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Modulus of Fourier Transform

Modulus of Fourier Transform en a two-dimensional Fourier transform, you can apply the inverse Fourier transform to generate an image. An image and its spectrum form a Fourier pair, each of which can be transformed into the other.

17.2.3 Properties of the Fourier Transform

The Fourier transform has properties that make it uniquely useful for processing images. These properties are linearity, symmetry, and inverse scaling. Not only do they apply to the Fourier transform of one-dimensional functions, but also to two-dimensional ones such as sampled images.

Linearity. The linearity property says that the sum of the Fourier transforms of the two functions equals the Fourier transform of the sum of the two functions. In other words, if the Fourier transform of s(x) is S(f), and the Fourier transform of t(x) is T(f), then the Fourier transform of s(x) + t(x) is S(f) + T(f).

This reiterates a key point: the Fourier spectrum of frequencies is the sum of the functions in spatial coordinates.

The linearity property also says that the Fourier transform of a constant times a function equals a constant times the Fourier transform of the function. Given a function s(x) with Fourier transform of S(f), the Fourier transform of cs(x) is cS(f). In other words, if we multiply the amplitude of a signal by a factor of c, the Fourier transform increases by a factor of c.

Symmetry. For real-valued functions such as images, Fourier transforms are symmetrical around zero, so that if you reflect an image about its axis, its Fourier transform stays the same. If the Fourier transform of s(x) is S(f), then the symmetry property says that the Fourier transform of s(-x) is S(f), and the inverse Fourier

Image (circle) Modulus of the Fourier Transform

Figure 17.9 The Fourier transform of a circle is a familiar pattern: the diffraction disk and rings. The large "aperture" circle has a low fundamental frequency and produces a compact transform, while the small "aperture" circle has a higher fundamental frequency and produces a larger pattern.

Image (circle) Modulus of the Fourier Transform

Figure 17.9 The Fourier transform of a circle is a familiar pattern: the diffraction disk and rings. The large "aperture" circle has a low fundamental frequency and produces a compact transform, while the small "aperture" circle has a higher fundamental frequency and produces a larger pattern.

transform of S(-f) is s(x). This is borne out when you see Fourier transforms of images: they are always symmetrical about the origin.

Inverse Scaling. If the Fourier transform of s(x) is S(f), then for a real-valued function, the scaling property says that the Fourier transform of s(cx) is:

The inverse scaling property means that if you double the size of an object in an image, the amplitude and frequency of its spectrum are cut in half; in other words, the wavelengths that make up the image (i.e., the size of the image) are the inverses of the frequencies. The inverse relationship is also true: given a Fourier transform S(cf), the inverse Fourier transform is:

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