Distortions and changes of the view cone

It is obvious that the variation of the focal length implies a decrease of the solid angle of view, when the canera is immersed. This variation is directly proportional to the index because the image size (and hence the CCD size) is constant. (C.f. Figure 3)

Figure 3: variation of the field of view, between air and water.

What can be said about distortion ? How are air and water distortions related ?

Up to now we have not been able to determine in a theoretical way the mathematical relation between the two distortions, moreover it is doubtful if such a relation exists which does not involve the complete description of the lens systems. Even without any distortion, the image must be magnified with a factor $1.333$.

Let $u$ be the distorted image of a point in the air medium and $du$ the distortion correction to obtain the perfect perspective projection. If now, in the same way $u'$ is the distorted image of a point in the water medium and $du'$ the new distortion correction, we must have:

$$1.333(u+du) = u' + du'$$ (12)