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

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 S(k) decays as k-2.5 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

Weak Statistical Constraints for Variational Stereo Imaging of Oceanic Waves

We develop an observational technique for the stereoscopic reconstruction of the wave form of oceanic sea states via a variational stereo method. In the context of active surfaces, the shape and radiance of the wave surface are obtained as minimizers of an energy functional that combines image observations and smoothness priors. To obey the quasi Gaussianity of oceanic waves observed in nature, a given statistical wave law is enforced in the stereo variational framework as a weak constraint. Multigrid methods are then used to solve the partial differential equations derived from the optimality conditions of the augmented energy functional. An application of the developed method to two sets of experimental stereo data is finally presented

A Variational Wave Acquisition Stereo System for the 3-D Reconstruction of Oceanic Sea States

We propose a novel remote sensing technique that infers the three-dimensional wave form and radiance of oceanic sea states via a variational stereo imagery formulation. In this setting, the shape and radiance of the wave surface are minimizers of a composite cost functional which combines a data fidelity term and smoothness priors on the unknowns. The solution of a system of coupled partial differential equations derived from the cost functional yields the desired ocean surface shape and radiance. The proposed method is naturally extended to study the spatio-temporal dynamics of ocean waves, and applied to three sets of video data. Statistical and spectral analysis are carried out. The results shows evidence of the fact that the omni-directional wavenumber spectrum S(k) of the reconstructed waves decays as k^{-2.5} in agreement with Zakharov's theory (1999). Further, the three-dimensional spectrum of the reconstructed wave surface is exploited to estimate wave dispersion and currents

Variational Stereo Imaging of Oceanic Waves with Statistical Constraints

An image processing observational technique for the stereoscopic reconstruction of the wave form of oceanic sea states is developed. The technique incorporates the enforcement of any given statistical wave law modeling the quasi Gaussianity of oceanic waves observed in nature. The problem is posed in a variational optimization framework, where the desired wave form is obtained as the minimizer of a cost functional that combines image observations, smoothness priors and a weak statistical constraint. The minimizer is obtained combining gradient descent and multigrid methods on the necessary optimality equations of the cost functional. Robust photometric error criteria and a spatial intensity compensation model are also developed to improve the performance of the presented image matching strategy. The weak statistical constraint is thoroughly evaluated in combination with other elements presented to reconstruct and enforce constraints on experimental stereo data, demonstrating the improvement in the estimation of the observed ocean surface

Space-time Reconstruction of Oceanic Sea States via Variational Stereo Methods

We present a remote sensing observational method for the measurement of the spatio-temporal dynamics of ocean waves. Variational techniques are used to recover a coherent space-time reconstruction of oceanic sea states given stereo video imagery. The stereoscopic reconstruction problem is expressed in a variational optimization framework. There, we design an energy functional whose minimizer is the desired temporal sequence of wave heights. The functional combines photometric observations as well as spatial and temporal regularizers. A nested iterative scheme is devised to numerically solve, via 3-D multigrid methods, the system of partial differential equations resulting from the optimality condition of the energy functional. The output of our method is the coherent, simultaneous estimation of the wave surface height and radiance at multiple snapshots. We demonstrate our algorithm on real data collected off-shore. Statistical and spectral analysis are performed. Comparison with respect to an existing sequential method is analyzed

Wave modelling in coastal and inner seas

In the long term development of the research on wind waves and their modelling, in particular of the inner and coastal seas, the present situation is framed with a short look at the past, a critical analysis of the present capabilities and a foresight of where the field is likely to go. After a short introduction, Chapter 2 deals with the basic processes at work and their modelling aspects. Chapter 3 highlights the interaction with wind and currents. Chapter 4 stresses the need for a more complete, spectral, approach in data assimilation. Chapter 5 summarizes the situation with a discussion on the present status in wave modelling and a look at what we can expect in the future.Peer ReviewedPostprint (author's final draft

Offshore stereo measurements of gravity waves

Stereo video techniques are effective for estimating the space-time wave dynamics over an area of the ocean. Indeed, a stereo camera view allows retrieval of both spatial and temporal data whose statistical content is richer than that of time series data retrieved from point wave probes. To prove this, we consider an application of the Wave Acquisition Stereo System (WASS) for the analysis of offshore video measurements of gravity waves in the Northern Adriatic Sea. In particular, we deployed WASS at the oceanographic platform Acqua Alta, off the Venice coast, Italy. Three experimental studies were performed, and the overlapping field of view of the acquired stereo images covered an area of approximately 1100 m2. Analysis of the WASS measurements show that the sea surface can be accurately estimated in space and time together, yielding associated directional spectra and wave statistics that agree well with theoretical models. From the observed wavenumber-frequency spectrum one can also predict the vertical profile of the current flow underneath the wave surface. Finally, future improvements of WASS and applications are discussed

