Image Representation Using Fast Algorithms Based on the Zak Transform

摘要

Flight simulator imagery is often made up of natural scenes whose characteristicsare not constant across the image. This property suggests that such imagery can be most efficiently represented by spectral techniques that use spatially localized basis functions. This report describes techniques for decomposing full gray-scale images into a joint position/spatial-frequency domain using bases derived from various window functions. The first set of window functions consists of the hermite functions which are related to gaussian derivatives. The second set is based on a new window function that is obtained from a weighted ZAK transform and that provides good localization properties and stable computation. The third set is based on a localized cosine function and allows images to be decomposed using real numbers only. All of the techniques described provide a framework for filtering images in a position-varying manner. For all the basis functions described here, image generation from combined position and spatial-frequency information involves a computationally intensive four-dimensional summation. By an application of the Zak Transform, we are able to replace this summation with a fast Fourier transform, this significantly reducing the complexity of the computation. (AN).