Highly accurate schemes for PDE-based morphology with general structuring elements

The two fundamental operations in morphological image processing are dilation and erosion. These processes are defined via structuring elements. It is of practical interest to consider a variety of structuring element shapes. The realisation of dilation/erosion for convex structuring elements by use of partial differential equations (PDEs) allows for digital scalability and subpixel accuracy. However, numerical schemes suffer from blur by dissipative artifacts. In our paper we present a family of so-called flux-corrected transport (FCT) schemes that addresses this problem for arbitrary convex structuring elements. The main characteristics of the FCT-schemes are: (i) They keep edges very sharp during the morphological evolution process, and (ii) they feature a high degree of rotational invariance. We validate the FCT-scheme theoretically by proving consistency and stability. Numerical experiments with diamonds and ellipses as structuring elements show that FCT-schemes are superior to standard schemes in the field of PDE-based morphology

Morphological PDE and dilation/erosion semigroups on length spaces

International audienceThis paper gives a survey of recent research on Hamilton-Jacobi partial dierential equations (PDE) on length spaces. This theory provides the background to formulate morphological PDEs for processing data and images supported on a length space, without the need of a Riemmanian structure. We first introduce the most general pair of dilation/erosion semigroups on a length space, whose basic ingredients are the metric distance and a convex shape function. The second objective is to show under which conditions the solution of a morphological PDE in the length space framework is equal to the dilation/erosion semigroups

A realistic modeling of fluid infiltration in thin fibrous sheets

In this paper, a modeling study is presented to simulate the fluid infiltration in fibrous media. The Richards’ equation of two-phase flow in porous media is used here to model the fluid absorption in unsaturated/partially saturated fibrous thin sheets. The required consecutive equations, relative permeability, and capillary pressure as functions of medium’s saturation are obtained via fiber-level modeling and a long-column experiment, respectively. Our relative permeability calculations are based on solving the Stokes flow equations in partially saturated three-dimensional domains obtained by imaging the sheets’ microstructures. The Richards’ equation, together with the above consecutive correlations, is solved for fibrous media inclined with different angles. Simulation results are obtained for three different cases of upward, horizontal, and downward infiltrations. We also compared our numerical results with those of our long-column experiment and observed a good agreement. Moreover, we establish empirical coefficients for the semianalytical correlations previously proposed in the literature for the case of horizontal and downward infiltrations in thin fibrous sheets

PDE-based Group Equivariant Convolutional Neural Networks

We present a PDE-based framework that generalizes Group equivariant Convolutional Neural Networks (G-CNNs). In this framework, a network layer is seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients become the layer's trainable weights. Formulating our PDEs on homogeneous spaces allows these networks to be designed with built-in symmetries such as rotation in addition to the standard translation equivariance of CNNs. Having all the desired symmetries included in the design obviates the need to include them by means of costly techniques such as data augmentation. We will discuss our PDE-based G-CNNs (PDE-G-CNNs) in a general homogeneous space setting while also going into the specifics of our primary case of interest: roto-translation equivariance. We solve the PDE of interest by a combination of linear group convolutions and non-linear morphological group convolutions with analytic kernel approximations that we underpin with formal theorems. Our kernel approximations allow for fast GPU-implementation of the PDE-solvers, we release our implementation with this article in the form of the LieTorch extension to PyTorch, available at https://gitlab.com/bsmetsjr/lietorch . Just like for linear convolution a morphological convolution is specified by a kernel that we train in our PDE-G-CNNs. In PDE-G-CNNs we do not use non-linearities such as max/min-pooling and ReLUs as they are already subsumed by morphological convolutions. We present a set of experiments to demonstrate the strength of the proposed PDE-G-CNNs in increasing the performance of deep learning based imaging applications with far fewer parameters than traditional CNNs.Comment: 27 pages, 18 figures. v2 changes: - mentioned KerCNNs - added section Generalization of G-CNNs - clarification that the experiments utilized automatic differentiation and SGD. v3 changes: - streamlined theoretical framework - formulation and proof Thm.1 & 2 - expanded experiments. v4 changes: typos in Prop.5 and (20) v5/6 changes: minor revisio

Advances in Reduced-Order Modeling Based on Proper Orthogonal Decomposition for Single and Two-Phase Flows

This thesis presents advances in reduced-order modeling based on proper orthogonal decomposition (POD) for single and two-phase flows. Reduced-order models (ROMs) are generated for two-phase gas-solid flows. A multiphase numerical flow solver, MFIX, is used to generate a database of solution snapshots for proper orthogonal decomposition. Time-independent basis functions are extracted using POD from the data, and the governing equations of the MFIX are projected onto the basis functions to generate the multiphase POD-based ROMs. Reduced-order models are constructed to simulate multiphase two-dimensional non-isothermal flow and isothermal flow particle kinetics and three-dimensional isothermal flow. These reduced-order models are applied to three reference cases. The results of this investigation show that the two-dimensional reduced-order models are capable of producing qualitatively accurate results with less than 5 percent error with at least an order of magnitude reduction of computational costs. The three-dimensional ROM shows improvements in computational costs. This thesis also presents an algorithm based on mathematical morphology used to extract discontinuities present in quasi-steady and unsteady flows for POD basis augmentation. Both MFIX and a Reynolds Average Navier-Stokes (RANS) flow solver, UNS3D, are used to generate solution databases for feature extraction. The algorithm is applied to bubbling uidized beds, transonic airfoils, and turbomachinery seals. The results of this investigation show that all of the important features are extracted without loss in accuracy