Image Focusing in Space and Time

Abstract : The integral form of the instrument transmission function for a one-dimensional pixel in a two-dimensional optical system is presented. The integral is solved explicitly in the paraxial ray approximation for a single spatial Fourier component of a Lambertian object. The difference between signals from adjacent pixels is derived. It is shown to have zero derivative with respect to focusing error when the focusing error is zero, i.e., it is a weak source of range-from-focus information. Describing the instantaneous focusing error as the sum of a fixed offset and a time-domain sinusoidal dither, the power spectrum of the signal from each individual pixel is shown to large first and second harmonic terms for physically reasonable values of the parameters. The first harmonic signal is proportional to the product of the dither amplitude and the offset. The second harmonic signal is proportional to the square of the dither amplitude and is independent of offset. The two coefficients are identical except for an integral numerical factor. It is suggested that the ratio of second harmonic to first harmonic signals is thus potentially a powerful measure of offset, i.e., of focusing error in the limit of zero dither, and thus of range-from-focus pixel-by-pixel. Extending the model to three dimensions, removing the approximations, extending the model to natural scenes, and verifying and implementing the results experimentally are outlined briefly. Keywords: Image processing; Pixels(picture elements). (JHD)