A Variational Stereo Method for the Three-Dimensional Reconstruction of Ocean Waves

Abstract

We develop a novel remote sensing technique for the observation of waves on the ocean surface. Our method infers the 3-D waveform and radiance of oceanic sea states via a variational stereo imagery formulation. In this setting, the shape and radiance of the wave surface are given by minimizers of a composite energy functional that combines a photometric matching term along with regularization terms involving the smoothness of the unknowns. The desired ocean surface shape and radiance are the solution of a system of coupled partial differential equations derived from the optimality conditions of the energy functional. The proposed method is naturally extended to study the spatiotemporal dynamics of ocean waves and applied to three sets of stereo video data. Statistical and spectral analysis are carried out. Our results provide evidence that the observed omnidirectional wavenumber spectrum <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> ) decays as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2.5</sup> is in agreement with Zakharov's theory (1999). Furthermore, the 3-D spectrum of the reconstructed wave surface is exploited to estimate wave dispersion and currents.