Fourier Transforms and Spatial Filtering

Introduction

Spatial filtering is a method of image processing in which spatial frequencies, analogous to the more commonly considered temporal frequencies, are filtered in ways conceptually similar to the filtering of temporal frequencies. Like an electronic signal, which can be considered in terms of either its temporal shape or in terms of the frequencies that make it up, a graphical image can be considered in terms of either its spatial form or the spatial frequencies that compose it.

Fourier Transforms

An imageís information is translated to and from the frequency domain by a Fourier transform. The Fourier transform g(k) of a function f(x) is commonly defined by the following relation:

The Fourier transform can be performed either digitally, calculating the amplitude function using the intensity of individual pixels, or optically, using a light source and lenses. Several of the links below include explanations of some of the ways this is done, but briefly, a converging lens effectively performs a Fourier transform of light incident parallel to the lens' axis.

Filtering

Once a two-dimensional Fourier transform has been performed, regions of k-space are attenuated (removed) or phase-shifted. Attenuation in the case of digital processing means zeroing the amplitude function in those regions, and in the case of optical filtering it means blocking the light's path in the transform plane. Phase-shifting can be done optically with crystals or digitally by switching the real and imaginary parts of the amplitude.

When the image is recomposed (by performing the inverse Fourier transform) the image lacks those spatial frequencies that were filtered out, emphasizing variations on remaining length scales.

Applications

Finding defects in periodic microstructures A research group using spatial filtering optics to bring out abnormalities in structures such as CCD and memory microchips. They use birefringent and liquid crystals for the apertures. Looking for fine structures in galaxies An astrophysics lab using digital spatial filtering to enhance photographs of galaxies, trying to analyze their structural details. Check out Part 3 "Image Enhancement" to see the photographs produced. Scanning Probe Image Processor Software designed for digitally manipulating images acquired through Scanning Probe Microscopy. Under the Fourier Analysis link is a discussion with cool pictures of spatial filtering techniques. Characterizing nasal spray plumes A short discussion of the use of fourier transform optics to measure the size of nasal spray mist droplets. Fourier transform profilometry A mathematically intense report on a technique of using spatial filtering and phase shift information to produce a precise 3-D image with a single digital camera. Making multiple optical tweezers Describes the use phase shift (holographic) spatial filtering to create an array of optical tweezers. Satellite photographs Brief discussion, with images, of spatial filtering uses in extracting information from satellite photographs. More remote sensing photographs Examples of several kinds of digital spatial filtering applied to remote sensing images. Political Geography A site on processing geographically-associated health statistics. Spatial filtering is used to focus on variations on a particular length scale. More Political Geography A paper on an investigation that used spatial filtering to analyze spatial dependencies of employment rates.

More Instructional Pages

Fourier Optics Quite illuminating set of optical engineering lecture notes on Fourier transform methods and filtering, with samples of digitally processed images. Interactive Spatial Filter A java applet that performs four different smoothing algorithms on a variety of images. Notes on digital filtering A discussion of some of the kinds of noise and kinds of digital filters implemented to remove that noise. Image Processing Lecture Notes part I Image Processing Lecture Notes part II In-depth treatment of digital smoothing and sharpening methods. Note: These are PDF-format documents, not html. ECE Lecture Notes A good introduction to Fourier transforms and spatial filtering, dealing with both intensity and phase-shift filtering, with both digital and optical transforms. Note: This is also a PDF document.