# American Institute of Mathematical Sciences

April  2018, 12(2): 261-280. doi: 10.3934/ipi.2018011

## Reconstruction of cloud geometry from high-resolution multi-angle images

 1 Departments of Statistics and Mathematics and CCAM, University of Chicago, Chicago, IL 60637, USA 2 Department of Mathematical Sciences, Rensselear Polytechnic Institute, Troy, NY 12180, USA 3 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA

Received  November 2015 Revised  October 2016 Published  February 2018

We consider the reconstruction of the interface of compact, connected "clouds" from satellite or airborne light intensity measurements. In a two-dimensional setting, the cloud is modeled by an interface, locally represented as a graph, and an outgoing radiation intensity that is consistent with a diffusion model for light propagation in the cloud. Light scattering inside the cloud and the internal optical parameters of the cloud are not modeled explicitly. The main objective is to understand what can or cannot be reconstructed in such a setting from intensity measurements in a finite (on the order of 10) number of directions along the path of a satellite or an aircraft. Numerical simulations illustrate the theoretical predictions. Finally, we explore a kinematic extension of the algorithm for retrieving cloud motion (wind) along with its geometry.

Citation: Guillaume Bal, Jiaming Chen, Anthony B. Davis. Reconstruction of cloud geometry from high-resolution multi-angle images. Inverse Problems and Imaging, 2018, 12 (2) : 261-280. doi: 10.3934/ipi.2018011
##### References:

show all references

##### References:
Geometry of cloud interface
Left: A cloud model. Right: Simulated radiances $u_j(X) : = u(X,Z,\theta_j)$ for that cloud using (7) with $j = 1,\dots,J = 5$ (specifically, $\theta \in \{90,90\pm26.1,90\pm45.6\}$ in degrees clockwise from the positive $x$ axis) for a uniform $\alpha$ and $\beta = \sin\phi$.
True angular radiation function $\beta(\phi) = \sin\phi$ (in green), reconstructed function (in blue), and initial guess (in red).
 [1] Carlos E. Kenig, Mikko Salo, Gunther Uhlmann. Reconstructions from boundary measurements on admissible manifolds. Inverse Problems and Imaging, 2011, 5 (4) : 859-877. doi: 10.3934/ipi.2011.5.859 [2] Guillaume Bal, Ian Langmore, François Monard. Inverse transport with isotropic sources and angularly averaged measurements. Inverse Problems and Imaging, 2008, 2 (1) : 23-42. doi: 10.3934/ipi.2008.2.23 [3] Gérard Gagneux, Olivier Millet. A geological delayed response model for stratigraphic reconstructions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 457-474. doi: 10.3934/dcdss.2016007 [4] Nuutti Hyvönen, Lassi Päivärinta, Janne P. Tamminen. Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps. Inverse Problems and Imaging, 2018, 12 (2) : 373-400. doi: 10.3934/ipi.2018017 [5] Jutta Bikowski, Jennifer L. Mueller. 2D EIT reconstructions using Calderon's method. Inverse Problems and Imaging, 2008, 2 (1) : 43-61. doi: 10.3934/ipi.2008.2.43 [6] Melody Alsaker, Jennifer L. Mueller. Use of an optimized spatial prior in D-bar reconstructions of EIT tank data. Inverse Problems and Imaging, 2018, 12 (4) : 883-901. doi: 10.3934/ipi.2018037 [7] Corinna Burkard, Roland Potthast. A time-domain probe method for three-dimensional rough surface reconstructions. Inverse Problems and Imaging, 2009, 3 (2) : 259-274. doi: 10.3934/ipi.2009.3.259 [8] Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems and Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107 [9] Klas Modin. Geometry of matrix decompositions seen through optimal transport and information geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 335-390. doi: 10.3934/jgm.2017014 [10] Robert J. McCann. A glimpse into the differential topology and geometry of optimal transport. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1605-1621. doi: 10.3934/dcds.2014.34.1605 [11] Sarah J. Hamilton, David Isaacson, Ville Kolehmainen, Peter A. Muller, Jussi Toivanen, Patrick F. Bray. 3D Electrical Impedance Tomography reconstructions from simulated electrode data using direct inversion $\mathbf{t}^{\rm{{\textbf{exp}}}}$ and Calderón methods. Inverse Problems and Imaging, 2021, 15 (5) : 1135-1169. doi: 10.3934/ipi.2021032 [12] Guillaume Bal, Alexandre Jollivet. Stability estimates in stationary inverse transport. Inverse Problems and Imaging, 2008, 2 (4) : 427-454. doi: 10.3934/ipi.2008.2.427 [13] Brittan Farmer, Cassandra Hall, Selim Esedoḡlu. Source identification from line integral measurements and simple atmospheric models. Inverse Problems and Imaging, 2013, 7 (2) : 471-490. doi: 10.3934/ipi.2013.7.471 [14] Guillaume Bal, Alexandre Jollivet. Boundary control for transport equations. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022014 [15] Giovanni Alessandrini, Elio Cabib. Determining the anisotropic traction state in a membrane by boundary measurements. Inverse Problems and Imaging, 2007, 1 (3) : 437-442. doi: 10.3934/ipi.2007.1.437 [16] Deyue Zhang, Yukun Guo, Fenglin Sun, Hongyu Liu. Unique determinations in inverse scattering problems with phaseless near-field measurements. Inverse Problems and Imaging, 2020, 14 (3) : 569-582. doi: 10.3934/ipi.2020026 [17] Mourad Sini, Nguyen Trung Thành. Inverse acoustic obstacle scattering problems using multifrequency measurements. Inverse Problems and Imaging, 2012, 6 (4) : 749-773. doi: 10.3934/ipi.2012.6.749 [18] De-Han Chen, Daijun Jiang, Irwin Yousept, Jun Zou. Addendum to: "Variational source conditions for inverse Robin and flux problems by partial measurements". Inverse Problems and Imaging, 2022, 16 (2) : 481-481. doi: 10.3934/ipi.2022003 [19] De-Han Chen, Daijun Jiang, Irwin Yousept, Jun Zou. Variational source conditions for inverse Robin and flux problems by partial measurements. Inverse Problems and Imaging, 2022, 16 (2) : 283-304. doi: 10.3934/ipi.2021050 [20] Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems and Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003

2020 Impact Factor: 1.639