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  • Topic: Fourier transform, Convolution, Laplace transform
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Final Examination

1. It is often useful in computer vision to represent and analyze image content by means of complex variables, even though an image itself is defined as an array of real numbers. Give at least two examples of useful operations in computer vision based on complex variables, identifying clearly the mathematical domain in which the complex variables exist. Explain in each case what is achieved by adopting such a representation. Answer:

One example of the use of complex variables in computer vision is the Fourier transform. The Fourier transform is a representation of an image as a sum of complex exponentials of varying magnitudes, frequencies, and phases. The Fourier transform plays a critical role in a broad range of image processing applications, including enhancement, analysis, restoration, and compression. Definition of Fourier Transform:

If f(m, n) is a function of two discrete spatial variables m and n, then the two-dimensional Fourier transform of f(m, n) is defined by the relationship [pic]
The variables ω1 and ω2 are frequency variables; their units are radians per sample. F(ω1, ω2) is often called the frequency-domain representation of f(m, n). F(ω1, ω2) is a complex-valued function that is periodic both inω1andω2, with period 2π. Because of the periodicity, usually only the range [pic] is displayed. Note that F(0,0) is the sum of all the values of f(m, n). For this reason, F(0,0) is often called the constant component or DC component of the Fourier transform. (DC stands for direct current; it is an electrical engineering term that refers to a constant-voltage power source, as opposed to a power source whose voltage varies sinusoidally.) The inverse of a transform is an operation that when performed on a transformed image produces the original image. The inverse two-dimensional Fourier transform is given by [pic]

Roughly speaking, this equation means that f(m, n) can be represented as a sum of an infinite number of complex...
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