1. Field of the Invention
This invention relates to a device for detecting relative displacement of an optical image on an array of photosensitive elements as a phase change of an electrical output based on the output of the array, and more particularly to a device for converting the optical image on the array of photosensitive elements into an electrical signal meaning a two-dimensional vector, and for detecting the relative displacement of the optical image as a phase change of the vector thereof.
2. Description of the Prior Art
Detection of displacement of an optical image of an object relative to an array of photosensitive elements enables measurement of movement and velocity of the object and further enables discrimination between focus and defocus of a focusing lens as disclosed in U.S. Pat. No. 4,002,899.
According to the spatial shifting theorem Fourier transforms (see "Introduction to Fourier Optics" J. W. Goodman, McGraw-Hill), translation of a function in the space domain introduces a linear phase shift in the frequency domain. Fourier transform Io(k) of an optical image I(x), with respect to a spatial frequency k, is given by ##EQU1## Here, the Fourier transform Ih(k) when the optical image I(x) is displaced by h, is given as follows: ##EQU2##
It is seen from this theorem that knowing the phase term kh of Fourier transform of an optical image leads to knowing the amount of displacement h of the optical image.
In this theorem, attention is drawn to the fact that the integrating section is [-.infin..multidot..infin.].
However, when it is tried to obtain a certain Fourier transform, namely a certain spatial frequency component of an optical image by the use of an array of photosensitive elements, displacement of the optical image cannot be detected from the phase change of the Fourier transform where the optical image is of a special illumination distribution, because the length of the array is not infinite, but finite.