Stereo measurement of water surface waves

Image Analysis techniques are used for retrieving water surface wave field spatially and temporally from CCD-images and CCD-image-sequences. The technique developed utilizes binocular stereogrammetry to recover topographic information from a sequence of synchronous, overlapping video images. The complete geometrical method is given, including the derivation of a relationship relating the geometry of the stereo rig and the expected quantization errors. At the end, the 3-D calculated scattered points give the complete spatio-temporal distribution of the water surface elevations. The measurable length-scales depend on the pixel resolution, “filtered” by the triangulation process, and the acquisition frame rate. The results are compared to in-situ sensor for validation. Measurements in field campaigns indicate that the approach is both accurate and applicable for measuring surface water waves. Moreover, in the water wave measurement scenario stereo analysis shows new potentialities to investigate the shape of the water surface and the dynamics of the oceanographic phenomena.

In the oceanographic measurements field, images of the water surface in the visible range of the electromagnetic spectrum have for many years provided quantitative information on a number of physical parameters associated with the waves which ripple such surface. The application of stereography (a classical method to infer the depth of opaque surfaces) in such a field of measurements created the basis for inferring the 3-D shape of the water waves. Merging this kind of information with the time coordinate it is possible to thoroughly go through the complete spatio-temporal development of the water surface waves.
The application of stereography to measure the water surface topography starts from the conventional stereographic technique algorithms used to survey geodetical surfaces or static objects (Ma et al., 2004). The major differences depend on the fact that the water surface is a specular object in (rapid) movement. For this reason each stereo-pair is acquired simultaneously and the geometry of the stereo system is such that the matching errors are minimized. In such a way, the first adaptations were implemented in an experiment with cameras mounted on an ocean going ship (Schumacher, 1939). More recently Coté et al. (1960) demonstrated, for long ocean waves, the use of stereophotography to measure sea surface topography. The effort required to extract 3-D elevation maps from an image pair limited the use of this technique in studying the dynamics of oceanographic phenomena until late 70s and early 80s (Sugimori, 1975; Holthuijsen, 1983). Nevertheless, the need to know directional information of the waves allowed the stereography to remain one of the investigation tools in the oceanographic studies. Shemdin et al. (1988) proposed the directional measurements of short ocean waves applying stereography. This experiment used a pair of cameras mounted on an oceanographic offshore tower near San Diego (USA) to create the 3-D shape model of the sea surface and then, via spectral analysis, to extract directional information of the waves under inspection. In the 1989, Banner et al. applied stereographic measurements in order to study wavenumber spectra of short gravity waves. The most recent integration of stereographic techniques into the field of oceanography has been the WAVESCAN project (Santel et al., 2004) which contributed the important extension to the time domain of the stereographic pairs.

3-D reconstruction is one of the most popular problems in Computer Vision (Ma et al., 2004). One of the earliest methods in this field is stereo analysis, which uses two cameras to obtain two images of the scene. Stereo analysis can be static, if two still images are acquired and used for reconstruction, or dynamic, when a movie of the scene is acquired. Usually the two cameras are fixed and precalibrated and search of correspondences is limited to epipolar lines (Zhang, 2000).
Usually stereo matching works properly if the surface is not reflective and has Lambertian properties (Ma et al., 2004). However it can be shown (Jähne, 1993) that some conditions in the position of cameras and characteristics of the surface allow the reflections to be considered as texture on the sea surface, in fact the main problem in stereo analysis is how to find, given a point in the first image, the corresponding point in the second image. This problem is commonly known as the correspondence problem (Klette et al., 1998). After stereo matching and triangulation, height of the waves is determined with a bias, which is due to the specular properties of the surface. Such bias depends on both the wavelength of the waves and their steepness. It is small for steep waves, i.e. when the angle formed by the slope with the horizontal plane is much larger than the inclination of the lines of sight of the cameras. This error can be reduced using a small angle between the lines of sight of the cameras; however, this increases the quantization error. A trade-off has to be found: the geometry of the camera set-up used for the work here showed is characterized by a base to height ratio of about 0.1, and the inclination of the line of sight of the cameras is about 0.05 radiants. This is a very small value, if compared with typical maximum slopes of waves: therefore waves can be reconstructed with accuracy.
In my PhD work a pixel-based stereo matching algorithm is used. Cross-correlation is used to find correspondences. A pyramidal search is useful for computational reasons: the finding correspondences process starts from few points on a sparse grid on the surface, and then iterates the process refining the grid in different steps, to obtain a dense disparity map. From the second iterative step, it is exploited the hypothesis that locally the water surface can be approximate by a planar surface. Only at the first level, the correspondence is searched over all the epipolar line. From the second level, the region of interest is reduced over a small part of the image, around the estimated point. It reduces the cross-correlations that must be computed and saves computational time.

Special tools were deveoped to calibrate the stereo system and to aligneate the z-axis with the vertical direction.

Figure - Comparison between water elevations measured with Ultrasonic probe and Stereo system.

Figure - Example of 3-D reconstruction.

Main references

Benetazzo, A., 2006. Measurements of short water waves using stereo matched image sequences. Coastal Engineering, Volume 53, Issue 12, Pages 1013-1032.

Wanek, J. and Wu, C.H., 2006. Automated trinocular stereo imaging system for three-dimensional surface wave measurements, Ocean Engineering, 33(5-6), 723-747.