Observation of extreme sea waves in a space-time ensemble

In this paper, an observational space-time ensemble of sea surface elevations is investigated in search of the highest waves of the sea state. Wave data were gathered by means of a stereo camera system, which was installed on top of a fixed oceanographic platform located in the Adriatic Sea (Italy). Waves were measured during a mature sea state with an average wind speed of 11 m s-1. By examining the space-time ensemble, the 3D wave groups have been isolated while evolving in the 2D space and grabbed "when and where" they have been close to the apex of their development, thus exhibiting large surface displacements. The authors have selected the groups displaying maximal crest height exceeding the threshold adopted to define rogue waves in a time record, that is, 1.25 times the significant wave height (Hs). The records at the spatial positions where such large crests occurred have been analyzed to derive the empirical distributions of crest and wave heights, which have been compared against standard statistical linear and nonlinear models. Here, the maximal observed wave crests have resulted to be outliers of the standard statistics, behaving as isolated members of the sample, apparently uncorrelated with other waves of the record. However, this study has found that these unexpectedly large wave crests are better approximated by a space-time model for extreme crest heights. The space-time model performance has been improved, deriving a second-order approximation of the linear model, which has provided a fair agreement with the empirical maxima. The present investigation suggests that very large waves may be more numerous than generally expected.In this paper, an observational space-time ensemble of sea surface elevations is investigated in search of the highest waves of the sea state. Wave data were gathered by means of a stereo camera system, which was installed on top of a fixed oceanographic platform located in the Adriatic Sea (Italy). Waves were measured during a mature sea state with an average wind speed of 11 m s(-1). By examining the space-time ensemble, the 3D wave groups have been isolated while evolving in the 2D space and grabbed "when and where" they have been close to the apex of their development, thus exhibiting large surface displacements. The authors have selected the groups displaying maximal crest height exceeding the threshold adopted to define rogue waves in a time record, that is, 1.25 times the significant wave height (H-s). The records at the spatial positions where such large crests occurred have been analyzed to derive the empirical distributions of crest and wave heights, which have been compared against standard statistical linear and nonlinear models. Here, the maximal observed wave crests have resulted to be outliers of the standard statistics, behaving as isolated members of the sample, apparently uncorrelated with other waves of the record. However, this study has found that these unexpectedly large wave crests are better approximated by a space-time model for extreme crest heights. The space-time model performance has been improved, deriving a second-order approximation of the linear model, which has provided a fair agreement with the empirical maxima. The present investigation suggests that very large waves may be more numerous than generally expected

Offshore remote sensing of the ocean by stereo vision systems

In recent years, remote sensing imaging systems for the measurement of oceanic sea states have attracted renovated attention. Imaging technology is economical, non-invasive and enables a better understanding of the space-time dynamics of ocean waves over an area rather than at selected point locations of previous monitoring methods (buoys, wave gauges, etc.). We present recent progress in space-time measurement of ocean waves using stereo vision systems on offshore platforms, which focus on sea states with wavelengths in the range of 0.01 m to 1 m. Both traditional disparity-based systems and modern elevation-based ones are presented in a variational optimization framework: the main idea is to pose the stereoscopic reconstruction problem of the surface of the ocean in a variational setting and design an energy functional whose minimizer is the desired temporal sequence of wave heights. The functional combines photometric observations as well as spatial and temporal smoothness priors. Disparity methods estimate the disparity between images as an intermediate step toward retrieving the depth of the waves with respect to the cameras, whereas elevation methods estimate the ocean surface displacements directly in 3-D space. Both techniques are used to measure ocean waves from real data collected at offshore platforms in the Black Sea (Crimean Peninsula, Ukraine) and the Northern Adriatic Sea (Venice coast, Italy). Then, the statistical and spectral properties of the resulting observed waves are analyzed. We show the advantages and disadvantages of the presented stereo vision systems and discuss future lines of research to improve their performance in critical issues such as the robustness of the camera calibration in spite of undesired variations of the camera parameters or the processing time that it takes to retrieve ocean wave measurements from the stereo videos, which are very large datasets that need to be processed efficiently to be of practical usage. Multiresolution and short-time approaches would improve efficiency and scalability of the techniques so that wave displacements are obtained in feasible times