A certain number of scanned lines (line images) in a real-life image may display sinusoidal intensity profiles, but not all line images along the x or y axis are likely to contain sinusoidal or periodic intensity variations. Despite the nonperiodic nature of line images, and hence the whole image frame, the frequency spectrum of an image or an image segment is derived by extending the Fourier series to nonperiodic waveforms. Much of the 1D Fourier transform mathematics developed in Appendix A is extended to two dimensions in frequency-based image analysis and filtering. In image processing, the 2D image frequency spectrum is commonly used to enhance the visual appearance or accentuate frequency-related features, to correct distortions introduced during the capturing process, or to retrieve image information from noise. At a very basic level, all operations are related to removing, highlighting, or estimating certain frequency components to derive a filtered frequency spectrum, which generates the desired image after inversion. Figure 13.1 illustrates this concept with a collection of edge images created by the summation of cosinusoidal line images. When viewed from the bottom up, the collection shows the construction of a sharp edge image with the successive addition of sinusoidal line images. When viewed from the top down, it shows the smoothing or blurring of a sharp edge by successive removal of the high-frequency line images. Smoothing or low-pass filtering is used to remove high-frequency noise. In contrast, higher-frequency components are gradually added to create a sharp edge, and the edge area dominates the image. This edge enhancement, image sharpening, or high-pass filtering shows the other side of filtering when image edges are required - for example, to separate subimages in image segmentation.