Getting a good grip on the frequency domain is usually not an easy job! So we have decided to give the issue a complete review here. Convolution in the frequency domain (see Convolution theorem) heavily relies on the concepts of Fourier transform (Fourier transform) and Fourier series (Fourier series) so we will be investigating these important operations first. It has become something of a cliché for people to say that the Fourier series “is a way to represent a (wave-like) function as the sum of simple sine waves” (from Wikipedia). However, sines themselves are abstract functions, so this statement really adds no extra layer of physical insight.
Before jumping head-first into the equations and proofs, we will begin with a historical background to see how the importance of frequencies actually roots in our ancient desire to see everything in terms of circles. A short review of how the complex plane should be interpreted is then given. Having paved the way with these two basics, we define the Fourier series and subsequently the Fourier transform. The final aim is to explain discrete Fourier transform, however some very important concepts need to be solidified first: The Dirac comb, convolution theorem and sampling theorem. So each of these topics are explained in their own separate sub-sub-section before going on to the discrete Fourier transform. Finally we revisit (after Edges in the spatial domain) the problem of convolution on the edges, but this time in the frequency domain. Understanding the sampling theorem and the discrete Fourier transform is very important in order to be able to pull out valuable science from the discrete image pixels. Therefore we have included the mathematical proofs and figures so you can have a clear understanding of these very important concepts.
|• Fourier series historical background:||Historical background.|
|• Circles and the complex plane:||Interpreting complex numbers.|
|• Fourier series:||Fourier Series definition.|
|• Fourier transform:||Fourier Transform definition.|
|• Dirac delta and comb:||Dirac delta and Dirac comb.|
|• Convolution theorem:||Derivation of Convolution theorem.|
|• Sampling theorem:||Sampling theorem (Nyquist frequency).|
|• Discrete Fourier transform:||Derivation and explanation of DFT.|
|• Fourier operations in two dimensions:||Extend to 2D images.|
|• Edges in the frequency domain:||Interpretation of edge effects.|