10Magnitude and Phase of DFT (2/2) Reconstructed image usingmagnitude only(i.e., magnitude determines thecontribution of each component!)Reconstructed image usingphase only(i.e., phase determineswhich components are present!)
11Why is FT Useful? Easier to remove undesirable frequencies. Faster perform certain operations in the frequency domain than in the spatial domain.
12Removing undesirable frequencies frAequenciesnoisy signalTo remove certainfrequencies, set theircorresponding F(u)coefficients to zero!remove highfrequenciesreconstructedsignal
13How do frequencies show up in an image? Low frequencies correspond to slowly varying information (e.g., continuous surface).High frequencies correspond to quickly varying information (e.g., edges)Original ImageLow-passed
15Frequency Filtering Steps 1. Take the FT of f(x): 2. Remove undesired frequencies: 3. Convert back to a signal:
16Fast Fourier Transform (FFT) The FFT is an efficient algorithm for computing the DFTThe FFT is based on the divide-and-conquer paradigm:If n is even, we can divide a polynomialinto two polynomialsand we can write