstruction algorithm, this leads to a visually noticeable average gain in PSNR of 0.74 dB.

eess.IV↗

arXiv2022

Reducing Randomness of Non-Regular Sampling Masks for Image Reconstruction

Increasing spatial image resolution is an often required, yet challenging task in image acquisition. Recently, it has been shown that it is possible to obtain a high resolution image by covering a low resolution sensor with a non-regular sampling mask. Due to the masking, however, some pixel information in the resulting high resolution image is not available and has to be reconstructed by an efficient image reconstruction algorithm in order to get a fully reconstructed high resolution image. In this paper, the influence of different sampling masks with a reduced randomness of the non-regularity on the image reconstruction process is evaluated. Simulation results show that it is sufficient to use sampling masks that are non-regular only on a smaller scale. These sampling masks lead to a visually noticeable gain in PSNR compared to arbitrary chosen sampling masks which are non-regular over the whole image sensor size. At the same time, they simplify the manufacturing process and allow for efficient storage.

eess.IV↗

arXiv2022

Reconstruction of Videos Taken by a Non-Regular Sampling Sensor

Recently, it has been shown that a high resolution image can be obtained without the usage of a high resolution sensor. The main idea has been that a low resolution sensor is covered with a non-regular sampling mask followed by a reconstruction of the incomplete high resolution image captured this way. In this paper, a multi-frame reconstruction approach is proposed where a video is taken by a non-regular sampling sensor and fully reconstructed afterwards. By utilizing the temporal correlation between neighboring frames, the reconstruction quality can be further enhanced. Compared to a state-of-the-art single-frame reconstruction approach, this leads to a visually noticeable gain in PSNR of up to 1.19 dB on average.

eess.IV↗

arXiv2021

A neural network multigrid solver for the Navier-Stokes equations

We present the deep neural network multigrid solver (DNN-MG) that we develop for the instationary Navier-Stokes equations. DNN-MG improves computational efficiency using a judicious combination of a geometric multigrid solver and a recurrent neural network with memory. DNN-MG uses the multi-grid method to classically solve on coarse levels while the neural network corrects interpolated solutions on fine ones, thus avoiding the increasingly expensive computations that would have to be performed there. This results in a reduction in computation time through DNN-MG's highly compact neural network. The compactness results from its design for local patches and the available coarse multigrid solutions that provides a "guide" for the corrections. A compact neural network with a small number of parameters also reduces training time and data. Furthermore, the network's locality facilitates generalizability and allows one to use DNN-MG trained on one mesh domain also on different ones. We demonstrate the efficacy of DNN-MG for variations of the 2D laminar flow around an obstacle. For these, our method significantly improves the solutions as well as lift and drag functionals while requiring only about half the computation time of a full multigrid solution. We also show that DNN-MG trained for the configuration with one obstacle can be generalized to other time dependent problems that can be solved efficiently using a geometric multigrid method.

physics.comp-ph↗

arXiv2021

Steady azimuthal flow field induced by a rotating sphere near a rigid disk or inside a gap between two coaxially positioned rigid disks

Geometric confinements play an important role in many physical and biological processes and significantly affect the rheology and behavior of colloidal suspensions at low Reynolds numbers. On the basis of the linear Stokes equations, we investigate theoretically and computationally the viscous azimuthal flow induced by the slow rotation of a small spherical particle located in the vicinity of a rigid no-slip disk or inside a gap between two coaxially positioned rigid no-slip disks of the same radius. We formulate the solution of the hydrodynamic problem as a mixed-boundary-value problem in the whole fluid domain, which we subsequently transform into a system of dual integral equations. Near a stationary disk, we show that the resulting integral equation can be reduced into an elementary Abel integral equation that admits a unique analytical solution. Between two coaxially positioned stationary disks, we demonstrate that the flow problem can be transformed into a system of two Fredholm integral equations of the first kind. The latter are solved by means of numerical approaches. Using our solution, we further investigate the effect of the disks on the slow rotational motion of a colloidal particle and provide expressions of the hydrodynamic mobility as a function of the system geometry. We compare our results with corresponding finite-element simulations and observe very good agreement.

physics.flu-dyn↗

arXiv2021

On temporal homogenization in the numerical simulation of atherosclerotic plaque growth

A temporal homogenization approach for the numerical simulation of atherosclerotic plaque growth is extended to fully coupled fluid-structure interaction (FSI) simulations. The numerical results indicate that the two-scale approach yields significantly different results compared to a simple heuristic averaging, where only stationary long-scale FSI problems are solved, confirming the importance of incorporating stress variations on small time-scales. In the homogenization approach, a periodic fine-scale problem, which is periodic with respect to the heart beat, has to be solved for each long-scale time step. Even if no exact initial conditions are available, periodicity can be achieved within only 2-3 heart beats by simple time-stepping.

