Transforming a signal from a function of time or space to frequency(ω) which holds within it’s unique (ω) key also the respected phase(φ) and a respected amplitude/magnitude (A), is called a fourier transform.
This is possible because any signal can be represented as a sum of sinusoids.
f(x) –> Transform –> (A)sin(ω(x) +φ) = F(ω) = Real(ω) + Imaginary(i)(ω)
A =amplitude/magnitude = √(Real(ω) ² + Imaginary(i)(ω)²)
φ = Phase = tan¯¹(Imaginary(i)(ω)/Real(ω) )
Imaginary(i)(ω) = sin part of the F(ω)
Real(ω) = cos part of the F(ω)
F(ω) is a basis set
Discrete Fourier transform – in computer vision there are mostly pixels which are discrete and non continuous one to the other.
Therefore the following formula takes this into account:
F(k) = 1/N ∑(x=0 until x=n-1) for ƒ(x)e¯i(-2pkx/n)
