Frequency analysis

Methods that focus on analyzing oscillations in an image.

Most often, this involves some version of the Fourier transform. The Fourier transform gives the “frequencies” for a function; given a function $f(x):\mathbb{R}\to \mathbb{R}$, the Fourier transform $\hat{f}(\xi )$ is equal to

$$\hat{f}(\xi )={\int }_{-\infty }^{\infty }f(x)\cdot e^{-2\pi ix\cdot \xi }dx,$$

a complex number whose magnitude and argument indicate the amplitude and phase of the corresponding wave.

The intuition for the above formula is that while the frequency at $\xi$ is transformed into the frequency at $0$ and thus can be integrated, all other frequencies become nonzero and thus integrate over complete cycles to zero.

The multivariable formulation of this, commonly used in computer vision, is

$$\hat{f}({\xi }_1,{\xi }_2,\dots )={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\cdots {\int }_{-\infty }^{\infty }f(x_1,x_2,\dots )\cdot e^{-2\pi i(x_1{\xi }_1+x_2{\xi }_2+\dots )}d{x}_n\cdots d{x}_{2}d{x}_1.$$

The Fourier transform $\mathcal{F}$ also has the neat property that ${\mathcal{F}}^2(f)=-f$, i.e. the Fourier transform of a function’s Fourier transform is (roughly) itself.