We solve the classic albedo and Milne problems of plane-parallel illumination of an isotropically-scattering half-space when generalized to a Euclidean domain @math for arbitrary @math . A continuous family of pseudo-problems and related @math functions arises and includes the classical 3D solutions, as well as 2D "Flatland" and rod-model solutions, as special cases. The Case-style eigenmode method is applied to the general problem and the internal scalar densities, emerging distributions, and their respective moments are expressed in closed-form. Universal properties invariant to dimension @math are highlighted and we find that a discrete diffusion mode is not universal for @math in absorbing media. We also find unexpected correspondences between differing dimensions and between anisotropic 3D scattering and isotropic scattering in high dimension.
For the case of bounded domains, isotropic scattering in a Flatland half space has been solved in a number of works @cite_15 @cite_62 @cite_77 @cite_9 . Slab geometry @cite_79 and layered problems @cite_67 in Flatland have also been solved. The study of inverse problems in plane-parallel domains of general dimension has been considered in a number of works (see, for example, @cite_92 ). We expand on these solutions by considering general dimension and by producing the singular eigenfunctions, whose orthogonality properties allow derivation of the moments of the internal scalar flux and angular distributions. These moment derivations complement the mean, variance and general moments previously produced for 3D @cite_69 @cite_8 and are useful for forming approximate searchlight approximations @cite_56 and for guiding Monte Carlo estimators towards zero-variance @cite_17 .
@cite_67 @cite_62 @cite_69 @cite_8 @cite_92 @cite_9 @cite_56 @cite_77 @cite_79 @cite_15 @cite_17
2227626537 2035369868 326163211 2010548775
Abstract Two classic problems of radiative transfer and neutron transport are solved for a spatially-uniform semi-infinite medium with isotropic scattering. General analytical equations are derived for (1) angular moments of the outward current and (2) spatial moments of the total flux within the half-space. Such moments, for example, can provide analytically explicit equations for determining the surface albedo of the medium as well as the mean depth and mean square distance of travel within the medium. The analysis is done with the Case-style eigenmode method as expressed in terms of the Chandrasekhar H -function and its moments. We study the flatland (two dimensional) linear transport equation, under an angular2@pperiodicity assumption both on particle density function @j(x,y,@q) and on the differential scattering @s"s(@q). We consider the beam problem, with a forward peaked source on phase-space, and derive P"1 approximation with a diffusion coefficient of 1 2@s"t"r, (versus 1 3@s"t"r of the three dimensional problem), where @s"t"r is the transport cross section. Further assumptions as peaked @s"s(@q) near @q=0 (small angle of scattering), and small angle of flight (@q 0) yield Fokker-Planck and Fermi approximations with the diffusion coefficients @s"t"r (rather than @s"t"r 2 of the three dimensional case). We discretize the problem using four different Galerkin schemes and justify the results through some numerical examples. We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a|<M, arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards. This paper describes a generalized axiomatic scale-space theory that makes it possible to derive the notions of linear scale-space, affine Gaussian scale-space and linear spatio-temporal scale-space using a similar set of assumptions (scale-space axioms). The notion of non-enhancement of local extrema is generalized from previous application over discrete and rotationally symmetric kernels to continuous and more general non-isotropic kernels over both spatial and spatio-temporal image domains. It is shown how a complete classification can be given of the linear (Gaussian) scale-space concepts that satisfy these conditions on isotropic spatial, non-isotropic spatial and spatio-temporal domains, which results in a general taxonomy of Gaussian scale-spaces for continuous image data. The resulting theory allows filter shapes to be tuned from specific context information and provides a theoretical foundation for the recently exploited mechanisms of shape adaptation and velocity adaptation, with highly useful applications in computer vision. It is also shown how time-causal spatio-temporal scale-spaces can be derived from similar assumptions. The mathematical structure of these scale-spaces is analyzed in detail concerning transformation properties over space and time, the temporal cascade structure they satisfy over time as well as properties of the resulting multi-scale spatio-temporal derivative operators. It is also shown how temporal derivatives with respect to transformed time can be defined, leading to the formulation of a novel analogue of scale normalized derivatives for time-causal scale-spaces. The kernels generated from these two types of theories have interesting relations to biological vision. We show how filter kernels generated from the Gaussian spatio-temporal scale-space as well as the time-causal spatio-temporal scale-space relate to spatio-temporal receptive field profiles registered from mammalian vision. Specifically, we show that there are close analogies to space-time separable cells in the LGN as well as to both space-time separable and non-separable cells in the striate cortex. We do also present a set of plausible models for complex cells using extended quasi-quadrature measures expressed in terms of scale normalized spatio-temporal derivatives. The theories presented as well as their relations to biological vision show that it is possible to describe a general set of Gaussian and or time-causal scale-spaces using a unified framework, which generalizes and complements previously presented scale-space formulations in this area.