We introduce a new method for computing the linear velocity and angular velocity of an unmanned air vehicle (UAV) using only the information obtained from image sequences. We show that it is possible to build a strap-down type inertial navigation system [that we call an optical inertial navigation system (ONS)] using a simple apposition eye, that is a type of compound eye found in insects. In Part I, we use a recently proposed model for a lens and optical fiber system and show through computations that the model can predict well the angular sensitivity of a single ommatidium of a worker bee and an artificial eye. We develop the optical transfer function of the lens-fiber system for quasi-monochromatic, incoherent excitation, and study the properties of the kernel function. We study the cross-talk between neighboring fibers of the lens-fiber system for a worker bee and an artificial eye, and show that it is not significant. The above work allows us to consider in Part II, a mathematical idealization of a corneal surface as a continuum of lens-fiber systems. We show in Part II that the ego-motion estimation problem is well-posed for sufficiently rich quasi-monochromatic, incoherent excitation on an allowable, regular corneal surface, and further show that the solution does not depend on the parameterization of the surfaces or the parameters of the aircraft (such as mass and inertia matrix) on which the ONS is mounted.