High-frequency asymptotic compression of dense BEM matrices for general geometries without ray tracing

Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green's function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green's function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the ...Comment: 24 pages, 13 figure

On the eigenmodes of periodic orbits for multiple scattering problems in 2D

Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the oscillatory properties of the solution. However, the high-frequency wave pattern becomes very complicated in the presence of multiple scattering obstacles. We consider a boundary integral equation formulation of the Helmholtz equation in two dimensions involving several obstacles, for which ray tracing schemes have been previously proposed. The existing analysis of ray tracing schemes focuses on periodic orbits between a subset of the obstacles. One observes that the densities on each of the obstacles converge to an equilibrium after a few iterations. In this paper we present an asymptotic approximation of the phases of those densities in equilibrium, in the form of a Taylor series. The densities represent a full cycle of reflections in a periodic orbit. We initially exploit symmetry in the case of two circular scatterers, but also provide an explicit algorithm for an arbitrary number of general 2D obstacles. The coefficients, as well as the time to compute them, are independent of the wavenumber and of the incident wave. The results may be used to accelerate ray tracing schemes after a small number of initial iterations.Comment: 24 pages, 9 figures and the implementation is available on https://github.com/popsomer/asyBEM/release

Analysis of geologic terrain models for determination of optimum SAR sensor configuration and optimum information extraction for exploration of global non-renewable resources. Pilot study: Arkansas Remote Sensing Laboratory, part 1, part 2, and part 3

Computer-generated radar simulations and mathematical geologic terrain models were used to establish the optimum radar sensor operating parameters for geologic research. An initial set of mathematical geologic terrain models was created for three basic landforms and families of simulated radar images were prepared from these models for numerous interacting sensor, platform, and terrain variables. The tradeoffs between the various sensor parameters and the quantity and quality of the extractable geologic data were investigated as well as the development of automated techniques of digital SAR image analysis. Initial work on a texture analysis of SEASAT SAR imagery is reported. Computer-generated radar simulations are shown for combinations of two geologic models and three SAR angles of incidence

Sparse Bases and Bayesian Inference of Electromagnetic Scattering

Many approaches in CEM rely on the decomposition of complex radiation and scattering behavior with a set of basis vectors. Accurate estimation of the quantities of interest can be synthesized through a weighted sum of these vectors. In addition to basis decompositions, sparse signal processing techniques developed in the CS community can be leveraged when only a small subset of the basis vectors are required to sufficiently represent the quantity of interest. We investigate several concepts in which novel bases are applied to common electromagnetic problems and leverage the sparsity property to improve performance and/or reduce computational burden. The first concept explores the use of multiple types of scattering primitives to reconstruct scattering patterns of electrically large targets. Using a combination of isotropic point scatterers and wedge diffraction primitives as our bases, a 40% reduction in reconstruction error can be achieved. Next, a sparse basis is used to improve DOA estimation. We implement the BSBL technique to determine the angle of arrival of multiple incident signals with only a single snapshot of data from an arbitrary arrangement of non-isotropic antennas. This is an improvement over the current state-of-the-art, where restrictions on the antenna type, configuration, and a priori knowledge of the number of signals are often assumed. Lastly, we investigate the feasibility of a basis set to reconstruct the scattering patterns of electrically small targets. The basis is derived from the TCM and can capture non-localized scattering behavior. Preliminary results indicate that this basis may be used in an interpolation and extrapolation scheme to generate scattering patterns over multiple frequencies

Computational Electromagnetism and Acoustics

It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems

Visual Prototyping of Cloth

Realistic visualization of cloth has many applications in computer graphics. An ongoing research problem is how to best represent and capture appearance models of cloth, especially when considering computer aided design of cloth. Previous methods can be used to produce highly realistic images, however, possibilities for cloth-editing are either restricted or require the measurement of large material databases to capture all variations of cloth samples. We propose a pipeline for designing the appearance of cloth directly based on those elements that can be changed within the production process. These are optical properties of fibers, geometrical properties of yarns and compositional elements such as weave patterns. We introduce a geometric yarn model, integrating state-of-the-art textile research. We further present an approach to reverse engineer cloth and estimate parameters for a procedural cloth model from single images. This includes the automatic estimation of yarn paths, yarn widths, their variation and a weave pattern. We demonstrate that we are able to match the appearance of original cloth samples in an input photograph for several examples. Parameters of our model are fully editable, enabling intuitive appearance design. Unfortunately, such explicit fiber-based models can only be used to render small cloth samples, due to large storage requirements. Recently, bidirectional texture functions (BTFs) have become popular for efficient photo-realistic rendering of materials. We present a rendering approach combining the strength of a procedural model of micro-geometry with the efficiency of BTFs. We propose a method for the computation of synthetic BTFs using Monte Carlo path tracing of micro-geometry. We observe that BTFs usually consist of many similar apparent bidirectional reflectance distribution functions (ABRDFs). By exploiting structural self-similarity, we can reduce rendering times by one order of magnitude. This is done in a process we call non-local image reconstruction, which has been inspired by non-local means filtering. Our results indicate that synthesizing BTFs is highly practical and may currently only take a few minutes for small BTFs. We finally propose a novel and general approach to physically accurate rendering of large cloth samples. By using a statistical volumetric model, approximating the distribution of yarn fibers, a prohibitively costly, explicit geometric representation is avoided. As a result, accurate rendering of even large pieces of fabrics becomes practical without sacrificing much generality compared to fiber-based techniques

Human Skin Modelling and Rendering

Creating realistic-looking skin is one of the holy grails of computer graphics and is still an active area of research. The problem is challenging due to the inherent complexity of skin and its variations, not only across individuals but also spatially and temporally among one. Skin appearance and reflectance vary spatially in one individual depending on its location on the human body, but also vary temporally with the aging process and the body state. Emotions, health, physical activity, and cosmetics for example can all affect the appearance of skin. The spatially varying reflectance of skin is due to many parameters, such as skin micro- and meso-geometry, thickness, oiliness, and pigmentation. It is therefore a daunting task to derive a model that will include all these parameters to produce realistic-looking skin. The problem is also compounded by the fact that we are very well accustomed to the appearance of skin and especially sensitive to facial appearances and expressions. Skin modelling and rendering is crucial for many applications such as games, virtual reality, films, and the beauty industry, to name a few. Realistic-looking skin improves the believability and realism of applications. The complexity of skin makes the topic of skin modelling and rendering for computer graphics a very difficult, but highly stimulating one. Skin deformations and biomechanics is a vast topic that we will not address in this dissertation. We rather focus our attention on skin optics and present a simple model for the reflectance of human skin along with a system to support skin modelling and rendering