Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms

We propose strongly consistent algorithms for reconstructing the characteristic function 1_K of an unknown convex body K in R^n from possibly noisy measurements of the modulus of its Fourier transform \hat{1_K}. This represents a complete theoretical solution to the Phase Retrieval Problem for characteristic functions of convex bodies. The approach is via the closely related problem of reconstructing K from noisy measurements of its covariogram, the function giving the volume of the intersection of K with its translates. In the many known situations in which the covariogram determines a convex body, up to reflection in the origin and when the position of the body is fixed, our algorithms use O(k^n) noisy covariogram measurements to construct a convex polytope P_k that approximates K or its reflection -K in the origin. (By recent uniqueness results, this applies to all planar convex bodies, all three-dimensional convex polytopes, and all symmetric and most (in the sense of Baire category) arbitrary convex bodies in all dimensions.) Two methods are provided, and both are shown to be strongly consistent, in the sense that, almost surely, the minimum of the Hausdorff distance between P_k and K or -K tends to zero as k tends to infinity.Comment: Version accepted on the Journal of the American Mathematical Society. With respect to version 1 the noise model has been greatly extended and an appendix has been added, with a discussion of rates of convergence and implementation issues. 56 pages, 4 figure

Convergence of algorithms for reconstructing convex bodies and directional measures

We investigate algorithms for reconstructing a convex body $K$ in $\mathbb {R}^n$ from noisy measurements of its support function or its brightness function in $k$ directions $u_1,...,u_k$. The key idea of these algorithms is to construct a convex polytope $P_k$ whose support function (or brightness function) best approximates the given measurements in the directions $u_1,...,u_k$ (in the least squares sense). The measurement errors are assumed to be stochastically independent and Gaussian. It is shown that this procedure is (strongly) consistent, meaning that, almost surely, $P_k$ tends to $K$ in the Hausdorff metric as $k\to\infty$. Here some mild assumptions on the sequence $(u_i)$ of directions are needed. Using results from the theory of empirical processes, estimates of rates of convergence are derived, which are first obtained in the $L_2$ metric and then transferred to the Hausdorff metric. Along the way, a new estimate is obtained for the metric entropy of the class of origin-symmetric zonoids contained in the unit ball. Similar results are obtained for the convergence of an algorithm that reconstructs an approximating measure to the directional measure of a stationary fiber process from noisy measurements of its rose of intersections in $k$ directions $u_1,...,u_k$. Here the Dudley and Prohorov metrics are used. The methods are linked to those employed for the support and brightness function algorithms via the fact that the rose of intersections is the support function of a projection body.Comment: Published at http://dx.doi.org/10.1214/009053606000000335 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Efficacy of Off-Label Anti-Amoebic Agents to Suppress Trophozoite Formation of Acanthamoeba spp. on Non-Nutrient Agar Escherichia Coli Plates

Acanthamoeba keratitis (AK) is a dangerous infectious disease, which is associated with a high risk of blindness for the infected patient, and for which no standard therapy exists thus far. Patients suffering from AK are thus treated, out of necessity, with an off-label therapy, using drugs designed and indicated for other diseases/purposes. Here, we tested the capability of the off-label anti-amoebic drugs chlorhexidine (CH; 0.1%), dibromopropamidine diisethionate (DD; 0.1%), hexamidine diisethionate (HD; 0.1%), miltefosine (MF; 0.0065%), natamycin (NM; 5%), polyhexamethylene biguanide (PHMB; 0.02%), povidone iodine (PVPI; 1%), and propamidine isethionate (PD; 0.1%) to suppress trophozoite formation of Acantamoeba castellanii and Acanthamoeba hatchetti cysts on non-nutrient agar Escherichia coli plates. Of the eight off-label anti-amoebic drugs tested, only PVPI allowed for a complete suppression of trophozoite formation by drug-challenged cysts for all four Acanthamoeba isolates in all five biological replicates. Drugs such as NM, PD, and PHMB repeatedly suppressed trophozoite formation with some, but not all, tested Acanthamoeba isolates, while other drugs such as CH, DD, and MF failed to exert a relevant effect on the excystation capacities of the tested Acanthamoeba isolates in most, if not all, of our repetitions. Our findings suggest that pre-testing of the AK isolate with the non-nutrient agar E. coli plate assay against the anti-amoebic drug intended for treatment should be performed to confirm that the selected drug is cysticidal for the Acanthamoeba isolate

Tensor valuations and their applications in stochastic geometry and imaging

The purpose of this volume is to give an up-to-date introduction to tensor valuations and their applications. Starting with classical results concerning scalar-valued valuations on the families of convex bodies and convex polytopes, it proceeds to the modern theory of tensor valuations. Product and Fourier-type transforms are introduced and various integral formulae are derived. New and well-known results are presented, together with generalizations in several directions, including extensions to the non-Euclidean setting and to non-convex sets. A variety of applications of tensor valuations to models in stochastic geometry, to local stereology and to imaging are also discussed