Multidimensional Processing

Learning Objectives

• Continuous and Discrete Fourier Transform
• Sampling and aliasing
• Interpolation and resampling
• 2D convolution

Fourier Transform

This is where signals are represented as a linear combination of infinitesimally related complex exponentials. In 1D, a rect function (time domain) has a Fourier transform of a sinc function (frequency domain). Similarly in 2D, a circ function (time domain) has a Fourier transform of a line function (frequency domain).

Fourier Transform

$ F(u) \ = \int_{-\infty}^{\infty} f(t)e^{-j2\pi ut} dt $

Inverse Fourier Transform

$ f(t) \ = \int_{-\infty}^{\infty} F(u)e^{j2\pi ut} du $

Function of two variables

A function of two variables can be defined by the formulae $ f(x,y) $ and assign a real or a complex value to each pair $(x,y) $.

A function is separable if $ f(x,y) \ = \ f_1(x)f_2(y) $

A function is circularly symmetric if $ f(x,y) $ depends only on $ r \ = \ \sqrt{x^2 + y^2} $ i.e. can be expressed as $ f(r) $.

Examples

Here a few examples of functions of two variables:

separable circularly symmetric

$ Gauss(x,y) \ = \ e^{-\pi (x^2 + y^2)} \ = \ e^{-\pi r^2} $ $ Rect(x,y) \ = \ $