Extrinsic Calib

The processing of finding R and C is called extrinsic calibration. This can only be done after camera installation and requires estimation of the three viewing angles (tilt, azimuth and roll) that make up R as well as the x,y,z location of the camera in C. This is a total of 6 unknowns.

It is possible to directly measure some or all of these unknowns. For instance, for fixed cameras located on a structure, it is usually possible to survey the camera location with sufficient accuracy to be useful, reducing the number of unknowns to 3. While it is impossible to place a survey instrument in the middle of a camera body, the location the equivalent "pin hole" would be, you can survey to the top of the camera and estimate the vertical offset between this and the axis of the lens. Experience has shown us that the equivalent "pin hole" location for lenses is approximately 1" toward the camera from the front surface of the lens.

It is much harder to measure the viewing angles for a camera to sufficient accuracy. To scale the problem, a 0.25 degree error in estimated viewing error looking 300 m from the camera using a 40 m tower would have a cross-shore error of 1.3 m and 10 m longshore error. Instead, these extrinsic calibration values are usually found using Ground Control Points (GCPs), points whose world coordinates are known by survey and whose image coordinates can be accurately digitized from an image. Combining equations (1) and (2) in the photogrammetry section and applying these to a set of such points, the only unknowns will be the 6 camera parameters so these can be found by a standard nonlinear solver (comparing measured and predicted image coordinates for a guess at the 6 unknowns then searching for their optimum values that minimize the squares of their differences). These values should be stored in a database or file. For a fixed camera, these need only, in principle, be found once. However, our experience show that no camera is truly fixed but moves slightly due to wind, thermal effects or structure aging (we used to be able to distinguish sunny from cloudy days at Duck by variations in the camera tilt). Thus users should be sensitive to geometry variations at a station, especially when newly installed.

At the CIL, the solution for extrinsic variables is carried out using a routine called geomTool6.m that makes use of many CIL database conventions. For the UAV toolbox, the solution is found in the routine

Since there are 6 unknowns, we need at least 6 knowns for a solution. Each control point contributes 2 values (U and V coordinates) so at least three points are needed. We prefer to be over-determined so will use at least four points in the following tests.

For terrestrial applications it is typically easy to find or place an abundance of GCPs throughout the view to allow solution of camera extrinsic geometries. This is at the heart of Structure from Motion algorithms like those from Agisoft or Pix4D. However, surf zone images usually contain only a minimum amount of land by design, so GCP options are often limited and poorly distributed over the image, often lying in a line along the dune crest, a configuration that makes the inverse solution ill-posed. For these cases, common for nearshore studies, we must rely on alternate sources of information to reduce the number of degrees of freedom and the requirements on GCP layout.

For a fixed station, the locations of each camera are usually known by survey so that only the 3 rotation angles are still unknown. These can be solved by nonlinear fit to at least two, but preferably more GCPs.

For a UAV, discussed in Holman, Brodie and Spore [2017], it is rare to find sufficiently accurate information of the azimuth and tilt of an airborne camera so these variables almost always must be solved for. However the camera location is often available in the imagery, based on an onboard GPS system, and can be extracted, for example by using exiftool or other image information packages. Vertical position is also often returned in the image metadata and could be used if no better GCPs are available. However, it is usually less accurate than horizontal position data. For example, altitude may be expressed relative to the takeoff point, rather than in a global coordinate system or it may be a low-quality uncorrected GPS measurement. Finally, it is reasonable to assume for a good stabilized gimbal as on the Phantom 3 that roll is stable and can perhaps taken as equal to 0° for a reasonable approximation. Thus it is possible to reduce to as low as two unknowns which can be solved in a least squares sense with just two GCPs anywhere on the image or in a non-least squares sense with just a single GCP. The relative accuracy of these alternate assumptions is tested in the Holman et al UAV paper.