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.
Going beyond 3D, higher-order dimensions occasionally find application in practice. Exact time-dependent solutions in 2D and 4D have been combined to approximate the unknown 3D solution for the isotropic point source in infinite media @cite_63 , and later applied to a time-dependent searchlight problem using the method of images @cite_23 . In the study of cosmic microwave background radiation it has even been considered to change dimension over the course of a single random flight @cite_83 .
@cite_23 @cite_63 @cite_83
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In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian computational geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green functions of Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the new method is that the Huygens–Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The second novelty is that a butterfly algorithm is adapted to carry out the matrix–vector products induced by the Huygens–Kirchhoff integration in O(NlogN) operations, where N is the total number of mesh points, and the proportionality constant depends on the desired accuracy and is independent of the frequency parameter. To reduce the storage of the resulting traveltime and amplitude tables, we compress each table into a linear combination of tensor-product based multivariate Chebyshev polynomials so that the information of each table is encoded into a small number of Chebyshev coefficients. The new method enjoys the following desired features: (1) it precomputes a set of local traveltime and amplitude tables; (2) it automatically takes care of caustics; (3) it constructs Green functions of the Helmholtz equation for arbitrary frequencies and for many point sources; (4) for a specified number of points per wavelength it constructs each Green function in nearly optimal complexity in terms of the total number of mesh points, where the prefactor of the complexity only depends on the specified accuracy and is independent of the frequency parameter. Both two-dimensional (2-D) and three-dimensional (3-D) numerical experiments are presented to demonstrate the performance and accuracy of the new method. We present a novel paradigm to deal with depth reconstruction from 4D light fields in a variational framework. Taking into account the special structure of light field data, we reformulate the problem of stereo matching to a constrained labeling problem on epipolar plane images, which can be thought of as vertical and horizontal 2D cuts through the field. This alternative formulation allows to estimate accurate depth values even for specular surfaces, while simultaneously taking into account global visibility constraints in order to obtain consistent depth maps for all views. The resulting optimization problems are solved with state-of-the-art convex relaxation techniques. We test our algorithm on a number of synthetic and real-world examples captured with a light field gantry and a plenoptic camera, and compare to ground truth where available. All data sets as well as source code are provided online for additional evaluation. In this paper we give a unified proof to the soliton resolution conjecture along a sequence of times, for the semilinear focusing energy critical wave equations in the radial case and two dimensional equivariant wave map equations, including the four dimensional radial Yang Mills equation, without using outer energy type inequalities. Such inequalities have played a crucial role in previous works with similar results. Roughly speaking, we prove that along a sequence of times @math (the maximal time of existence), the solution decouples to a sum of rescaled solitons and a term vanishing in the energy space, plus a free radiation term in the global case or a regular part in the finite time blow up case. The main difficulty is that in general (for instance for the radial four dimensional Yang Mills case and the radial six dimensional semilinear wave case) we do not have a favorable outer energy inequality for the associated linear wave equations. Our main new input is the simultaneous use of two virial identities. The Green’s function of the time dependent radiative transfer equation for the semi-infinite medium is derived for the first time by a heuristic approach based on the extrapolated boundary condition and on an almost exact solution for the infinite medium. Monte Carlo simulations performed both in the simple case of isotropic scattering and of an isotropic point-like source, and in the more realistic case of anisotropic scattering and pencil beam source, are used to validate the heuristic Green’s function. Except for the very early times, the proposed solution has an excellent accuracy (>98 for the isotropic case, and >97 for the anisotropic case) significantly better than the diffusion equation. The use of this solution could be extremely useful in the biomedical optics field where it can be directly employed in conditions where the use of the diffusion equation is limited, e.g. small volume samples, high absorption and or low scattering media, short source-receiver distances and early times. Also it represents a first step to derive tools for other geometries (e.g. slab and slab with inhomogeneities inside) of practical interest for noninvasive spectroscopy and diffuse optical imaging. Moreover the proposed solution can be useful to several research fields where the study of a transport process is fundamental.