General theory of three-dimensional radiance measurements with optical microprobes

Measurements of the radiance distribution and fluence rate within turbid samples with fiber-optic radiance microprobes contain a large variable instrumental error caused by the nonuniform directional sensitivity of the microprobes. A general theory of three-dimensional radiance measurements is presented that provides correction for this error by using the independently obtained function of the angular sensitivity of the microprobes. © 1997 Optical Society of America

Theory of equidistant three–dimensional radiance measurements with optical microprobes

Fiber-optic radiance microprobes, increasingly applied for measurements of internal light fields in living tissues, provide three-dimensional radiance distribution solids and radiant energy fluence rates at different depths of turbid samples. These data are, however, distorted because of an inherent feature of optical fibers: nonuniform angular sensitivity. Because of this property a radiance microprobe during a single measurement partly underestimates light from the envisaged direction and partly senses light from other directions. A theory of three-dimensional equidistant radiance measurements has been developed that provides correction for this instrumental error using the independently obtained function of the angular sensitivity of the microprobe. For the first time, as far as we know, the measurements performed with different radiance microprobes are comparable. An example of application is presented. The limitations of this theory and the prospects for this approach are discussed. © 1996 Optical Society of America

On well-rounded ideal lattices - II

We study well-rounded lattices which come from ideals in quadratic number fields, generalizing some recent results of the first author with K. Petersen. In particular, we give a characterization of ideal well-rounded lattices in the plane and show that a positive proportion of real and imaginary quadratic number fields contains ideals giving rise to well-rounded lattices.Comment: 13 pages; to appear in the International Journal of Number Theor

Parametric Polyhedra with at least kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup \mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{ Sg}(A) such that $P_A(b) \cap {\mathbb Z}^n$ has at least $k$ solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ are fixed natural numbers, one can compute in polynomial time an encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives

Solution of the inverse problem of radiative transfer on the basis of measured internal fluxes

A method for the solution of the inverse problem of radiative transfer is presented which utilizes the internal fluxes measured at different depths and in different directions with optical radiance microprobes in dense multiple scattering media. The method yields optical cross-sections and the phase function for the sample even when these parameters are depth dependent. The sensitivity analysis shows that the theoretical errors caused by the finite number of measurements as well as by the non-uniform directional sensitivity of the microprobes can be held on a low level; even the fourth Legendre coefficient of the unknown phase function can be recovered to the accuracy of 10%. Copyright (C) 1998 Elsevier Science Ltd. All rights reserved