10 Magnitude 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!)
11 Why is FT Useful? Easier to remove undesirable frequencies. Faster perform certain operations in the frequency domain than in the spatial domain.
12 Removing undesirable frequencies frAequenciesnoisy signalTo remove certainfrequencies, set theircorresponding F(u)coefficients to zero!remove highfrequenciesreconstructedsignal
13 How 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
15 Frequency Filtering Steps 1. Take the FT of f(x): 2. Remove undesired frequencies: 3. Convert back to a signal:
16 Fast 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
17 The FFT AlgorithmThe running time is O(n log n)
18 ConclusionFourier Transform has multitude of applications in all the field of engineering but has a tremendous contribution in image processing fields like image enhancement and restoration.
19 ReferencesImage Processing, Analysis and Machine Vision, chapter Chapman and Hall, 1993The Image Processing Handbook, chapter 4. CRC Press, 1992Fundamentals of Electronic Image Processing, chapter 8.4. IEEE Press, 1996