Why are magnitudes normalised during synthesis (IDFT), not analysis (DFT)?

1. What is the difference between DFT and IDFT?
2. Why we use FFT instead of DFT?
3. What is the purpose of DFT?
4. What is DFT in digital image processing?

What is the difference between DFT and IDFT?

The DFT allows one to convert a set of digital time samples to its frequency domain representation. In contrast, the IDFT can be used to invert the DFT samples, allowing one to reconstruct the signal samples x(k) directly from its frequency domain form, X(m).

Why we use FFT instead of DFT?

FFT is an implementation of the DFT used for used for fast computation of the DFT. In short, FFT can do everything a DFT does, but more efficiently and much faster than a DFT. It's an efficient way of computing the DFT.

What is the purpose of DFT?

The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers. Since it deals with a finite amount of data, it can be implemented in computers by numerical algorithms or even dedicated hardware.

What is DFT in digital image processing?

As we are only concerned with digital images, we will restrict this discussion to the Discrete Fourier Transform (DFT). The DFT is the sampled Fourier Transform and therefore does not contain all frequencies forming an image, but only a set of samples which is large enough to fully describe the spatial domain image.