The Application of Linear Systems Analysis to Image Processing. Some Notes.

Abstract

The Fourier transform is a convenient tool for analyzing the performance of an image-forming system, but must be treated with caution. One of its major uses is turning convolutions into products. It is also used to transform a problem that is more naturally thought of in terms of frequency than time or space. We define the point-spread function and modulation transfer function in a two-dimensional linear system as analogues of the one-dimensional impulse response and its Fourier transform, the frequency response, respectively. For many imagine devices, the point-spread function is rotationally symmeteric. Useful tranforms developed for the special cases of a "pill box,", a gaussian blob, and an inverse scatter function. Fourier methods are appropriate in the analysis of a defocused imaging system. We define a focus function as a weighted sum of high frequency terms in the spectrum of the system. This function will be a maximum when the image is in focus, and we can hill-climb on it to determine the best focus. We compare this function against two others, the sum of squares of intensities, and the sum of square of first differences, and show it to be superior. Another use of the Fourier transform is in optimal filtering, that is, of filtering to separate additive noise from a desired signal. We discuss the theory for the two-dimensional case, which is actually easier than for a single dimension since causality is not an issue. We show how to consumerist a linear, shift-invariant filter for imaging systems given only the input power spectrum and cross-power spectrum of input versus desired output. Finally, we present two ways to calculate the line-spread function given the point-spread function.