
Refraction at the WaterAir Interface Refraction of the light beam occurs at the waterair interface (precisely, waterglassair interface) due to the difference in density between the water and the air (Figure 1). Regardless of the type of the recording system, such as waterproof camera housing, underwater viewing window, or inverse periscope system, there always is a water/air interface. As shown in the figure, marker M maps to image point I on the image plane (film plane) and the refraction occurs at refraction point R on the interface plane. Point I' is the nonrefracted image point and the distance between I and I' is the error due to refraction.
Incident angle j and emergent angle j' suffice the Snell's Law, i.e.:
where r = the refractive index (approximately 1.3330 for the waterair interface). For the waterair interface, the emergent angle (j') is larger than the incident angle (j) and this is why the water looks shallower than it is or the underwater image looks wider than it is. Points M, R & N are not collinear any more, and as a result, the underwater image shows deformation similar to pincushion distortion. Figure 2 shows the refracted and nonrefracted coordinates of the control points marked on three square rims. (The control plane is parallel to the image plane and the camera axis passes through the center of the control plane.) As the distance from the center of the rim to the control point increases the degree of deformation increases as well.
As a result, if one fits the refracted imageplane coordinates to the objectplane coordinates, the mismatch error occurs as shown in Figure 3. The x's shown in the figure are the reallife coordinates of the control points while the boxes are the reconstructed coordinates. The control points at the edges generally show overestimation errors while the rest of them show a trend of underestimation of the distance from the center. The maximum calibration error occurs at the outermost edge.
Refraction vs. Pincushion Distortion The 16parameter DLT reported by Marzan & Karara (1975) equips with an optical distortion correction algorithm. Unfortunately, one can not use this algorithm to correct the refraction error since refraction and optical distortion are different in nature. In a normal situation, the optical distortion error involved in the image coordinates can be modeled as:
where
(u, v) = the imageplane coordinates, and (u_{o}, v_{o}) = the principal coordinates. (See DLT Method for the details.) [2] basically says that the distortion error is a function of the image coordinates. On the other hand, it is not possible to express the refraction error as a function of the image coordinates alone. Figure 2 explains why. Markers M_{1} and M_{2} shown in Figure 2 both map to image point I, but the refraction error involved in the image coordinates (DI_{1} vs. DI_{2}) varies depending on the actual position of the marker. It is why the 16parameter DLT does not provide satisfactory error correction in underwater motion analysis. Therefore, it is necessary to develop a special refraction correction algorithm to completely solve the refraction problem (Kwon, 1999a).
During the camera calibration based on the DLT method, the objectspace coordinates of the control points are forced to fit to the erroneous (deformed) imageplane coordinates. In other words, the mismatch error must be distributed evenly throughout the control volume. Here are some of the important observations (Kwon, 1999b):
There are two possible solutions for the refraction problem: (1) develop a new calibration algorithm that has the refraction correction capability, and (2) modify existing algorithms to reduce the error due to refraction. See Kwon (1999a) for the first approach and Kwon & Lindley (2000) for the second approach. References & Related Literature Kwon, Y.H. (1999a). A camera calibration algorithm for the underwater motion analysis. In R.H. Sanders & B.J. Gibson (Eds.), Scientific Proceedings of the XVII International Symposium on Biomechanics in Sports (pp. 257260). Perth, Australia: Edith Cowan University. Kwon, Y.H. (1999b). Object plane deformation due to refraction in 2dimensional underwater motion analysis. Journal of Applied Biomechanics, 15, 396403. Kwon, Y.H., & Lindley, S.L. (2000). Applicability of the localizedcalibration methods in underwater motion analysis. Submitted to the XVIII International Symposium on Biomechanics in Sports. Marzan, G.T. & Karara, H.M. (1975). A computer program for direct linear transformation solution of the collinearity condition, and some applications of it. Proceedings of the Symposium on CloseRange Photogrammetric Systems (pp. 420476). Falls Church, VA: American Society of Photogrammetry.

© YoungHoo Kwon, 1998 