math.NA↗

arXiv2021

Deep neural networks for geometric multigrid methods

We investigate scaling and efficiency of the deep neural network multigrid method (DNN-MG). DNN-MG is a novel neural network-based technique for the simulation of the Navier-Stokes equations that combines an adaptive geometric multigrid solver, i.e. a highly efficient classical solution scheme, with a recurrent neural network with memory. The neural network replaces in DNN-MG one or multiple finest multigrid layers and provides a correction for the classical solve in the next time step. This leads to little degradation in the solution quality while substantially reducing the overall computational costs. At the same time, the use of the multigrid solver at the coarse scales allows for a compact network that is easy to train, generalizes well, and allows for the incorporation of physical constraints. Previous work on DNN-MG focused on the overall scheme and how to enforce divergence freedom in the solution. In this work, we investigate how the network size affects training and solution quality and the overall runtime of the computations. Our results demonstrate that larger networks are able to capture the flow behavior better while requiring only little additional training time. At runtime, the use of the neural network correction can even reduce the computation time compared to a classical multigrid simulation through a faster convergence of the nonlinear solve that is required at every time step.

math.NA↗

arXiv2021

Using a deep neural network to predict the motion of under-resolved triangular rigid bodies in an incompressible flow

We consider non-spherical rigid body particles in an incompressible fluid in the regime where the particles are too large to assume that they are simply transported with the fluid without back-coupling and where the particles are also too small to make fully resolved direct numerical simulations feasible. Unfitted finite element methods with ghost-penalty stabilisation are well suited to fluid-structure-interaction problems as posed by this setting, due to the flexible and accurate geometry handling and for allowing topology changes in the geometry. In the computationally under resolved setting posed here, accurate computations of the forces by their boundary integral formulation are not viable. Furthermore, analytical laws are not available due to the shape of the particles. However, accurate values of the forces are essential for realistic motion of the particles. To obtain these forces accurately, we train an artificial deep neural network using data from prototypical resolved simulations. This network is then able to predict the force values based on information which can be obtained accurately in an under-resolved setting. As a result, we obtain forces on very coarse and under-resolved meshes which are on average an order of magnitude more accurate compared to the direct boundary-integral computation from the Navier-Stokes solution, leading to solid motion comparable to that obtained on highly resolved meshes that would substantially increase the simulation costs.

physics.flu-dyn↗

arXiv2021

Error estimation and adaptivity for differential equations with multiple scales in time

We consider systems of ordinary differential equations with multiple scales in time. In general, we are interested in the long time horizon of a slow variable that is coupled to solution components that act on a fast scale. Although the fast scale variables are essential for the dynamics of the coupled problem, they are often of no interest in themselves. Recently we have proposed a temporal multiscale approach that fits into the framework of the heterogeneous multiscale method and that allows for efficient simulations with significant speedups. Fast and slow scales are decoupled by introducing local averages and by replacing fast scale contributions by localized periodic-in-time problems. Here, we generalize this multiscale approach to a larger class of problems but in particular, we derive an a posteriori error estimator based on the dual weighted residual method that allows for a splitting of the error into averaging error, error on the slow scale and error on the fast scale. We demonstrate the accuracy of the error estimator and also its use for adaptive control of a numerical multiscale scheme.

math.NA↗

arXiv2020

Structure Preservation for the Deep Neural Network Multigrid Solver

The simulation of partial differential equations is a central subject of numerical analysis and an indispensable tool in science, engineering and related fields. Existing approaches, such as finite elements, provide (highly) efficient tools but deep neural network-based techniques emerged in the last few years as an alternative with very promising results. We investigate the combination of both approaches for the approximation of the Navier-Stokes equations and to what extent structural properties such as divergence freedom can and should be respected. Our work is based on DNN-MG, a deep neural network multigrid technique, that we introduced recently and which uses a neural network to represent fine grid fluctuations not resolved by a geometric multigrid finite element solver. Although DNN-MG provides solutions with very good accuracy and is computationally highly efficient, we noticed that the neural network-based corrections substantially violate the divergence freedom of the velocity vector field. In this contribution, we discuss these findings and analyze three approaches to address the problem: a penalty term to encourage divergence freedom of the network output; a penalty term for the corrected velocity field; and a network that learns the stream function, i.e. the scalar potential of the divergence free velocity vector field and which hence yields by construction divergence free corrections. Our experimental results show that the third approach based on the stream function outperforms the other two and not only improves the divergence freedom but in particular also the overall fidelity of the simulation.

math.NA↗