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

Estimating residual biomass of olive tree crops using terrestrial laser scanning

[EN] Agricultural residues have gained increasing interest as a source of renewable energy. The development of methods and techniques that allow to inventory residual biomass needs to be explored further. In this study, the residual biomass of olive trees was estimated based on parameters derived from using a Terrestrial Laser Scanning System (TLS). To this end, 32 olive trees in 2 orchards in the municipality of Viver, Central Eastern Spain, were selected and measured using a TLS system. The residual biomass of these trees was pruned and weighed. Several algorithms were applied to the TLS data to compute the main parameters of the trees: total height, crown height, crown diameter and crown volume. Regarding the last parameter, 4 methods were tested: the global convex hull volume, the convex hull by slice volume, the section volume, and the volume measured by voxels. In addition, several statistics were computed from the crown points for each tree. Regression models were calculated to predict residual biomass using 3 sets of potential explicative variables: firstly, the height statistics retrieved from 3D cloud data for each crown tree, secondly, the parameters of the trees derived from TLS data and finally, the combination of both sets of variables. Strong relationships between residual biomass and TLS parameters (crown volume parameters) were found (R2 = 0.86, RMSE = 2.78 kg). The pruning biomass pre- diction fraction was improved by 6%, in terms of R2, when the variance of the crown-point elevations was selected (R2 = 0.92, RMSE = 2.01 kg). The study offers some important insights into the quantification of residual biomass, which is essential information for the production of biofuel.Fernández-Sarría, A.; López- Cortés, I.; Estornell Cremades, J.; Velázquez Martí, B.; Salazar Hernández, DM. (2019). Estimating residual biomass of olive tree crops using terrestrial laser scanning. International Journal of Applied Earth Observation and Geoinformation. 75:163-170. https://doi.org/https://doi.org/10.1016/j.jag.2018.10.019S1631707

An Edge-based Interface Tracking (EBIT) Method for Multiphase-flows Simulation with Surface Tension

We present a novel Front-Tracking method, the Edge-Based Interface Tracking (EBIT) method for multiphase flow simulations. In the EBIT method, the markers are located on the grid edges and the interface can be reconstructed without storing the connectivity of the markers. This feature makes the process of marker addition or removal easier than in the traditional Front-Tracking method. The EBIT method also allows almost automatic parallelization due to the lack of explicit connectivity. In a previous journal article we have presented the kinematic part of the EBIT method, that includes the algorithms for interface linear reconstruction and advection. Here, we complete the presentation of the EBIT method and combine the kinematic algorithm with a Navier--Stokes solver. To identify the reference phase and to distinguish ambiguous topological configurations, we introduce a new feature: the Color Vertex. For the coupling with the Navier--Stokes equations, we first calculate volume fractions from the position of the markers and the Color Vertex, then viscosity and density fields from the computed volume fractions and finally surface tension stresses with the Height-Function method. In addition, an automatic topology change algorithm is implemented into the EBIT method, making it possible the simulation of more complex flows. A two-dimensional version of the EBIT method has been implemented in the open-source Basilisk platform, and validated with five standard test cases: (1) translation with uniform velocity, (2) single vortex, (3) capillary wave, (4) Rayleigh-Taylor instability and (5) rising bubble. The results are compared with those obtained with the Volume-of-Fluid (VOF) method already implemented in Basilisk