Fourier transform

Fourier transform numerically describes the texture and the pattern in the picture. This information may be important for certain computer vision tasks.

Fourier transform breaks a function into a combination of simpler functions. It's similar to the Taylor series but instead of polynomials it uses trigonometric functions.

In particular, a gray scale image (as a function of two variables) is decomposed into a combination of "sinusoidal" images. Those are represented by sine functions of various periods. The collection of these periods captures the texture - larger periods for larger patterns. Instead of periods it is common to speak in terms of frequencies = reciprocals of the periods. In other words, FT captures the translation symmetry of the image, on all possible scales.

There is also a discrete version of FT which is computed by the Fast Fourier Transform (FFT) algorithm.

A couple of facts:

• If there is no clear pattern in the image, the data produced by Fourier transform will tell you very little. For example, it'll tell you how "busy" the image is. (In this case, Fourier transform remains useful as an image processing tool.)
• Fourier transform processes the image "globally" as a certain kind of averaging over the whole image. As a result, the output is highly affected by an even partial image degradation.

Given a continuous function f, the Fourier transform $F(s)$ of $f(t)$ is defined as follows: $$ F(s)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ist}f(t)dt.$$ In this case, there exists the inverse of the Fourier transform: $$f(t)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{ist}F(s)ds.$$

Given $f$, we denote the result of the Fourier transform of $f$, i.e., function $F$, as $\mathcal{F}(f).$

Properties.

• $\mathcal{F}(af+bg)=a\mathcal{F}(f)+b\mathcal{F}(g), a,b\in {\bf R}.$.
• $\mathcal{F}\left(\frac{\partial}{\partial x}f\right)=\frac{\partial}{\partial x}\mathcal{F}(f).$

If $f(t)$ is an image or another signal then the frequency domain of $f$ is given by $\mathcal{F}(f)$. Then the energy of the signal $f$ is: $$E=\int_{-\infty}^{\infty}|\mathcal{F}(f)(s)|^2ds.$$

Besides image analysis Fourier transform is used for solving differential equations.