Refraction of the light beam occurs at the water-air interface (precisely, water-glass-air 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 non-refracted 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 water-air interface). For the water-air 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 pin-cushion distortion. Figure 2 shows the refracted and non-refracted 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 image-plane coordinates to the object-plane coordinates, the mismatch error occurs as shown in Figure 3. The x's shown in the figure are the real-life 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 outer-most edge.
The 16-parameter 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:
(u, v) = the image-plane coordinates, and (uo, vo) = the principal coordinates. (See DLT Method for the details.)  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 M1 and M2 shown in Figure 2 both map to image point I, but the refraction error involved in the image coordinates (DI1 vs. DI2) varies depending on the actual position of the marker. It is why the 16-parameter 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 object-space coordinates of the control points are forced to fit to the erroneous (deformed) image-plane 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.
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. 257-260). Perth, Australia: Edith Cowan University.
Kwon, Y.-H. (1999b). Object plane deformation due to refraction in 2-dimensional underwater motion analysis. Journal of Applied Biomechanics, 15, 396-403.
Kwon, Y.-H., & Lindley, S.L. (2000). Applicability of the localized-calibration 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 Close-Range Photogrammetric Systems (pp. 420-476). Falls Church, VA: American Society of Photogrammetry.
© Young-Hoo Kwon, 1998-