# 4.1 Frequency Spectrum of Images

Since the computational model in spectral domain is rooted in frequency spectrum of the image and the input image is generally in the spatial domain, the transform from spatial domain to frequency domain should be considered first. Images can often be divided into two categories: natural images and man-made object images. Natural images include natural objects (animals, flowers etc.), landscapes (forests, rivers, beaches, mountains etc.) that are commonly outdoor images. Man-made object images involve man-made objects, indoor scenes, streets, city views and so on. Whatever the image is, it always contains some real signal contents and significations that are different from other random signals. Therefore, their frequency spectra consist of some unique statistical properties. In this section, we will review the acquisition of the frequency spectrum of the image, the property of frequency spectrum and the statistical rule.

## 4.1.1 Fourier Transform of Images

To illustrate the computational model in the frequency domain, first let us review the properties of the image frequency spectrum. If I (x, y) is an M-by-N array that is obtained by sampling a continuous 2D image at equal intervals on a rectangular grid, then its discrete frequency spectrum is the array given by the 2D discrete Fourier transform (DFT):

(4.1)

and the inverse DFT is

(4.2)

where ( ...